Rational Proofs with Multiple Provers

Interactive proofs model a world where a verifier delegates computation to an untrustworthy prover, verifying the prover's claims before accepting them. These proofs have applications to delegation of computation, probabilistically checkable proofs, crowdsourcing, and more. In some of these applications, the verifier may pay the prover based on the quality of his work. Rational proofs, introduced by Azar and Micali (2012), are an interactive proof model in which the prover is rational rather than untrustworthy--he may lie, but only to increase his payment. This allows the verifier to leverage the greed of the prover to obtain better protocols: while rational proofs are no more powerful than interactive proofs, the protocols are simpler and more efficient. Azar and Micali posed as an open problem whether multiple provers are more powerful than one for rational proofs. We provide a model that extends rational proofs to allow multiple provers. In this model, a verifier can cross-check the answers received by asking several provers. The verifier can pay the provers according to the quality of their work, incentivizing them to provide correct information. We analyze rational proofs with multiple provers from a complexity-theoretic point of view. We fully characterize this model by giving tight upper and lower bounds on its power. On the way, we resolve Azar and Micali's open problem in the affirmative, showing that multiple rational provers are strictly more powerful than one (under standard complexity-theoretic assumptions). We further show that the full power of rational proofs with multiple provers can be achieved using only two provers and five rounds of interaction. Finally, we consider more demanding models where the verifier wants the provers' payment to decrease significantly when they are lying, and fully characterize the power of the model when the payment gap must be noticeable (i.e., at least 1/p where p is a polynomial).