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 $r \in [0,c]$ 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 $f \in \#P$, then $f$ is computable by a one-round rational Merlin-Arthur 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.