Supply Function Competition, Market Power and theGeneralised Winner’s Curse: A Laboratory Study.

Anna Bayona∗, Jordi Brandts†and Xavier Vives‡§

April 2016

Abstract

We test in the laboratory whether in the context of supply function competitionwith private information, as predicted in Bayesian equilibrium, positively correlatedcosts lead to steeper supply functions and less competitive outcomes than uncorrelatedcosts. We find that the majority of subjects bid in accordance with the equilibriumprediction when the strategic environment is simple (uncorrelated costs treatment) butfail to do so in a more complex strategic environment (positively correlated costs treat-ment). We do not find statistically significant differences between treatments in averagebehaviour and outcomes, but there are significant differences in the distribution of sup-ply functions. The results are consistent with the presence of sophisticated agents thatbest respond to a large proportion of subjects who fall prey to the generalised winner’scurse. Experimental welfare losses in both treatments are higher than the equilibriumprediction due to a substantial degree of productive inefficiency.

Keywords: Experiment; Correlation Neglect; Private Information; Electricity Mar-ket.

JEL Codes: C92, D43, L13.

∗ESADE Business School.†Institut d’Anàlisi Econòmica (CSIC) and Barcelona GSE.‡IESE Business School.§We thank Rosemarie Nagel, José Apesteguia, Maria Bigoni, Colin Camerer, Margaret Meyer, Cristina

Lopez-Mayan, Albert Satorra, Arthur Schram and Jack Stecher for useful comments and discussions. JordiBrandts acknowledges the financial support of the Spanish Ministry of Economics and Competitiveness(Grant ECO2014-59302-P) and the Generalitat de Catalunya (AGAUR Grant 2014 SGR 510). Xavier Vivesacknowledges the financial support of the Generalitat de Catalunya (AGAUR Grant 2014 SGR 1496).

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1 Introduction

We conduct a laboratory experiment to study the effects of informational frictions in a mar-ket where bidders compete in supply functions to meet the demand of a divisible good. Eachseller has incomplete information about his costs and receives a private signal. Costs may bepositively correlated or uncorrelated among bidders. We investigate whether a higher cost cor-relation leads to enhanced market power as predicted by Bayesian equilibrium. This settingis relevant in wholesale electricity bidding, treasury auctions, and central bank open-marketoperations.1 We provide experimental evidence of a generalised winner’s curse (Ausubel etal. (2014)) in a multi-unit divisible good auction with interdependent values.

We consider a market where firms compete in supply functions (see Klemperer and Meyer(1989)) and with incomplete cost information (Vives (2011)). The latter paper finds thatin a unique linear Bayesian equilibrium private information with cost correlation generatesmarket power over and above the full information benchmark. With positively correlatedcosts, the model predicts that the supply function’s slope is steeper and the intercept islower than when costs are uncorrelated. The mechanism which explains these results is asfollows. A seller receives a private signal which is informative about his random costs. Afully rational seller who is strategic must also realise that when costs are positively correlated,a high price conveys the information that costs are high, and therefore he should competeless aggressively than if costs were uncorrelated in order to protect himself from adverseselection. As a consequence, the main theoretical prediction when all sellers are fully rationalis that private information and strategic behaviour lead to a greater degree of market power,resulting in larger expected prices and profits when costs are correlated than when they areuncorrelated. In positively correlated costs environments, if sellers fail to understand that themarket price is informative about costs then behaviour and outcomes will be indistinguishablefrom environments where costs are uncorrelated. The mechanism which relates higher costcorrelation to enhanced market power is related to a generalised winner’s curse (Ausubel etal. (2014)) which extends the concept of the winner’s curse to multi-unit demand auctions.

In our experiment we compare behaviour and outcomes in two treatments: positivelycorrelated costs and uncorrelated costs. In each treatment, subjects were randomly assignedto independent groups of twelve subjects, each consisting of four markets of three sellers.Within each group, we applied random matching between rounds to keep the one-shot natureof the theoretical model. The buyer was simulated and subjects were given the role of sellers.Subjects received a private signal about the uncertain cost and were subsequently asked to

1Holmberg and Wolak (2015) argue the importance of cost uncertainty among bidders in wholesale elec-tricity markets. Cassola et al. (2013) and Ewerhart et al. (2010) provide the rationale of this setting fortreasury auctions where bidders compete in demand schedules.

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submit a (linear) supply function. Like in the theoretical model, and in contrast to most ofthe experimental literature, we used a normally distributed information structure which maybe a good approximation of the distribution of values in naturally occurring environments.After all decisions had been made, the uniform market price was calculated and each subjectreceived detailed feedback about his own performance, the market price, and the behaviourand performance of rivals in the same market. At the end of the experiment, we conducteda post-experimental questionnaire that asked about participant’s demographic information,bidding behaviour and understanding of the game. Subjects were given incentives whichconsisted of fixed and variable parts. The variable part depended on each participant’sperformance during the game.

Our experimental data conforms to some of the theoretical predictions. First, we confirmthat from the beginning of the experiment, average behaviour in the uncorrelated costs treat-ment is close to the theoretical prediction, and that over time, the average supply functionfurther tends towards the equilibrium supply function. This result is important since theuncorrelated costs treatment provides us with a benchmark with which to compare beha-viour in the positively correlated costs treatment. Second, we find that the features of theequilibrium that are common in both treatments are observed in the data. In particular,we observe that the supply function’s intercept is increasing in a bidder’s signal realisation.This is consistent with subjects understanding that a higher signal implies a higher averageintercept of the marginal cost and that therefore they should set a higher ask price for thefirst unit offered, leading to a higher supply function intercept. We also observe that thesupply function slope is unrelated to the signal received.

However, we cannot reject the hypothesis that on average supply functions are the samein both treatments. In the positively correlated costs treatment we observe that the averagesupply function is substantially flatter and has a higher intercept than predicted by theequilibrium (consistent with subjects being too strongly guided by the signal received). Thisdivergence does not die out as subjects gain bidding experience. Furthermore, we find that thevariances of the supply function slope and intercept are larger in the positively correlated coststreatment than in the uncorrelated costs treatment, leading to differences in the distributionsof the supply function slope and intercept between treatments. We also find insignificantdifferences in market prices, profits and efficiency of the allocations between treatments.In the positively correlated costs treatment, bidders forgo a large percentage of profits, acommon trait in auctions where bidders ignore the adverse effects of the correlation amongcosts. We find that experimental welfare losses in both treatments are larger than predictedby the equilibrium due to the substantial degree of productive inefficiency. These resultssuggest that subjects in the positively correlated costs treatment fall prey to the generalised

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winner’s curse and, hence, compete too aggressively in relation to the equilibrium prediction.The larger heterogeneity in behaviour in the positively correlated costs treatment comparedto the uncorrelated costs treatment deserves to be further explored.

We provide a detailed analysis of behaviour which consists of three parts. First, wetheoretically analyse the strategic incentives of a subject in each treatment. We note thatthe positively correlated costs treatment presents a higher degree of strategic complexity thanthe uncorrelated costs treatment. This is because when costs are correlated the market priceis informative about the level of costs, and therefore, the equilibrium logic does not onlyrequire that subjects form correct beliefs about the economic environment but also that theyform the correct higher order beliefs. Furthermore, a (sophisticated) subject in the positivelycorrelated costs treatment who best responds to the rivals’ average choices has an incentiveto bid a supply function between the equilibrium of the uncorrelated costs treatment and theequilibrium of the positively correlated costs treatment. Therefore, behaviour and outcomesbetween treatments and types of subjects (sophisticated who best respond and naïve whofall prey to the generalised winner’s curse) are less differentiated than predicted.

Second, we descriptively organise the heterogeneity in individual level behaviour withcluster analysis and compare the clusters to the various theoretical benchmarks. We useaverage behaviour in blocks of five rounds. In the uncorrelated costs treatment, we endogen-ously find two clusters: one cluster assembles choices which are close to the equilibrium andtheoretical best response to the average choice, which includes 58% of the subjects in thefirst five rounds, and this percentage increases to 72% in the last five rounds of bidding. Theother cluster groups subjects with supply functions that are steeper and with a lower inter-cept than the equilibrium of the uncorrelated costs treatment (42% and 28% of the subjectsin the first 5 and last 5 rounds, respectively). In the positively correlated costs treatment,we find three clusters: one cluster groups subjects whose supply functions are close to thebenchmark for subjects that fall prey to the generalised winner’s curse, which includes 58%of the subjects in the first 5 rounds and 50% in the last 5 rounds. Another cluster gatherssubjects whose bidding behaviour is not inconsistent with sophisticated behaviour since theirsupply functions are close to the theoretical best response to the average supply functionand it includes 36% in the first five rounds and 42% in the last 5 rounds of bidding. Finally,the rest of the subjects bid a very steep supply function with a low intercept (6% and 8%in the first five and last 5 rounds, respectively). The categorisation of subjects into clusterssuggests that the differences in the heterogeneity of behaviour are driven by the differentlevel of strategic complexity of the two treatments, which lead to a distinct composition oftypes of subjects (naïve and sophisticated) in each treatment.

Third, we conduct a dynamic analysis and find differences in the determinants of the

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evolution of behaviour between treatments: best response dynamics has a more prominentrole in the uncorrelated costs than in the positively correlated costs treatment, while imitationof the best has a smaller role in the uncorrelated than in the positively correlated coststreatment.2 In both treatments, reinforcement learning and imitation of the average areimportant factors for explaining the evolution of behaviour across rounds. The combinationof initial conditions, determinants of the evolution of behaviour, strategic complexity andincentives explain why behaviour evolves towards the equilibrium in the uncorrelated coststreatment while it does not in the positively correlated costs treatment.

The paper is organised as follows. Section 2 reviews the related literature. Section 3 ex-plains the theoretical model. Section 4 lays out the experimental design. Section 5 presentsthe main experimental results. Section 6 provides a detailed analysis of the observed beha-viour during the experiment. Section 7 provides a robustness check and Section 8 concludes.The instructions of the experiment can be found in Appendix C.

2 Related Literature

Our experimental paper studies competition in supply schedules with an information en-vironment that encompasses both positively correlated costs and uncorrelated costs. Ourinformation environment is reminiscent of Goeree and Offerman (2003) since we also usenormally distributed values and error terms. However, their information environment com-pares behaviour in common and uncorrelated private values environments in a single unitsecond-price auction, while ours compares behaviour with correlated and uncorrelated costsin a supply function uniform price auction.

With respect to the competition environment, some early experiments used bid functionsin auctions with incomplete information (e.g. Selten and Buchta (1994)), but few experimentshave tried to analyse competition in supply functions in the laboratory. Exceptions are thework of Bolle et al. (2013) which focuses on the testable predictions of the Supply FunctionEquilibrium concept, and Brandts et al. (2014) which compare the testable predictions ofalternative models of how pivotal suppliers affect supply function bidding. Our experimentfocuses on a framework where market power is driven by a small number of firms, increasingmarginal costs and private information about costs. Outside the laboratory, Hortaçsu andPuller (2008) provide an empirical evaluation of strategic bidding behaviour in multi-unitauctions using data from the Texas electricity market. They find evidence that large firmsbid according to the theoretical benchmark, while smaller firms significantly deviated from

2Best response dynamics and imitation of the average are collinear, and therefore, we cannot disentangletheir separate effect.

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this benchmark.To the best of our knowledge, we have conducted the first laboratory experiment that

finds evidence of the generalised winner’s curse in a multi-unit divisible good auction withinterdependent values. The generalised winner’s curse (Ausubel et al. (2014)) essentiallyreflects the idea that "winning" a larger quantity is worse news than "winning" a smallerquality since winning a larger quantity implies a higher expected cost for a bidder (whencompeting in upward sloping supply functions). Therefore, rational bidders refrain fromcompeting too aggressively. Due to similarities in the information environment, our resultsare related to the findings of the literature on the winner’s curse, a prevalent, consistent androbust phenomenon in common value (or correlated values) auctions (e.g. Kagel and Levin(1986); Goeree and Offerman (2003)), where bidders ignore the adverse selection problem.This large body of evidence, which has mainly focused on single-unit auctions, is summarisedby Kagel and Levin (forthcoming). Our market structure is reminiscent of multi-unit uniformprice auctions, where there is evidence of demand reduction in independent private valuesindivisible good demand auctions experiments (e.g. Kagel and Levin (2001)) and in the field(e.g. List and Lucking-Reiley (2000); Engelbrecht-Wiggans et al. (2006)). In contrast tothis literature, we focus on an interdependent values divisible good uniform price auction.There is also a connection between our experiment and Sade et al. (2006) which tests thetheoretical predictions of a divisible good multi-unit auction model under different auctiondesigns and also find some inconsistencies between actual experimental behaviour and theequilibrium strategies.

There is also an experimental literature on correlation neglect in various strategic contexts,such as in bilateral negotiations (e.g. Samuelson and Bazerman (1985)); trade with adverseselection (e.g. Holt and Sherman (1994)); social learning (e.g. Weizsacker (2010)); bilateralbargaining (e.g. Carrillo and Palfrey (2011)); voting (e.g. Esponda and Vespa (2014); Levyand Razin (2015)) and belief formation (e.g. Enke and Zimmermann (2013); Koch andPenczynski (2015)). Our results are also consistent with these experimental findings sincethere is a substantial proportion of our subjects that ignore the correlation among costs and,therefore, its adverse effects (the generalised winner’s curse).

The more detailed analysis of our results is related to the analysis of other strategic gameswith private information, such as Brocas et al. (2014), who also conduct a similar clusteranalysis, and Carrillo and Palfrey (2011). They both find that there is a large proportionof subjects that behave as in the equilibrium when subjects play simple strategic privateinformation games but this proportion is substantially reduced when the strategic complexityof the game increases. Furthermore, the evolution of choices across rounds is connected tothe work of Huck et al. (1999) and Bigoni and Fort (2013), who analyse learning in a Cournot

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setting. Similar to the results of these papers, we also find that learning is a composite processwith elements of adaptive learning and reinforcement learning, even though our environmentis more complex, since subjects set supply functions with incomplete information instead ofsubjects choosing quantities with full information.

