Feasible best-response correspondences and quadratic scoring rules

The rational choice paradigm in game theory and other fields of economics has agents best-responding to beliefs about factors that are outside their control. And making certain options a best response is a common problem in mechanism design and information elicitation. But not every correspondence can be made into a best-response correspondence. So what characterizes a feasible best-response correspondence? And once we know that, can we find some or even all utility functions that give rise to this best-response correspondence? We answer these three questions for an expected-utility maximizing agent with finitely many actions and probabilistic beliefs over finitely many states or opponents' strategies. We apply our results to information elicitation problems where contracts (scoring rules) are designed to financially reward an expected-payoff maximizing agent to truthfully reveal a property of her belief by sending a report from some finite set of messages. This leads to a number of new insights: firstly, we characterize exactly which properties can be elicited using scoring rules; secondly, we show that in this class of problems quadratic scoring rules are both necessary and sufficient methods of doing so.