Algorithmic Collusion For IO Reading Group

Transcription

1 Algorithmic Collusion For IO Reading Group Chris Doyle, Department of Economics, Warwick University, 14 March 2018

2 Papers Review paper: Algorithmic Collusion: Problems and Counter-Measures by Ezrachi and Stucke (2017) for OECD at te=daf/comp/wd%282017%2925&doclanguage=en Research paper: Algorithmic Collusion in Cournot Duopoly Market: Evidence from Experimental Economics by Zhou et al (2018) at Related paper (theory): Bruno Calcedo (2015) Pricing Algorithms and Tacit Collusion Background: FTC When algorithms set prices: winners and losers Oxera, June When_algorithms_set_prices-winners_and_losers.pdf Algorithmic Collusion for IO Reading Group, Slide 2 of 26

3 Algorithmic Pricing (or dynamic pricing algorithms) Algorithm: A process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer Sellers delegate price setting to computer algorithms Designed to match supply and demand, facilitate finer price discrimination/personalised offers, monitor competition, etc. Used in e-commerce and other electronically mediated products like airline tickets, concert tickets, etc. Examples: Uber s Surge Pricing algorithm Amazon Marketplace Airbnb Algorithmic Collusion for IO Reading Group, Slide 3 of 26

4 Airbnb Millions of properties for rental how do sellers set their prices? If a potential seller (renter) believes there are significant costs to establishing a price at which to sell, this may result in fewer sellers giving the platform less profit and giving demanders less choice (and welfare) 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, where the programme learns to update price tips (the tool is called Aerosolve and is publicly available) The machine learning components in Aerosolve represent an advance over standard dynamic pricing algorithms historically used by airlines and hotel groups Algorithmic Collusion for IO Reading Group, Slide 4 of 26

5 InCompetitor Algorithmic Collusion for IO Reading Group, Slide 5 of 26

6 Omnia Algorithmic Collusion for IO Reading Group, Slide 6 of 26

7 Collusion Collusion is an agreement between two or more parties, sometimes illegal but always secretive to limit open competition by deceiving, misleading, or defrauding others of their legal rights, or to obtain an objective forbidden by law typically by defrauding or gaining an unfair market advantage A cartel is a group of apparently independent producers whose goal is to increase their collective profits by means of price fixing, limiting supply, or other restrictive practices which may be secretive and hence collusive or transparent and open (e.g. OPEC) Algorithmic Collusion for IO Reading Group, Slide 7 of 26

8 Algorithmic Collusion The design of processes or sets of rules supported by computers that facilitate anti-competitive prices as an equilibrium in a secretive manner A pricing algorithm instructs the computer to set the price of an item for sale, and can be written to rely on competitors prices and demographic or other information about the customer AI or machine learning could lead to outcomes not explicitly given in the programming 2015 case FTC fined Topkins who working for a company adopted specific pricing algorithms to sell posters with the goal of coordinating changes in their respective prices and wrote computer code that instructed algorithm-based software to set prices in conformity with this agreement Algorithmic Collusion for IO Reading Group, Slide 8 of 26

9 Zhou et al (2018) Theoretical Background Algorithmic Collusion for IO Reading Group, Slide 9 of 26

10 Iterated Prisoner s Dilemma One shot game, defection dominant strategy and NE play Repeated (or iterated) PD game, folk theorems show any feasible and individually rational outcome can be sustained as an equilibrium if the probability 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 the focus are the properties of strategies (e.g. ESS, Maynard Smith (1972)) In the repeated PD game defect is ESS (all ESS are Nash equilibria, but not 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? Algorithmic Collusion for IO Reading Group, Slide 10 of 26

11 Iterated Prisoner s Dilemma: Zero Determinant Strategies Press and Dyson (2012) Iterated Prisoner s Dilemma contains strategies that dominate any evolutionary opponent See also Zero-determinant strategies in finitely repeated games Ichinose and Masuda (2018) Jnl Theoretical Biology Recognising a property of memory-one strategies (stationary Markov), the property of the expected payoffs are such that a player can unilaterally choose a strategy that enforces a linear relationship between the players payoffs To do this a players choose what are called ZD (zero determinant) strategies, and one of the players could then determine the expected payoff of the other player Background stuff on Markov chains and properties: Press and Dyson apply results from matrix algebra given the stationary Markov chain and show that a player can choose a vector of probabilities that will determine the other players expected payoff irrespective of that player s vector of probabilities See See Hilbe et al Partners or Rivals? Strategies for the iterated prisoner s dilemma 2015, Games and Economic Behavior See Algorithmic Collusion for IO Reading Group, Slide 11 of 26

