  Interactive proofs (IP) model a world where a verifier delegates computation
to an untrustworthy prover, verifying the prover's claims before accepting
them. IP protocols have applications in areas such as verifiable computation
outsourcing, computation delegation, cloud computing. In these applications,
the verifier may pay the prover based on the quality of his work. Rational
interactive proofs (RIP), introduced by Azar and Micali (2012), are an
interactive-proof system with payments, in which the prover is rational rather
than untrustworthy---he may lie, but only to increase his payment. Rational
proofs leverage the provers' rationality to obtain simple and efficient
protocols. Azar and Micali show that RIP=IP(=PSAPCE). They leave the question
of whether multiple provers are more powerful than a single prover for rational
and classical proofs as an open problem.
  In this paper, we introduce multi-prover rational interactive proofs (MRIP).
Here, a verifier cross-checks the provers' answers with each other and pays
them according to the messages exchanged. The provers are cooperative and
maximize their total expected payment if and only if the verifier learns the
correct answer to the problem. We further refine the model of MRIP to
incorporate utility gap, which is the loss in payment suffered by provers who
mislead the verifier to the wrong answer.
  We define the class of MRIP protocols with constant, noticeable and
negligible utility gaps. We give tight characterization for all three MRIP
classes. We show that under standard complexity-theoretic assumptions, MRIP is
more powerful than both RIP and MIP ; and this is true even the utility gap is
required to be constant. Furthermore the full power of each MRIP class can be
achieved using only two provers and three rounds. (A preliminary version of
this paper appeared at ITCS 2016. This is the full version that contains new
results.)
