Approximate Nash Equilibria with Near Optimal Social Welfare
Authors: Artur Czumaj, Michail Fasoulakis, Marcin Jurdzinski
IJCAI 2015 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | In this paper, we show that for every fixed ε > 0, every bimatrix game (with values in [0, 1]) has an ε-approximate Nash equilibrium with the total payoff of the players at least a constant factor, (1 1 ε)2, of the optimum. Furthermore, our result can be made algorithmic in the following sense: for every fixed 0 ε < ε, if we can find an ε -approximate Nash equilibrium in polynomial time, then we can find in polynomial time an ε-approximate Nash equilibrium with the total payoff of the players at least a constant factor of the optimum. Our analysis is especially tight in the case when ε 1 2. In this case, we show that for any bimatrix game there is an ε-approximate Nash equilibrium with constant size support whose social welfare is at least 2 ε ε 0.914 times the optimal social welfare. Furthermore, we demonstrate that our bound for the social welfare is tight, that is, for every ε 1 2 there is a bimatrix game for which every ε-approximate Nash equilibrium has social welfare at most 2 ε ε times the optimal social welfare. |
| Researcher Affiliation | Academia | Department of Computer Science Centre for Discrete Mathematics and its Applications (DIMAP) University of Warwick, United Kingdom |
| Pseudocode | Yes | ε-APPROXIMATE NASH (R, C, ε) Find i, j such that Rij + Cij is maximized. Find r, c such that Rrj is maximized and Cic is maximized. If Rrj Rij ε and Cic Cij ε, then return strategy profile (i, j). If Rrj Rij < ε and Cic Cij > ε, then set p = ε Cic Cij and return strategy profile (i, pj + (1 p)c). If Rrj Rij > ε and Cic Cij < ε, then set p = ε Rrj Rij and return strategy profile (pi + (1 p)r, j). |
| Open Source Code | No | The paper does not contain any statements about releasing source code or links to a code repository. |
| Open Datasets | No | This is a theoretical paper and does not describe empirical experiments with datasets. Thus, there is no mention of dataset availability for training. |
| Dataset Splits | No | This is a theoretical paper and does not describe empirical experiments with datasets, and therefore no dataset split information for validation is provided. |
| Hardware Specification | No | This is a theoretical paper that focuses on algorithm design and proofs, not empirical experiments. Therefore, no hardware specifications are mentioned. |
| Software Dependencies | No | This is a theoretical paper focused on mathematical proofs and algorithms, not practical implementation or empirical experiments. Therefore, no software dependencies with version numbers are mentioned. |
| Experiment Setup | No | This is a theoretical paper and does not describe empirical experiments, thus no experimental setup details like hyperparameters or training settings are provided. |