Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

A learning-based approach to multi-agent decision-making

We propose a learning-based methodology to reconstruct private information held by a population of interacting agents in order to predict an exact outcome of the underlying multi-agent interaction process, here identified as a stationary action profile. We envision a scenario where an external observer, endowed with a learning procedure, is allowed to make queries and observe the agents' reactions through private action-reaction mappings, whose collective fixed point corresponds to a stationary profile. By adopting a smart query process to iteratively collect sensible data and update parametric estimates, we establish sufficient conditions to assess the asymptotic properties of the proposed learning-based methodology so that, if convergence happens, it can only be towards a stationary action profile. This fact yields two main consequences: i) learning locally-exact surrogates of the action-reaction mappings allows the external observer to succeed in its prediction task, and ii) working with assumptions so general that a stationary profile is not even guaranteed to exist, the established sufficient conditions hence act also as certificates for the existence of such a desirable profile. Extensive numerical simulations involving typical competitive multi-agent control and decision making problems illustrate the practical effectiveness of the proposed learning-based approach

Learning convex partitions and computing game-theoretic equilibria from best-response queries

Suppose that an m-simplex is partitioned into n convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance ε from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant m uses poly(n, log (1/ε)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly(m, log (1/ε)) queries. We show via Kakutani’s fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.</p

Learning convex partitions and computing game-theoretic equilibria from best response queries

Suppose that an m-simplex is partitioned into n convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance ε from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant m uses poly(n,log(1ε)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly(m,log(1ε)) queries. We show via Kakutani’s fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies

Learning convex partitions and computing game-theoretic equilibria from best response queries

Suppose that an m-simplex is partitioned into n convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance ε from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant m uses poly(n,log(1ε)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly(m,log(1ε)) queries. We show via Kakutani’s fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies

The complexity of solution concepts in Lipschitz games

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 via the probabilistic method that n-player Lipschitz games with m strategies per player have pure -approximate Nash equilibria, for ≥ λ√8n log(2mn). They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop an efficient reduction from more general games to Lipschitz games. We use this reduction to study both the query and computational complexity of algorithms finding λ-approximate pure Nash equilibria of λ-Lipschitz games and related classes. We show a query lower bound exponential in nλ/ against randomized algorithms finding - approximatepure Nash equilibria of n-player, λ-Lipschitz games. We additionally present the first PPAD-completeness result for finding pure Nash equilibria in a class of finite, non-Bayesian games (we show this for λ-Lipschitz polymatrix games for suitable pairs of values and λ) in which both the proof of PPAD-hardness and the proof of containment in PPAD require novel approaches (in fact, our approach implies containment in PPAD for any class of Lipschitz games in which payoffs from mixed-strategy profiles can be deterministically computed), and present a definition of “randomized PPAD”. We define and subsequently analyze the class of “Multi-Lipschitz games”, a generalization of Lipschitz games involving player-specific Lipschitz parameters in which the value of interest appears to be the average of the individual Lipschitz parameters. We discuss a dichotomy of the deterministic query complexity of finding -approximate Nash equilibria of general games and, subsequently, a query lower bound for λ-Lipschitz games in which any non-trivial value of requires exponentially-many queries to achieve. We examine which parts of this extend to the concepts of approximate correlated and coarse correlated equilibria, and in the process generalize the edge-isoperimetric inequalities to generalizations of the hypercube. Finally, we improve the block update algorithm presented by Goldberg and Marmolejo to break the potential boundary of a 0.75-approximation factor, presenting a randomized algorithm achieving a 0.7368-approximate Nash equilibrium making polynomially-many profile queries of an n-player 1/n−1 -Lipschitz game with an unbounded number of actions