7. On Symmetric Games

Gale, David; Kuhn, Harold W.; Tucker, A. W.

doi:10.1515/9781400881727-008

Cited by 27 publications

(31 citation statements)

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“…Proposition 2.1 (Gale et al (1950)). Suppose A and B are m × n matrices whose entries are all positive, and let…”

Section: Symmetric Games and Linear Complementarity Problemsmentioning

confidence: 93%

“…Gale et al (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game.…”

Section: December 31 2007mentioning

confidence: 99%

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Imitation games and computation

McLennan¹,

Tourky²

2010

Games and Economic Behavior

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An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al.(1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) Imitation Games and ComputationAndrew McLennan * and Rabee Tourky † December 31, 2007Abstract: An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.

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“…Proposition 2.1 (Gale et al (1950)). Suppose A and B are m × n matrices whose entries are all positive, and let…”

Section: Symmetric Games and Linear Complementarity Problemsmentioning

confidence: 93%

Section: December 31 2007mentioning

confidence: 99%

Imitation games and computation

McLennan¹,

Tourky²

2010

Games and Economic Behavior

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“…We should also add that in [14,27], a similar oneto-one correspondence and polynomial-time reduction is established between finding Nash equilibria of a game, finding symmetric equilibria of a symmetric game and a solution to an instance of LCP2. Using those results, it is possible to give a shorter, but less self-contained, proof for Theorem 3.1.…”

Section: Bimatrix Games Encode the (Pairing) Leontief Economymentioning

confidence: 99%

Leontief economies encode nonzero sum two-player games

Codenotti

Saberi

Varadarajan

et al. 2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria.We give a reduction from two-player games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash equilibria of any game are in one-to-one correspondence with the equilibria of the corresponding economy.Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies (where an equilibrium is guaranteed to exist) is at least as hard as finding a Nash equilibrium for two-player nonzero sum games.As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [17]. In particular, among other results, we show that it is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium.Perhaps more importantly, we also prove that it is NPhard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known PPAD-completeness result of [30], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer's Fixed Point Theorem.On the algorithmic side, we present an algorithm for finding an approximate equilibrium for some special Leontief economies, which achieves quasi-polynomial time whenever each trader does not demand too much more of any good than some other good.

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“…Structural Result 1: The Interplay Between Symmetries and Randomization. Since the inception of Game Theory scientists were interested in the implications of symmetries in the structure of equilibria [15,6,20]. In his seminal paper [20], Nash showed a rather interesting structural result, informally reading as follows: "If a game has any symmetry, there exists a Nash equilibrium satisfying that symmetry."…”

Section: Overview Of Our Approachmentioning

confidence: 99%

On optimal multidimensional mechanism design

2011

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We efficiently solve the optimal multi-dimensional mechanism design problem for independent bidders with arbitrary demand constraints when either the number of bidders is a constant or the number of items is a constant. In the first setting, we need that each bidder's values for the items are sampled from a possibly correlated, item-symmetric distribution, allowing different distributions for each bidder. In the second setting, we allow the values of each bidder for the items to be arbitrarily correlated, but assume that the distribution of bidder types is bidder-symmetric. These symmetric distributions include i.i.d. distributions, as well as many natural correlated distributions. E.g., an item-symmetric distribution can be obtained by taking an arbitrary distribution, and "forgetting" the names of items; this could arise when different members of a bidder population have various sorts of correlations among the items, but the items are "the same" with respect to a random bidder from the population.For all > 0, we obtain a computationally efficient additive -approximation, when the value distributions are bounded, or a multiplicative (1− )-approximation when the value distributions are unbounded, but satisfy the Monotone Hazard Rate condition, covering a widely studied class of distributions in Economics. Our running time is polynomial in max{#items,#bidders}, and not the size of the support of the joint distribution of all bidders' values for all items, which is typically exponential in both the number of items and the number of bidders. Our mechanisms are randomized, explicitly price bundles, and in some cases can also accommodate budget constraints.Our results are enabled by establishing several new tools and structural properties of Bayesian mechanisms. In particular, we provide a symmetrization technique that turns any truthful mechanism into one that has the same revenue and respects all symmetries in the underlying value distributions. We also prove that item-symmetric mechanisms satisfy a natural strong-monotonicity property which, unlike cyclic-monotonicity, can be harnessed algorithmically. Finally, we provide a technique that turns any given -BIC mechanism (i.e. one where incentive constraints are violated by ) into a truly-BIC mechanism at the cost of O( √ ) revenue. We expect our tools to be used beyond the settings we consider here. Indeed there has already been follow-up research [8,9] making use of our tools.

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7. On Symmetric Games

Cited by 27 publications

References 0 publications

Imitation games and computation

Imitation games and computation

Leontief economies encode nonzero sum two-player games

On optimal multidimensional mechanism design

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