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An Experimental Study of the Centipede Game with Aggregate Information

An Experimental Study of Aggregate Information Revelation in the Centipede Game.

Zacharias Maniadis1

Abstract: Is aggregate information about the behavior of people beneficial to society? We performed several experiments of the centipede game with a new treatment, in which subjects received information about aggregate play, and the answer we found was surprisingly negative. Based on the results of previous experimental studies of the centipede game by McKelvey and Palfrey , and drawing from the insights of Fudenberg-Levine we expected that revealing aggregate information would encourage more trusting behavior and would increase subjects’ payoffs. Our results show that, contrary to expectations, aggregate information is detrimental for the evolution of trust. Our experiments contribute in answering other important questions as well, such as why the equilibrium prediction fails in the previous centipede game experiments, and what is the relative importance of conformity preferences when aggregate information is provided.

(1992)(1997)

Keywords: Game theory, Experimental economics. 1 Economics PhD student, UCLA.

1


1. Introduction

The information that an agent receives about how other people behave matters for the

agent’s own behavior.2 Accordingly, information revelation of aggregate data may be a

powerful tool, in the hands of those who possess it, typically governments and special

interests, to influence the behavior of the public.3 In Maniadis ( we argue that it is

useful to think of the government as a benevolent social planner who reveals selectively

information to maximize the social surplus. Hence, for economic policy, it is important

to know whether aggregate information is beneficial or detrimental for society. We

show with an experiment that aggregate information works in unexpected ways, and its

release may be harmful for society even if there is evidence for the contrary.

2007)

In this study, we focus on an evolutionary game with anonymous matching. Our

experiments are devised accordingly, with each player interacting with each opponent

exactly once. There is a large population of agents in the role of a single ‘player’ in the

game, and we shall consider the effects of selective information revelation without

reputation effects.4 To test how information revelation affects the evolution of play, we

perform a series of experimental sessions of the four-move centipede game with

2 Theories of learning in games support this idea. The notion of self-confirming equilibrium (Fudenberg-Levine, 1993a) is sensitive to aggregate information revelation. Thus, even equilibrium behavior could be affected by aggregate information. There is a vast literature in social psychology regarding social influence, conformity, social norms and cognitive dissonance. See Cialdini and Goldstein ( ), Cialdini et al , Burger (2001).

2004)1999(

3 This issue is very significant for economic policy, in many different policy areas such as corruption control, crime prevention, social discrimination and others. See Maniadis (2007) for a more thorough discussion. 4 We have not been able to rule out repeated game effects totally, however, because of the potential that players will realize they can affect the aggregate information revealed.

2


information revelation. As we shall explain in part , the centipede game is a

particularly interesting one to test the effects of information.

2

Our main result is that revealing aggregate information leads to a reduction in the

subjects’ payoffs, and hence to a reduction in social welfare in our model society.

Furthermore, our experiments offer some insight to different questions. Why do

subjects typically fail to play according to the Nash equilibrium prediction in the

centipede game experiments? The results suggest that the relatively small number of

repetitions is responsible, to a large degree, for this.5 Does aggregate information lead to

Nash equilibrium? Our answer is that under some conditions yes, but not always. Which

theory explains better the results in the centipede experiments? Our result cannot be

captured by theories based on pure altruism, so alternative approaches are required.

What is the relative importance of social norms and strategic information in achieving

these results? It seems that conformity preferences play a major role in explaining our

results.

Part discusses the centipede game with exponentially increasing payoffs and the

experimental literature of this game. Part 3 reviews the experimental literature on

aggregate information treatments. Part introduces the experiments. The results are

presented in part and discussed in part . We conclude in part .

2

4

5 6 7

5 The players never actually get to know the true distribution of play.

3


2. Previous Experiments of the Centipede Game and Theoretical Explanations.

In the two-player centipede game (Figure1), a monetary amount is divided between two

players into a large and a small pile, in a predetermined way for each terminal node. In

each decision node, the player who moves can either “take” the large pile of money and

the games ends, or “pass” for the next round. A player should always “take” now, if he

expects that the other player will “take” in the subsequent move, but each player is

better off passing now, if it is expected that the other player will also pass when her turn

comes. In its finite version, the centipede game has an obvious candidate for a

prediction of how it will be played: backward induction shows that in all Nash

equilibria of the game, player one “takes” in the first move.

1 2 1 2

T4 T3 T2T1

P1 P2 P3 P4

The Two-Player Centipede Game

with Geometrically Increasing Payoffs

)4.2,6.9(

)15.0,6.0( )2.1,3.0( )6.0,4.2( )8.4,.2.1(

Figure 1

Experimental studies have found little support in favor of the Nash prediction, and it

seems that subjects do not exclusively use backward induction and do not assume full

4


rationality of others when they try to predict other people’s behavior. Most early

experiments of the centipede game found very little support for the theoretical

equilibrium outcome. (Note that here and in the remaining of the paper we shall mainly

refer to the last five rounds of experiments, where play is more likely to have converged

to equilibrium). McKelvey and Palfrey, in their experimental study of four-round and

six-round centipede games , find that no more than 8% of the total number of

subjects choose “take” in the first decision node which corresponds to the Nash

equilibrium outcome. Fey, McKelvey and Palfrey find that, even in a setting of

constant social payoffs, where the predictions of Nash, fairness and focal point theories

agree in the same predicted outcome (player one takes at stage one), players fail to

achieve the equilibrium outcome to 80 percent of the time, depending on the

version of the game.

(1992)

(1996)

30

Nagel and Tang , using the equivalent normal form of the game, find relative

frequencies of equilibrium play not exceeding . Other authors find more support for

the equilibrium by changing the basic features of the game, usually confounding more

than one such change in the same experiment. Stein, Rappoport, Parco, and

Nicholas ( , find that equilibrium play is chosen to percent of the time in an

experiment where each “inning” of choices involved three, rather than two, players,

stakes was significantly higher relative to MP and the last node gave zero

payoffs to all players. Murphy, Rappoport, Parco, used a discrete time version of

the centipede game with three players, and they showed that games end earlier in later

(1998)

5%

2003) 30 40

(1992)

(2006)

5


rounds, which is evidence of convergence to equilibrium. With seven players,

convergence is complete in all sessions.

