Bayesian proportional resource allocation games

We consider a proportional allocation mechanism that gives to each user an amount of a resource proportional to the user´s bid. We study a corresponding Bayesian game in which each user has incomplete information on the state or type of the other users. We prove the existence of a Bayes-Nash equilibrium. Furthermore, under some mild assumptions, we establish asymptotic efficiency: we show that the per user social welfare achieved at any Bayes-Nash equilibrium is asymptotically equal to the maximum possible.