Algorithmic Collusion 140318 [Read-Only] · Algorithmic Collusion for IO Reading Group, Slide 7 of...

Algorithmic CollusionFor IO Reading Group

Chris Doyle, Department of Economics, WarwickUniversity, 14 March 2018

Algorithmic Collusion for IO Reading Group, Slide 2 of 26

Chris Doyle, Department of Economics, March 2018

Papers

– Review paper: “Algorithmic Collusion: Problems and Counter-Measures”by Ezrachi and Stucke (2017) for OECD athttps://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=DAF/COMP/WD%282017%2925&docLanguage=En

– Research paper: “Algorithmic Collusion in Cournot Duopoly Market:Evidence from Experimental Economics” by Zhou et al (2018) athttps://arxiv.org/pdf/1802.08061.pdf

– Related paper (theory): Bruno Calcedo (2015) “Pricing Algorithms andTacit Collusion”

– Background: “FTC” https://www.ftc.gov/system/files/attachments/us-submissions-oecd-other-international-competition-fora/algorithms.pdf

– “When algorithms set prices: winners and losers” Oxera, June 2017https://www.regulation.org.uk/library/2017-Oxera-When_algorithms_set_prices-winners_and_losers.pdf



Algorithmic Pricing (or dynamic pricingalgorithms)

– Algorithm: A process or set of rules to be followed in calculations or otherproblem-solving operations, especially by a computer

– Sellers delegate price setting to computer algorithms

– Designed to match supply and demand, facilitate finer pricediscrimination/personalised offers, monitor competition, etc.

– Used in e-commerce and other electronically mediated products likeairline tickets, concert tickets, etc.

– Examples:

• Uber’s Surge Pricing algorithm

• Amazon Marketplace

• Airbnb



Airbnb

– Millions of properties for rental – how do sellers’ set their prices?

– If a potential seller (renter) believes there are significant costs toestablishing a price at which to sell, this may result in fewer sellers –giving the platform less profit and giving demanders less choice (andwelfare) – classic two-sided market indirect externality effects

– Algorithm designed by platform to lower burden of price setting

– Airbnb started building pricing tools in 2012, see IEEE

– Since 2014 Airbnb has been making its pricing algorithm dynamic, wherethe programme learns to update price tips (the tool is called Aerosolveand is publicly available)

– The machine learning components in Aerosolve represent an advanceover standard dynamic pricing algorithms historically used by airlines andhotel groups



InCompetitor



Omnia



Collusion

– Collusion is an agreement between two or more parties, sometimesillegal–but always secretive–to limit open competition by deceiving,misleading, or defrauding others of their legal rights, or to obtain anobjective forbidden by law typically by defrauding or gaining an unfairmarket advantage

– A cartel is a group of apparently independent producers whose goal is toincrease their collective profits by means of price fixing, limiting supply,or other restrictive practices–which may be secretive and hencecollusive–or transparent and open (e.g. OPEC)



Algorithmic Collusion

– The design of processes or sets of rules supported by computers thatfacilitate anti-competitive prices as an equilibrium in a secretive manner

– A pricing algorithm instructs the computer to set the price of an item forsale, and can be written to rely on competitors’ prices and demographicor other information about the customer

• AI or machine learning could lead to outcomes not explicitly given in theprogramming

– 2015 case FTC fined Topkins who working for a company adoptedspecific pricing algorithms to sell posters with the goal of coordinatingchanges in their respective prices and wrote computer code thatinstructed algorithm-based software to set prices in conformity with thisagreement

– https://www.justice.gov/sites/default/files/opa/press-releases/attachments/2015/04/06/topkins_information.pdf



Zhou et al (2018) Theoretical Background



Iterated Prisoner’s Dilemma

– One shot game, defection dominant strategy and NE play

– Repeated (or iterated) PD game, folk theorems show any feasible andindividually rational outcome can be sustained as an equilibrium if theprobability of a further round is sufficiently large

– In evolutionary game theory, the focus is on strategies and their success

– Players in evolutionary games play pre-programmed strategies and thefocus are the properties of strategies (e.g. ESS, Maynard Smith (1972))

– In the repeated PD game defect is ESS (all ESS are Nash equilibria, butnot conversely)

– In the infinitely repeated PD there are infinite contingent strategies

– Three simple contingency plans have received substantial attention:Always Defect, Always Cooperate, and Tit-for-Tat (TFT)

– What is the best strategy in an iterated PD?