To explain their results, McKelvey and Palfrey6 and Camerer and Weigelt

suggested that subjects do not perceive the other player’s payoffs the way the

experimenters would like them to, but they assign some probability that their opponent

is a different type, for example an “altruist” who passes at every opportunity. Fudenberg

and Levine’s

(1992)

(1988)

7 analysis differs in some important aspects from the other

approaches. Players do not know the payoffs of their opponents, and they do not even

have assessments or “home-made priors” about opponents’ payoffs, but they only have

beliefs about the distribution of actions in the population. The authors argue that much

of the behavior in centipede games, mainly in the results of MP ( , can be

explained as equilibrium behavior. Actions are optimal with respect to beliefs, about the

distribution of opponents’ actions, which need not be correct for those nodes that are

not reached given the subject’s strategy

(1997)

1992)

8.

We wish to examine this interpretation and we follow FL’s suggestion to compare

treatments with full information revelation of aggregate play with treatments where

people only observe play in their own matches.9 If people are rational, but do not

6 The authors will be referred to as MP. 7 The authors will be referred to as FL. 8 The only part of the data that cannot be explained according to this behavior is the choice of “pass” at the last node. 9 FL’s theory clearly implies that selective information revelation of aggregate data matters even in equilibrium. If people are “trapped” in a specific strategy and wrong beliefs, due to their strong priors and lack of experimentation, then, in the face of information revelation about the aggregate statistics, their

6


experiment actively, then we would expect that information revelation in the four-move

centipede of MP will lead to an increase in social welfare. To understand why, a

careful look at figure1 reveals that at each node, the player who moves would maximize

her payoffs by choosing “pass” if she knew that the probability that her opponent will

choose “take” in the next round is less than

(1992)

76 . In the data of MP , the aggregate

fraction of “take” was less than

)1992(

76 in almost all decision nodes, in all four-move

centipede treatments. Consequently, FL argued that if the players knew these

fractions, they would optimize by passing all the way at least until the last decision

node. Hence, players would achieve high payoffs.

(1997)

3. The Experimental Literature on Aggregate Information

As far as we know, there have been few economic experiments that directly examine

the possibility that the experimenter can manipulate the equilibrium behavior of people.

Roth and Schoumaker revealed “manipulated” private histories in the ultimatum

game by having subjects play with computer opponents. This treatment had significant

and lasting effects, as subjects continued to make agreements in the same range of

offers and proposals as they were used to in the computer rounds. Harrison and

McCabe ( use both the computer-subject treatment of Roth-Schoumaker, and

information revelation of aggregate data

(1983)

1996)

10 to try to manipulate subjects’ expectations in

expectations could change in a predictable way9. This leaves the door open to manipulation of people’s behavior by those who posses the aggregate information. 10 This data was on contingent strategies, not actions.

7


the ultimatum game. They found that the release of information about the aggregate

data had a major effect on play, relative to the results of the control treatment, because it

allowed for the consistency of expectations. In the aggregate information revelation

treatment, where subjects were able to see the “contingent plans” of all senders and

responders after each round, the offers gradually declined and signs of convergence to

the subgame perfect equilibrium outcome of “zero offers” appeared.

Berg, Dickhaut and McCabe performed experiments of one-round trust games(1995) 11,

and found some support for the notion that information revelation of aggregate data can

‘push’ the economy to desirable equilibria. They found that when subjects received

information about the actions of different subjects who played the game, social surplus

increased. There was a large increase in the relative amount which receivers returned.

However, the increase in the amount sent was small and not statistically significant.

Dufwenberg and Gneezy ( reported the results of experimental auctions that

resemble Bertrand price competition

2002)

12, and they found that treatments with information

about the entire vector of bids had a very different steady state than treatments where

subjects received less information13. In the full information case, winning bids remained

much higher than the theoretical prediction of zero, whereas in the other treatment bids

11 Each “sender” had 10$ that he could send to the receiver. The amount sent tripled, and then the receiver decided how much money to send back to the sender. 12 Each subject was coupled with another subject and each chose a bid in integers between 2 and 100. The subject that submitted the lowest bid won the auction and received a monetary amount times the winning bid. The subject with the losing bid won zero, and in cases of a tie the won amount is split. The fact that subjects were randomly matched is good for our comparisons. However, the small number of participants make signaling possible, where the ability of evolutionary approaches to explain the data is limited. 13 In one treatment they received either no feedback at all, and in the other they got to see the winning bids always, even if they lost.

8


converged towards zero. Hargreaves–Heap and Varoufakis used a hawk-dove

(symmetric) game where each subject was assigned to one of two groups using the

random choice of two arbitrary colors. They showed that revealing the aggregate play of

the two groups had a very significant impact in the evolution of the distributions of

play. Groups, whose members started by playing “hawkish” or “dovish” more,

invariably ended up playing this strategy exclusively. Finally, Friedman , and

Cheung and Friedman ( compare different information conditions in several

normal-form evolutionary games. They typically find that aggregate information can

change the rate of convergence to the behavioral equilibrium, but not the equilibrium

itself.

(2002)

(1996)

1997)

4. The Experiment.

Seven experimental sessions were conducted at the California Social Science Lab

(CASSEL) at UCLA. All the subjects were UCLA students. In all sessions the number

of subjects was thirty, except in two (twenty-eight). Each subject played fifteen (or, in

the -subject treatments, fourteen) rounds of the four-move centipede game28 14

allowing for many repetitions and learning. Subjects also had the chance to gain

experience with the game during three practice sessions. The relatively large number of

subjects mitigated the effects of repeated games and signaling that information

14 The game played in all sessions is exactly the one described in figure 1. Payoffs were 50% higher than MP (1992) in all nodes. Given the time lapsed from these experiments, the stakes are similar.

