Sampled Fictitious Play is Hannan Consistent

TL;DR

This paper proves that sampled fictitious play, a variant of the adaptive heuristic for repeated games, is Hannan consistent using anti-concentration techniques from Littlewood-Offord theory.

Contribution

It introduces and proves Hannan consistency for sampled fictitious play using a novel anti-concentration approach, differing from traditional concentration-based proofs.

Findings

Sampled fictitious play is Hannan consistent.

Uses Bernoulli sampling in the analysis.

Employs Littlewood-Offord anti-concentration results.

Abstract

Fictitious play is a simple and widely studied adaptive heuristic for playing repeated games. It is well known that fictitious play fails to be Hannan consistent. Several variants of fictitious play including regret matching, generalized regret matching and smooth fictitious play, are known to be Hannan consistent. In this note, we consider sampled fictitious play: at each round, the player samples past times and plays the best response to previous moves of other players at the sampled time points. We show that sampled fictitious play, using Bernoulli sampling, is Hannan consistent. Unlike several existing Hannan consistency proofs that rely on concentration of measure results, ours instead uses anti-concentration results from Littlewood-Offord theory.