IA Scholar Query: Games of fixed rank: A hierarchy of bimatrix games
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Internet Archive Scholar query results feedeninfo@archive.orgThu, 01 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games
https://scholar.archive.org/work/hmyda665rzasbd5fl4iisdauqa
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [Tsaknakis and Spirakis, 2008], with an approximation guarantee of (0.3393+δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a (1/3+δ)-Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [Tsaknakis and Spirakis, 2008], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakis, Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, Grzegorz Hermanwork_hmyda665rzasbd5fl4iisdauqaThu, 01 Sep 2022 00:00:00 GMTA Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games
https://scholar.archive.org/work/6mgvyrav5raipcpofl3toqro4i
Since the seminal PPAD-completeness result for computing a Nash equilibrium even in two-player games, an important line of research has focused on relaxations achievable in polynomial time. In this paper, we consider the notion of ε-well-supported Nash equilibrium, where ε∈ [0,1] corresponds to the approximation guarantee. Put simply, in an ε-well-supported equilibrium, every player chooses with positive probability actions that are within ε of the maximum achievable payoff, against the other player's strategy. Ever since the initial approximation guarantee of 2/3 for well-supported equilibria, which was established more than a decade ago, the progress on this problem has been extremely slow and incremental. Notably, the small improvements to 0.6608, and finally to 0.6528, were achieved by algorithms of growing complexity. Our main result is a simple and intuitive algorithm, that improves the approximation guarantee to 1/2. Our algorithm is based on linear programming and in particular on exploiting suitably defined zero-sum games that arise from the payoff matrices of the two players. As a byproduct, we show how to achieve the same approximation guarantee in a query-efficient way.Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakiswork_6mgvyrav5raipcpofl3toqro4iThu, 14 Jul 2022 00:00:00 GMTA Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games
https://scholar.archive.org/work/aitb5zhy4vgmhkrmrr6heut3sm
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis, with an approximation guarantee of (0.3393+δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a (1/3+δ)-Nash equilibrium, for any constant δ>0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of Tsaknakis and Spirakis, and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakiswork_aitb5zhy4vgmhkrmrr6heut3smThu, 19 May 2022 00:00:00 GMTSimilarity Suppresses Cyclicity: Why Similar Competitors Form Hierarchies
https://scholar.archive.org/work/eyvvo4cxy5g7jhpqlkdhhb5x5i
Competitive systems can exhibit both hierarchical (transitive) and cyclic (intransitive) structures. Despite theoretical interest in cyclic competition, which offers richer dynamics, and occupies a larger subset of the space of possible competitive systems, most real-world systems are predominantly transitive. Why? Here, we introduce a generic mechanism which promotes transitivity, even when there is ample room for cyclicity. Consider a competitive system where outcomes are mediated by competitor attributes via a performance function. We demonstrate that, if competitive outcomes depend smoothly on competitor attributes, then similar competitors compete transitively. We quantify the rate of convergence to transitivity given the similarity of the competitors and the smoothness of the performance function. Thus, we prove the adage regarding apples and oranges. Similar objects admit well ordered comparisons. Diverse objects may not. To test that theory, we run a series of evolution experiments designed to mimic genetic training algorithms. We consider a series of canonical bimatrix games and an ensemble of random performance functions that demonstrate the generality of our mechanism, even when faced with highly cyclic games. We vary the training parameters controlling the evolution process, and the shape parameters controlling the performance function, to evaluate the robustness of our results. These experiments illustrate that, if competitors evolve to optimize performance, then their traits may converge, leading to transitivity.Christopher Cebra, Alexander Strangwork_eyvvo4cxy5g7jhpqlkdhhb5x5iMon, 16 May 2022 00:00:00 GMTAlgorithmic aspects of resource allocation and multiwinner voting: theory and experiments
https://scholar.archive.org/work/ya63ds7t4rdc7fplipudhvkkym
This thesis is concerned with investigating elements of computational social choice in the light of real-world applications. We contribute to a better understanding of the areas of fair allocation and multiwinner voting. For both areas, inspired by real-world scenarios, we propose several new notions and extensions of existing models. Then, we analyze the complexity of answering the computational questions raised by the introduced concepts. To this end, we look through the lens of parameterized complexity. We identify different parameters which describe natural features specific to the computational problems we investigate. Exploiting the parameters, we successfully develop efficient algorithms for spe- cific cases of the studied problems. We complement our analysis by showing which parameters presumably cannot be utilized for seeking efficient algorithms. Thereby, we provide comprehensive pictures of the computational complexity of the studied problems. Specifically, we concentrate on four topics that we present below, grouped by our two areas of interest. For all but one topic, we present experimental studies based on implementations of newly developed algorithms. We first focus on fair allocation of indivisible resources. In this setting, we consider a collection of indivisible resources and a group of agents. Each agent reports its utility evaluation of every resource and the task is to "fairly" allocate the resources such that each resource is allocated to at most one agent. We concentrate on the two following issues regarding this scenario. The social context in fair allocation of indivisible resources. In many fair allocation settings, it is unlikely that every agent knows all other agents. For example, consider a scenario where the agents represent employees of a large corporation. It is highly unlikely that every employee knows every other employee. Motivated by such settings, we come up with a new model of graph envy-freeness by adapting the classical envy-freeness notion to account for social relations [...]Andrzej Kaczmarczyk, Technische Universität Berlin, Rolf Niedermeierwork_ya63ds7t4rdc7fplipudhvkkymFri, 10 Dec 2021 00:00:00 GMTO(1/T) Time-Average Convergence in a Generalization of Multiagent Zero-Sum Games
https://scholar.archive.org/work/wzhbszocnzb35lw6payjlxijpm
We introduce a generalization of zero-sum network multiagent matrix games and prove that alternating gradient descent converges to the set of Nash equilibria at rate O(1/T) for this set of games. Alternating gradient descent obtains this convergence guarantee while using fixed learning rates that are four times larger than the optimistic variant of gradient descent. Experimentally, we show with 97.5 time-averaged strategies that are 2.585 times closer to the set of Nash equilibria than optimistic gradient descent.James P. Baileywork_wzhbszocnzb35lw6payjlxijpmWed, 06 Oct 2021 00:00:00 GMT