We consider the rational versions of two of the classical problems in foundations of cryptography: secret sharing and multiparty computation, suggested by Halpern and Teague (STOC 2004). Our goal is to design games and fair strategies that encourage rational participants to exchange information about their inputs for their mutual benefit, when the only mean of communication is a broadcast channel.We show that protocols for the above information exchanging tasks, where players' values come from a bounded domain, cannot satisfy some of the most desirable properties. In contrast, we provide a rational secret sharing scheme with simultaneous broadcast channel in which shares are taken from an unbounded domain, but have finite (and polynomial sized) expectation.Previous schemes (mostly cryptographic) have required computational assumptions, making them inexact and susceptible to backward induction, or used stronger communication channels. Our scheme is non-cryptographic, immune to backward induction, and satisfies a stronger rationality concept (strict Nash equilibrium). We show that our solution can also be used to construct an ε-Nash equilibrium secret sharing scheme for the case of a non-simultaneous broadcast channel.

The goal of this paper is nding fair protocols for the secret sharing and secure multiparty computation (SMPC) problems, when players are assumed to be rational.It was observed by Halpern and Teague (STOC 2004) that protocols with bounded number of iterations are susceptible to backward induction and cannot be considered rational. Previously suggested cryptographic solutions all share the property of having an essential exponential upper bound on their running time, and hence they are also susceptible to backward induction.Although it seems that this bound is an inherent property of every cryptography based solution, we show that this is not the case. We suggest coalition-resilient secret sharing and SMPC protocols with the property that after any sequence of iterations it is still a computational best response to follow them. Therefore, the protocols can be run any number of iterations, and are immune to backward induction.The mean of communication assumed is a broadcast channel, and we consider both the simultaneous and non-simultaneous cases.

We study the interactive channel capacity of an -noisy channel. The interactive channel capacity C( ) is defined as the minimal ratio between the communication complexity of a problem (over a non-noisy channel), and the communication complexity of the same problem over the binary symmetric channel with noise rate , where the communication complexity tends to infinity. Our main result is the upper bound C( ) ≤ 1 − Ω H( ) . This compares withShannon's non-interactive channel capacity of 1 − H( ). In particular, for a small enough , our result gives the first separation between interactive and non-interactive channel capacity, answering an open problem by Schulman [6].We complement this result by the lower bound C( ) ≥ 1 − O H( ) , proved for the case where the players take alternating turns.

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We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k), and distributional communication complexity ≥ 2 k . This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [Bra12b], our gap is the largest possible. By a result of Braverman and Rao [BR11], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity cannot hold.