There are several alternative explanations of the winner’s curse. Eyster and Rabin (2005)propose the cursed equilibrium concept, where players incorrectly assess the relationshipbetween the rivals’ strategy and their private information. In our setting, a player in thepositively correlated costs treatment is fully cursed if he ignores that the price conveys in-formation about costs. Except for the two extreme cases in which all players are fully cursedor all players are fully rational, there are several analytical difficulties in computing the cursedequilibrium in our complex setting, and therefore, we do not further relate our results to thecursed equilibrium. A non-equilibrium explanation of the winner’s curse has been providedby the level-k model of strategic thinking (e.g. Nagel (1995); Crawford and Iriberri (2007)).The level-k model does not provide a good description of our experimental choices since thetheoretical level-k predictions do not correspond to the observed peaks in the distribution ofchoices and in addition, the different levels are not sufficiently differentiated to clearly identifya subjects’ degree of strategic thinking. Given these limitations, we focus on analysing thetheoretical and empirical distribution of best replies, which provides a categorisation of sub-jects into naïve (subjects that fall prey to the generalised winner’s curse) and sophisticated(subjects that best respond to the rivals’ average strategy).

3 Theoretical Background

We use the framework of Vives (2011) to guide us with the experimental design. There area finite number of sellers, n, who compete simultaneously in a uniform price auction. Eachseller submits a supply function. Seller i’s profits are

πi = (p− θi)xi −λ

2x2i , (1)

where xi are the units sold, θi is a random cost parameter, p is a uniform market price andλ > 0 represents a parameter which measures the strength of transaction costs. The marketclearing condition allows us to find the uniform market price, p. The random cost parameterθi is normally distributed with θi ∼ N(θ, σ2

θ). The demand is inelastic and equal to q.3

The information structure is as follows. A seller does not know the value of the costshock, θi, before setting the supply schedule and receives a signal with si = θi + εi, where the

3Since q > 1 this is a multi-unit auction.

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error term is distributed εi ∼ N(0, σ2ε ). Seller’s random costs parameters may be correlated

with corr(θi, θj) = ρ for i 6= j. When ρ = 1, the model is equivalent to a common costsmodel; when ρ = 0 to an uncorrelated costs model; when 0 < ρ < 1 to a correlated costsmodel. Error terms are uncorrelated with the random cost shocks and among themselves. Inthe experiment our treatment variable is the correlation among costs, ρ.

Because of both the quadratic payoff function and the normally distributed informationstructure, we focus on linear supply schedules. Linear supply functions are an approximationof the types of supply functions that bidders may submit in real markets.4 Given the signalreceived, a strategy for seller i is to submit a price contingent schedule, X(si, p), which is ofthe form

X(si, p) = b− asi + cp. (2)

The three coefficients (a, b, c) determine the supply function. The interpretation of thesecoefficients is as follows: a is a bidder’s response to the private signal; b is the fixed part ofthe supply function’s intercept, where the supply function’s intercept, f , is: f = b− asi; andc is the supply function’s slope.

Vives (2011) finds a unique supply function equilibrium and describes how the equilibriumparameters (a, b, c) depend on the information structure (θ, σ2

θ , σ2ε , ρ) and on the market

structure (n, q, λ). Refer to Appendix A for the formulae that characterise the equilibriumsupply function and outcomes. Figure 1 summarises graphically the comparative statics ofthe unique symmetric linear Bayesian Nash supply function equilibrium. The equilibriumprediction for a fully rational bidder’s behaviour can be summarised as follows. When costsare positively correlated, a high price conveys the bad news that the bidder’s costs are high.Therefore, in equilibrium, bidders submit steeper schedules than when costs are uncorrelatedto protect themselves from adverse selection. Additionally, for the same signal realisation,the equilibrium supply function’s intercept will be lower when costs are positively correlatedthan when costs are uncorrelated. As a result, the model also predicts that equilibriummarket outcomes will be less competitive when costs are positively correlated than when costsare uncorrelated: the expected market price and profits are larger with positively correlatedcosts than with uncorrelated costs. Additionally, outcomes in both treatments lie between theCournot and competitive benchmarks since supply functions have positive and finite slopes inboth treatments. In terms of welfare, we note that the equilibrium allocation is inefficient dueto distributive inefficiency. Since the demand is inelastic, there is no aggregate inefficiencyat the equilibrium allocation. At the equilibrium allocation, sellers supply quantities whichexhibit too little dispersion in relation to the efficient benchmark.

The comparative statics of the unique Bayesian Nash equilibrium, which assume that all4See, for example, Baldick et al. (2004).

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sellers are fully rational and ex-ante symmetric, allow us to derive some testable predictionsfrom which we formulate the following six hypotheses. They focus on the general predictionsof the model and on the comparative statics with respect to the correlation among costs sinceρ is our treatment variable.

(A) In each treatment the supply function slope is positive and unrelated to a bidder’ssignal realisation.

(B) In each treatment the supply function intercept is non-zero and increasing in a bidder’ssignal realisation.

(C) The supply function is steeper in the positively correlated costs treatment than inthe uncorrelated costs treatment.

(D) For a given signal realisation, the supply function’s intercept is lower in the positivecorrelated costs treatment than in the uncorrelated costs treatment, and hence, the expectedsupply function intercept is lower in the positive correlated costs treatment than in theuncorrelated costs treatment.

(E) The expected market price and profits are larger in the positively correlated coststreatment than in the uncorrelated costs treatment.

(F) The expected deadweight loss is larger in the uncorrelated costs treatment than inthe positively correlated costs treatment.5

If subjects ignore the correlation among costs, and therefore, do not understand that whencosts are positively correlated winning a larger quantity is worse news than winning a smallerquantity then subjects fall prey to the winner’s curse in a multi-unit auction context withinterdependent values. This term has been defined by Ausubel et al. (2014)) as the generalisedwinner’s curse. Therefore, the benchmark for subjects that fall prey to the generalisedwinner’s curse is the equilibrium of the uncorrelated costs treatment. If all subjects fallprey to the generalised winner’s curse then we would expect behaviour and market outcomesin both treatments to be indistinguishable and predictions (C), (D), (E) and (F) may notbe satisfied.6 The hypotheses above pertain to point estimates and expected behaviour.Experimental variation in behaviour and outcomes may also be of interest. We explore theseissues in the results sections.

5This is so given the chosen experimental parameter constellation (see Table 2 of Section 4) although thetheoretical prediction asserts that the expected deadweight loss can increase or decrease with the correlationamong costs.

6Since costs and signals follow a normal distribution we may expect that a subject can fall prey to thenews curse, whereby he ignores the prior information and takes the signal at face value (e.g. Goeree andOfferman (2003)). In our model, we can show that a bidder that falls prey to the news curse sets the samesupply function slope as that of a bidder that falls prey to the generalised winner’s curse, and therefore, thesetwo curses are not easily distinguished.

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Figure 1: Bayesian Nash Equilibrium supply function predictions for the uncorrelated andpositively correlated costs treatments when all agents are fully rational.

Notes. The theoretical predictions for each treatment have been computed for three different signal realisations.A high signal with value 1.200; a signal equal to the ex-ante value of θi with value 1.000 and a low signalwith value 800.

4 Experimental Design

Sessions were conducted in the LINEEX laboratory of the University of Valencia. The par-ticipants were undergraduate students in economics, finance, business, engineering and nat-ural sciences. All sessions were computerised.7 Instructions were read aloud, questions wereanswered in private and subjects were not allowed to communicate throughout the sessions.Instructions explained all the details of the market rules, the distributional assumptions ofthe random costs, signals and the correlation among costs. Before starting the experiment,we tested participant’s understanding. Refer to Appendix C for the instructions of the ex-periment and to Appendix D (first part) for the comprehension quiz.

We ran the experiment with 144 participants, half of which participated in the uncorrel-ated costs treatment and half in the positively correlated costs treatment.8 Each treatmenthad 6 independent groups of 12 members each. Students competed for 2 trial rounds fol-lowed by 25 rounds. In each round, each independent group had 4 markets of 3 sellers ineach market. We chose a market size of three since this is the minimum market size thatdoes not lead to collusion in other similar environments (e.g. Dufwenberg and Gneezy (2000)

7Using z-tree (Fischbacher (1999)).8Before running the experiment we conducted two pilots (one in each treatment).

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in a Bertrand game, and Huck, Normann and Oechssler (2004) in a Cournot market). Inorder to keep the spirit of the one-shot nature of the theoretical model, we applied randommatching between rounds.9 Thus, the composition of each of the four markets varied eachround within a group. Table 1 summarises the structure of our experimental design.

Table 1: Experimental design.

Refer to Appendix D (second part) for the screenshots used for running the experiment.In each round, all subjects received a private signal and were subsequently asked to choosetwo ask prices: one for the first unit and one for the second unit offered. With these twoask prices, we constructed a linear supply schedule, which was depicted graphically on eachsubject’s screen. The participant could then revise the ask prices several times until theparticipant was satisfied with the decision. The buyer was simulated.

Once all supply schedules had been submitted, each bidder received feedback on theuniform market price, his own performance (revenues, production costs, transaction costs,units sold and profits), the performance of the other two market participants (units sold,profits and supply functions), and the values of the random variables drawn (his own costand the costs of the other two participants in the same market).10 Participants could consultthe history of their own performance. Other experiments show that feedback affects behaviourin the laboratory. For example, Offerman, Potters, and Sonnemans (2002) show that in aCournot game, different feedback rules can generate outcomes which go from competitive tocollusive. Given the complexity of the experiment, we chose to give maximal feedback aftereach round in order to maximise the potential learning of participants. After each participanthad checked his feedback, a new round of the game would start. Note that in each market andeach round, we generated three random unit costs from a multivariate normal distribution.In each round and for each participant, unit costs and signals were independent draws fromprevious and future rounds.

9Notice that random matching within a group means that we have less independent observations than ifwe had chosen fixed markets. However, we prioritised this in order to keep the spirit of the theoretical model.

10Subjects did not receive feedback on the signal received since we would not expect this to occur in reality,i.e. after trading, firms may observe the actual costs of competitors but it is very unlikely that a firm alsoobserves the private signals that competitors had at the time of the decision.

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In order to implement the experiment, we specified numerical values for the parametersof the theoretical model, as shown in Table 2 below. In doing so, we applied three maincriteria: (1) the existence of a unique equilibrium; (2) sufficiently differentiated behaviourand outcomes between the two treatments; and (3) simplification of the participant’s compu-tational requirements. Importantly, notice that the uncorrelated costs treatment had ρ = 0,the control treatment, and the positively correlated costs treatment had ρ = 0.6.11 Referto Appendix B for the equilibrium supply function and outcomes given the experimentalparameters of Table 2 and for a statistical description of the distribution of the random costsand errors that were used in the experiment.

Table 2: Experimental parameters.

We imposed certain market rules which were inspired by the theoretical model and madethe experiment implementable. First, we asked each seller to offer all the 100 units for sale.Second, we asked sellers to enter a non-decreasing and linear supply function. Third, askprices had to be zero or positive. Fourth, we told bidders that the simulated buyer wouldnot purchase any unit at a price above the price cap of 3,600. We imposed this price cap inorder to limit the potential gains of sellers in the experimental sessions. This price cap wasnot present in the theoretical model but we chose its value high enough so that it did notdistort equilibrium behaviour. The only difference between treatments was the correlationamong costs and consequently the distribution of random costs and signals.

At the end of the experiment, participants answered a questionnaire with personal in-formation and questions about each subject’s reflections after playing the game. Refer toAppendix E for the questionnaire. Once the questionnaire was completed, each participantwas paid in private.

Regarding incentives, each participant started with 50,000 experimental points.12 During11In order to have maximally differentiated predictions, we would have liked to set a higher correlation

among unit costs. However, with an inelastic demand, the range for which a unique equilibrium exists isreduced and ρ = 0.6 was the maximal correlation which satisfied our implementation criteria. We used aninelastic demand in order to simplify participants’ computational requirements.

12Or equivalently 5 Euros.

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the experiment, subjects won or lost points. At the end of the experiment, points were ex-changed for Euros at the rate 10,000 experimental points to 1 Euro. In addition, participantsreceived a 10 Euros show-up fee. The average payment per subject varied from 10 to 27.8,with an average of 20.8. Session length was between 2 and 3 hours.

5 Experimental Results

First, we present an overview of average bidders’ behaviour and outcomes. Second, weexamine behaviour conditional on the private signal. Third, we examine time trends. Fourth,we analyse the distribution of individual behaviour. Fifth, we conduct a panel data analysis.Sixth, we evaluate the hypotheses formulated in Section 3. The statistical tests of 5.1 use thegroup average as unit of analysis, while the statistical tests of 5.5 use the individual choiceor outcome over rounds.

We shall discuss the experimental numerical results in terms of the inverse supply func-tion since it reflects more clearly how participants made their decisions. From each par-ticipant’s two dimensional decision, (AskPrice1, AskPrice2), we can infer the slope andintercept of each participant’s inverse supply function, p = f + cX(si, p). The inverse supplyfunction slope is c, where c = AskPrice(2) − AskPrice(1) and the intercept is f , wheref = AskPrice1− c. The coefficients of the inverse supply function can be related to the coef-ficients of the supply function (2) as: b = −b

c; a = a

c; c = 1

cfor c 6= 0, where the inverse supply

function’s intercept is f = b + asi. We will omit the word inverse throughout the text. Weshall use InterceptPQ as the empirical counterpart of f , and SlopePQ as the empirical coun-terpart of c. For the graphs, we plot the supply function in the usual (Quantity, AskPrice)

space.13

5.1 Average behaviour and outcomes

We first present bidders’ behaviour and outcomes in each treatment, when decisions andoutcomes are aggregated across all rounds. Figure 2 shows, for each treatment, the observedaverage supply function and the corresponding equilibrium prediction. Table 3 displays, foreach treatment, the average experimental supply function slope and intercept. Table 3 alsoshows outcomes and the corresponding predictions as summarised by market price, profitsand efficiency levels of the allocations, as captured by deadweight losses, together with thecorresponding theoretical values. We employ both parametric and non-parametric tests. Inthis section, we use the group average as the unit of analysis (which aggregates individual

13Note that a steep supply function in the (Quantity,AskPrice) space has a high c and low c.

