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The Computational Complexity of Games and Markets: an Introduction for Economists

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The Computational Complexity of Games and Markets: an Introduction for Economists The Computational Complexity of Games and Markets: An Introduction for Economists Andrew McLennan∗ University of Queensland December 2011 Abstract This is an expository survey of recent results in computer science related to the computation of fixed points, with the central one beingthat the problemof finding an approximate Nash equilibrium of a bimatrix game is PPAD-complete. This means that this problem is, in a certain sense, as hard as any fixed point problem. Subsequently many other problems have been shown to be PPAD- complete, including finding Walrasian equilibria in certain simple exchange economies. We also comment on the scientific consequences of complexity as a barrier to equilibration, and other sorts of complexity, for our understanding of how markets operate. It is argued that trading in complex systems of markets should be analogized to games such as chess, go, bridge, and poker, in which the very best players are much better than all but a small number of competitors. These traders make positive rents, and their presence is a marker of complexity. Consequences for the efficient markets hypothesis are sketched. Running Title: Complexity of Games and Markets Journal of Economic Literature Classification Numbers G12 and G14. Keywords: Computational complexity, two person games, Nash equilibrium, Scarf algorithm, NP, TFNP, PPAD, FPTAS, Lemke-Howson algorithm, Walrasian equilibrium, arbitrage, asset trading, efficient market hypothesis. ∗School of Economics, Level 6 Colin Clark Building, University of Queensland, QLD 4072 Australia, [email protected]. McLennan’s work was funded in part by Australian Research Council grant DP0773324. I would like to gratefully acknowledge the comments received at Games Toulouse 2011, and at seminar pre- sentations at the Australian National University, the Institute for Social and Economic Research at Osaka University, and the Kyoto Institute for Economic Research. Tim Roughgarden provided some especially useful comments and advice. All errors are the responsibility of the author. With a project such as this it is inevitable that the some work that should be covered will be missed; I sincerely regret such oversights. 1 1 Introduction With the advent of the internet, computer science has become a social science. In addition, com- putational issues related to the mathematics at the heart of noncooperative game theory and general equilibrium theory have become increasingly prominent. As a result of this, following the large scale research program sketched by Papadimitriou (2001), during the last decade computer science has seen an explosion of interest in game theory and other aspects of theoretical economics. This paper provides an exposition of some of the central results of this literature, and a survey of many others, that is in- tended for theoretical economists and other game theorists who have little or no prior background in computer science. In addition, it attempts to stake out a position concerning how these developments, and the larger phenomenon of complexity in economic affairs, should affect the scientific outlook of economists. In a sense this can be thought of as a progress report updating McKelvey and McLennan (1996) and von Stengel (2002). However, those surveys focused on algorithms, and while there is some discussion of algorithms here, our main focus is on computational complexity. In particular, we do not attempt to provide a comprehensive overview of new algorithms for computing Nash equilibrium, its refinements, and correlated equilibrium. Datta (2010) and Herings and Peeters (2010) are recent surveys covering certain types of algorithms, and other papers in the January 2010 issue of Economic Theory provide perspectives on recent developments. Nor do we cover the very extensive literature in economics that is concerned with computation of equilibrium, which is largely rooted in economic theory and concrete computational experience. Judd et al. (2003) surveys a portion of this literature that is relatively close to the focus here, because it considers general equilibrium models with multiple agents. Other references for our main topic aiming at a broader audience (at least within the computer science community) include Papadimitriou (2007), Daskalakis et al. (2009b), Daskalakis (2009), and Goldberg (2011). Longer overviews of the subject, with detailed descriptions of some of the central arguments, are given by Chen et al. (2009b) and Daskalakis et al. (2009a). In comparison with our treatment, these assume little prior background in game theory but at least some background and fa- miliarity with the elements of computational complexity. We should also mention some very useful references that reach out to readers from economics. Roughgarden (2010b) describes the central result exposited here in the context of a broader survey of the computational complexity of equilibrium com- putation, and Roughgarden (2010a) is a still more expansive overview of work in computer science related to game theory and economics. On a larger scale, the essays in Nisan et al. (2007) provide many perspectives on areas in which economics and computer science are in vigorous interaction. All of these are highly recommended as next steps into the subject. Our central topic is a result, due to Daskalakis et al. (2006a) and Chen and Deng (2006b), which states that the problem of finding a Nash equilibrium of a finite two player normal form game is “com- plete” for PPAD, which is a class of computational problems that encompasses search problems in which a fixed point of a function of correspondence is sought. In computer science it is a landmark 2 result resolving a well known longstanding open problem. It has been awarded distinguished prizes, and has been followed up in the research of some of the leading figures of that discipline. In my opinion this is also one of the premier theorems of game theory, and one of the major contributions of game theory to science. In addition to its conceptual importance, the proof is quite beautiful, with several striking ideas that deserve to be widely appreciated. A distinctive feature of our exposition is that we provide an overview of the proof that is complete and hopefully compelling, in the sense that the reader will find it easy to believe that missing details can be filled in, but also quite informal and (I hope) not at all hard to read. This is possible because the main ideas have geometric descriptions that are intuitive and immediately accessible. What we omit are the nuts and bolts of certain algorithms and the verifications of supporting results that are, for the most part, intuitively plausible, even if the arguments are in some cases quite challenging. Of course our approach requires firm foundations, so the relevant background material in computer science is introduced from the very beginning. Again, the treatment is informal and much less detailed than what one would find in texts such as Papadimitriou (1994b) and Arora and Boaz (2007), but nonetheless rigorous within the bounds it sets for itself. Indeed, quite aside from the particular subject matter, we hope to provide an introduction to computational complexity that is succinct yet precise, and which conveys some sense of the extent to which this theory is transforming our understanding of computation, and beyond that the larger landscape of mathematics. What is the conceptual significance of the main result? Very roughly, it says that a wide variety of computational problems related to games, Walrasian equilibrium of exchange economies, and other fixed point problems, are “equally complex.” In somewhat more detail, there is a method of passing from any problem in this large class to a two player normal form game whose Nash equilibria are associated with solutions of the original problem. If there was an efficient algorithm for finding a Nash equilibrium of a two player game, then for any problem in this class we would have an efficient three step method for solving the problem: a) pass to the two player game; b) find a Nash equilibrium; c) go from there to a solution of the given problem. There are good reasons to believe that some problems in this class are very hard to solve, so this result constitutes extremely compelling evidence that there will never be an algorithm that can find a Nash equilibrium of a large two player game quickly and reliably. We should stress that this result does not say that computation of a Nash equilibrium of a large two player game is impossibly hard. There are computational methods that are reliable, but sometimes quite slow, and there are other methods that are quick when they work, but not reliable. You just can’t have both. More crucially, the most common pattern of software development (start with something that can be proven to work in a practically finite amount of time, then make it better) is infeasible. You must begin with lower aspirations and some more sophisticated view of what you might hope to accomplish. In addition, the central result does not say that all the problems in this large class are “the same” from the point of view of practical computation. Common sense and practical experience suggest that 3 this can hardly be the case, and that there should be some sense in which some algorithms are better than others. We will have little to say about this later, but not because the issue is uninteresting. Rather, these are issues for which robust, theoretically tractable concepts are currently unavailable. (Analysis of mean running times can provide some useful insights, but it is often mathematically challenging, and it is highly dependent on the distribution of problem instances. For these reasons there is as yet little in the way of useful general theory in this direction.) Another way in which we can lower our computational ambitions is that we can ask for approxi- mate, rather than exact, solutions. There is an important improvement of the main result that addresses this issue.
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