The paper relates and unifies a number of extant results in the literature of multi-stage two person games of incomplete information on one side, where uncertainty concerns either the payoffs, such as in repeated games with incomplete information, or the strategies, such as in perturbation games and reputation phenomena. The unification is achieved by a general methodology, applying a few basic properties of a couple of notions of merging of conditional probability distributions related to a stochastic process.
Let \(Q\) denote the true probability distribution followed by a discrete stochastic process, while \(P\) denotes an a priori probability held by an observer on the process. This results in a probability distribution, conditional on past observations of the outcomes of the process, which can be compared with that given by \(Q\). The author makes use of two concepts of merging, i.e. convergence of the conditional probabilities. One is due to Blackwell and Dubins, which is asymptotic in nature, requiring the two probabilities to be close for all future events and is used in games with undiscounted payoffs. The other, used in discounted games where only events in the near future matter, uses a notion of proximity of probabilities for such events. This is the concept of weak merging due to Kalai and Lehrer, albeit strengthened to require a uniform rate of merging.
The general method used, which unifies and simplifies the proofs of the different results, involves studying the uninformed Player 2’s payoffs, who is playing a best reply in equilibrium, to obtain a bound on the payoff under the distribution \(P\) and hence, by merging, similar results under \(Q\). This provides conditions satisfied by the probability \(Q\), and thus a bound on the informed Player 1’s payoff while using his strategy generating \(Q\), under the given strategy of Player 2.
The methodology allows one to replace a series of specific Martingale convergence results, adapted to each particular case, by a general property which is independent of the probability space, easier to use, shortens proofs, and extends results. It reveals hidden connections between incomplete information games with known payoffs and reputation phenomena. The author expects it to have potentially many applications, since it relies on a general bound on the number of bad observations in a merging framework, a property which is independent of the length of the players, of the type of signals and of the nature of perturbations.