We consider the classical secret sharing problem in the case where all agents are selfish but rational. In recent work, Kol and Naor show that, when there are two players, in the non-simultaneous communication model, i.e. when rushing is possible, there is no Nash equilibrium that ensures both players learn the secret. However, they describe a mechanism for this problem, for any number of players, that is an epsilon-Nash equilibrium, in that no player can gain more than epsilon utility by deviating from it. Unfortunately, the Kol and Naor mechanism, and, to the best of our knowledge, all previous mechanisms for this problem require each agent to send O(n) messages in expectation, where n is the number of agents. This may be problematic for some applications of rational secret sharing such as secure multi-party computation and simulation of a mediator. We address this issue by describing mechanisms for rational secret sharing that are designed for large n. Both of our results hold for n > 2, and are Nash equilbria, rather than just epsilon-Nash equilbria. Our first result is a mechanism for n-out-of-n rational secret sharing that is scalable in the sense that it requires each agent to send only an expected O(log n) bits. Moreover, the latency of this mechanism is O(log n) in expectation, compared to O(n) expected latency for the Kol and Naor result. Our second result is a mechanism for a relaxed variant of rational m-out-of-n secret sharing where m = Theta(n). It requires each processor to send O(log n) bits and has O(log n) latency. Both of our mechanisms are non-cryptographic, and are not susceptible to backwards induction.
The impossibility of a Nash equilibrium for two players carries over to the setting with secure private channels, since there is no difference between private channels and broadcast channels when there are only two players. However, one might hope that the algorithm of Kol and Naor @cite_10 could be simulated over secure private channels to give an everlasting @math -Nash equilibrium. Unfortunately, simulation of broadcast over private channels is expensive, requiring each player to send @math messages per round.
@cite_10
2001578254 2809099106 2094570745 1969882998
We consider the classical secret sharing problem in the case where all agents are selfish but rational. In recent work, Kol and Naor show that in the non-simultaneous communciation model (i.e. when rushing is possible), there is no Nash equilibrium that ensures all agents learn the secret. However, they describe a mechanism for this problem that is an e-Nash equilibrium, i.e. it is close to an equilibrium in the sense that no player can gain more than e utility by deviating from it. Unfortunately, the Kol and Naor mechanism, and, to the best of our knowledge, all previous mechanisms for this problem require each agent to send O(n) messages in expectation, where n is the number of agents. This may be problematic for some applications of rational secret sharing such as secure multiparty computation and simulation of a mediator. We address this issue by describing a mechanism for rational n-out-of-n secret sharing that is an e-Nash equilibrium, and is scalable in the sense that it requires each agent to send only an expected O(1) bits. Moreover, the latency of our mechanism is O(log n) in expectation, compared to O(n) expected latency for the Kol and Naor result. We also design mechanisms for a relaxed variant of rational m-out-of-n secret sharing where m = Θ(n) that require each processor to send O(log n) bits and have O( n) latency. Our mechanisms are non-cryptographic, and are not susceptible to backwards induction. A set of n players, each holding a private input bit, communicate over a noisy broadcast channel. Their mutual goal is for all players to learn all inputs. At each round one of the players broadcasts a bit to all the other players, and the bit received by each player is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed? This problem was first suggested by El Gamal in 1984. In 1988, Gallager gave an elegant noise-resistant protocol requiring only O(n loglogn) rounds. The problem got resolved in 2005 by a seminal paper of Goyal, Kindler, and Saks, proving that Gallager’s protocol is essentially optimal. We revisit the above noisy broadcast problem and show that O(n) rounds suffice. This is possible due to a relaxation of the model assumed by the previous works. We no longer demand that exactly one player broadcasts in every round, but rather allow any number of players to broadcast. However, if it is not the case that exactly one player chooses to broadcast, each of the other players gets an adversely chosen bit. We generalized the above result and initiate the study of interactive coding over the noisy broadcast channel. We show that any interactive protocol that works over the noiseless broadcast channel can be simulated over our restrictive noisy broadcast model with constant blowup of the communication. Our results also establish that modern techniques for interactive coding can help us make progress on the classical problems. We consider the problem of two wireless networks operating on the same (presumably unlicensed) frequency band. Pairs within a given network cooperate to schedule transmissions, but between networks there is competition for spectrum. To make the problem tractable, we assume transmissions are scheduled according to a random access protocol where each network chooses an access probability for its users. A game between the two networks is defined. We characterize the Nash Equilibrium behavior of the system. Three regimes are identified: one in which both networks simultaneously schedule all transmissions, one in which the denser network schedules all transmissions and the sparser only schedules a fraction, and one in which both networks schedule only a fraction of their transmissions. The regime of operation depends on the path loss exponent α, the latter regime being desirable but attainable only for α > 4. This suggests that in certain environments, rival wireless networks may end up naturally cooperating. To substantiate our analytical results, we simulate a system where networks iteratively optimize their access probabilities in a greedy manner. We also discuss a distributed scheduling protocol that employs carrier sensing and demonstrate via simulations that again a near cooperative equilibrium exists for sufficiently large α. 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 e-Nash equilibrium secret sharing scheme for the case of a non-simultaneous broadcast channel.