Signaling With Commitment

We study the canonical signaling game, endowing the sender with commitment power: before learning the state, sender designs a strategy, which maps the state into a probability distribution over actions. We provide a geometric characterization of the sender's attainable payoffs, described by the topological join of the graphs of the interim payoff functions associated with different sender actions. We then incorporate payoff irrelevant messages into the game, characterizing the attainable payoffs when sender commits to (i) both messages and actions, and (ii) only messages. By studying the tradeoffs between these model variations, we highlight the power of commitment to actions. We apply our results to the design of adjudication procedures and rating systems.