Rational Proof Systems and Scoring Rules

We study a new type of proof system, where an unbounded prover and a polynomial time verifier interact, on inputs a string x and a function f , so that the Verifier may learn f(x). The novelty of our setting is that there no longer are “good” or “malicious” provers, but only rational ones. In essence, the Verifier has a budget c and gives the Prover a reward r ∈ [0, c] determined by the transcript of their interaction; the prover wishes to maximize his expected reward; and his reward is maximized only if he the verifier correctly learns f(x). Rational proof systems are as powerful as their classical counterparts for polynomially many rounds of interaction, but are much more powerful when we only allow a constant number of rounds. Indeed, we prove that if f ∈ #P , then f is computable by a one-round rational MerlinArthur game, where, on input x, Merlin’s single message actually consists of sending just the value f(x). Further, we prove that CH, the counting hierarchy, coincides with the class of languages computable by a constant-round rational Merlin-Arthur game. Our results rely on a basic and crucial connection between rational proof systems and proper scoring rules, a tool developed to elicit truthful information from experts. Guided by our interest in verifiers who are computationally limited, we make two contributions to the theory of proper scoring rules. First, we prove that any deterministic, bounded scoring rule must make a number of queries to its input distribution that is linear in the number of states of the world. When this number of states is large, this is computationally infeasible. Second, we show a new generalization of scoring rules, sampling scoring rules, which allow the verifier to compute them using only two queries to the input distribution. The proof of our lower bound on the number of samples leverages Turan’s theorem, a classical result in extremal graph theory.