Players who have a common interest are engaged in a game with incomplete information. Before playing they get differential signals that stochastically depend on the actual state of nature. These signal not only provide the players with partial information about the state of nature but also serve as a correlation means.Different information structures induce different outcomes. An information structure is better than another, with respect to a certain solution concept, if the highest solution payoff it induces is at least that induced by the latter structure. This paper fully characterizes when one information structure is better than another with respect to various solution concepts. The solution concepts we refer to differ from each other in the scope of communication allowed between the players. The characterizations are phrased in terms of maps that take signals of one structure and (stochastically) translate them to signals of another structure.
The Lipschitz constant of a finite normal-form game is the maximal change in some player's payoff when a single opponent changes his strategy. We prove that games with small Lipschitz constant admit pure {\epsilon}-equilibria, and pinpoint the maximal Lipschitz constant that is sufficient to imply existence of pure {\epsilon}-equilibrium as a function of the number of players in the game and the number of strategies of each player. Our proofs use the probabilistic method.Comment: minor changes, forthcoming in Mathematics of Operations Researc
A sender persuades a receiver to accept a project by disclosing information about a payoff‐relevant quality. The receiver has private information about the quality, referred to as his type. We show that the sender‐optimal mechanism takes the form of nested intervals: each type accepts on an interval of qualities and a more optimistic type's interval contains a less optimistic type's interval. This nested‐interval structure offers a simple algorithm to solve for the optimal disclosure and connects our problem to the monopoly screening problem. The mechanism is optimal even if the sender conditions the disclosure mechanism on the receiver's reported type.
A theorem of Blackwell about comparison between information structures in classical statistics is given an analogue in the quantum probabilistic setup. The theorem provides an operational interpretation for tracepreserving completely positive maps, which are the natural quantum analogue of classical stochastic maps. The proof of the theorem relies on the separation theorem for convex sets and on quantum teleportation.
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