Iterated Prisoner’s Dilemma:Zero Determinant Strategies

– Press and Dyson (2012) “Iterated Prisoner’s Dilemma contains strategies thatdominate any evolutionary opponent”

• See also “Zero-determinant strategies in finitely repeated games” Ichinose and Masuda (2018) JnlTheoretical Biology

– Recognising a property of memory-one strategies (stationary Markov), the propertyof the expected payoffs are such that a player can unilaterally choose a strategy thatenforces a linear relationship between the players’ payoffs

• To do this a players choose what are called ZD (zero determinant) strategies, and one of the playerscould then determine the expected payoff of the other player

• Background stuff on Markov chains and properties: https://brilliant.org/wiki/stationary-distributions/

• Press and Dyson apply results from matrix algebra given the stationary Markov chain and show that aplayer can choose a vector of probabilities that will determine the other players expected payoffirrespective of that player’s vector of probabilities

• See https://golem.ph.utexas.edu/category/2012/07/zerodeterminant_strategies_in.html

• See Hilbe et al “Partners or Rivals? Strategies for the iterated prisoner’s dilemma” 2015, Games andEconomic Behavior

• See http://s3.boskent.com/prisoners-dilemma/ipd.html



(A) Markov matrix for the memory-one game shown in Fig. 1C.

William H. Press, and Freeman J. Dyson PNAS2012;109:26:10409-10413

©2012 by National Academy of Sciences



stochastic matrix (Markov chain)

S stationary probability distribution (stationary vector)

Stochastic matrix has unit eigenvalue r r

Each row of sums to 1 final column

( )

[ 1]

M

vM = v

v

M = M- I vM = 0

M vM v

M

n

ijj

Let be an n n matrix

The determinant of the submatrix n n obtained by eliminating

the ith row and jth column

s

Unit eigenvalue of not invertibl

of is the i j minor

e and

of a

1

1

1

det 0

( )1 1

,

(

–

)

M M

A

M

A A

i j

ij

i j i j

nd denoted minor

The scalar minor is called the i j cofactor of and denoted by cof1( –) ,

A

A A A



n

n nn

cof cof cof

cof

Adj cof

cof cof

11 21 1

12

13

1

A A A

A

A A

A A



1 2

( -1)

( )

( )

( , , , ) 0n n nn

Rank of is n

Every row of Adj is proportional to

and Adj

Take the final row M M M for some scalar

Press and Dyson manipulate the matrix to show for a

M

M v

vM = 0 M M = 0

v

M v x

1 1 2 2( x , x , , x )n n nn n

n arbitrary vector equals

M M M

x

v x



4

det

det i

Using P&D notation

Subtract column 1 from columns 2 and 3

Replace the final column with an arbitrary vector

Call the manipulated matrix

Take by Laplace expansion from final column

s

f

M

M

M4

4 1 14 2 24 3 34 4 441

1

)

ii

C f C f C f C f C

D

v f

v f (p,q,f



( ) ( )

( ) ( )Y Y

X Y

In the stationary state the normalized payoffs for players X and Y

Notation: =(R,S,T,P) and =(R,T,S,P)

D Ds = = and s = =

D D

Any linear combination of scores

X Y

X X

S S

v S p,q,S v S p,q,S

v 1 p,q,1 v 1 p,q,1

:

( )1

( )

1 0

YX Y

Y

X Y

also holds

Ds s

D

If X chooses a such that determinant is zero, so

s s

These are

X

X

p,q, S S 1

p,q,1

p p S S 1

Zero Determinant strategies



1 0

Y

Y

X enforcer - chooses Y expected stationary payoff

If X chooses a such that =0 so that determinant is zero, so

s

Other possibilities arise: X could play extortionate strategy and achie

p p S 1

ve more

than Y and essentially get Y to play C but occasionally (probabilistically)

X plays D

X cannot set his/her own score unilaterally



Zero Determinant Strategies:Other Interesting Results

– Adami and Hintze (2013, Nature Communications) “Evolutionary instability of zero-determinant strategies demonstrates that winning is not everything”

• Suggests that ZD strategies fail ESS (Maynard Smith)

– “New Winning Strategies for the Iterated Prisoner’s Dilemma” Mathieu andDelahaye (2017, Journal of Artificial Societies and Social Simulation)

• Using agent based simulations to test a large array of different strategies, concluding thatTFT and Grim (spiteful) are among the most successful

– “On Regulation with Zero–Determinant Strategies” Barlo and Imren (2016)

• Provides more detail on the Press and Dyson result and applies this to a setting where a regulatorchooses emission levels for a country



Zhou et al (2018) paper

– Apply ZD strategies as a pricing algorithm and see how this performsagainst human participants in experiments using Cournot duopoly

– The ZD strategy cam force the human player to collude

– Various claims are made about LECA (Linear Extortion to CollusionAlgorithm) which seems to satisfy memory-one ZD strategy requirements

– The authors then use a simple Cournot duopoly model with specifiedparameter values to test how human subjects would play against LECA



Possible Research Topics

– Replicate Zhou type experiments

• Should result in similar outcomes

– Experiment with InCompetitor/Omnia against humans

• Examine how fast or slow markets might evolve towards higher prices

– Experiment with incomplete information environment – typesalgorithmic or not

• What fraction of sellers tips market towards ‘as if’ collusion

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