7 research outputs found
Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries
Suppose that an -simplex is partitioned into 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 from that point.
We present two algorithms for this task: Constant-Dimension Generalised Binary
Search (CD-GBS), which for constant uses queries, and Constant-Region Generalised Binary
Search (CR-GBS), which uses CD-GBS as a subroutine and for constant uses
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 Īµ 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/Īµ)) 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/Īµ)) queries.
We show via Kakutaniās 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 Īµ 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(1Īµ)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly(m,log(1Īµ)) queries. We show via Kakutaniās 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 Īµ 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(1Īµ)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses poly(m,log(1Īµ)) queries. We show via Kakutaniās 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 Ī»-Lipschitz games in which every
playerās payoff function is Ī»-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 ā„ Ī»ā8n 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 Ī»-approximate pure Nash equilibria of Ī»-Lipschitz games and related
classes.
We show a query lower bound exponential in nĪ»/ against randomized algorithms finding -
approximatepure Nash equilibria of n-player, Ī»-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 Ī»-Lipschitz polymatrix games for suitable pairs of values and Ī») 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 ārandomized
PPADā. We define and subsequently analyze the class of āMulti-Lipschitz gamesā, 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 Ī»-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/nā1 -Lipschitz game with an unbounded number of actions