9


revelation made possible.15 The matching scheme was the same as in MP’s

experiments. A rotating matching scheme was used, and subjects were separated into

two groups of fifteen

(1992)

16, the composition of which was fixed throughout the

experiment.17 Each participant was matched with each subject that belongs to the other

group exactly once.

All the information about the structure of the game and the matching details was made

public knowledge to the subjects, since the instructions were read in public. The

subjects did not seem to have particular difficulties understanding the game, and also

had many opportunities to learn during the practice rounds and the repetitions of the

game. Subjects were not allowed to talk or otherwise interact, except through the

computers. Appendix two contains the instructions for treatment FIR.

In two of the sessions, the treatment was called ‘No Information Revelation’ (NIR1 and

NIR2). This was essentially the same treatment as in the four-move centipede

experiments of MP , only with higher dollar payoffs. All the other sessions

involved information revelation. In sessions FIR1 and FIR the treatment was the “Full

Information Revelation” treatment, in which, subjects received information about how

the members of both groups played in the previous round. In particular, during any

round, all subjects saw the fractions of the players in each group that chose “pass” and

(1992)

2

15 Such effects are possible because if a particular node is reached only a few times in a given round, the choices of one player in one single match have a large impact on the fractions of play as the subjects will see it in the next round. We shall discuss the implications of this more thoroughly. 16 Or, for the unique 28-subject session, two groups of 14. 17 For subjects, the groups were labeled “the GREEN group” and the “YELLOW” group. The members of the GREEN group always had the role of player 1 in the centipede game and the members of the YELLOW group always had the role of player 2.

10


“take”, in each of the decision nodes of the game, in the previous round.18 For example,

during the tenth round, in the first decision node, all subjects saw the fraction of the

members of the GREEN group that chose “pass” or “take”, in this particular node,

during the ninth round. In the second decision node, all subjects were shown the

fractions of the members of the YELLOW group that chose “pass” and “take”, in this

node, during the ninth round. Similarly, the subjects saw the responding information for

all other decision nodes.19 In sessions PIR1, PIR 2 and PIR , the treatment provided

“Partial Information Revelation”. The same information as in treatment FIR was

provided, but only for the “other” group. For example, all GREEN subjects in round

five, would be able to see the fractions of the YELLOW group of subjects that choose

“pass” or “take” in the fourth round in all nodes where YELLOW moves. Subjects

could not see the fractions of choices in nodes where their own group moved. Figure

illustrates the main features of the seven sessions. We will use the convention of calling

sessions FIR1, FIR2, PIR1, PIR3 and PIR3 ‘information sessions’ and NIR and FIR

‘information treatments’. The rest, including the sessions in MP ( , will be called

‘non-information sessions’ belonging to the ‘non-information treatment’ NIR.

3

2

1992)

18Note that each node belongs to members of one group only. 19 Of course, since not all nodes were reached in each match, subjects saw information only about those subjects who moved in each particular node in the previous round.

11


Session Number

of

subjects

Aggregate Information Number of

Matches

Game

NIR 30 NO 225 4 -Move Centipede

NIR 28 NO 196 4 -Move Centipede

FIR1 30 FULL 225 4 -Move Centipede

FIR2 30 FULL 225 4 -Move Centipede

PIR1 30 OTHER GROUP ONLY 225 4 -Move Centipede



Figure 2

5. Results

Appendix 1 contains descriptive data for all sessions.20 Our data have some of the main

features of other experiments of the centipede game. In particular, one major stylized

fact from previous experiments is that the conditional “take” probabilities21 increase as

we move from the first to the last decision node of the game. This was true for our data,

for all sessions and all decision nodes. However, the data in the FIR and PIR treatments

have some substantial new features. First of all, in all information treatments except

20 We use the notation in figure1 to describe the data. The terminal nodes are denoted T1, T , T 3 , T and P 4 .

2 4

21 For a decision node, the ‘conditional “take” probability’ is the fraction of people who chose “take” in this node in the experiment.

12


one, convergence to the Nash equilibrium outcome (T1) was very strong in the late

rounds, much stronger than the results of MP . Accordingly, the differences in

play between the early and the last rounds of the experiment were very large in all

information sessions except one, PIR1. Furthermore, in the control sessions, without

aggregate information release, a significant fraction of matches in the last five rounds

ends in the Nash equilibrium outcome ( and37% ). This is much larger than the 8%

found in MP ( . This indicates that the number of rounds or the subject pool may

have played a role in the results. Another interesting feature of the data is that in the

‘information treatments’ very few people chose “pass” in the last decision node. Finally,

the large difference in aggregate play between session PIR1 and PIR and PIR3 is also

interesting and warrants an explanation. As we will argue, one explanation is that

round-per-round information revelation causes play to be path-dependent.

(1992)

29%

1992)

2

Figures display the fraction of total matches that end in each of the five terminal

nodes in our three treatments, NIR, PIR and FIR. There are 225 matches in each

fifteen-round session and 196 matches in the fourteen-round session. The data from the

all the sessions of a given treatment are pooled. Thus, there are observations for the

NIR treatment, observations for the FIR treatment, and observations for the

PIR treatment. The differences in aggregate play in all rounds between information and

non-information sessions, displayed in figures3

3 5−

421

450 646

5− , are not very large. Figures 6 8− ,

which show the respective data for the last five rounds of play, tell a different story. The

fraction of matches with information revelation that end in the Nash outcome is

about50% . The results are even more extreme if we consider the PIR treatment

13


excluding session PIR, shown in figure . We are not arguing here that PIR1 should be

discarded. It seems that the distributions in all PIR sessions were significantly different

that the control sessions, but PIR1 was different ‘in the opposite direction’ than PIR2

and PIR3.