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choices/outcomes within the group and across rounds).Figure 2 and Table 3 show that in the uncorrelated costs treatment the average supply

function is close to the theoretical prediction. We cannot reject the hypothesis that thesupply function slope is the same as the theoretical prediction (Wilcoxon signed-rank test,two-sided, n1 = 6, n2 = 6, p = 0.600; two-sided t-test, n1 = 6, n2 = 6, p = 0.914). Theaverage intercept is lower than expected (950.11 vs. 1,000). The evaluation test of thehypothesis that the average intercept is equal to its expected value gives us mixed results atthe 5% significance level (Wilcoxon signed-rank test, two-sided, n1 = 6, n2 = 6, p = 0.046;two-sided t-test, n1 = 6, n2 = 6, p = 0.054). The finding that behaviour in the uncorrelatedcosts treatment is on average close to the theoretical prediction is important since it willallow us to use the control treatment as benchmark for our analysis.

Comparing the average supply function of the two treatments, we notice that as predictedby the Bayesian Nash Equilibrium, the average supply function of the positively correlatedcosts treatment has a higher slope (7.79 in the positively correlated costs treatment and6.05 in the uncorrelated costs treatment) and lower intercept than in the uncorrelated coststreatment (899.25 in the positively correlated costs treatment and 950.11 in the uncorrelatedcosts treatment). The difference between the average supply functions in the two treatmentsis qualitatively in accordance with the theoretical model. However, differences between treat-ments are substantially smaller than predicted.14 In fact this difference is not statisticallysignificant in terms of supply function intercept (Mann-Whitney U-test, one-sided, n1 = 6,n2 = 6, p = 0.556; one-sided t-test with unequal variance, n1 = 6, n2 = 6, p = 0.198) norslope (Mann-Whitney U-test, one-sided, n1 = 6, n2 = 6, p = 0.667; one-sided t-test withunequal variance, n1 = 6, n2 = 6, p = 0.120).

14The tests reported for the rest of the section are as follows. The null hypothesis for all of them is:H0 : µ0 − µ1 = 0, where µ1 is the mean for the positively correlated costs treatment and µ0 is the mean forthe uncorrelated costs treatment. The alternative hypothesis for the supply function slope (SlopePQ), marketprice and profits is: H1 : µ0 − µ1 < 0. For the supply function intercept (InterceptPQ) and deadweight lossthe alternative hypothesis is H1 : µ0 − µ1 > 0. The reported p-values refer to the corresponding one-sidedtests.

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Figure 2: Average experimental and equilibrium supply functions in each treatment.

Notes. SF refers to supply function, T0 to the uncorrelated costs treatment and T1 to the positively correlatedcosts treatment.

We also notice that the heterogeneity of behaviour is larger in the positively correlatedcosts treatment than in the uncorrelated costs treatment. In Table 14 of Appendix F wepresent an analysis of variance for the different levels of aggregation and the correspondingstatistical tests. The variances of both the supply function slope and intercept are sub-stantially larger in the positively correlated costs treatment than in the uncorrelated coststreatment. This difference is mainly driven by the large heterogeneity between subjects andbetween groups rather than by the heterogeneity between rounds. As a result, even thoughthere are no differences in average behaviour between treatments, there are differences inthe variance of behaviour between treatments. This difference in the level of heterogeneitymay be the expression of differences in the interaction of distinct kinds of players (e.g. somesubjects fall prey to the winner’s curse while others may not). We further explore this findingin 5.4 using individual choices and we provide a more detailed explanation in section 6.2.

Turning to average market prices, we note that the difference in average market pricesbetween treatments is statistically insignificant (Mann-Whitney U-test, one-sided n1 = 6,n2 = 6, p = 0.556; one-sided t-test with unequal variance, n1 = 6, n2 = 6, p = 0.210).In both treatments, the average market prices are lower than their corresponding theoreticalpredictions: 92.7% and 72.6% of the theoretical predictions in the uncorrelated and positivelycorrelated costs treatment, respectively.

15

Table 3: Average behaviour and outcomes and their corresponding theoretical predictions ineach treatment.

Notes. 1. Theoretical predictions refer to the equilibrium prediction for the supply function slope andexpected intercept. The theoretical predictions for outcomes and efficiency refer to ex-ante expected marketprices, ex-ante expected profits and ex-ante expected deadweight loss at the equilibrium allocation. 2. s.d.refers to standard deviation and is in parenthesis below the average. For all variables except for market priceand deadweight loss, the standard deviation is at the individual level. For market price and deadweight loss,the standard deviations are at the market level.

Average profits in each treatment are substantially lower than their corresponding ex-anteequilibrium predictions: average profits are 40.2% and 11.3% of the theoretical predictionsin the uncorrelated and positively correlated costs treatment, respectively.15 We also notethat in the correlated costs treatment, bidders forgo a large percentage of ex-ante expectedprofits, a common trait in auctions where bidders ignore the correlation among costs (e.g.Kagel and Levin (1986)). The average difference in profits between the two treatments isnot statistically significant (Mann-Whitney U-test, one-sided, n1 = 6, n2 = 6, p = 0.361;one-sided t-test with unequal variance, n1 = 6, n2 = 6, p = 0.757). These differences inexperimental and theoretical profits are mainly driven by the fact that market prices arelower than their corresponding theoretical predictions since subjects submit flatter supplyfunctions than the equilibrium supply function, and the difference is more pronounced in thepositively correlated costs treatment as reflected in Table 3.16

With regards to the efficiency of the experimental allocations there is a difference betweentreatments in the direction predicted by Hypothesis F but it is not statistically significant

15The theoretical predictions assume that all subjects play the Bayesian Nash Equilibrium in the corres-ponding treatment. Note that if one subject realises that the opponents are not playing the Bayesian NashEquilibrium then it is not optimal for him to play the Bayesian Nash Equilibrium and therefore, ex-anteexpected profits are different. Section 6 discusses this further.

16From the analysis of the variance of experimental outcomes presented in Appendix F, we note thatvariances of market price, profits and deadweight loss are not driven by the heterogeneity between subjectsor between groups but are mainly driven by the heterogeneity across rounds.

16

(Mann-Whitney U-test, one-sided, n1 = 6, n2 = 6, p = 0.583; one-sided t-test with un-equal variance, n1 = 6, n2 = 6, p = 0.242). In order to further understand why the largerheterogeneity of behaviour in the positively correlated costs treatment compared to the un-correlated costs treatment does not lead to differences in the efficiency of the allocations, wedecompose deadweight loss into three components as follows

dwl = (λ

2)(

n∑i=1

(xei −q

n)2 + (xoi −

q

n)2 − 2(xei −

q

n)(xoi −

q

n)), (3)

where xoi is the efficient allocation and xei is the experimental allocation (superscript e is todifferentiate it from the equilibrium allocation, xi). The first term of (3) can be associatedwith the variance of the experimental allocation; the second with the variance of the efficientallocation; and the third with the covariance of the experimental and efficient allocations.Refer to Appendix A for further details of these formulae.

Comparing the three components of deadweight loss across treatments, we find the fol-lowing. First, there is no difference in variances of the experimental allocations betweentreatments. Second, the variance of the efficient allocations is larger in the uncorrelatedcosts treatment than in the positively correlated costs treatment. This result correspondsto the theoretical prediction using the parameter configuration of the experimental design.Third, the covariance of the experimental and efficient allocations is substantially larger inthe uncorrelated costs treatment than in the positively correlated costs treatment. Combin-ing these three components with the relevant coefficients, we find that the difference betweentreatments in the variance of the efficient allocations is offset by the difference in the cov-ariance term. For a graphic display of these components refer to Figure 9 and Table 15 inAppendix F.

Furthermore, we compare the deadweight losses of the experimental and equilibrium alloc-ations. As shown in Table 3, in both treatments, the average experimental deadweight lossesare substantially larger than those of the equilibrium allocations. The reasons are as follows.In both treatments, the deadweight losses at the equilibrium allocations are due to too littledispersion and too little covariation with respect to the corresponding efficient allocations.In the uncorrelated costs treatment, the deadweight loss at the experimental allocation islarger than at the equilibrium due to too little covariation between the experimental andefficient allocations which does not sufficiently compensate the sum of the variances of theexperimental and efficient allocations (these two variances are very similar). In contrast,in the positively correlated costs treatment, the difference between the deadweight losses ofthe experimental and equilibrium allocations is mainly driven by the comparatively largervariance of the experimental allocation in relation to the equilibrium one.

17

5.2 Behaviour conditional on the private signal

The previous analysis has not conditioned choices on the private signal and therefore leftout an important aspect of behaviour. In Section 3 we have seen that the theoretical frame-work predicts in both treatments that the supply function slope is independent of the signalreceived, while the supply function intercept increases with the signal realisation. This isbecause a higher signal implies a higher average intercept of the marginal cost and, there-fore, the bidder should set a higher ask price for the first unit offered (AskPrice1 ), leadingto a higher supply function intercept. We first present evidence of average behaviour andthen analyse individual choices. Figure 3 illustrates the average supply function slope andintercept for each signal quartile using all the choices from both treatments.

In both treatments, we find that the average supply function intercept increases withthe signal in each quartile (a regression of the supply function intercept on signal gives acoefficient of 0.856 with p = 0.000 in the uncorrelated costs treatment and a coefficient of0.926 with p = 0.000 in the positively correlated costs treatment), while the supply functionslope remains approximately constant in each signal quartile (in both treatments a regressionof the supply function slope on signal gives an insignificant coefficient with p = 0.703 in theuncorrelated costs treatment and p = 0.655 in the positively correlated costs treatment).17

This evidence suggests that bidding behaviour is qualitatively consistent with the most generaltestable predictions of the theoretical model which are common in both treatments.

17The regressions of this sub-section have used the group across rounds as unit of observation. The numberof observations is equal to 150 (6 groups over the course of 25 rounds) in each treatment.

18

Figure 3: Experimental supply function intercept and slope in each signal quartile and thecorresponding theoretical prediction in each treatment.

Notes. T0 refers to the uncorrelated costs treatment and T1 to the positively correlated costs treatment.

Next, we further disaggregate choices conditional on the private signal and analyse in-dividual ask prices of the first unit offered (which essentially is equivalent to the supplyfunction intercept) in each treatment.18 Figure 4 displays the equilibrium and experimentalratio AskPrice1/Signal in each treatment using disaggregated individual choices during allrounds of the experiment.

In the uncorrelated costs treatment, the Bayesian Nash Equilibrium predicts that theratio of AskPrice1/Signal is between 0.92 and 1.2, with the mean and median equal to1.01. In other words, in the uncorrelated costs treatment the Bayesian Nash Equilibriumpredicts that AskPrice1 is on average equal to the signal received. Experimental choices ofthe uncorrelated costs treatment displayed in Figure 4 (bottom left panel) illustrate thatthe AskPrice1/Signal ratio has a similar mean to its theoretical counterpart, average ratio is0.96, but a larger standard deviation due to the heterogeneity in individual choices.

In the positively correlated costs treatment, the theoretical prediction sets the ratio18The relationship between AskPrice1 and InterceptPQ is AskPrice1=InterceptPQ+slopePQ. In each round

each subject had to decide (AskPrice1, AskPrice2 ) and thus Ask Price1 more clearly reflects how participantstook their decisions.

19

AskPrice1/Signal between 0.58 and 0.73, with the mean and median equal to 0.68, mean-ing that the equilibrium predicts that the AskPrice1 is lower than the signal received. Theexperimental distribution has a mean of 0.91 and median of 0.99, meaning that, for a givensignal, the AskPrice1 is on average larger than predicted by the Bayesian Nash Equilibriumin this treatment and meaning that subjects in the positively correlated costs treatment arestrongly guided by the signal received when deciding on AskPrice1 (refer to the bottom rightpanel of Figure 4) and may choose a number close to the signal received since it acts as afocal point.

Figure 4: Histogram of the theoretical and experimental ratio of AskPrice1 over Signal receivedfor individual choices in all rounds of the experiment.

Notes. For the elaboration of this histogram, we have not plotted the two observations that had an Experi-mental AskPrice1/Signal which was greater than 2.

The previous analysis illustrates systematic divergences between behaviour of how sub-jects respond to the private signal (through the supply function intercept) and Bayesian Nashequilibrium predictions. In the positively correlated costs treatment, for any signal received,the experimental distribution of the AskPrice1 has a higher mean than predicted and it is notsubstantially different than in the uncorrelated costs treatment. Therefore, we interpret thisresult of further evidence that subjects do not take into account the effects of the correlationamong costs when setting the supply function intercept and consequently over-respond tothe signal received.

20

5.3 Time trends

Figure 5 shows the evolution across rounds of the supply function intercept and slope in eachtreatment and the corresponding theoretical predictions.

Figure 5: Evolution across rounds of the supply function slope and intercept in each treatment.

Notes. SF refers to supply function, T0 to the uncorrelated costs treatment and T1 to the positively correlatedcosts treatment. The theoretical supply function intercept has been calculated using the average signalreceived in each round for each treatment. The theoretical predictions use the actual draws of the signalreceived to calculate the theoretical predictions. Refer to Appendix B for how actual draws compare to thetheoretical distribution.

The panel on the left of Figure 5 shows that the supply function intercept increases acrossrounds in the uncorrelated costs treatment (a regression of the supply function intercept onround gives a coefficient of 2.28, with p = 0.002), while in the positively correlated coststreatment it increases during the first 10 rounds of play and then it decreases across rounds(a regression of the supply function intercept on round gives an insignificant coefficient withp = 0.927). The panel on the right shows that the average supply function slope becomessmaller across rounds in the uncorrelated costs treatment (a linear regression of the supplyslope on round gives a coefficient of -0.096 with p = 0.000), while there is no significant timetrend in the positively correlated costs treatment (a linear regression of supply function slopeon round gives an insignificant coefficient with p = 0.212).19 Therefore, we find that the

19The regressions of this sub-section have used the group across rounds as the unit of observation. The

21

evolution across rounds is different in each treatment.In addition, we observe that the change in behaviour (both for the supply function in-

tercept and slope) is more pronounced in the first ten rounds of play, and more specificallythe change in the first five rounds is particularly marked. In the last five rounds of play, weobserve that the average supply function slope is stable, while the supply function interceptshows a marked increase in the uncorrelated costs treatment and it decreases in the positivelycorrelated costs treatment.

We find that in the uncorrelated costs treatment, both the supply function intercept andslope tend towards the theoretical prediction as the number of rounds increases. However, inthe positively correlated costs treatment, we do not observe that the difference between theaverage supply function and the theoretical prediction becomes smaller across rounds duringthe 25 rounds of play, indicating that naïve behaviour persists.

We now turn to the evolution of market outcomes across rounds. Table 4 below showsthe evolution of the average market price and profits in blocks of five rounds.

Table 4: Evolution across rounds of average market price and profits in each treatment.

