Sampled Fictitious Play is Hannan Consistent
This provides a new Hannan-consistent algorithm for game theory, addressing a known limitation in adaptive heuristics for repeated games.
The paper tackles the problem of ensuring Hannan consistency in repeated games by proposing sampled fictitious play with Bernoulli sampling, and proves that this variant achieves Hannan consistency, unlike standard fictitious play.
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.