Lipschitz Games

Azrieli, Yaron; Shmaya, Eran

doi:10.1287/moor.1120.0557

Cited by 34 publications

(92 citation statements)

References 19 publications

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“…Any λ-Lipschitz k-strategy anonymous game is guaranteed to have an ǫ-approximate pure Nash equilibrium, with ǫ = O(λk) [2,15]. The convergence rate to a Nash equilibrium of best-reply dynamics in the context of two-strategy Lipschitz anonymous games is studied by [3,24].…”

Section: Related Workmentioning

confidence: 99%

“…Given this, we must set p r = 4 5 and p c = 2 3 in order for r and c to be in equilibrium between each other. Player m's expected payoff for playing 1 is 2 …”

Section: Exact Nash Equilibriamentioning

confidence: 99%

“…where the first inequality used Equation (2). We should perhaps note that in summing over big-oh expressions, the hidden constant in the notation is the same for all terms in the summation (namely, one that applies for k − 1 strategy games, which exists by inductive hypothesis).…”

Section: Proofmentioning

confidence: 99%

“…Below, we also discuss why Theorem 5.2 does not extend straightforwardly to more than two strategies. [2] ) be a two-strategy anonymous game, and let G := (n, 2, {ū i j } i∈[n],j∈ [2] ) be the following self-anonymous game.ū i 2 (x + 1) is set equal tō u i 1 (x) for all x, as required forḠ to be self-anonymous. Then,…”

Section: Proofmentioning

confidence: 99%

“…The following algorithms exploit our knowledge of the existence of pure approximate equilibria in Lipschitz games [2,15]. Algorithm 2 is subsequently used in Section 7 as a subroutine of an algorithm for general (not necessarily Lipschitz) anonymous games.…”

Section: Two-strategy Lipschitz Gamesmentioning

confidence: 99%

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Query complexity of approximate equilibria in anonymous games

Goldberg

Turchetta²

2017

Journal of Computer and System Sciences

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Abstract. We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is query complexity, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-complexity bounds are attainable if either further symmetries of the utility functions are assumed or we focus on approximate equilibria. We investigate four sub-classes of anonymous games previously considered by [5,15]. Our main result is a new randomized query-efficient algorithm that finds a O(n −1/4 )-approximate Nash equilibrium queryingÕ(n 3/2 ) payoffs and runs in timeÕ(n 3/2 ). This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be utilized as an efficient polynomial-time approximation scheme (PTAS). Furthermore, we prove that Ω(n log n) payoffs must be queried in order to find any ǫ-well-supported Nash equilibrium, even by randomized algorithms.

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Section: Related Workmentioning

confidence: 99%

“…Given this, we must set p r = 4 5 and p c = 2 3 in order for r and c to be in equilibrium between each other. Player m's expected payoff for playing 1 is 2 …”

Section: Exact Nash Equilibriamentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Two-strategy Lipschitz Gamesmentioning

confidence: 99%

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Query complexity of approximate equilibria in anonymous games

Goldberg

Turchetta²

2017

Journal of Computer and System Sciences

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Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games

Goldberg

Katzman

2021

Algorithmic Game Theory

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Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every player's payoff function is λ-Lipschitz with respect to the actions of the other players. They showed that such games admit -approximate pure Nash equilibria for certain settings of and λ. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding -approximate pure Nash equilibria of n-player, binaryaction, λ-Lipschitz games that is exponential in nλ/ . In addition, we introduce "Multi-Lipschitz games," a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the average of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding -approximate Nash equilibria of n-player, m-action, λ-Lipschitz games for strong values of , motivating the consideration of explicitly randomized algorithms in the above results.

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PPAD-Complete Pure Approximate Nash Equilibria in Lipschitz Games

Goldberg

Katzman

2022

Algorithmic Game Theory

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Lipschitz Games

Cited by 34 publications

References 19 publications

Query complexity of approximate equilibria in anonymous games

Query complexity of approximate equilibria in anonymous games

Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games

PPAD-Complete Pure Approximate Nash Equilibria in Lipschitz Games

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