In each treatment, the table shows that there is no time trend in the average market price(in both treatments a regression of market price on round gives an insignificant coefficient withp = 0.561 in the uncorrelated costs treatment and p = 0.817 in the positively correlated coststreatment). In addition, we find that profits increase in the uncorrelated costs treatment afterthe 10th round but this trend is not observed in the positively correlated costs treatment(in both treatments a regression of profits on round gives an insignificant coefficient withp = 0.103 in the uncorrelated costs treatment and p = 0.556 in the positively correlated coststreatment).20 Deadweight losses decrease over time in both treatments (in both treatments aregression of deadweight loss on round gives a significant coefficient -56.40 with p = 0.003 in

number of observations is equal to 150 in each treatment.20Despite the fact that we observe a time trend in profits in the uncorrelated costs treatment after the

10th round, this time trend is not statistically significant when data is aggregated at the group level. Section5.4 will show that this time trend is statistically significant when we consider choices at the individual levelacross rounds.

22

the uncorrelated costs treatment and -56.29 with p = 0.000 in the positively correlated coststreatment). In the uncorrelated costs treatment, deadweight losses decrease with the numberof rounds since the covariance between the experimental and efficient allocations increasessubstantially. In the positively correlated costs treatment, deadweight losses decrease for adifferent reason. The variance of the experimental allocations decreases significantly withthe number of rounds, thus reducing the difference between the variance of the experimentaland efficient allocations.

5.4 Individual Behaviour: distribution of supply functions

The analysis of section 5.1 shows that even if differences in the average supply function of eachtreatment are insignificant, the variance of the supply function slope and intercept is largerin the positively correlated costs than in the uncorrelated costs treatment. This sub-sectionprovides further evidence of the differences in the distribution of behaviour between treat-ments which are a result of the different levels of strategic complexity in the two treatments.We use individual choices as unit of analysis.

Figure 6 illustrates the empirical cumulative distribution function (hereafter ECDF) forthe supply function slope and intercept in each treatment using disaggregated individualchoices during the first and last five rounds of bidding. These two time periods are importantsince they allow us to summarise behaviour at the beginning of the experiment when subjectshave no experience and at the end of the experiment when they have bid for 20 rounds.

We notice that there is a difference in the ECDFs of supply function slopes betweentreatments both in the first five rounds and last five rounds of bidding. The ECDF of supplyfunction slopes of the positively correlated costs treatment is on the right of the ECDF ofsupply function slopes of the uncorrelated costs treatment in both the first and last fiverounds. In fact, we can reject the hypothesis that the distribution of slopes is the same inthe two treatments during the first five rounds of bidding (Kolmogorov-Smirnov equality ofdistributions test, n1 = 360, n2 = 360, p = 0.023) and during the last five rounds of bidding(Kolmogorov-Smirnov equality of distributions test, n1 = 360, n2 = 360, p = 0.000).21

With regards to the supply function intercepts, we notice that the differences in ECDFsbetween treatments are very small in the first five rounds of bidding and are not statisticallysignificant (Kolmogorov-Smirnov equality of distributions test, n1 = 360, n2 = 360, p =

0.689). In the last five rounds of bidding, differences in the distribution of supply functionintercepts are more substantial and are driven by a few subjects in the positively correlated

21Repeating the same Kolmogorov-Smirnov equality of distributions test for the intermediate periods ingroups of five rounds, we find that the distribution of slopes in the two treatments are significantly differentin all intermediate time periods considered with a p-value of 5%.

23

costs treatment that bid a supply function with a very low intercept. In fact, we reject thehypothesis that the ECDFs of supply function intercepts between treatments are the same inthe last five rounds of bidding (Kolmogorov-Smirnov equality of distributions test, n1 = 360,n2 = 360, p = 0.009).22

Figure 6: Empirical Cumulative Distribution Function (ECDF) for the supply function slopeand intercept in the first and last five rounds of bidding.

5.5 Panel Data Analysis

We present the results of panel data analysis, which uses the individual choice or outcomeacross rounds. We use a random effects panel data approach.23

First, we evaluate whether there is a difference in market behaviour and outcomes in thepositively correlated costs treatment in relation to the control treatment. Table 5 reports thepanel data random effects regression results of estimating the equation for the supply functionslope and intercept jointly, using a seemingly unrelated regression approach for panel data,

22Repeating the same Kolmogorov-Smirnov equality of distributions test for the intermediate periods ingroups of five rounds, we find that the distribution of intercepts in the two treatments are not significantlydifferent in the second and third groups of five rounds (Rounds [6,10] and Rounds [11,15]) but the distributionbecomes significantly different after the 15th round.

23For market price and deadweight loss the unit of analysis is the market across rounds.

24

which assumes that disturbances across equations may be correlated and gives us efficientestimates.24 In Table 5(1), we jointly estimate the following set of equations:

SlopePQit = β0S + β1SD Treatmenti +∑j

βjDGroupij + νi + uit

InterceptPQit = β0I + β1ID Treatmenti +∑j

βiDGroupj + wi + vit,

where subscript it denotes the choice of subject i at round t, and j is an index for the Groupthat a subject belongs to. D_Treatment is a treatment dummy which takes value 0 for theuncorrelated costs treatment and 1 for the positively correlated costs treatment; D−Groupijare a set of dummies which take value 1 if subject i belongs to group j and 0 otherwise. Theerrors uit, vit are observation specific errors and νi, wi are unobserved individual effects. InTable 5(2), we show the second set of regressions augmenting the number of regressors andestimate:

SlopePQit = β0S + β1SD Treatmenti +∑j

βjDGroupij + β2SRound+

+β3S(D Treatment ∗Round)it + β4SSignalit + νi + uit

InterceptPQit = β0I + β1ID Treatmenti +∑j

βjD Groupij + β2IRoundt +

+β3I(D Treatment ∗Round)it + β4ISignalit + wi + vit,

where Round is a variable for the round number, D_Treatment*Round is the interactionterm between the round number and treatment, Signal is the signal received by subject i atround t. Errors uit, vit are correlated across equations.

In Table 5(1) and 5(2) we find that D_Treatment is not statistically significant, bothwhen it is the only regressor and once we have controlled for additional variables. FromTable 5(2), in the equation for the supply function slope, we note that: Round has a signi-ficant and negative coefficient, meaning that supply functions become flatter as the numberof rounds increases; D_Treatment*Round has a significant and positive coefficient, meaningthat the decrease of the supply function slope is less pronounced in the positively correlatedcosts treatment. We also find that Signal is not statistically significant in determining thesupply function slope. Turning to the regression for the supply function intercept, the res-ults show that: Round has a significant and positive coefficient, meaning that the supplyfunction intercept increases as the number of rounds increases; D_Treatment*Round with a

24Refer to Baltagi (2008) for the theoretical background regarding this estimation.

25

significant and negative coefficient that is nearly identical to coefficient of Round, meaningthat the supply function intercept does not vary with Round in the positively correlatedcosts treatment; and Signal with a significant and positive coefficient, as predicted by thetheoretical model.

Table 5: Seemingly Unrelated Panel Data Random Effects Regression.

Notes. Each equation was estimated with group dummies. Standard errors (s.e.) are provided in parenthesis.*, **, *** denote the 10%, 5% and the 1% significance levels, respectively.This estimation has been conducted using the Stata xtsur command, Nguyen (2010), which conducts one-wayrandom effect estimation of seemingly-unrelated regressions (SUR) in a panel data set. The command xtsurdoes not allow for clustered standard errors.

Second, we estimate a panel with random effects with standard errors clustered at thegroup level for the various dependent variables (market prices, profits and deadweight losses).In Table 6(1) we estimate the following equation for market price:

MarketPricemt = β0 + β1D Treatmentm + νm + εmt,

where the subscript mt denotes market m at round t and εmt is the error, νm is the randomeffect which is uncorrelated with the regressor, and D_Treatment is the treatment dummydefined earlier. In Table 6(2), we augment the regression to include further controls and weestimate

MarketPricemt = β0+β1D Treatmentm+β2Periodt+β3(D Treatment∗Period)mt+β4Signalmt+νm+εmt,

where the unit of observation is the market across time and the variables are defined previ-ously.

The results reported in Table 6(1) show that D_Treatment is not statistically significant,both when the treatment dummy is used as the only regressor and also when we include

26

the additional controls. In Table 6(2), we notice that the signal received is the only variablewhich has a positive and significant coefficient in determining market price.

Table 6: Panel Data Random Effects Regression for Market Outcomes and Deadweight Loss.

Notes. dwl refers to deadweight loss. Standard errors (s.e.) have been clustered at the group level and areprovided in parenthesis. *, **, *** denote the 10%, 5% and the 1% significance levels, respectively.The estimation has been conducted using the Stata xtreg, re command.

Third, for profits we use the same panel data approach as for market price, with thedifference that the unit of observation is the subject over round.25 The results show thatD_Treatment is not significant in the results of Table 6(3) and neither in the results of Table6(4). The signal received has a negative and significant coefficient, reflecting the negativecorrelation between signal (and also unit costs) and profits. We also find that profits increaseslightly across rounds in the uncorrelated costs treatment, but we do not find any evolutionacross rounds of profits in the positively correlated costs treatment.

Fourth, for deadweight loss we use the same same panel data approach as for marketprice.26 The results presented in Table 6(5) and (6) show that D_Treatment is not stat-istically significant. We also find that in both treatments deadweight losses decrease acrossrounds, indicating that allocations become more efficient as subjects gain bidding experience.

To summarise, all panel data results are consistent with the stylised facts and tests thatwe have already presented in the previous sub-sections.

25In Table 6(3), we estimate Profitit = β0 + β1D Treatmenti + νi + εit, where the subscript it denotessubject i at round t and εit is the error. In Table 6(4), we augment the previous regression in the followingway Profitit = β0 + β1D Treatmenti + β2Roundt + β3(D Treatment ∗Round)it + β4Signalit + νi + εit.

26In Table 6(5) we estimate the following equation for deadweight loss: dwlmt = β0 + β1DTreatmentm +νm+εmt, where the subscriptmt denotes marketm at round t and εmt is the error. In Table 6(6), we augmentthe regression to include further controls and we estimate dwlmt = β0 + β1DTreatmentm + β2Roundt +β3(DTreatment ∗Round)mt + β4Signalmt + νm + εmt.

27

5.6 Evaluation of hypotheses

We are now in a position to state our findings in relation to hypotheses A-F and also to thequestion formulated at the end of Section 3. With regards to hypotheses A and B, we findthat:

RESULT 1 (Common predictions in both treatments):

As predicted by the theoretical model, in each treatment the average supply func-tion’s intercept is increasing in a bidder’s signal realisation, while the averagesupply function’s slope is positive and unrelated to a bidder’s signal realisation.

We interpret this result to mean that our data supports hypotheses A and B. The next resultpertains to hypotheses C and D:

RESULT 2 (Differences between experimental behaviour and theoretical predic-tions; Differences in the average supply function between treatments):

i) The average supply function in the uncorrelated costs treatment is close to thecorresponding equilibrium prediction, while we reject that the average supply func-tion in the positively correlated costs treatment is the same as the correspondingequilibrium prediction.

ii) Differences in the average supply function between treatments are not statist-ically significant.

Therefore, we do not find empirical support for hypotheses C and D. In addition, Result 2suggests that the generalised winner’s curse is a prevalent phenomenon in the positively cor-related costs treatment since: (1) behaviour in the uncorrelated costs treatment is consistentwith the theoretical prediction and therefore it can serve as a benchmark for subjects thatfall prey to generalised winner’s curse; (2) average behaviour in the positively correlated coststreatment is not different than average behaviour in the uncorrelated costs treatment. Wenow address the question posed at the end of Section 3 about the variation and distributionof behaviour.

RESULT 3 (Difference in the distributions of the supply function slope and inter-cept between treatments):

i) The variances of the supply function slope and intercept are larger in the pos-itively correlated costs treatment than in the uncorrelated costs treatment dueto large heterogeneity between subjects’ and groups’ behaviour (and not acrossrounds).

28

ii) Using individual choices, we reject the hypothesis that the distribution of supplyfunction slopes is the same in both treatments both in the first five and in the lastfive rounds of bidding. With respect to intercepts, we cannot reject the hypothesisthat the distribution of supply function intercepts is the same in both treatmentsin the first five rounds of bidding. However, we reject the hypothesis that thedistribution of supply function intercepts is the same in both treatments in thelast five rounds of bidding.

Result 3 indicates that there are differences in the distribution of supply functions betweentreatments. Furthermore, we examine the drivers of this difference in Section 6.2. Wenow evaluate hypotheses E and F regarding differences in experimental outcomes betweentreatments:

RESULT 4 (Differences in average market prices, profits and deadweight lossesbetween treatments):

There is no average difference in market prices, profits and deadweight lossesbetween treatments.

Result 4 implies that we do not find greater market power in the positively correlatedcosts treatment compared to the uncorrelated costs treatment, and therefore, we do not findsupporting evidence for hypotheses E and F. Next, we report an additional finding about theevolution of behaviour across rounds that has emerged from the analysis of our experimentaldata.

RESULT 5 (Evolution of supply functions and outcomes across rounds):

i) The evolution of supply functions across rounds is different in the two treat-ments. In the uncorrelated costs treatment, average behaviour starts close to theequilibrium and across rounds average behaviour moves even closer to the equi-librium prediction. In the positively correlated costs treatment, it starts far fromfrom the equilibrium and does not move much closer to the equilibrium prediction.

ii) In both treatments, deadweight losses decrease as subjects gain bidding exper-ience, while we do not find evidence that market prices evolve across rounds. Inthe uncorrelated costs treatment, there is some evidence that profits increase assubjects gain bidding experience; we do not find that profits evolve across roundsin the positively correlated costs treatment.

Result 5 suggests that the learning process may be different in the two treatments, and this isexplored further in Section 6.3. The finding that as the number of rounds increases behaviour

29

in the positively correlated costs treatment does not move much closer to the equilibriumprediction suggests that naïve behaviour does not die out in this treatment.

Overall, the results outlined above suggest that some subjects fall prey to the generalisedwinner’s curse since they ignore the correlation among costs and its adverse effects.27 Thedetailed explanation described in the next section uncovers whether these results are drivenby a small proportion of subjects or whether the generalised winner’s curse is a prevalentphenomenon in the positively correlated costs treatment.

