On Convergence and Optimality of Best-Response Learning with Policy Types in Multiagent Systems — Appendix

Proof. Kalai and Lehrer [2] studied a model which can be equivalently described as a single-state SBG (i.e. |S| = 1) with a pure type distribution and product posterior. They showed that, if the player’s assessment of future play is absolutely continuous with respect to the true probabilities of future play (i.e. any event that has true positive probability is assigned positive probability by the player), then (1) must hold. In our case, absolute continuity always holds by Assumption 5 and the fact that the prior probabilities Pj are positive, as well as the fact that the type distribution is pure (from which we can infer that the true types always have positive posterior probability). In this proof, we seek to extend the convergence result of Kalai and Lehrer (henceforth [2]) to multi-state SBGs with pure type distributions. Our strategy is to translate a SBG Γ into a modified SBG Γ̂ which is equivalent to Γ in the sense that the players behave identically, and which is