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43
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
 IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 39 (4 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any nontrivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot sum problem, as well as a more general arithmetic circuit decision problem that characterizes Ptime in a unitcost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
Complexity of computing optimal Stackelberg strategies in security resource allocation games
 In Proceedings of the National Conference on Artificial Intelligence (AAAI
, 2010
"... Recently, algorithms for computing gametheoretic solutions have been deployed in realworld security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strateg ..."
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Cited by 31 (9 self)
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Recently, algorithms for computing gametheoretic solutions have been deployed in realworld security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strategy, a socalled Stackelberg model. As pointed out by Kiekintveld et al. (Kiekintveld et al. 2009), in these applications, generally, multiple resources need to be assigned to multiple targets, resulting in an exponential number of pure strategies for the defender. In this paper, we study how to compute optimal Stackelberg strategies in such games, showing that this can be done in polynomial time in some cases, and is NPhard in others.
How hard is it to approximate the best Nash equilibrium?
, 2009
"... The quest for a PTAS for Nash equilibrium in a twoplayer game seeks to circumvent the PPADcompleteness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibri ..."
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Cited by 10 (0 self)
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The quest for a PTAS for Nash equilibrium in a twoplayer game seeks to circumvent the PPADcompleteness of an (exact) Nash equilibrium by finding an approximate equilibrium, and has emerged as a major open question in Algorithmic Game Theory. A closely related problem is that of finding an equilibrium maximizing a certain objective, such as the social welfare. This optimization problem was shown to be NPhard by Gilboa and Zemel [Games and Economic Behavior 1989]. However, this NPhardness is unlikely to extend to finding an approximate equilibrium, since the latter admits a quasipolynomial time algorithm, as proved by Lipton, Markakis and Mehta [Proc. of 4th EC, 2003]. We show that this optimization problem, namely, finding in a twoplayer game an approximate equilibrium achieving large social welfare is unlikely to have a polynomial time algorithm. One interpretation of our results is that the quest for a PTAS for Nash equilibrium should not extend to a PTAS for finding the best Nash equilibrium, which stands in contrast to certain algorithmic techniques used so far (e.g. sampling and enumeration). Technically, our result is a reduction from a notoriously difficult problem in modern Combinatorics, of finding a planted (but hidden) clique in a random graph G(n, 1/2). Our reduction starts from an instance with planted clique size k = O(log n). For comparison, the currently known algorithms due to Alon, Krivelevich and Sudakov [Random Struct. & Algorithms, 1998], and Krauthgamer and Feige [Random Struct. & Algorithms, 2000], are effective for a much larger clique size k = Ω(√n).
A Revealed Preference Approach to Computational Complexity in Economics
, 2010
"... One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally ..."
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Cited by 8 (1 self)
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One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally hard in the worstcase under reasonable assumptions. In this paper, we study the empirical consequences of strengthening consumer choice theory to enforce that utilities are computationally easy to maximize. We prove the possibly surprising result that computational constraints have no empirical consequences whatsoever for consumer choice theory. That is, a data set is consistent with a utility maximizing consumer if and only if a data set is consistent with a utility maximizing consumer having a utility function that can be maximized in strongly polynomial time. Our result motivates a general approach for posing questions about the empirical content of computational constraints: the revealed preference approach to computational complexity. The approach complements the conventional worstcase view of computational complexity in important ways, and is methodologically close to mainstream economics.
Reducibility among fractional stability problems
 in Proc. IEEE FOCS, 2009
"... Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particular ..."
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Cited by 5 (0 self)
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Abstract — In a landmark paper [32], Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [11], [16], [5], [6], [7], [8] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPADcomplete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications. Here is the list of problems we show to be PPADcomplete, along with the domains of practical significance: Fractional Stable
Market Equilibrium under Separable, PiecewiseLinear, Concave Utilities
"... We consider Fisher and ArrowDebreu markets under additivelyseparable, piecewiselinear, concave utility functions, and obtain the following results: • For both market models, if an equilibrium exists, there is one that is rational and can be written using polynomially many bits. • There is no simp ..."
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Cited by 5 (1 self)
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We consider Fisher and ArrowDebreu markets under additivelyseparable, piecewiselinear, concave utility functions, and obtain the following results: • For both market models, if an equilibrium exists, there is one that is rational and can be written using polynomially many bits. • There is no simple necessary and sufficient condition for the existence of an equilibrium: The problem of checking for existence of an equilibrium is NPcomplete for both market models; the same holds for existence of an ɛapproximate equilibrium, for ɛ = O(n −5). • Under standard (mild) sufficient conditions, the problem of finding an exact equilibrium is in PPAD for both market models. • Finally, building on the techniques of [CDDT09] we prove that under these sufficient conditions, finding an equilibrium for Fisher markets is PPADhard.
Metastability of Logit Dynamics for Coordination Games
, 2012
"... Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. ..."
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Cited by 3 (2 self)
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Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the longterm equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. In several cases it happens that on a timescale shorter than mixing time the chain is “quasistationary”, meaning that it stays close to some small set of the state space, while in a timescale multiple of the mixing time it jumps from one quasistationaryconfiguration to another; this phenomenon is usually called “metastability”. In this paper we give a quantitative definition of “metastable probability distributions ” for a Markov chain and we study the metastability of the Logit dynamics for some classes of coordination games. In particular, we study noriskdominant coordination games on the clique (which is equivalent to the wellknown Glauber dynamics for the Ising model) and coordination games on a ring (both the riskdominant and noriskdominant case). We also describe a simple “artificial” game that highlights the distinctive features of our metastability notion based on distributions.
2Player Nash and Nonsymmetric Bargaining Games: Algorithms and Structural Properties
"... The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2player game whose convex program has linear constraints, admits a rational solution and such a solution can be fo ..."
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Cited by 2 (2 self)
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The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are “small”, then its solution can be found in strongly polynomial time. We also give a nonsuccinct linear game whose solution can be found in strongly polynomial time. 1
A complementary pivot algorithm for market equilibrium under separable piecewiselinear concave utilities
 In ACM Symposium on the Theory of Computing
, 2012
"... Using the powerful machinery of the linear complementarity problem and Lemke’s algorithm, we give a practical algorithm for computing an equilibrium for ArrowDebreu markets under separable, piecewiselinear concave (SPLC) utilities, despite the PPADcompleteness of this case. As a corollary, we obt ..."
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Cited by 2 (1 self)
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Using the powerful machinery of the linear complementarity problem and Lemke’s algorithm, we give a practical algorithm for computing an equilibrium for ArrowDebreu markets under separable, piecewiselinear concave (SPLC) utilities, despite the PPADcompleteness of this case. As a corollary, we obtain the first elementary proof of existence of equilibrium for this case, i.e., without using fixed point theorems. In 1975, Eaves [10] had given such an algorithm for the case of linear utilities and had asked for an extension to the piecewiselinear, concave utilities. Our result settles the relevant subcase of his problem as well as the problem of Vazirani and Yannakakis of obtaining a path following algorithm for SPLC markets, thereby giving a direct proof of membership of this case in PPAD. We also prove that SPLC markets have an odd number of equilibria (up to scaling), hence matching the classical result of Shapley about 2Nash equilibria [25], which was based on the LemkeHowson algorithm. For the linear case, Eaves had asked for a combinatorial interpretation of his algorithm. We provide this and it yields a particularly simple proof of the fact that the set of equilibrium prices is convex. 1