6 A closer look at the data

In this section, we first study a subjects’ strategic incentives. Second, we conduct clusteranalysis to descriptively study how close subjects’ choices are to the various theoretical bench-marks. Third, we provide a description of the determinants of the evolution of behaviouracross rounds. Taken together, all the parts provide an explanation of the results that weobserve.

6.1 Best response analysis

Bidding in the positively correlated costs treatment involves a higher degree of strategic com-plexity than bidding in the uncorrelated costs treatment. This is because in the uncorrelatedcosts treatment the market price is not informative about the level of costs, while in thepositively correlated costs treatment equilibrium reasoning requires a subject to correctlyunderstand how the market price is informative about the average cost, which also involveshaving the correct higher order beliefs. In order to have a more complete understanding of abidder’s strategic incentives, we derive theoretically the best response strategy of seller i as-suming that he knows the average strategy of rivals, which determines his residual demand.We then compute the comparative statics of the best response with respect to the rivals’average strategy. For the mathematical derivations refer to Appendix G.

In Figure 7 we illustrate some features of the best response supply function in eachtreatment and compare them to the corresponding equilibrium supply functions. In eachtreatment, we first analyse the best response supply function to the rivals’ average strategyof the first five rounds of bidding.28

27Refer to Section 3 for an explanation of this phenomenon.28Given that the evolution of behaviour across rounds is small, the features of the best response supply

function would be similar had we considered alternative definitions of the rivals’ average strategy.

30

Figure 7: Equilibrium supply function, average supply function of the first five rounds ofbidding and best response supply function to the former.

Notes. The dashed blue line is the best response supply function to rivals that bid the average supply functionof the first five rounds of bidding in each treatment.

In the uncorrelated costs treatment, the average supply function of the first five roundsis steeper and has a lower intercept than the equilibrium. If we assume that rivals bid asthe “representative seller” in the first five rounds then seller i’s best response is to bid aflatter supply function with a higher intercept than rivals’ supply function, which is stillsteeper and has a higher intercept than the equilibrium supply function. In contrast, inthe positively correlated costs treatment, the average supply function of the first five roundsdeparts substantially from the equilibrium supply function since it is significantly flatter andhas a higher intercept than predicted. Seller i’s best response supply function is steeper thanthe rivals’ supply function and flatter than the equilibrium supply function. The intercept ofseller i’s best reply supply function is also between these two benchmarks.

We identify the drivers of these features and focus on the supply function slope sinceit is the most relevant strategic variable. There are two effects which characterise the bestresponse slope of seller i. First, there is the strategic effect which occurs regardless of thecorrelation among costs. If rivals bid a steep supply function then the slope of the inverseresidual demand increases and seller i also has an incentive to bid a steep supply function. Onthe contrary, if rivals bid a flat supply function then the slope of the inverse residual demand

31

decreases and seller i has an incentive to bid a flat supply function. We note the strategiceffect causes that the best response slope of seller i is increasing in the rivals’ average supplyfunction slope.

In addition to the strategic effect, when costs are positively correlated there is an inferenceeffect which is related to the information conveyed by the price. Seller i correctly thinks thata high price means that rivals’ average signal is high, and if costs are positively correlated,seller i infers that his costs must be high. Therefore, seller i has an incentive to bid a steepersupply function than if costs were uncorrelated. When costs are positively correlated, theinference effect causes that the best response slope of seller i to be decreasing in the rivals’average supply function slope. This is because when costs are positively correlated theinference effect is negative and the absolute size of the inference effect increases with therivals’ average supply function slope.

Consider seller i who bids in the positively correlated costs treatment. Suppose thatseller i’s rivals fall prey to the generalised winner’s curse and bid as in the equilibrium of theuncorrelated costs treatment. Then seller i’s best response supply function slope is increasingwith the supply function slope of the rivals, meaning that the strategic effect dominates theinference effect. However, the inference effect moderates the magnitude of the increaseof the best response supply function slope as a result of an increase in the rivals’ supplyfunction slope in relation to when costs are uncorrelated. Therefore, the optimal responsefor seller i is to bid a supply function with a slope which is between the supply function ofnaïve sellers and the equilibrium supply function. In other words, a sophisticated seller whois best responding to naïve rivals has an incentive to bid a flatter supply function than theequilibrium would predict, leading to less distinct behaviour between naïve and sophisticatedsellers. This type of result was first noted by Camerer and Fehr (2006) in the context of gamesof strategic complementarities with sophisticated and boundedly rational subjects.29 Figure10 in Appendix G depicts graphically the best response supply function slope as a functionof the rivals’ average supply function slope in each treatment.

In summary, the positively correlated costs treatment presents a higher degree of strategiccomplexity than the uncorrelated costs treatment, and therefore, it is not surprising that av-erage choices in the positively correlated costs treatment are more distant to the equilibriumthan in the uncorrelated costs treatment. In the positively correlated costs treatment, thebest response strategy of a sophisticated seller that best responds to the rivals’ actual choices

29We can also relate the best response and equilibrium strategies to models of strategic thinking. In thelanguage of the level-k model (e.g. Nagel (1995)), a level-1 subject would best respond to the strategies ofrivals believing that rivals are level-0 subjects (e.g. subjects that fall prey to the generalised winner’s curse).A level-k subject, for k ≥ 1, would best respond to rivals believing that all other players are of level k − 1.The Bayesian Nash equilibrium would be obtained as the limiting behaviour when all subjects do an infinitenumber of level-k iterations.

32

is between the equilibrium of the positively correlated costs treatment and the benchmarkof the generalised winner’s curse (equilibrium of the uncorrelated costs treatment). There-fore, behaviour between treatments and types of subjects (naïve and sophisticated) are lessdifferentiated than predicted by the equilibrium.

6.2 Cluster Analysis

We conduct a descriptive study of experimental choices using cluster analysis, which allowsus to organise the heterogeneity in behaviour and relate it to the various theoretical bench-marks. We present the results of analysing subjects’ choices during the first five rounds,when strategic thinking is most relevant, and the last five rounds of bidding, when subjectsare most experienced.

We use a model-based clustering technique which is based on mixture models for endogen-ously and simultaneously determining the number of clusters and the type of model whichis characterised by the properties of the underlying probability distributions (orientation,volume and shape). The Bayesian Information Criterion (BIC) allows us to select the mostappropriate clustering model. We implemented this procedure using the Mclust package in Rdeveloped by Fraley and Raftery (2006), which uses the Expectation-Maximisation algorithm(EM).

The data used for the cluster analysis is as follows. We first compute the average subjectchoice for each group of five rounds. Each choice is two dimensional because it consists ofthe supply function slope and intercept. We then compute the deviation of each averageindividual choice with respect to the corresponding equilibrium benchmark, and use thesedeviations to conduct cluster analysis. We run the cluster analysis separately for each treat-ment and each group of five rounds. For ease of comparability and interpretation within thesame treatment, we choose the same number of clusters and type of model in each group offive rounds.30

Table 7 provides a summary of the results of cluster analysis which displays the typeof model, the number of components, the frequency of subjects in each cluster, the averagesupply slope and intercept of the subjects belonging to a particular cluster. Appendix Hdisplays the cluster analysis results for the intermediate time periods in blocks of five rounds.

30In the uncorrelated costs treatment, the five clustering models presented in the text and Appendix Hhave the highest possible BIC or there is a difference in BIC of less than 2 to the optimal model. (Fraleyand Raftery (2006) argue that if the difference of BIC between two models is less than 2 then there is weakevidence that one model is better than the other). In the positively correlated costs treatment, there is nota single model which has the highest BIC for all five sets of data. We have considered the second best modelfor all five time periods which is a VEI (varying volume, equal shape and coordinate axes orientation withdiagonal distribution) type model with 3 components.

33

Figure 8 graphically displays these results.In the uncorrelated costs treatment, the BIC criteria leads to two clusters with a VEV

model (varying volume, equal shape and varying orientation with ellipsoidal distribution).In both time periods, cluster 1 groups subjects whose choices are close to the Bayesian Nashequilibrium choice and the best response to the average choice. Cluster 1 contains 58% ofthe subjects in the first five rounds, and this percentage rises to 72% in the last five rounds.Subjects in Cluster 2 bid a steeper supply function with a lower intercept than the equilibriumprediction. These results suggest that behaviour of most subjects in the uncorrelated coststreatment is approximately in accordance with the theoretical predictions, as already reflectedby the average results of Section 5. Furthermore, we notice that 14% of the subjects movefrom cluster 2 to cluster 1 from the beginning to the end of the experiment.

In the positively correlated costs treatment, the BIC criteria leads to three clusters witha VEI model (varying volume, equal shape and coordinate axes orientation with diagonaldistribution). In the first five rounds, no subject bids as in the equilibrium of this treatment.In the first five rounds of bidding, there are 58% of the subjects in cluster 3, which approx-imately corresponds to the benchmark of the generalised winner’s curse (equilibrium of theuncorrelated costs treatment); 36% of the subjects are in cluster 2, whose behaviour is notinconsistent with sophisticated behaviour (i.e.bidding a supply function which is close to thetheoretical best reply) and 6% of the subjects are in cluster 1, meaning that they choose asupply function with a very steep slope and a very low intercept (close to 0). Subjects incluster 1 may be attracted to the zero-intercept focal point.

34

Table 7: Descriptive statistics which characterise each cluster.

Notes. SF refers to supply function and s.d. refers to standard deviation (in parenthesis below the average).The equilibrium supply function intercept has been calculated using the average signal realisation.

In order to further explore the relationship between clusters and optimal behaviour, wepresent Table 17 in Appendix H which summarises the average difference between the ex-perimental supply function and the theoretical best response supply function in each cluster.In the uncorrelated costs treatment, cluster 1 groups subjects whose supply function is closeto the theoretical best response in both time periods, while in the positively correlated coststreatment cluster 2 assembles subjects whose choice has a small distance to both the the-oretical best response slope and intercept in both time periods. In the positively correlatedcosts treatment, subjects in cluster 1 bid supply functions which are too steep and have atoo low intercept in relation to the theoretical best reply and subjects in cluster 3 bid supplyfunctions which are too flat and have a too high intercept.31

31We may ask whether there are differences in the profits of subjects that belong to each cluster. In thepositively correlated costs treatment, there are no differences in the profits of subjects in the three clusters.In the uncorrelated costs treatment, we can only find significant differences in profits between cluster 1 andcluster 2 in the first five rounds of bidding (Mann-Whitney U-test, n1 = 42, n2 = 30, p = 0.041). The largerandom component in profits blurs the differences in profits between clusters, and therefore, we do not findstatistically significant differences.

35

In the positively correlated costs treatment, between the first five to the last five roundsof bidding, there is a move of subjects from cluster 3 to cluster 2, which might lead us toinfer that there is a greater percentage of subjects that bid a supply function with a steeperslope. However, Table 7 shows that the average supply function slope of clusters 2 and 3decreases from the first five to the last five rounds of bidding, and as a result, in the last fiverounds of bidding, the benchmark of the generalised winner’s curse is between clusters 2 and3. This evidence leads us to conclude that there are no important changes in the evolutionof clusters between the beginning and the end of the experiment.

We can summarise these results as follows:

RESULT 6 (Cluster analysis):

i) In the uncorrelated costs treatment the percentage of subjects that are in thesame cluster as the equilibrium is equal to 58% in the first five rounds, and thispercentage increases to 72% in the last five rounds. In addition, this cluster groupssubjects whose supply function is closest to the theoretical best response.

ii) In the positively correlated costs treatment no subject bids as predicted by theequilibrium of this treatment in the first or last five rounds of bidding. The per-centage of subjects that are in the same cluster as the benchmark of the generalisedwinner’s curse is 58% in the first five rounds and 50% in the last five rounds ofbidding; the percentage of subjects whose bidding behaviour is not inconsistent withsophisticated behaviour (best response to the average supply function) is equal to36% in the first 5 rounds and 42% in the last five rounds of bidding; the percentageof subjects that bid a very steep supply function with a low intercept is equal to6% in the first five rounds and 8% in the last five rounds of bidding.

Notice that in the positively correlated costs treatment it is individually optimal for asubject that is best responding to actual choices to bid a supply function with a flatter slopeand with a higher intercept than the equilibrium would predict. This explains why averagebehaviour and outcomes are not sufficiently differentiated between treatments and makes itmore complicated to empirically distinguish the behaviour of naïve and sophisticated sub-jects. Furthermore, if some subjects never learn, or learn very slowly, and remain in cluster3, then a subject that is best responding to actual choices may not have an incentive toincreasingly bid steeper supply functions during the course of the experiment. This may ex-plain why slopes do not evolve towards the equilibrium prediction in the positively correlatedcosts treatment. In the following sub-sections, we study the determinants of the evolution ofchoices across rounds in the two treatments.

36

Figure 8: Graphical Display of the Cluster Analysis Results.

Notes. Each data point corresponds to a subject’s average choice averaged in the first or last five rounds,which is two dimensional since it is characterised by the supply function slope and intercept. The graphs’horizontal axes represent the difference between theoretical and experimental supply function slopes. Thevertical axes represent the difference between theoretical and experimental supply function intercepts. Theblack dashed lines plot the equilibrium supply function slope and intercept in each treatment, and therefore,point (0,0) corresponds to the equilibrium. In the positively correlated treatment graphs, there is also a greendashed line which represents the benchmark of the generalised winner’s curse (equilibrium of the uncorrelatedcosts treatment).

6.3 Evolution of choices across rounds

Our goal is to provide a dynamic description of the experimental behaviour and to understandthe drivers of the different evolution of behaviour across rounds in the two treatments. Ourexperimental design provided subjects with very complete feedback after each round, whichincluded each subjects’ own choice, own profits and the behaviour and profits of the othertwo market participants. Subjects may be influenced by the average choices of rivals in thesame group since there we gave them substantial feedback after each round. In this high-information environment and with twenty-five rounds we aimed to foster learning about theequilibrium. A subject’s use of a particular learning model depends on: (i) the informationavailable and (ii) subjects’ cognitive abilities. Given the complexity of the experiment, weconsidered only those learning models in which subjects had the necessary information tolearn.

We consider that the evolution of behaviour across rounds can be described by threemain categories of learning models (for an excellent review see Camerer (2003)): experientiallearning, in which subjects learn from their own experience; imitation based learning, inwhich after the first round subjects choose a strategy which has been previously chosen byother players in the previous round; and belief learning, which are essentially adaptive sincesubjects update their beliefs based on the history of play. Unlike for Cournot competition,there are very few theoretical results on the convergence properties of learning models in thecontext of supply function competition with private information.32 As a result, the aim ofthis section is only to provide an informative description of actual behaviour across rounds.