12 (A) Markov matrix for the memory-one game shown in Fig. 1C. William H. Press, and Freeman J. Dyson PNAS 2012;109:26: Algorithmic Collusion for IO Reading Group, Slide 12 of by National Academy of Sciences

13 M stochastic matrix (Markov chain) vm = v v (S ) stationary probability distribution (stationary vector) M = M - I vm = 0 Stochastic matrix M has unit eigenvalue [ vm rv Each row of M sums to 1 r 1] n 1 final column 1 sij j 1 Unit eigenvalue of M M not invertible and det M 0 Let A be an n n matrix The determinant of the submatrix (n 1) (n 1) obtained by eliminating the ith row and jth column of A is the i, j minor of A and denoted minora ij The scalar ( 1) i j minora i j is called the i, j cofactor of A and denoted by cofa i j Algorithmic Collusion for IO Reading Group, Slide 13 of 26

14 cofa11 cofa 21 cofa n1 cofa 12 AdjA cofa13 cofa1n cofa nn Algorithmic Collusion for IO Reading Group, Slide 14 of 26

15 Rank of M is (n - 1) Every row of Adj (M ) is proportional to v vm = 0 and Adj (M )M = 0 ) for some scalar 0 Take the final row v (M1 n, M 2 n,, Mnn Press and Dyson manipulate the matrix M to show v x for an arbitrary vector x equals xn ) v x (M1 n x1, M2 n x 2,, Mnn Algorithmic Collusion for IO Reading Group, Slide 15 of 26

16 Using P&D notation Subtract column 1 from columns 2 and 3 Replace the final column with an arbitrary vector f Call the manipulated matrix M Take det M by Laplace expansion from final column 4 det M si 4 Ci 4 f1 C14 f2 C24 f3 C34 f4 C44 i 1 v f D(p,q,f ) Algorithmic Collusion for IO Reading Group, Slide 16 of 26 1 v f

17 In the stationary state the normalized payoffs for players X and Y Notation: SX =(R,S,T,P) and SY =(R,T,S,P) sx = v SX D(p,q,SX ) v SY D(p,q,SY ) = and sy = = v 1 D(p,q,1) v 1 D(p,q,1) Any linear combination of scores also holds : s X sy 1 D(p,q, S X SY 1) D(p,q,1) If X chooses a p such that p S X SY 1 determinant is zero, so s X sy 1 0 These are Zero Determinant strategies Algorithmic Collusion for IO Reading Group, Slide 17 of 26

18 X enforcer - chooses Y expected stationary payoff If X chooses a p such that =0 so that p SY 1 determinant is zero, so sy 1 0 Other possibilities arise: X could play extortionate strategy and achieve more than Y and essentially get Y to play C but occasionally (probabilistically) X plays D X cannot set his/her own score unilaterally Algorithmic Collusion for IO Reading Group, Slide 18 of 26

19 Zero Determinant Strategies: Other Interesting Results Adami and Hintze (2013, Nature Communications) Evolutionary instability of zerodeterminant 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 and Delahaye (2017, Journal of Artificial Societies and Social Simulation) Using agent based simulations to test a large array of different strategies, concluding that TFT 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 regulator chooses emission levels for a country Algorithmic Collusion for IO Reading Group, Slide 19 of 26

20 Zhou et al (2018) paper Apply ZD strategies as a pricing algorithm and see how this performs against 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 Collusion Algorithm) which seems to satisfy memory-one ZD strategy requirements The authors then use a simple Cournot duopoly model with specified parameter values to test how human subjects would play against LECA Algorithmic Collusion for IO Reading Group, Slide 20 of 26

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26 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 types algorithmic or not What fraction of sellers tips market towards as if collusion Algorithmic Collusion for IO Reading Group, Slide 26 of 26