Suppose that an @math -simplex is partitioned into @math 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 @math from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant @math uses @math queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant @math uses @math 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.
Best response queries are a weaker but natural query model which is powerful enough to implement fictitious play, a dynamic first proposed by Brown @cite_17 , and proven to converge by Robinson @cite_20 in two-player zero-sum games to an approximate NE. Fictitious play does not always converge for general games where both players have more than two strategies @cite_16 . Furthermore, Daskalakis and Pan have proven that the rate of convergence of the dynamic is quite slow in the worst case (with arbitrary tie-breaking) @cite_15 . Also, beyond non-convergence, the dynamic can have a poor approximation value for general games @cite_23 . In addition, the relationship between best responses and convex partitions of simplices has been studied by Von Stengel @cite_11 in the context of sequential games where one player has to commit to and announce a strategy before playing.
@cite_11 @cite_23 @cite_15 @cite_16 @cite_20 @cite_17
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Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by Brown [6], and shown to converge by Robinson [33]. Samuel Karlin conjectured in 1959 that fictitious play converges at rate O(t -- 1 2) with respect to the number of steps t. We disprove this conjecture by showing that, when the payoff matrix of the row player is the n × n identity matrix, fictitious play may converge (for some tie-breaking) at rate as slow as a#x3A9;(t -- 1 n). We initiate the study of game dynamics in the Sum Basic Network Creation Game, which was recently introduced by [SPAA'10]. In this game players are associated to vertices in a graph and are allowed to "swap" edges, that is to remove an incident edge and insert a new incident edge. By performing such moves, every player tries to minimize her connection cost, which is the sum of distances to all other vertices. When played on a tree, we prove that this game admits an ordinal potential function, which implies guaranteed convergence to a pure Nash Equilibrium. We show a cubic upper bound on the number of steps needed for any improving response dynamic to converge to a stable tree and propose and analyse a best response dynamic, where the players having the highest cost are allowed to move. For this dynamic we show an almost tight linear upper bound for the convergence speed. Furthermore, we contrast these positive results by showing that, when played on general graphs, this game allows best response cycles. This implies that there cannot exist an ordinal potential function and that fundamentally different techniques are required for analysing this case. For computing a best response we show a similar contrast: On the one hand we give a linear-time algorithm for computing a best response on trees even if players are allowed to swap multiple edges at a time. On the other hand we prove that this task is NP-hard even on simple general graphs, if more than one edge can be swapped at a time. The latter addresses a proposal by . Imitating successful behavior is a natural and frequently applied approach when facing scenarios for which we have little or no experience upon which we can base our decision. In this paper, we consider such behavior in atomic congestion games. We propose to study concurrent imitation dynamics that emerge when each player samples another player and possibly imitates this agents' strategy if the anticipated latency gain is sufficiently large. Our main focus is on convergence properties. Using a potential function argument, we show that these dynamics converge in a monotonic fashion to stable states. In such a state none of the players can improve their latency by imitating others. As our main result, we show rapid convergence to approximate equilibria. At an approximate equilibrium only a small fraction of agents sustains a latency significantly above or below average. In particular, imitation dynamics behave like fully polynomial time approximation schemes (FPTAS). Fixing all other parameters, the convergence time depends only in a logarithmic fashion on the number of agents. Since imitation processes are not innovative they cannot discover unused strategies. Furthermore, strategies may become extinct with non-zero probability. For the case of singleton games, we show that the probability of this event occurring is negligible. Additionally, we prove that the social cost of a stable state reached by our dynamics is not much worse than an optimal state in singleton congestion games with linear latency functions. While we concentrate on the case of symmetric network congestion games, most of our results do not explicitly use the network structure. They continue to hold accordingly for general symmetric and asymmetric congestion games when each player samples within his commodity. We consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion in the setting of stochastic search, with application to target localization. Each player yields a noisy response to a binary query governed by a certain error probability. First, we propose a sequential policy for constructing questions that queries each player in sequence and refines the posterior of the target location. Second, we consider a joint policy that asks all players questions in parallel at each time instant and characterize the structure of the optimal policy for constructing the sequence of questions. This generalizes the single player probabilistic bisection method for stochastic search problems. Third, we prove an equivalence between the two schemes showing that, despite the fact that the sequential scheme has access to a more refined filtration, the joint scheme performs just as well on average. Fourth, we establish convergence rates of the mean-square error and derive error exponents. Finally, we obtain an extension to the case of unknown error probabilities. This framework provides a mathematical model for incorporating a human in the loop for active machine learning systems.