Our empirical strategy follows the approach of Huck et al. (1999) and Bigoni and Fort(2013), whereby we consider a few representative models of each of the three categoriesdescribed above. Furthermore, in this sub-section, we consider that the supply functionslope is the relevant strategic variable which best summarises how a subject adjusts hisbehaviour across rounds. We do not analyse the dynamics of the supply function interceptsince we consider that subjects cannot learn about it because of lack of information: we didnot provide feedback on the signals received by the rivals. As a result, the supply functionintercept may be affected by confounding factors, such as how subjects respond to the privatesignal. In addition, supply function intercepts are practically identical in the two treatmentsand do not evolve across rounds in the positively correlated costs treatment.

In terms of experiential learning, we focus on reinforcement learning, whereby a playerincreases the probability of playing a strategy (i.e. the strategy is reinforced) according to pre-vious own profits. The propensity of a strategy is the cumulative sum of the previous profitsobtained with such strategy, while the relative propensity of a strategy is the propensity of

32For some general considerations, refer to the introduction of Chapter 7 of Vives (2008).

38

a strategy divided by the aggregate propensities of all strategies in a given round (refer toErev and Roth (1998)). Our implementation of reinforcement learning assumes that a playerwill bid the supply function slope with the highest relative propensity.

We then consider two models based on learning by imitation. First, a payoff-independentmodel, imitation of the average, where subjects imitate the rivals’ average strategy of theprevious round, and second a payoff-dependent model, imitation of the best, where subjectsimitate the strategy of the subject with highest profits in the previous round, and the mostsuccessful player imitates himself.33

As a representative of belief learning models, we consider best response dynamics, wherebysubjects best respond to the rivals’ previous round strategies. The features of the bestresponse strategy in each treatment have been described in Section 6.1. 3435

We estimate a dynamic panel regression which studies the determinants of each subject’ssupply function slope across rounds with respect to the various learning models describedabove in the following equation:

SlopePQit = γSlopePQit−1 + βERerit−1 + βIAiait−1 + βIBibit−1 + βBRbrit−1 + νi + εit, (4)

where SlopePQit is subject i’s supply function slope in round t; SlopePQit−1 is the sup-ply function slope of subject i in round t − 1 and provides an indicator of the persistencein behaviour; erit−1 is the supply function slope which corresponds to the highest relativepropensity based on player i’s profits in round t − 1 (reinforcement learning); iait−1 is theaverage supply function slope of i’s rivals in round t− 1 (imitation of the average); ibit−1 isthe supply function slope of the subject with highest profits in the previous round (imitationof the best); and brit−1 is subject i’s theoretical best reply to the rivals’ supply function slopein round t − 1 (best response dynamics), εit are error terms and vi the panel-level randomeffects.36

33We have not considered learning models such as imitate the exemplary firm, such as in Offerman et al.(2002), since it would require subjects to calculate the joint maximisation of profits if the three firms produceda given level of output, and this is not trivial in our supply function with private information environment.

34The difference between the best response analysis of this section and the one of the previous ones (Sections6.1, 6.2 and Appendices G and H) is that here we consider the best response to the rivals’ strategy of theprevious round while in the ones presented earlier we considered the best response to the rivals’ strategy ofthe contemporaneous round.

35We do not consider “fictitious play” since it is cognitively more demanding that best response dynamics.Subjects must remember the average strategies of all previous rounds of play, while best response dynamicsonly requires subjects to remember the strategies of the other two players in the previous round.

36In relation to Huck et al. (1999) we have added the reinforcement learning term. In relation to Bigoni andFort (2013) we have replaced the ’trial and error’ experiential learning model by the reinforcement learningmodel since it has a wider acceptance in the literature.

39

If a learning model provides a full explanation of how a subject adjusts the supply func-tion slope across rounds then the corresponding coefficient would be equal to 1 and theother coefficients would be insignificant. If the explanation is partial then we expect thatcorresponding coefficient to be significantly different from zero and positive.

Table 8 presents the estimation results of equation (3) for individual choices across roundsin each treatment using the approach proposed by Arellano and Bover (1995) and Blundelland Bond (1998).37 Columns (1)-(2) of Table 8 present the results of estimating equation(3) for the uncorrelated costs treatment. We notice that we cannot estimate the full learningmodel of equation (3) since the variable Best response, brit−1, is perfectly collinear withimitation of the average, iait−1 since there exists a functional relationship between the two.38

Therefore, we cannot disentangle the effects of these two learning models in the uncorrelatedcosts treatment. The results of estimating equation (3) for the uncorrelated costs treatmentare presented in column Table 8(1), where imitation of the average is excluded, and in Table8(2) where best response dynamics is excluded. In both equations all significant coefficientshave the expected signs. We find that reinforcement learning is an important component ofthe adjustment dynamics while imitation of the best is not significant. In column (1), bestresponse dynamics is significant and has the largest coefficient, and in column (2) we findthat imitation of the average has a positive and significant coefficient. To summarise, wefind that reinforcement learning, imitation of the average or best response dynamics jointlycontribute to explain the evolution of behaviour across rounds in the uncorrelated coststreatment. Behaviour exhibits some persistence over time.

In Table 8(3) we display the results for the positively correlated costs treatment whichexclude 64 observations from the estimation since these have a theoretical best response whichwas unfeasible given our experimental design.39 Behaviour also exhibits some persistence overtime. Furthermore, reinforcement learning, imitation of the best and imitation of the averagejointly provide a partial description of how subjects adjust their slope across rounds, while

37The estimation has been conducted using the Stata xtdpdsys command using the two-step estimator.The regressors corresponding to each of the learning models are considered to be predetermined since theymay be correlated to past errors but are not correlated with future errors. With regards to the restrictionson instruments these are as follows: (1) we use at most 1 lag of the dependent variable as instrument; (2)Each regressor appears as contemporaneous and at most 1 lag is used as instrument.In all the equations presented, the post-estimation specification test for autocorrelation of the error termsgives no evidence of autocorrelation of order higher than one at the 10% significance level. In addition, wecannot reject the null hypothesis that the over-identifying restrictions are valid (Sargan test) for all regressionsdisplayed.

38Refer to Appendix G and equation (17) for the details.39This is so since the theoretical best response would have required a negative supply function slope which

was not allowed. The average supply function slope from these 64 excluded observations does not differsignificantly from the average supply function slope of the rest. If we include the 64 observations in theestimation of equation (3) then the coefficient are very similar to the ones presented in Table 8(3).

40

best response dynamics does not contribute in explaining how subjects’ behaviour evolves.

Table 8: Dynamic panel estimation results of the evolution of behaviour across rounds.

Notes. Standard error are robust and provided in parenthesis below the coefficient. *, **, *** denote the10%, 5% and the 1% significance levels, respectively.

The following statement summarises these findings:

RESULT 7 (Determinants of evolution of behaviour over rounds):

There are significant differences between how subjects learn in the two treatments:best response dynamics is a more prominent determinant of the evolution of beha-viour across rounds in the uncorrelated costs treatment compared to the positivelycorrelated costs treatment. Imitation of the best is a significant factor in the posit-ively correlated costs treatment while it is not in the uncorrelated costs treatment.In both treatments, reinforcement learning and imitation of the average help toexplain the evolution of behaviour across rounds.

Result 7 shows that the evolution of behaviour across rounds is a composite process withelements consisting of the various learning models considered, which have different weights ineach treatment. The fact that the best response dynamics has a significant effect on learningwith uncorrelated costs but not with correlated costs is consistent with the notion that theenvironment of the uncorrelated costs treatment is cognitively simpler for the participants inthe experiment.

6.4 Post-Experimental Questionnaire Analysis

The main result of the experiment is that most subjects in the positively correlated treatmentbehave as if they did not understand that the market price is informative about the level

41

of costs. The analysis of the answers of the post-experimental questionnaire provide furtherevidence of this interpretation. We asked: (1) “Do you think that a high market pricegenerally means good/mixed/bad news about the level of your costs?”; (2) “Explain youranswer”. With these questions we aimed to check whether subjects understood that the priceis informative of the level of costs in the positively correlated costs treatment after takingdecisions during 25 rounds, and whether there were differences between the two treatments.We analyse the explanations given in the second part of the question since these are morerevealing about the reasoning processes that subjects used. In Table 9, we classify the answersof the post-experimental questionnaire into types and provide an example of a typical answerin each category.

Table 9: Classification of post-questionnaire answers.

Notes. The answers correspond to the explanation part of: (1) “Do you think that a high market pricegenerally means good/mixed/bad news about the level of your costs?”; (2) “Explain your answer”.

The answers classified as market price is informative about costs reflect the logic of theequilibrium reasoning of the positively correlated costs treatment, while the answers classi-fied as market price is not informative about costs reflect the equilibrium reasoning of theuncorrelated costs treatment. The answers classified as high costs imply a high market priceare somewhat ambiguous since they could either reflect: (1) understanding of the equilibriumreasoning of the positively correlated costs treatment, or (2) an understanding of the commonprediction in both treatments that a high signal implies high costs and therefore subjects bida supply function with a higher intercept.40 Therefore we are not going to focus on theinterpretation of this category of answers nor on those classified as other factors. Table 10below quantifies for each treatment the percentage of answers that correspond to each of thecategories analysed above.

40We note, however, that this is the first reasoning step towards understanding that the Market price isinformative about costs.

42

Table 10: Answers of post-experimental questionnaire in each group of each treatment.

The classification shows that there are approximately twice the percentage of subjects inthe category market price is informative about costs in the positively correlated treatmentcompared to the uncorrelated treatment (17% vs. 8%), reflecting that a few subjects inthe positively correlated costs treatment understand that the market price is informativeabout the level of costs. In addition, we note that the answer market price is not inform-ative about costs is the most prevalent answer in both treatments with no major differencesbetween how subjects answered this question in the uncorrelated and positively correlatedcosts treatments. The answers of the post-experimental questionnaire are consistent withour interpretation that there is a large percentage of subjects that fall prey to the general-ised winner’s curse in the positively correlated costs treatment and a smaller percentage ofsubjects that are sophisticated (whose best response is more similar to those that that fallprey to the generalised winner’s curse than the equilibrium would predict).

7 Robustness check: Bayesian Updating

It could be argued that subjects fail to conduct Bayesian updating and this is the reasonwhy subjects in the positively correlated costs treatment fail to understand that the marketprice is informative about the level of costs. In order to further explore this possibility andaid subjects in their decision making process, we conducted an additional session with twogroups in the positively correlated costs treatment. These sessions had the same experimentaldesign features as in the baseline treatment except for: (1) in addition to the signal received,each subject received the expected value of his own costs and his rivals’ costs conditional onthe signal received (subject i received a signal si and was also given E[θi | si] and E[θj | si],for i 6= j)41; (2) we explicitly asked each subject to think about what his rivals would do, andaccording to these beliefs, we provided a simulation tool where a subject could tentativelyenter a decision and visualise the provisional market price. The subject could then revise

41The instructions of these additional treatments are available upon request. In order to explain theconditional expectations, we told subjects that in each round an expert would give them the expected valueof their unit cost and the unit cost of their rivals.

43

his decision; (3) The experiment lasted for 15 rounds instead of 25 rounds.42 Table 11presents the summary statistics of outcomes and choices of these two groups in the positivelycorrelated costs treatment.

Table 11: Robustness Session: behaviour and outcomes in the positively correlated coststreatment.

We find that the average supply function and outcomes of the robustness session are verysimilar to the averages in the baseline treatments presented in Table 3 which correspond to thepositively correlated costs treatment. In other words, aiding subjects in their decision makingprocess does not seem to affect the behaviour and outcomes of the positively correlated coststreatment in a significant way.

This seems to suggest that our results in the positively correlated costs treatment are notdue to a bias related to simple Bayesian updating. Our interpretation that subjects in thepositively correlated costs treatment fail to understand that the market price is related tothe level of their costs and therefore should bid less aggressively is robust.

8 Concluding Remarks

The experiment presented in this paper has analysed bidding behaviour in the laboratory,in a market where bidders compete in supply functions, have incomplete information abouttheir costs and receive a private signal. We have used the unique Bayesian Nash equilibriumprediction and comparative statics as the benchmark for contrasting the results of the ex-periment. Our experiment uses a between subject design with two treatments: uncorrelatedcosts and positively correlated costs. The uncorrelated costs treatment serves as the con-trol treatment. In implementing the experiment, we have chosen numerical values for the

42We shortened the number of rounds since these modifications made the experiment longer and sinceSection 5.3 has shown that subjects do not adjust much their behaviour in the last few rounds of bidding.

44

parameters of the model so that the theoretical predictions for behaviour and outcomes weresufficiently differentiated between the two treatments.

We find that most subjects bid in accordance with the equilibrium predictions in a simplestrategic environment (uncorrelated costs treatment) but the majority of subjects fail to do soin a more complex strategic environment (positively correlated costs treatment) where theyare required to extract information from the market price about the level of costs. Subjectsin the positively correlated costs treatment bid on average flatter supply functions thanpredicted by the equilibrium. Our findings are consistent with the experimental literature oncorrelation neglect (Section 2), a phenomenon which in our setting is intertwined with thegeneralised winner’s curse. Indeed, to the best of our knowledge we are the first to documentthis phenomenon in an environment with interdependent values and multiple units of adivisible good being auctioned.

The mitigation of market power has been the concern of regulators in electricity markets.43

When costs are positively correlated, our experiment shows that competition in supply func-tions with a large proportion of subjects that fall prey to the generalised winner’s curse leadsto more competitive market outcomes than in the equilibrium. Despite being more compet-itive, allocations present a higher productive inefficiency than the corresponding equilibriumallocations.

This experiment suggests a few open questions for future research, both experimental andtheoretical. Future work could explore whether these results are robust to a subject poolconsisting of professional subjects. Our experimental results also call for the developmentof theoretical models which analyse market competition with subjects that present differentdegrees of strategic sophistication, and also for the study of the convergence properties ofdifferent types of learning models in markets characterised by supply function competitionand private information.

43See, for example, Hortaçsu and Puller (2008) and Holmberg and Wolak (2015).

