Lipschitz Continuity and Approximate Equilibria

  title={Lipschitz Continuity and Approximate Equilibria},
  author={Argyrios Deligkas and John Fearnley and Paul G. Spirakis},
In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty… 
Efficient algorithms for computing approximate equilibria in bimatrix, polymatrix and Lipschitz games
This thesis designs algorithms for computing approximate equilibria that beat the cur- rent best algorithms for these problems and constructs an approximation-preserving reduction from the problem of computing an approximate Bayesian Nash equilibrium (e-BNE) for a two-player Bayesian game to the problemof computing an e-NE of a polymatrix game and shows that the algorithm of Chapter 4 can be applied to two- player Bayesian games.
Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games
This work develops a query-efficient reduction from more general games to Lipschitz games, and provides an exponential lower bound on the deterministic query complexity of finding -approximate correlated equilibria of n-player, m-action, λ-Lipschitzer games for strong values of , motivating the consideration of explicitly randomized algorithms in the above results.
Computing Equilibria in Atomic Splittable Polymatroid Congestion Games with Convex Costs
It is shown that there is a polynomial time transformation to atomic splittable polymatroid congestion games implying that one can compute $\epsilon$-approximate Cournot-Nash equilibria within pseudo-polynomial time.
G T ] 8 A ug 2 01 8 Equilibrium Computation in Resource Allocation Games
We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion
Equilibrium Computation in Atomic Splittable Singleton Congestion Games
We devise the first polynomial time algorithm computing a pure Nash equilibrium for atomic splittable congestion games with singleton strategies and player-specific affine cost functions. Our
Distance-based Equilibria in Normal-Form Games
We propose a simple uncertainty modification for the agent model in normal-form games; at any given strategy profile, the agent can access only a set of “possible profiles” that are within a certain
Distance-Based Equilibria in Normal-Form Games
We propose a simple uncertainty modification for the agent model in normal-form games; at any given strategy profile, the agent can access only a set of “possible profiles” that are within a certain
Approximating the Existential Theory of the Reals
The main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games, that states that if an ETR problem has an exact solution, then it has a k-uniform approximate solution, where k depends on various properties of the formula.
Algorithms and complexity of problems arising from strategic settings
This thesis deals with an evolutionary setting where it is shown that for a wide range of symmetric bimatrix games, deciding ESS existence is intractable, and presents a general framework for constructing approximation schemes for problems that can be written as an Existential Theory of the Reals formula with variables constrained in a bounded convex set.
Uniqueness and computation of equilibria in resource allocation games
The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.


An Optimization Approach for Approximate Nash Equilibria
An efficient algorithm is provided that computes 0.3393- approximate equilibria, the best approximation till now, based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players' payoffs from the best payoff each player could achieve given the strategy chosen by the other.
New algorithms for approximate Nash equilibria in bimatrix games
A note on approximate Nash equilibria
Well Supported Approximate Equilibria in Bimatrix Games
This paper provides the first polynomial time algorithms constructing ε-SuppNE for normalized or win lose bimatrix games, for any nontrivial constant 0≤ε<1, bounded away from 1.
Playing large games using simple strategies
The existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies is proved and it is proved that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support.
Simple approximate equilibria in large games
It is proved that in every normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium in which each player randomizes uniformly among a set of O(log m + log n) pure actions, and an inverse connection between the entropy of Nash equilibria in the game, and the time it takes to find such an approximation using the random sampling method is established.
Settling the Complexity of Computing Approximate Two-Player Nash Equilibria
  • A. Rubinstein
  • Computer Science
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2016
We prove that there exists a constant ε > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an ε-approximate Nash equilibrium in a two-player (n × n) game requires
Distributed Methods for Computing Approximate Equilibria
A new, distributed method to compute approximate Nash equilibria in bimatrix games that first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players.
Settling the complexity of computing two-player Nash equilibria
We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by