9

We tested for statistical significance of the differences in the distributions in the last

five rounds across treatments. We assumed that in the last five rounds play has

converged, and therefore each observation is independent of the others. A chi-square

test of homogeneity of all three treatments, using all seven sessions, gave Chi-square

value 14.92 of with a p-value . Session PIR1was responsible, to a large degree, for

this result. We have seen that if this session is excluded, the pooled results for PIR are

very different. However, the Chi-square test in this case is certainly inappropriate with

no observations in the last two nodes of the PIR distribution. A simple test of

differences in the proportions of equilibrium play (all the other nodes were pooled into

one category called T) between NIR and FIR gave z= and a p-value of less that

. Testing for the same difference in proportions between NIR and PIR we found

z=1.67 , with a p-value , and between FIR and PIR we found z=1.55 , with p-

value of 0.061 . It should be noted that the very low observed frequency of play in the

node P creates a problem regarding the appropriateness of the chi-square test of

homogeneity.

0.061

2.93

0.003

0.0475

4

14


NIR Treatment, Aggregate Play in all Rounds (N=421)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

No

de Series1

Series1 0.266 0.375 0.228 0.102 0.029

T1 T2 T3 T4 P4

Figure 3

PIR Treatment, Aggregate Play in all Rounds (N=646)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

No

de Series1

Series1 0.27 0.383 0.252 0.085 0.0077

T1 T2 T3 T4 P4

Figure 4

15


FIR Treatment, Aggregate Play in all Rounds (N=450)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

No

de Series1

Series1 0.302 0.34 0.242 0.108 0.0066

T1 T2 T3 T4 P4

Figure 5

NIR Treatment, Aggregate Play in the Last Five Rounds(N=145)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

No

de Series1

Series1 0.331 0.386 0.186 0.076 0.02

T1 T2 T3 T4 P4

Figure 6

16


PIR Treatment, Aggregate Play in the Last Five Rounds (N=220)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

No

de Series1

Series1 0.418 0.359 0.159 0.055 0.009

T1 T2 T3 T4 P4

Figure 7

FIR Treatment, Aggregate Play in the Last Five Rounds (N=150)

0

0.1

0.2

0.3

0.4

0.5

0.6

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

N

ode

Series1

Series1 0.5 0.353 0.126 0.02 0

T1 T2 T3 T4 P4

Figure 8

17


PIR2 and PIR3, Aggregate Play in the Last Five Rounds (N=145)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Terminal Node

Frac

tion

of a

ll M

atch

es E

ndin

g in

this

Nod

e

Series1

Series1 0.593 0.331 0.076 0 0

T1 T2 T3 T4 P4

Figure 9

To illustrate more the convergence over time that occurred in most ‘information’

sessions, figures display the path of play in sessions NIR1, PIR , PIR3 and

FIR . The fraction of matches ending in each terminal node is displayed, round by

round. In session NIR (figure10 ), no obvious trends are apparent through rounds,

except perhaps a small tendency for the frequency of T1 to increase. Session FIR

(figure1 ) is particularly interesting. Within seven rounds, play had already shown

strong signs of convergence, and the fraction of equilibrium play reached 80% .

However, at this point, some subjects may have realized that signaling is possible and

passed as a means to induce more passing in the future.

1310 − 2

2

2

1

22 Passing behavior increased for

a few rounds and then fell again until play returned to equilibrium. In PIR , a tendency

for equilibrium play to increase through time is clear, and signaling does not seem to

2

22 In the nodes where play has converged, some nodes are never reached or very seldom reached. This implies that a subject could behave altruistically and almost single-handedly determine the fractions of play at these nodes. If other subjects are slow learners, they will no realize that these data are due to single decision, and this may ignite more passing. Knowing that, the subject that played altruistically may have been maximizing self-interest.

18


have been of great importance. In PIR3 the fractions fluctuate, but eventually

convergence is almost complete.

NIR1, Dynam cs of Play

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 9 10 11 12 13 14 15

Rou ber

Frac

tion

of th

e R

ound

M

atch

es th

at e

nd in

Eac

h No

de

i

8

nd num

T1T2T3T4P4

Figure 10

FIR2, Dynamics of play

00.10.20.30.40.50.60.70.80.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Round number

Frac

tion

of th

e ro

und

mat

ches

that

end

in e

ach

node

T1T2T3T4P4

Figure 11

19


PIR2, Dynamics of play

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Round number

Fr

Figure 12

PIR3, Dynamics of Play

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Round Number

Frac

tion

of th

e Ro

und

Mat

ches

that

End

in E

ach

Nod

e

T1T2T3T4P4

Figure 13

actio

tou

mat

ches

that

end

in e

ach

node

nd

her

n of

T1T2T3T4P4

20


Another important result is that relatively few matches ended in the last node (P 4 ) in

the information treatments. Figure 14 shows the fractions of total matches that end in

the final node. To compare, we also use the data from MP ( of the four-move

centipede experiments.

1992)

23 Clearly, the fractions for the non-information sessions are on

average higher than for the information sessions.24 Figure 15 shows the “take”

probabilities, conditional that the last decision node was reached, in all information and

non-information sessions, including the sessions of McKelvey and Palfrey . The

differences are more important than they seem. It is worth pointing out that the

‘threshold value’ of this probability, bellow which it is worthwhile for a GREEN

subject to pass in the node before (the third decision node) is . In almost all non-

information sessions the “take” probability is smaller than the threshold value, which

implies that all selfish players should pass at all nodes except the last one.

(1992)

0.857

25 In all

‘information sessions’ the “take” probability in the last decision node is larger than the

threshold value, which implies that if the game were played long enough it should

(theoretically) unfold to the equilibrium outcome. Hence, the observed differences are

important.

To test whether these results are statistically significant, we make the strong assumption

that the behavior in the last decision node does not depend on the round of the game.

23 MP had 4 four-move sessions without information revelation. Two of them had subjects from the Pasadena Community College (PCC1 and PCC ) and the other two had Caltech students (CIT1 and CIT ).

22

24 The last node was reached 31in out of the802 matches in the treatments without information revelation, and in just 8 out of the109 matches, in the treatment with information revelation. The last decision node was reached 11 times in the information treatments and 124 times in the no information treatments.

