For a constant , we prove a polyN lower bound on the (randomized) communication complexity of -Nash equilibrium in two-player N N games. For n-player binary-action games we prove an expn lower bound for the (randomized) communication complexity of , -weak approximate Nash equilibrium, which is a profile of mixed actions such that at least 1 -fraction of the players are -best replying. PPAD x P); even though these assumptions are widely accepted by computer scientists, they make these theorems less accessible to game theorists and economists. Query complexity lower bounds hold only against a fairly restricted model of accessing data, where the algorithm must pay for querying the utility at each strategy profile; what if, for example, we instead give the algorithm access to a best-response oracle? In particular, no lower bounds were known for convergence to approximate Nash equilibrium via randomized uncoupled dynamics (see Subsection 1.1). It is thus of great interest to prove unconditional lower bounds on the much more general model of communication complexity, where each player has unrestricted access to information about her own utility.While computational and query complexity of approximate Nash equilibrium are quite well understood, for communication complexity, only results on pure Nash equilibrium or exact Nash equilibria were known: The communication complexity of pure Nash equilibrium in twoplayer N ! N game is polyN [CS04], and in n-player games it is expn [HM10]. The communication complexity of exact Nash equilibrium in n-player games is also expn [HM10] 1 . No communication complexity lower bounds were known for approximate Nash equilibria. In fact, even for an approximate equilibrium for approximation of value 1~polyN , no bounds were known, see [Nis09a]. In this paper we prove the hardness of approximate Nash equilibria in the randomized 2 communication complexity model.Theorem (Main Theorem, informal). There exists a constant e 0, such that:2-player -Nash equilibrium in two-player N ! N games requires polyN communication.n-player -Nash equilibrium in n-player binary-action games require 2 Ωn communication.In fact, we prove the exponential lower bound even for a weaker notion of , -weak approximate Nash equilibrium, where it is allowed that -fraction of the players will play an arbitrary action (not necessarily an -best-reply).
For a constant , we prove a polyN lower bound on the (randomized) communication complexity of -Nash equilibrium in two-player N N games. For n-player binary-action games we prove an expn lower bound for the (randomized) communication complexity of , -weak approximate Nash equilibrium, which is a profile of mixed actions such that at least 1 -fraction of the players are -best replying.
We show that in an n-player m-action strategic form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three notions of equilibrium: Nash, correlated and coarse correlated. For each one of them we obtain upper and lower bounds on the asymptotic (where max(m, n) → ∞) worst-case number of samples required for the empirical frequency of the sampled action profiles to form an approximate equilibrium with probability close to one.These bounds imply that using a small number of samples we can test whether or not players are playing according to an approximate equilibrium, even in games where n and m are large. In addition, our results include a substantial improvement over the previously known upper bounds on the existence of a small-support approximate equilibrium in games with many players. For all three notions of equilibrium, we show the existence of approximate equilibrium with support size polylogarithmic in n and m, whereas the best previously known results were polynomial in n [8,6,7].
Bayesian experts who are exposed to different evidence often make contradictory probabilistic forecasts. An aggregator, ignorant of the underlying model, uses this to calculate her own forecast. We use the notions of scoring rules and regret to propose a natural way to evaluate an aggregation scheme. We focus on a binary state space and construct low regret aggregation schemes whenever there are only two experts which are either Blackwell-ordered or receive conditionally i.i.d. signals. In contrast, if there are many experts with conditionally i.i.d. signals, then no scheme performs (asymptotically) better than a (0.5, 0.5) forecast.
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