Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited

A probability distribution governing the evolution of a stochastic process has infinitely many Bayesian representations of the form μ=∫μdλ(θ). Among these, a natural representation is one whose components ( μ's) are ‘learnable’ (one can approximate μ by conditioning μ on observation of the process) and ‘sufficient for prediction’ (μ's predictions are not aided by conditioning on observation of the process). We show the existence and uniqueness of such a representation under a suitable asymptotic mixing condition on the process. This representation can be obtained by conditioning on the tail-field of the process, and any learnable representation that is sufficient for prediction is asymptotically like the tail-field representation. This result is related to the celebrated de Finetti theorem, but with exchangeability weakened to an asymptotic mixing condition, and with his conclusion of a decomposition into i.i.d. component distributions weakened to components that are learnable and sufficient for prediction.     

Equilibrium and potential in coalitional congestion games

The model of congestion games is widely used to analyze games related to traffic and communication. A central property of these games is that they are potential games and hence posses a pure Nash equilibrium. In reality, it is often the case that some players cooperatively decide on their joint action in order to maximize the coalition’s total utility. This is modeled by Coalitional Congestion Games. Typical settings include truck drivers who work for the same shipping company, or routers that belong to the same ISP. The formation of coalitions will typically imply that the resulting coalitional congestion game will no longer posses a pure Nash equilibrium. In this paper, we provide conditions under which such games are potential games and posses a pure Nash equilibrium.