62

25 Taking for granted that the “take” probabilities in all previous nodes are larger.

21


Given the fact that subjects did not use signaling in the information treatments, and by

inspection of the data, this seems reasonable. Hence, we pooled the data from all rounds

and all sessions in a given treatment. We performed a simple test of differences in the

proportions in our two categories, P and a ‘pooled’ category with all other terminal

nodes. Comparing the proportions in NIR (N= ) and FIR (N= ) gives z=

with a p-value . Comparing the proportions in the NIR (N= ) and PIR

(N= 646 ) gives z= with a p-value . Finally, comparing proportions in the

FIR (N= ) and PIR (N= ) gives z= with a p-value larger than . These

results indicate that the effects are significant.

4

421 450 2.47

0.0068 421

2.647 0.004

450 646 0.205 0.4

However, we have to take into account the fact that the observed frequency of the last-

node ending is very low in the FIR and PIR treatments. Hence, we take the previous

results only as an indication. Accordingly, we also performed Fisher’s exact test. For

this, we pool all the observations of nodes T1 to T into one category again, and we test

the equality of the proportions in pairs of treatments. This gives us

4

2 2 2x contingency

tables, for which the calculations are not too bothersome. Comparing the last-node

proportions in the NIR (N= ) and PIR (N= ) gives a left p-value of 0.0088 and a

two-tail p-value of 0.011. Comparing the last-node proportions in the NIR (N= )

and FIR (N= ) gives a left p-value of 0.012 and a two-tail p-value of .

Comparing the last-node proportions in the NIR (N= ) and both information

treatments FIR and PIR (N=1096) gives a left p-value of 0.0023 and a two-tail p-value

of .

421 646

421

450 0.017

421

0.0037

22


Why did so few people pass in the last decision node in the FIR sessions and

consequently made convergence to the subgame perfect equilibrium possible? If

anything, we would expect more people to pass, conditional that they have reached the

last decision node, given the strong incentive of subjects to signal in order to induce

cooperative play in the future. It seems that subjects did not appreciate their

opportunities for signaling. This was critical for the results. Given the conditional

“Take” probabilities of MP ( experiments, it would be profitable, even for a selfish

player, to “Pass” all the way in the hope of meeting an altruistic opponent. Hence, with

information revelation, we would expect converge to node T , so convergence to the

equilibrium play was not guaranteed at all. Convergence occurred because information

revelation somehow affected the willingness of subjects of the YELLOW group to pass

in the last decision node. This is an interesting fact, because it contradicts MP’s original

explanation of their ( data, that is, that some people are “pure altruists”, who pass

at every node.

1992)

4

1992)

Another ‘difficult to explain’ result is the difference in the aggregate play in session

PIR1 and in sessions PIR 2 and PIR . These sessions use the same treatment, but in

session PIR1 of the matches in the last 5 rounds ended in the first node, whereas

more than was the respective fraction for the sessions PIR 2 and PIR3 . As we

explain in part 6 , there were reasons to believe that in treatment PIR more last-node

“Pass” behavior will be observed. This was true only for session PIR1. Figure 13 shows

the evolution of play in that session.

3

8%

%59

23


PIR1, Dynamics of play

00.10.20.30.40.50.60.70.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Round number

Frac

tion

of th

e ro

und

mat

ches

that

end

in th

is

node

T1T2T3T4P4

Figure 14

Figure 15

Comparison of the Fractions of Total Matches that Ended in the Last Terminal Node (P4)

00.010.020.030.040.050.060.070.080.090.1

Name of Session ("MP" Denotes Session of McKelvey and Palfrey, 1992)

Frac

tion

of M

atch

es T

hat E

nded

in

the

Last

Ter

min

al N

ode

Series1

Series1 0.04 0.01 0.09 0.05 0.031 0.026 0.009 0.004 0.018 0.005 0

MP PCC1

MP PCC2

MP CIT1

MP CIT2

NIR1 NIR2 FIR1 FIR2 PIR1 PIR2 PIR3

24


Comparison, Across Sessions, of the Probability of "Take" in the Last Decision Node

0

0.2

0.4

0.6

0.8

1

1.2

Name of Session ("MP" Denotes Session of McKelvey and Palfrey, 1992)

Prob

abili

tiy o

f "Ta

ke"

in th

e La

st

Deci

sion

Nod

e

Series1

Series1 0.83 0.9 0.61 0.69 0.77 0.8 0.95 0.92 0.91 0.87 1

MP PCC1

MP PCC2

MP CIT1

MP CIT2

NIR1 NIR2 FIR1 FIR2 PIR1 PIR2 PIR3

Figure 16

6. Discussion-Interpretation

The fact that almost all information sessions showed strong signs of convergence is

important but not very surprising, given the low percentage of people who passed in the

last decision node. However, our results are very different from the results of

MP where, even in the last five rounds, equilibrium play was around . This is

important, because there have been many efforts to explain the frequency of equilibrium

play in the early experiments of the centipede game. As we have discussed before,

researchers have performed experiments where they modified the number of players,

the size of the payoffs, the structure of the payoffs,

(1992) 8%

26 even the discrete nature of the

game, to check if the divergence from equilibrium play is robust to all these changes.

Here we show that in exactly the same game, with a different information structure, 26 By “structure” we mean some basic features of the payoffs of MP , such as the fact that the taker takes80% of the pie and payoffs double after every node.

(1992)

25


equilibrium play is much more common. Moreover, it seems that the many repetitions

our treatments used allowed for more learning, and this may explain the relatively high

frequency of equilibrium play even in the last nodes of the control treatments.

As we have already argued, a more puzzling feature of the data is that very few matches

exhibit passing in the last decision node.27 Levine ( proposed a model where

subjects tend to be generous when they interact with “altruistic” people and to be mean

towards “spiteful” opponents. Without aggregate information, people have prior beliefs

about the distribution of altruism in the population, and they play according to the type

of player they expect they are matched with. It is plausible that some altruistic subjects’

priors overestimate the probability that an opponent is an altruist.

1998)

28 If this is true,

information revelation of aggregate play shows to such altruistic persons that the truth is

different that they think, and they adjust their actions accordingly. We call that the

‘reciprocity interpretation’.