45

Appendix A. Theoretical Considerations

This section uses results from Vives (2011). When demand is inelastic and equal q and if−1n−1

< ρ < 1, σ2ε

σ2θ<∞ and λ > 0, the model described in Section 3 has a unique linear Supply

Function Equilibrium (SFE) given as

X(si, p) = b− asi + cp, (5)

where

a =(1− ρ)σ2

θ

(1− ρ)σ2θ + σ2

ε

(d+ λ)−1, (6)

b =1

1 +M

(qM

n− σ2

ε θ(d+ λ)−1

(1 + (n− 1)ρ)σ2θ + σ2

ε

), (7)

and

c =n− 2−M

λ(n− 1)(1 +M), (8)

where M = ρnσ2ε

(1−ρ)((1+(n−1)ρ)σ2θ+σ2

ε)represents an index of adverse selection and d = 1

(n−1)cis

the slope of the inverse residual demand. The expected intercept is equal to E[f ] = b− aθ.The expected market price is equal to

E[p] = θ +(d+ λ)q

n. (9)

Ex-ante expected profits of seller i at the SFE given the predicted values with full informationare equal to:44

E[π(t; d)] = (d+λ

2)

(q2

n2+

(1− ρ)2(n− 1)σ4θ

n(σ2ε + σ2

θ(1− ρ))

1

(d+ λ)2

), (10)

where t=(E [θ1 | s] , E [θ2 | s] , ..., E [θn | s]), s = (s1, s2, ..., sn) and π(t; d) = 1n

∑i πi(t; d). We

note that xi(t; d) = x(t; d)+ t−ti(d+λ)

. Therefore, the first term of expected profits corresponds to

expected profits at the average quantity since they are equal to (p− θ) qn− λ

2( qn)2 = (d+ λ

2) q

2

n2 .The second term is related to the dispersion of the predicted values. For each subject andround, we compute profits conditional on the private signal. We then calculate the averageprofits in each treatment, which give us an estimate for ex-ante expected profits.

44At the SFE the market price and the signal provide sufficient information on the joint information of themarket.

46

In terms of the efficiency at the equilibrium allocation, we note the following. First, there is noaggregate inefficiency since the demand is inelastic. Second, the ex-ante expected deadweightloss, E[DWL], at the equilibrium allocation is the difference between expected total surplusat the efficient allocation and at the equilibrium allocation. We can write it as

E[DWL] =nλ

2E[(xi − xoi )2]. (11)

Vives (2011) shows that the efficient allocation is equal to

xoi =q

n+

(1− ρ)σ2θ

λ((1− ρ)σ2θ + σ2

ε )(s− si), (12)

and that expected deadweight loss at the equilibrium allocation can be then written as

E[DWL] =λ

2(1

λ− 1

λ+ d)2 (1− ρ)2(n− 1)σ4

θ

(σ2ε + σ2

θ(1− ρ)). (13)

For each market we compute the empirical counterpart of the (interim) deadweight loss atthe experimental allocation as follows

dwl = (nλ

2)(

1

n

n∑i=1

(xei − xoi )2), (14)

where xei is the experimental allocation (superscript e is to differentiate it from the equilibriumallocation, xi) and xoi is the efficient allocation as defined above. In order to calculate theaverage deadweight loss in each treatment, we average the interim deadweight losses forthe 600 markets in each treatment, which gives us an estimate for the ex-ante expecteddeadweight loss.45

Appendix B. Equilibrium predictions, ex-ante expected

outcomes and statistical distribution of random costs and

errors used in the experiment.

In our experiment, given the experimental parameters of Table 2, the numerical equilibriumsupply function and ex-ante expected outcomes in the two treatments can be summarised inthe table below.

45There are 600 markets in each treatment since we have 6 groups, which consist of 4 markets each for 25rounds.

47

Table 12: Equilibrium supply function & ex-ante expected outcomes given the experimentalparameters.

Notes. The numerical value for the intercept corresponds to the expected intercept in each treatment. Supplyfunction (PQ) refers to the supply function viewed in the usual (Quantity, Ask Price) space. SFE refers tosupply function equilibrium.

The next table reports the summary statistics of the draws of the random variablesused in the experiment. The theoretical model assumes that in each market we draw ann-dimensional multivariate normal distribution with θ ∼ N(µ,Σθ), where the covariancematrix Σθ is such that the variance of each component, θi, is equal to σ2

θ and the covariancebetween any i 6= j is equal to ρσ2

θ , where ρ is the correlation coefficient. In Table 2 we havespecified the following parameter configuration: n = 3, θ = 1, 000, σ2

θ = 10, 000; ρ = 0 for theuncorrelated costs treatment and ρ = 0.6 in the positively correlated costs treatment. Thefirst part of the table below reports the statistics of the actual draws for random costs foreach of the three sellers, where subscripts 1, 2 and 3 correspond to seller 1, seller 2 and seller3, respectively. Because of our random matching protocol, in each round we have a differentsubject to take the role of sellers 1, 2 and 3. The model also assumes that in each marketwe draw an n-dimensional multivariate normal distribution of signals such that s = θ + ε,where the signals’ errors are distributed ε ∼ N(0,Σs). The variance of each component, εi,is equal to σ2

ε and the covariance between any i 6= j is equal to zero. In Table 2, we havespecified that σ2

ε = 3, 600. The second part of the table below reports the statistics of theactual draws for the signals’ errors for each of the three sellers. The theoretical model alsospecifies that error terms in the signals are uncorrelated with the θi’s. We check this in thethird part of the table below. There are no statistically significant differences between thedraws used in the experiment and the theoretically specified distributions.

48

Table 13: Statistical Distribution of Random Costs and Errors used in the Experiment.

Appendix C. Instructions of the Experiment.

These instructions are for the treatment with positively correlated costs and have been trans-lated from Spanish (except from figures).

INSTRUCTIONS

You are about to participate in an economic experiment. Your profits depend on yourdecisions and on the decisions of other participants. Read the instructions carefully. You canclick on the links at the bottom of each page to move forward or backward. Before startingthe experiment, we will give a summary of the instructions and there will be two trial rounds.

THE EXPERIMENT

You will earn 10 Euros for participating in the experiment regardless of your performance inthe game. You will gain (lose) points during the experiment. At the end of the experiment,points are exchanged for euros. 10,000 points are equivalent to 1 Euro. Each player willstart with an initial capital of 50,000 points. Gains (losses) that you accumulate during theexperiment will be added (subtracted) to the initial capital. Players who have accumulatedlosses at the end of the experiment will receive 10 Euros for participating. Players with gainswill receive their gains converted to Euros plus the 10 Euro participation fee.

The experiment will last 25 rounds. In the experiment you will participate in a market.You will be a seller of a fictitious good. Each market will have 3 sellers. Market participantswill change randomly from round to round. At any given time, no one knows who he ismatched with. We guarantee anonymity. The buying decisions will be made by the computer

49

and not a participant of the experiment. In each round and market, the computer will buyexactly 100 units of the good.

YOUR PROFITS

In each round, your profits are calculated as shown in the figure below:

Your profits are equal to the income you receive from selling units minus total costs(consisting of production and transaction costs).

Some details to keep in mind: you only pay the total costs of the units that you sell. Ifyou sell zero units in a round, your profits will also be zero in this round. You can make losseswhen your income is less than the total costs (production and transaction). The cumulativeprofits are the sum of the profits (losses) on each round. Losses will be deducted from theaccumulated profits. Throughout the experiment, a window in the upper left corner of yourscreen will show the current round and accumulated profits.

YOUR DECISION

In each round, you have to decide the minimum price that you are willing to sell each unitfor. We call these Ask Prices.

THE MARKET PRICE

Once the three sellers in a given market have entered and confirmed their decisions, thecomputer calculates the market price as follows.

1. In each market, the computer observes the 300 Ask Prices introduced by the sellers ofyour market.

2. The computer ranks the 300 Ask Prices from the lowest to the highest.3. The computer starts buying the cheapest unit at the lowest Ask Price, then it buys

the next unit, etc. until it has purchased exactly 100 units. At this time the computer stops.

50

4. The Ask Price of the 100th unit purchased by the computer is the market price (theprice of the last unit purchased by the computer).

The market price is the same for all units sold in a market. In other words, a sellerreceives a payment, which is equal to the market price for each unit he sells. If more thanone unit is offered at the market price, the computer calculates the difference:

Units Remaining= 100- Units that are offered at prices below the market price.The Units Remaining are then split proportionally among the sellers that have offered

them at an Ask Price equal to the market price.

UNITS SOLD

In each round and market, the three sellers offer a total of 300 units. The computer purchasesthe 100 cheapest units. Each seller sells those units that are offered at lower Ask Prices thanthe market price. Note that those units that are offered at higher Ask Prices than the marketprice are not sold. Those units offered at an Ask Price which is equal to the market pricewill be divided proportionally among the sellers that have offered them.

MARKET RULES

In each round and market, the computer buys exactly 100 units of the good at a price notexceeding 3.600. In order to simplify the task of entering all Ask Prices in each round, werequest that you to enter:

• Ask Price for Unit 1

• Ask Price for Unit 2

Ask Prices can be different for different units. To find Ask Prices for the other units, we willjoin the Ask Price for Unit 1 and the Ask Price for Unit 2 by a straight line. In this way,we find the Ask Prices for all the 100 units. In the experiment, you will be able to see thisgraphically and try different values until you are satisfied with your decision.

We apply the following five market rules.1. You must offer all the 100 units for sale.2. Your Ask Price for one unit must always be greater than or equal to the Ask Price of

the previous unit. Therefore, the Ask Price for the second unit cannot be less than the AskPrice for the first unit. You can only enter integers for your decisions.

3. Both Ask Prices must be zero or positive.4. The buyer will not purchase any unit at a price above the price cap of 3,600.5. The Ask Price for some units may be lower your unit cost, since unit costs are unknown

at the time when you decide the Ask Prices. You may have losses.

51

EXAMPLE

This example is illustrative and irrelevant to the experiment itself. We give the example onpaper. Here you can see how the computer determines the market price and units sold byeach seller in a market.

UNIT COST

In each round the unit cost is random and unknown to you at the time of the decision. Theunit cost is independent of previous and future round. Your unit cost is different from theunit cost of other participants. However, your unit cost is related to the unit costs of theother market participants. Below we explain how unit costs are related and we give a figureand explanation of the possible values of unit costs and their associated frequencies. Thisfigure is the same for all sellers and all round.

The horizontal displays the unit cost while the vertical axis shows the frequency withwhich each unit cost occurs (probability). This frequency is indicated by the length of thecorresponding bar.

In the figure you can see that the most frequent unit cost is 1,000. We obtain 1,000 asunit cost with a frequency of 0.35%. In general terms, we would obtain a unit cost of 1,000in 35 of 1,000 cases.

In 50% of the cases (50 of 100 cases), the unit cost will be between 933 and 1,067.In 75% of the cases (75 of 100 cases), the unit cost will be between 885 and 1,115 .In 95% of the cases (95 of 100 cases), the unit cost will be between 804 and 1,196 .There is a very small chance that the unit cost is less than 700. This can occur in 1 of

1,000 cases approximately. Similarly, there is a very small chance that the unit cost is greaterthan 1,300. This occurs can occur in 1 of 1,000 cases, approximately.

For participants with knowledge of statistics: the unit cost is normally distributed with

52

mean 1,000 and standard deviation 100.

INFORMATION ABOUT YOUR UNIT COST (YOUR SIGNAL)

In each round, each participant receives information on his unit costs. This information isnot fully precise. The signal that you receive is equal to:

Signal = UnitCost+ Error

The error is independent of your unit cost, it is also independent from the unit costs ofother participants and it is independent from past and future errors. The following figuredescribes the possible values of the error term and an indication of how likely each error islikely to occur. This graph is the same for all sellers and rounds.

On the horizontal axis you can observe the possible values of the error terms. On thevertical axis, you can observe the frequency with which each error occurs (probability). Thisfrequency is indicated by the length of the corresponding bar.

In the figure you can see that the most common error is 0. The frequency of error 0 is0.66 %. In general terms, this means that in approximately 66 of 10.000 cases you would getan error equal to 0.

In 50% of the cases (50 of 100 cases), the error term is between -40 and 40.In 75% of the cases (75 of 100 cases), the error is between -69 and 69.In 95% of the cases (95 of 100 cases), the error is between -118 and 118.There is a very small chance that the error is less than -200. This occurs in 4 out of

10,000 cases. Similarly, there is a very small probability that the error is greater than 200.This occurs in 4 out of 10,000 cases.

For participants with knowledge of statistics: the error has a normal distribution withmean 0 and standard deviation 60.

53

HOW YOUR COST IS RELATED TO THE COSTS OF THE OTHER SELLERS

The unit cost is different for each seller and your unit cost is related to the unit cost of theother sellers in your market. The association between your unit cost and unit cost of anotherseller in your market follows the trend:

• The higher your unit cost, the higher will be the unit cost of the other sellers.

• The lower your unit cost, the lower the unit cost of the other sellers.

The strength of the association between your unit cost and unit cost of another seller ismeasured on a 0 to 1 scale. The strength of the association between your unit cost and unitcost of the other seller is +0.6 .

Graphically we can see the relationship between your unit cost (horizontal axis) and theunit cost of another seller (vertical axis) for some strengths of association. The figure thathas a red frame corresponds to an intensity of association of +0.6.

For participants with knowledge of statistics: the correlation between your unit cost andunit cost of any other player is +0.6 .

END OF ROUND FEEDBACK

At the end of each round, we will give you information about:

• Your profits (losses) and its components (Revenue-Cost of Production - Cost of trans-action)

• Market price

• Your units sold

54

• Other market participants feedback: decisions; profits and unit costs.

You can also check your historical performance in a window in the upper right corner of yourscreen. During the experiment the computer performs mathematical operations to calculatethe market price, units sold, Ask Prices for intermediate units, etc. For these calculation weuse all available decimals. However, we show all the variables rounded to whole numbers,except from the market price.

THE END

This brings us to the end of the instructions. You can take your time to re-read the in-structions by pressing the BACK button. When you understand the instructions you canindicate it to us by pressing the OK button at the bottom of the screen. Next you haveto answer a questionnaire about the instructions, unit cost distributions and signals. Whenall participants have taken the questionnaire and indicated OK, we will start the practicerounds. Your profits or losses of the practice rounds will not be added or subtracted to yourearnings during the experiment.

Appendix D. Comprehension Questionnaire & Screenshots

of the Experiment

The first part of this appendix reports the comprehension questionnaire which was admin-istered before the trial rounds.

