Simple approximate equilibria in large games

Abstract

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].

Achieving target equilibria in network routing games without knowing the latency functions

“…For each edge e ′ , let α e ′ := 1 e ′ κ(f min /2). By Corollary 3.19, f (τ (1) + α e ′ ) > f min /2 for each edge e ′ . Then for each edge e ∈ E, define β e,e ′ = f e (τ (1) + α e ′ ) − f e (τ (1) ) /κ(f min /2).…”

“…An oblivious algorithm that does not depend on player utilities, and instead uses best-responses to compute a pure Nash equilibrium in bimatrix games was given by Sureka and Wurman [32]. Starting with [28], various papers have studied the complexity of computing an exact or approximate correlated equilibrium in multi-player games using both pure-and mixed-strategy queries [1,14,15]. More recently, Fearnley et al [9] study algorithms in the empiricalgame-theory model for bimatrix games, congestion games, and graphical games, and obtain various bounds on the number of queries required for equilibrium computation.…”

“…1 We assume the delay function on any edge e is l e (x) = a e x + b e . Define l max (x) := max e∈E a e x + b e , and κ(x) = x 2 /Kd.…”

“…While a similar result on the linearity of the equilibrium flow was shown in [7], Lemma 3.22 shows how to obtain the coefficients of the linear map. (1) , let f (τ (1) ) > 0. Then there exist coefficients (β e,e ′ ) e,e ′ ∈E so that for any tolls τ ,…”

“…Given tolls τ , let g := f (τ (1) ) + e ′ β e,e ′ τ e ′ . In general, g may be negative on some edges.…”