Another possible explanation is that people tend to conform to the behavior of the

public. If a social norm evolves that player 2 ’s do not pass in the last node, then the

others follow this. We call this the conformity interpretation. The choice of YELLOW

subjects in the last node cannot be affected by any strategic information if preferences

are selfish or purely altruistic. If pure altruism were the unique reason for last-node

passing, and if people had a fixed preference for altruism, we would expect the same 27 This is even more surprising if one considers the fact that of subjects could pass in the last node to signal for the future. This gives some additional incentives to pass to the last node. 28 The notion of “false consensus” in psychology describes people’s tendency to believe that other people are similar to them. See the survey by Marks and Miller (1 and the criticisms by Engelmann and Strobel and Dawes and Mulford .

987))2000( )1996(

26


conditional “take” probability in the last decision node in all treatments. Consequently,

a ‘reciprocal altruism’, ‘conformity’ or analogous ‘social preferences’ interpretation

needs to be invoked in order to explain the behavioral change of people in the last

decision node.

The treatments FIR and PIR differed from each other, in order to examine more

carefully the non-strategic reasons for the change in subjects’ behavior when aggregate

information is provided. Assuming the “conformism” interpretation is valid, play in the

last decision node should be significantly affected by information about what other

people, in the same group, do at this decision node. The results in session PIR1 offered

some support for the conformity view, but not the results in PIR 2 and PIR . For,

according to the conformity view, convergence to equilibrium play should be weaker in

the PIR session because with information about the ‘other’ group only, conformity

could not possibly discourage altruistic moves at the fourth decision node. The

‘reciprocity explanation’ survives the results in the PIR sessions, because a person may

earn information about the aggregate distribution of altruism and spite in a population

even if she only observes the behavior of her group - assuming that the members of the

two groups come from the same population.

3

Moreover, our results offer some support to the view that that both an increase in the

number of repetitions, and the provision of aggregate information as a devise that helps

agents to have consistent expectations, are conducive to achieving results closer to Nash

Equilibrium. However, we believe that this is not a general proposition, especially with

27


respect to the effects of providing aggregate information. As session PIR1indicates,

aggregate information can work both ways. It may attenuate the influence of social

preferences or it may accentuate it. We hypothesize that the aggregate information we

introduced in our experiments will generally either cause convergence or cause

divergence from the Nash equilibrium. That is, we do not believe that the distribution in

the late rounds of session PIR1 is the result of noise.

It is worth emphasizing that although signaling is possible in our experiments, this does

not reduce the strength of our generalizations in an evolutionary setting with large

populations. There are many reasons for this: first of all, signaling effects, if any, would

tend to decrease the occurrence of equilibrium behavior. Hence, in these sessions where

convergence is strong, our results only underestimate the importance of information

revelation for convergence to equilibrium play. In the session where convergence is not

strong (PIR1), signaling is difficult because of partial information (subjects do not

know how many people in their group are reaching each node). Furthermore, subjects

do not seem to have used much signaling. Even following the successful signaling effort

of one subject in session FIR 2 , no other signaling efforts were made. The very few

instances of passing in the last decision node in sessions FIR1 and FIR 2 , where

signaling was relatively easy,29 offer strong evidence that signaling was not an

important factor.

29 Passing in the last node could effectively signal that it is profitable to subjects of the other group to pass earlier in order to reach that node, so that more cooperative behavior follows.

28


7. Conclusions.

We conducted an experiment of the centipede game with aggregate information

revelation and found that equilibrium play is much more common relative to previous

experiments of the centipede game without information revelation. This shows that

aggregate information is important for convergence to equilibrium. It also shows that

the agents who posses information have the power to use it to change the outcomes of

the social interactions. However, we underscore the general difficulty to predict a priori

how play will be affected by information revelation, and that this type of period-by-

period information revelation may be sensitive to the noise in early rounds. Hence, if

the State, who has special aggregate information would like to use a policy of aggregate

information revelation to increase trust in a society, it ought to do it with caution.

An interesting goal would be to design alternative treatments where information

revelation would frequently lead to higher social payoffs, not only lower social payoffs.

This can be achieved if, for example, subjects who pass in the last decision node earn

higher monetary payoffs.30 Moreover, it may be worthwhile to give subjects aggregate

information that refers to more than one round, for example letting them play the game

for 10 rounds, and then reveal the aggregate fractions of “Pass” and Take” in all the

previous rounds before subjects get to play for another 5 rounds. This would avoid

signaling altogether and perhaps would have very different results when it comes to

behavior in the last decision node. 30 We have tried a session where the payoffs in the last decision node will be and 9 instead of and 3 4.2

6.9 . The results were very promising, since games finished in the last terminal node, meaning that social surplus was very high.

20

29


Appendix 1. Aggregate Data for all Sessions.

Results from Session NIR 1

Number of matches that finished in the particular node T1 T2 T3 T4 P4 Sum of all rounds 52 90 53 23 7 Sum of last 5 22 29 14 8 2

Fraction of all games in rounds 1-15 ending in each node

T1 T2 T3 T4 P4 0.231 0.4 0.236 0.102 0.031

Fraction of all games in rounds 11-15 ending in each node T1 T2 T3 T4 P4

0.293 0.387 0.187 0.107 0.027 Implied TAKE probability given a node has been reached T1 T2 T3 T4 Rounds 1-15 0.231 0.52 0.639 0.767 Rounds 11-15 0.293 0.547 0.583 0.8

Results from Session NIR 2

Number of matches that finished in the particular node

T1 T2 T3 T4 P4

Sum of all rounds 60 68 43 20 5Sum of Last 5 26 27 13 3 1

Fraction of all games in rounds 1-14 ending in each node T1 T2 T3 T4 P4 0.306122449 0.346939 0.219388 0.102041 0.02551 Fraction of all games in rounds 10-14 ending in each node T1 T2 T3 T4 P4 0.371428571 0.385714 0.185714 0.042857 0.014286 Implied TAKE Probability given a node has been reached T1 T2 T3 T4 Rounds 1-14 0.306122 0.5 0.632353 0.8 Rounds 10-14 0.371429 0.613636 0.764706 0.75

