Bargaining networks model the behavior of a set of players that need to reach pairwise agreements for making profits. Nash bargaining solutions are special outcomes of such games that are both stable and balanced. Kleinberg and Tardos proved a sharp algorithmic characterization of such outcomes, but left open the problem of how the actual bargaining process converges to them. A partial answer was provided by who proposed a distributed algorithm for constructing Nash bargaining solutions, but without polynomial bounds on its convergence rate. In this paper, we introduce a simple and natural model for this process, and study its convergence rate to Nash bargaining solutions. At each time step, each player proposes a deal to each of her neighbors. The proposal consists of a share of the potential profit in case of agreement. The share is chosen to be balanced in Nash's sense as far as this is feasible (with respect to the current best alternatives for both players). We prove that, whenever the Nash bargaining solution is unique (and satisfies a positive gap condition) this dynamics converges to it in polynomial time. Our analysis is based on an approximate decoupling phenomenon between the dynamics on different substructures of the network. This approach may be of general interest for the analysis of local algorithms on networks.
Azar and co-authors @cite_6 first studied the question as to whether a balanced outcome can be produced by a local dynamics, and were able to answer positively. Their results left however two outstanding challenges: @math The bound on the convergence time proved in @cite_6 is exponential in the network size, and therefore does not provide a solid justification for convergence to NB solutions in large networks; @math The algorithm analyzed by these authors first selects a matching @math in @math @cite_10 , corresponding to the pairing of players that trade. In a second phase the algorithm determines the profit of each player. While such an algorithm can be implemented in a distributed way, point out that it is not entirely realistic. Indeed the rules of the dynamics change after the matching is found. Further, if the pairing is established at the outset, the players lose their bargaining power.
@cite_10 @cite_6
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We study the speed of convergence of decentralized dynamics to approximately optimal solutions in potential games. We consider α-Nash dynamics in which a player makes a move if the improvement in his payoff is more than an α factor of his own payoff. Despite the known polynomial convergence of α-Nash dynamics to approximate Nash equilibria in symmetric congestion games [7], it has been shown that the convergence time to approximate Nash equilibria in asymmetric congestion games is exponential [25]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with linear and polynomial delay functions, the convergence time of α-Nash dynamics to an approximate optimal solution is polynomial in the number of players, with approximation ratio that is arbitrarily close to the price of anarchy of the game. In particular, we show this polynomial convergence under the minimal liveness assumption that each player gets at least one chance to move in every T steps. We also prove that the same polynomial convergence result does not hold for (exact) best-response dynamics, showing the α-Nash dynamics is required. We extend these results for congestion games to other potential games including weighted congestion games with linear delay functions, cut games (also called party affiliation games) and market sharing games. Two players A and B are randomly placed on a line. The distribution of the distance between them is unknown except that the expected initial distance of the (two) players does not exceed some constant @math The players can move with maximal velocity 1 and would like to meet one another as soon as possible. Most of the paper deals with the asymmetric rendezvous in which each player can use a different trajectory. We find rendezvous trajectories which are efficient against all probability distributions in the above class. (It turns out that our trajectories do not depend on the value of @math ) We also obtain the minimax trajectory of player A if player B just waits for him. This trajectory oscillates with a geometrically increasing amplitude. It guarantees an expected meeting time not exceeding @math We show that, if player B also moves, then the expected meeting time can be reduced to @math The expected meeting time can be further reduced if the players use mixed strategies. We show that if player B rests, then the optimal strategy of player A is a mixture of geometric trajectories. It guarantees an expected meeting time not exceeding @math This value can be reduced even more (below @math ) if player B also moves according to a (correlated) mixed strategy. We also obtain a bound for the expected meeting time of the corresponding symmetric rendezvous problem. We initiate the study of game dynamics in the Sum Basic Network Creation Game, which was recently introduced by [SPAA'10]. In this game players are associated to vertices in a graph and are allowed to "swap" edges, that is to remove an incident edge and insert a new incident edge. By performing such moves, every player tries to minimize her connection cost, which is the sum of distances to all other vertices. When played on a tree, we prove that this game admits an ordinal potential function, which implies guaranteed convergence to a pure Nash Equilibrium. We show a cubic upper bound on the number of steps needed for any improving response dynamic to converge to a stable tree and propose and analyse a best response dynamic, where the players having the highest cost are allowed to move. For this dynamic we show an almost tight linear upper bound for the convergence speed. Furthermore, we contrast these positive results by showing that, when played on general graphs, this game allows best response cycles. This implies that there cannot exist an ordinal potential function and that fundamentally different techniques are required for analysing this case. For computing a best response we show a similar contrast: On the one hand we give a linear-time algorithm for computing a best response on trees even if players are allowed to swap multiple edges at a time. On the other hand we prove that this task is NP-hard even on simple general graphs, if more than one edge can be swapped at a time. The latter addresses a proposal by . Imitating successful behavior is a natural and frequently applied approach when facing scenarios for which we have little or no experience upon which we can base our decision. In this paper, we consider such behavior in atomic congestion games. We propose to study concurrent imitation dynamics that emerge when each player samples another player and possibly imitates this agents' strategy if the anticipated latency gain is sufficiently large. Our main focus is on convergence properties. Using a potential function argument, we show that these dynamics converge in a monotonic fashion to stable states. In such a state none of the players can improve their latency by imitating others. As our main result, we show rapid convergence to approximate equilibria. At an approximate equilibrium only a small fraction of agents sustains a latency significantly above or below average. In particular, imitation dynamics behave like fully polynomial time approximation schemes (FPTAS). Fixing all other parameters, the convergence time depends only in a logarithmic fashion on the number of agents. Since imitation processes are not innovative they cannot discover unused strategies. Furthermore, strategies may become extinct with non-zero probability. For the case of singleton games, we show that the probability of this event occurring is negligible. Additionally, we prove that the social cost of a stable state reached by our dynamics is not much worse than an optimal state in singleton congestion games with linear latency functions. While we concentrate on the case of symmetric network congestion games, most of our results do not explicitly use the network structure. They continue to hold accordingly for general symmetric and asymmetric congestion games when each player samples within his commodity.