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 work of Kol and Naor @cite_10 is closest to our own work. They show that in the non-simultaneous broadcast model ( , when rushing is possible), there is no Nash equilibrium that ensures all agents learn the secret, at least for the case of two players. They thus consider and solve the problem of designing an @math -Nash equilibrium for the problem in this communication model. An @math -Nash equilibrium is close to an equilibrium in the sense that no player can gain more than @math utility by unilaterally deviating from it. Furthermore, the equilibrium they achieve is in the sense that after any history that is consistent with all players following the protocol, following the protocol continues to be an @math -Nash equilibrium. As we have already discussed, our protocols make use of several clever ideas from their result.
@cite_10
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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. We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a non-transient Nash equilibrium, for any α with 1 < α ≤ √n 2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for α ≥ 12n[log n], every Nash equilibrium forms a tree.Without relying on the tree conjecture, proved an upper bound on the price of anarchy of O(√α), where α ∈ [2, n2]. We improve this bound. Specifically, we derive a constant upper bound for α ∈ O(√n) and for α ≥ 12n[log n]. For the intermediate values we derive an improved bound of O(1 + (min α2 n, n2 α )1 3).Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing. We study network design games where @math self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as @math . Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of @math for weighted games, where @math is the total weight of all the agents. This almost matches the known upper bound of @math . Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. This paper addresses the control of a team of autonomous agents pursuing a smart evader in a non-accurately mapped terrain. By describing the problem as a partial information Markov game, we are able to integrate map-learning and pursuit. We propose receding horizon control policies, in which the pursuers and the evader try to respectively maximize and minimize the probability of capture at the next time instant. Since this probability is conditioned to distinct observations for each team, the resulting game is nonzero-sum. When the evader has access to the pursuers' information, we show that a Nash solution to the one-step nonzero-sum game always exists. Moreover, we propose a method to compute the Nash equilibrium policies by solving an equivalent zero-sum matrix game. A simulation example shows the feasibility of the proposed approach.