30


Results from Session FIR 1

Number of matches that finished in each node T1 T2 T3 T4 P4 Sum of all rounds 34 76 76 37 2 Sum of last 5 27 32 14 2 0


0.151 0.338 0.338 0.165 0.009Fraction of all games in rounds 11-15 ending in each node T1 T2 T3 T4 P4

0.36 0.427 0.187 0.027 0 Implied TAKE probability given a node has been reached T1 T2 T3 T4 Rounds 1-15 0.151 0.398 0.660 0.949Rounds 11-15 0.36 0.667 0.875 1

Results from Session FIR 2

Number of matches that finish in each node T1 T2 T3 T4 P4 Sum of all rounds 102 77 33 12 1 Sum of last 5 48 21 5 1 0

Fraction of games in rounds 1-15 ending in each node T1 T2 T3 T4 P4

0.453 0.342 0.147 0.053 0.0044Fraction of games in rounds 11-15 ending in each node T1 T2 T3 T4 P4

0.64 0.28 0.067 0.013 0

Implied TAKE probability given a node has been reached T1 T2 T3 T4Rounds 1-15 0.453 0.626 0.72 0.923Rounds 11-15 0.64 0.778 0.833 1

31


Results from Session PIR1

Number of matches that finish in each node T1 T2 T3 T4 P4 Sum of all rounds 15 86 79 41 4 Sum of last 5 6 31 24 12 2


0.067 0.382 0.351 0.1822 0.018 Fraction of all games in rounds 11-15 ending in each node

T1 T2 T3 T4 P4 0.08 0.413 0.32 0.16 0.027

Implied TAKE probability given a node has been reached T1 T2 T3 T4 Rounds 1-15 0.067 0.41 0.637 0.91 Rounds 11-15 0.08 0.45 0.632 0.858

Results from session PIR2

Number of matches that finish in the particular node T1 T2 T3 T4 P4 SUM of all rounds 67 78 43 7 1 Sum of last 5 37 25 8 0 0 Sum of 9 first 30 53 35 7 1 Fraction of all games in rounds 1-14 ending in each node T1 T2 T3 T4 P4 0.341 0.398 0.22 0.036 0.005 Fraction of all games in rounds 11-14 ending in each node T1 T2 T3 T4 P4 0.529 0.358 0.114285714 0 0 Implied TAKE probability given a node has been reached T1 T2 T3 T4 Rounds 1-14 0.341 0.605 0.843 0.875 Rounds 10-14 0.529 0.758 1 NA

32


Results from session PIR 3

Number of matches that finish in each node T1 T2 T3 T4 P4 Sum of all rounds 93 84 41 7 0 Sum of last five 49 23 3 0 0 SUM OF FIRST 10 44 61 38 7 0

Fraction of all games in rounds 1-15 ending in each node T1 T2 T3 T4 P4 0.413 0.373 0.182 0.031 0 Fraction of all games in rounds 11-15 ending in each node T1 T2 T3 T4 P4 0.653 0.306 0.04 0 0 Implied TAKE probability given a node has been reached T1 T2 T3 T4 Rounds 1-15 0.4133 0.636 0.854 1 Rounds 11-15 0.653 0.8846 1 NA

33


Appendix 2. Instructions for Treatment FIR.

INSTRUCTIONS Introduction. Welcome to CASSEL. The policy in this lab is never to deceive participants. This is an experiment in group decision making, and you will be paid for your participation in cash, at the end of the experiment. Different participants may earn different amounts. What you earn depends partly on your decisions, partly on the decisions of others, and partly on chance. Please turn off all pagers and cell phones now. The entire experiment will take place through computer terminals, and all interaction between you will take place through the computers. It is important that you do not talk, or in any way try to communicate with others during the experiment. We will start with a brief instruction period. During the instruction period you will be given a description of the main features of the experiment and will be shown how to use the computers. If you have any questions during the instruction period, please raise your hand and your question will be answered so anyone can hear. If you have any difficulties after the experiment has begun, please raise your hand and an experimenter will come and assist you. You will be divided into two groups, each containing ___ people. The groups will be labeled the GREEN and the YELLOW group. The computer you are using will assign you to one of the two groups. If you are assigned to be GREEN you will be GREEN throughout the experiment. If you are assigned to be YELLOW you will be YELLOW throughout the experiment. A Decision Problem In this experiment, you will be participating in the following interaction, for real money. In each round you will be matched with a person of the other color. During each move of a particular round, either you or the person you are matched with makes an action. The payoffs for you, and for the person you are matched with, depend on the moves you both make. In pages 6 and 7 of the instructions you see an illustration of a specific round based on the experiment screen. There are two piles of money: a Large Pile and a Small Pile. At the beginning of the round, the Large Pile has 60 cents and the Small Pile has 15 cents. GREEN has the first move and can either pass or take the pile. If GREEN chooses “Take”, GREEN gets the Large Pile of 60 cents, YELLOW gets the Small Pile of 15