Comprehension Quiz

Questions. Answer True or False.

1. The unit cost has the same value for each of the participants in your market.

2. The unit cost has the same value for each of the participants in your market.

3. If my unit cost is high, it is rather likely that the unit cost of another seller is high.

4. Unit costs between 1000 and 1200 occur with the same frequency than unit costsbetween 1000 and 700.

5. Unit costs larger than 1000 occur with the same frequency as unit costs smaller than1000.

55

6. Errors larger than 0 occur more frequently than errors smaller than 0.

7. An error of 5 is the most frequent error.

8. The seller who sells most units will always have the highest profit.

9. If my unit cost is low, it is rather likely that the unit cost of another seller is high.TRUE FALSE NOTE – treatment 0.6: The lower your unit cost, the lower the unitcost of the other sellers will tend to be.

10. The market price is the same for all units and sellers.

Answers (True (T) and False (F)): Q1. F Q2. T Q3. T (treatment 0.6); F (treatment0) Q4. F Q5. T Q6. F Q7. F Q8. F Q9. F Q10. T

Notes: These notes appeared on the screen if a participant answered wrongly any of theprevious questions.

Q1. Treatment 0.6: Your unit cost is different from the unit cost of other participants butit is related. Treatment 0: Your unit cost is different from the unit cost of other participants.There is no relation between your unit cost and that of other participants.

Q2. In each round, the unit cost is random and independent from the unit cost of pastand future rounds.

Q3. Treatment 0.6: The higher your unit cost, the higher the unit cost of the other sellerswill tend to be. Treatment 0: There is no relation between your unit cost and that of otherparticipants. Therefore, if my unit cost is high, I can not deduce anything from the unit costof the other participants.

Q4. Unit costs between 1000 and 1200 occur with higher frequency than unit costsbetween 1000 and 700.

Q5. The unit cost of 1000 is the most frequent one. Unit costs larger than 1000 occurwith the same frequency as unit costs smaller than 1000.

Q6. Errors larger than 0 occur with the same frequency as errors smaller than 0.Q7. An error of 0 is the most frequent error.Q8. Profit does not only depend on the number of units sold. Remember that: Profit =

(MarketPrice− UnitCost)UnitsSold− 1.5UnitsSold2.Q9. Treatment 0.6: The lower your unit cost, the lower the unit cost of the other sellers

will tend to be. Treatment 0: There is no relation between your unit cost and that of otherparticipants. Therefore, if my unit cost is high, I can not deduce anything about the unitcost of the other participants.

56

Q10. The market price is the same for all units and sellers in a market.

The second part of this appendix reports the screenshots used during the experiment.

Screen 1: Signal Screen

Screen 2: Decision Screen

57

Screen 3: Feedback about a seller’s own performance

Screen 4: Feedback about market performance and other sellers’ in

the same market

Appendix E. Post-experimental questionnaire

After the rounds were completed, we asked for 3 demographic questions: age, gender anddegree studying.We then asked the following additional questions regarding understanding of the game.

58

1. Do you think that a high market price generally means GOOD/MIXED/BAD news aboutthe level of your costs?2. Explain your answer.3. Do you think that the other sellers have answered the same as you to the previous question?4. Explain your answer.

Appendix F. Analysis of Variance & Decomposition of

deadweight loss into its components

The following table decomposes the variance of behaviour and outcomes in each treatmentinto its components.

Table 14: Analysis of Variance

Notes. The number of observations for the standard deviations reported in the third and fourth columnsfor market price and deadweight losses should be divided by 3 with respect to the corresponding number ofObservations/Treatment since these variables should be at the market level rather than at the subject level.

We can show that the variance of the supply function slope is larger in the positivelycorrelated than in the uncorrelated treatment at the different levels of aggregation: betweengroups (Variance ratio test, one-sided, n1 = 6, n2 = 6, p = 0.027) and between subjects(Variance ratio test, one-sided, n1 = 72, n2 = 72, p = 0.0012). The same finding is also truefor the supply function intercept: between groups (Variance ratio test, one-sided n1 = 6,n2 = 6, p = 0.027) and between subjects (Variance ratio test, one-sided, n1 = 72, n2 =

59

72, p = 0.000). The differences across periods at different levels of aggregation are notsubstantial.

The following graph illustrates how the components of deadweight loss combine in eachtreatment.

Figure 9: Components of deadweight loss at the experimental allocation in each treatment.

Notes. This graph corresponds to the decomposition of deadweight loss, dwl = (λ2 )(∑ni=1(x

ei −

qn )

2 + (xoi −qn )

2 − 2(xei −qn )(x

oi −

qn )). The first term is variance of units sold, the second is the variance of the efficient

allocation, and the third is -2 multiplied by the covariance of units sold and the efficient allocation. The barsassociated with “Sum of components” refer to the sum (

∑ni=1(x

ei −

qn )

2+(xoi −qn )

2− 2(xei −qn )(x

oi −

qn )). dwl

refers to deadweight loss.

The next table displays the components of deadweight loss at both the equilibrium andexperimental allocations in each treatment.

60

Table 15: Components of Deadweight Loss at the equilibrium and experimental allocationsin each treatment.

Notes. Alloc. refers to allocation. In order to calculate dwl in rows (5) and (9), we have used the actualdistribution of draws of the random variables used in the experiment.

Appendix G. Best Response Analysis

This section uses the notation of the theoretical background presented in Section 3 andcomputes a seller’s best response strategy given arbitrary strategies of rivals. We assume thatan agent knows the strategies of the rivals, and that he forms correct beliefs about events inthe competitive and information environments. The best response strategy of seller i can bewritten as Xi(si, p) = bi − aisi + cip, and the actual strategies of the rivals as Xj(sj, p) =

bj−ajsj+cjp, where j 6= i. The rivals’ average supply function slope is c−i = 1n−1

∑i 6=j cj, and

the rivals’ average fixed part of the intercept is b−i = 1n−1

∑j 6=i bj. Furthermore, for simplicity,

we assume that all rivals set the same response to private information, i.e. aj = a−i for allj 6= i.

Lemma 1 determines agent i ’s best response strategy given arbitrary strategies of therivals. The assumptions required for Lemma 1 to hold are consistent with the features of ourexperimental design.

Lemma 1: Best Response Strategy

Assume that ρ ∈ [0, 1) and that the rivals’ average supply function is characterised

61

by (a−i, b−i,, c−i). Suppose that rivals set a supply function such that c−i > 0 andthat all rivals set the same response to private information such that aj = a−i > 0

for all j 6= i. The best response strategy for seller i is then given as:

ai =R

di + λ+ Ti, (15)

bi =(θ(R + Ti(n− 1)a−i − 1) + Ti(q − (n− 1)b−i))

di + λ+ Ti, (16)

ci =1− Ti(n− 1)c−idi + λ+ Ti

, (17)

where di = 1(n−1)c−i

, R =σ2θ(σ2

θ(1−ρ)(1+(n−1)ρ)+σ2ε )

(σ2θ(1−ρ)+σ2

ε )(σ2θ(1+(n−1)ρ)+σ2

ε ), and

Ti =ρσ2θσ

2ε

(σ2θ(1−ρ)+σ2

ε )(σ2θ(1+(n−1)ρ)+σ2

ε )a−i.

Proof: Given the strategies of the rivals, seller i maximises his profits and the first ordercondition is given as: Xi(si, p) = p−E[θ|si,p]

(di+λ). Market clearing implies that q =

∑nj=1 X(sj, p),

and given the definitions of b−i, c−i and the assumption that aj = a−i for all j 6= i, we can re-write the market clearing expression as: p(n−1)c−i = q− (n−1)b−i+a−i

∑i 6=j sj−xi. Then

define di = 1(n−1)c−i

and Ii = q − (n− 1)b−i + a−i∑

i 6=j sj so that we can write p = Ii − dixi.All the information contained in p is also contained in hi, where hi = a−i

∑j 6=i sj and can be

shown to be equal to hi = (n − 1)b−i − q + (n − 1)c−ip + xi. The second order condition issatisfied if 2di + λ > 0, which is always satisfied if c−i > 0.We can now find an expression for E[θi | si, p] = E[θi | si, hi]. The mean of the vector θi

si

hi

is equal to

θ

θ

a−i(n− 1)θ

and the variance-covariance matrix is:

σ2θ σ2

θ ρσ2θa−i(n− 1)

σ2θ σ2

θ + σ2ε ρσ2

θa−i(n− 1)

ρσ2θa−i(n− 1) ρσ2

θa−i(n− 1) a2−i(n− 1)((σ2

ε + σ2θ) + (n− 2)ρσ2

θ)

Using the expressions for conditional expectations of normally distributed random variables,we obtain:

E[θi | si, hi] = θ +R(si − θ) + Ti(hi − (n− 1)a−iθ), (18)

where

62

R =σ2θ(σ

2θ(1− ρ)(1 + (n− 1)ρ) + σ2

ε )

(σ2θ(1− ρ) + σ2

ε )(σ2θ(1 + (n− 1)ρ) + σ2

ε ), (19)

and

Ti =ρσ2

θσ2ε

(σ2θ(1− ρ) + σ2

ε )(σ2θ(1 + (n− 1)ρ) + σ2

ε )a−i. (20)

In order to obtain the best response strategy for seller i, we first equate the coefficient onthe signal, si and obtain: ai = R−Tiai

(di+λ), or equivalently: ai = R

di+λ+Ti. Second, we equate

the coefficient on the price and obtain: ci = 1−Ti(n−1)c−i−Tici(di+λ)

. Grouping terms with ci, weobtain: ci = 1−Ti(n−1)c−i

(di+λ+Ti). Third, we equate the coefficient on the constant and obtain:

bi = θ(R+Ti(n−1)a−i−1)+Ti(q−(n−1)b−i)(di+λ+Ti)

. �The next Lemma describes the comparative statics of agent i’s best response function.

Lemma 2: Strategic incentives

Assume that ρ ∈ [0, 1) and that the rivals’ average supply function is characterisedby (a−i, b−i,, c−i). Suppose that rivals bid a supply function such that c−i > 0 andthat all rivals set the same response to private information such that aj = a−i > 0

for all j 6= i.

If ρ = 0 then: ∂ai∂a−i|c−i=const= 0, ∂bi

∂b−i|a−i=const,c−i=const= 0 and ∂ci

∂c−i|a−i=const> 0.

If ρ > 0 then: ∂ai∂a−i|c−i=const> 0, ∂bi

∂b−i|a−i=const,c−i=const< 0,

∂ci∂c−i|a−i=const> 0 for 0 < c−i < c∗−i, where c∗−i is a positive number

and ∂ci∂c−i|a−i=const< 0 for c−i > c∗−i.

Proof: We first evaluate:

∂ci∂c−i

|a−i=const=−Ti(n− 1)2(λ+ Ti)(c−i)

2 − 2Ti(n− 1)c−i + 1

(n− 1)(c−i)2(di + λ+ Ti)2. (21)

since ∂di∂c−i

= −1(n−1)(c−i)2

. When ρ = 0 then Ti = 0 and ∂ci∂c−i|a−i=const> 0. When ρ > 0 then

Ti > 0 and the numerator is a polynomial of degree two in c−i, which can be written as:

f(c−i) = −Ti(n− 1)2(λ+ Ti)(c−i)2 − 2Ti(n− 1)c−i + 1 (22)

63

The maximum of this polynomial occurs at cmax−i = −1(n−1)(λ+Ti)

< 0 and the parabola opensdownwards since the coefficient on (c−i)

2 is negative. The polynomial has a positive (c∗−i)and a negative root. Since we have assumed that c−i > 0, we note that ∂ci

∂c−i|a−i=const> 0 for

0 < c−i < c∗−i and∂ci∂c−i|a−i=const< 0 for c−i > c∗−i.

Second, we take the derivative of ai with respect to a−i. When ρ = 0 then ∂ai∂a−i|b−i=const,c−i=const

=

0. When ρ > 0 then

∂ai∂a−i

|c−i=const=−R

(di + λ+ Ti)2

∂Ti∂a−i

> 0 (23)

since ∂Ti∂a−i

< 0.Third, we take the derivate of bi with respect to b−i. When ρ = 0 then ∂bi

∂b−i|a−i=const,c−i=const=

0. When ρ > 0 then

∂bi∂b−i

|a−i=const,c−i=const=−Ti(n− 1)

di + λ+ Ti< 0 (24)

�

In the experiment, for each two dimensional choice, we observe a subject’s supply func-tion slope and intercept. Concerning the intercept, we cannot separately observe the twocomponents of the supply function’s intercept: response to the private signal and fixed part.This issue has consequences for the computation of the best response to the rivals’ supplyfunction in the positively correlated costs treatment, where we need to have an estimateof the rivals’ response to the private signal, a−i. We make the assumption that all sellersequally and optimally respond to the private signal, i.e. aj = a−i for all j 6= i. Introducingthis assumption into the best response strategy, we obtain: aj = a−i =

(1−ρ)σ2θ

(1−ρ)σ2θ+σ2

ε(di + λ)−1.

The figure below illustrates the best response slope, ci, as a function of the rivals’ averageslope, c−i, in each treatment (with the supply function viewed in (Ask Price, Quantity)space).

64

Figure 10: Best response supply function slope as a function of the rivals’ average supplyfunction slope in each treatment and the corresponding equilibrium predictions in the (AskPrice, Quantity) space.

Notes. The supply function is viewed in the (Ask Price, Quantity) space, which inverts the axes in relationto most of the figures shown throughout the text. Note that a steep supply function has a low c whenrepresented in the (Ask Price, Quantity).

65

Appendix H. Cluster Analysis

The next table presents the results of cluster analysis in the intermediate time periods ingroups of five rounds.

Table 16: Cluster analysis for the intermediate time periods.

Notes. The numbers in parenthesis below the average correspond to standard deviations (s.d.). The equilib-rium SF has been calculated using the average signal realisation.

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The following table summarises the average difference between the experimental supplyfunction and the theoretical best response supply function in each cluster.

Table 17: Difference between the average Experimental and Best Response supply function ineach cluster.

Notes. The numbers in parenthesis below the average correspond to the standard deviation. The theoreticalbest response has been calculated for each individual supply function choice and then averaged for subjectsin a particular cluster and time period. In order to calculate these averages, we have eliminated the choiceswhich had a best response which was unfeasible for subjects in the experiment.

67

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