34


cents, and the round is over. If GREEN chooses “Pass”, both piles double and it is YELLOW’s turn. The Large Pile now contains 1.20 dollars and the Small Pile 30 cents. Now YELLOW can take or pass the pile. If YELLOW takes, YELLOW ends up with the Large Pile of 1.20 dollars and GREEN gets the Small Pile of 30 cents and the round is over. If YELLOW passes, both piles double and it is GREEN’s turn again. The Large Pile now contains 2.40 dollars and the Small Pile 60 cents. GREEN can again take or pass the pile. If GREEN takes, GREEN ends up with the Large Pile of 2.40 dollars and YELLOW ends up with the Small Pile of 60 cents and the round is over. If GREEN passes, both piles double and it is YELLOW’s turn again. The Large Pile now contains 4.80 dollars and the Small Pile 1.20 dollars. This is the last move, and it is YELLOW’s second choice. If YELLOW takes the pile, YELLOW ends up with the Large Pile of 4.80 dollars and GREEN gets the Small Pile of 1.20 dollars and the round is over. If YELLOW passes, then the piles double again. GREEN then gets the Large Pile of 9.60 dollars and YELLOW gets the Small Pile of 2.40 dollars. Note that this is not an actual move, since GREEN has only one choice. After the end of the first round, you will have the opportunity to get information about what all the YELLOW people and all the GREEN people chose in the previous round. In particular, for each of the moves, you will see the fraction of the people who chose “Take” and the fraction that chose “Pass” in the previous round. For example, during the third round, you will see information that refers to the behavior of participants in the second round. In the first box, the GREEN people move. The numbers under the word “History” represent the fractions of GREEN people who chose “Take” and the fraction of the GREEN people who chose “Pass”, in this move, in the previous round. Similarly, in the second box, the YELLOW people move. In the second box, the numbers under the word “History” represent the fractions of the YELLOW people who chose “Take” and the fraction of the YELLOW people who chose “Pass”, in this particular move, in the previous round. Note that not all the YELLOW people need have moved in this box in the previous round. Remember that all boxes, except the first one, are reached only if the other player chooses “Pass” in the previous box. The numbers under “History” have the same meaning in the other boxes. If a box does not have “History”, this implies that this box was never reached in the previous round. The experiment consists of ____ rounds. In each round you will interact with a person of the different color. So this person will be GREEN if you are YELLOW and YELLOW if you are GREEN. You will not be matched with the same person twice, as there are _____ people of the other color. So you will be matched with each person of the other color exactly once.

35


Practice Session. We will now start the instruction session. During the instruction session, we will teach you how to use the computers by going through three practice rounds. During the instruction period please do not hit any key unless you are instructed to. You will not be paid for the practice rounds. Please wait until we set up the experiment. Please double click on the small red icon labeled “MC”. When the computer prompts for you name, please type the number of the computer you are in, for example if you are at computer 14, type “SSEL 14”. Then, please hit the “SUBMIT” key. Now you should all have a window saying: “Please Wait. Connecting to Server”. Please do not close any windows. Now all of you should be able to see the experiment screen. The experiment screen should display five boxes. Remember that the last box does not describe a real move since GREEN can only choose “Take”. You see that the first match has begun. The box with the red color represents the current move, in which, one of the two participants has to make a choice. If it is your turn to move, you are given a description of the choices available to you. If you are told in the first box that this is your move, and you have the choice menu, you are a GREEN participant. If you are told to wait for your partner to make his/her decision, you are a YELLOW participant. You will have the same color throughout the experiment. Please record your color and computer number in your record sheet. You need to record your computer number since you will be paid according to this number. We will now start the first practice round. Will all the GREEN participants please choose PASS from your menu now? GREEN participants now receive a message that they have passed, and now the other person (YELLOW) will get the opportunity to take or pass the pile. YELLOW participants now receive a message that the person they are matched with (GREEN) has passed the pile, and now they will have the move. Please do not forget to click “OK” on your information icons each time. Since GREEN chose PASS, the second box now has the red color, and the YELLOW person now has the choice menu, indicating that it is YELLOW’S move. The GREEN participants are told that it is the other person’s turn to choose. Notice that there is now a large pile of 1.2 dollars and a small pile of 30 cents. Will all the YELLOW participants please choose TAKE from you menu now? Since YELLOW chose TAKE, the round has ended. A message informs that you or the other participant, depending on your color, has taken the pile, and tells you your

36


payoffs. Please record your payoffs to the record sheet provided. You must do so after every round in order to double-check your payoffs are correct. You are not being paid for the practice session, but if this was the real experiment, then the payoffs you have recorded would be money you have earned from the first round, and you would be paid this amount for that round at the end of the experiment. The total you earn over all the _____ real rounds, plus your guaranteed show up fee of five dollars, is what you will be paid for your participation in the experiment. We will now proceed to the second practice round. You now see that you have been matched with a new person of the opposite color and that the second round has begun. Does everyone see this? The rules for the second round are exactly like the first, but now you can observe the way participants played in the first practice round. The numbers at the lower part of the boxes, under the word “History”, represent the fractions of “Take” and “Pass” decisions of participant the previous match. In the first box, you are being informed that that all the GREEN persons have chosen “Pass” in their first decision in the previous round. Similarly, in the second box, which corresponds to the second move of the round, but only to the first decision of the YELLOW participants, you are informed that all the YELLOW people who moved chose “Take” in their first decision. The other boxes do not have numbers because there were no decisions at all to be revealed: no GREEN or YELLOW participants reached their second move. Remember that the fractions under “History” refer only to the preceding round, not all the previous rounds completed. Now you are free to choose whatever you want in the next two practice rounds. Please stop after you have completed the third practice round. Please record your payoffs to the record sheet provided, but remember you are not paid for the practice rounds. Please remember to record your payoffs after each real round. This concludes the practice session. In the actual experiment there will be ____ rounds instead of three, and of course, it will be up to you to make your own decisions. You will not see any history in the first round. Remember that you will meet each person of the other color exactly once. At the end of round ____, the experiment ends and we will pay each of you privately in cash, the total amount you have accumulated during all real rounds, plus your guaranteed five dollar participation fee. No other person will be told how much cash you earned in the experiment. You need not tell any other participants how much you earned. We will now begin with the actual experiment. If there are any problems from this point on, please raise you hand and an experimenter will come and assist you.

37


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Roth, Alvin E. and Schoumaker, Francoise. “Expectations and Reputations in Bargaining: An Experimental Study” The American Economic Review, Vol. 73, No. 3, June 1983, pp. 362-372. Weibull, Jorgen. “Evolutionary Game Theory”. The MIT Press, Cambridge, Massachusetts, 1997. Zauner, Klaus G. “A Payoff Uncertainty Explanation of Results in Experimental Centipede Games”. Games and Economic Behavior 26,1999, pp. 157-185.

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