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Network congestion games with player-specific delay functions do not possess pure Nash equilibria in general. We therefore address the computational complexity of the corresponding decision problem and prove that it is $NP$-complete to decide whether a pure Nash equilibrium exists. This result is true for games with directed edges as well as for networks with undirected edges, and still holds for games with two players only. In contrast to games with networks of arbitrary size, we present a polynomial-time algorithm deciding whether there exists a Nash equilibrium in games with networks of constant size.
Additionally, we introduce a family of player-specific network congestion games that are guaranteed to possess equilibria. In these games players have identical delay functions. However, each player may use only a certain subset of the edges. For this class of games we prove that finding a pure Nash equilibrium is $PLS$-complete. Again, this result is true for networks with directed edges as well as for networks with undirected edges, and still holds for games with three players only. In games with networks of constant size, however, we prove that pure Nash equilibria can be computed in polynomial time.
We investigate the existence of optimal tolls for atomic symmetric network congestion games with unsplittable traffic and arbitrary nondecreasing latency functions. We focus on pure Nash equilibria, and consider a natural toll mechanism, which we call cost-balancing tolls. A set of cost-balancing tolls turns every path with positive traffic on its edges into a minimum-cost path. Hence any given configuration is induced as a pure Nash equilibrium of the modified game with the corresponding cost-balancing tolls. We show how to compute in linear time a set of cost-balancing tolls for the optimal solution such that the total amount of tolls paid by any player in any pure Nash equilibrium of the modified game does not exceed the latency on the maximum-latency path in the optimal solution. Our main result is that for congestion games on series-parallel networks with strictly increasing latency functions, the optimal solution is induced as the unique pure Nash equilibrium of the game with the corresponding cost-balancing tolls. To the best of our knowledge, only linear congestion games on parallel links were known to admit optimal tolls prior to this work. To demonstrate the difficulty of computing a better set of optimal tolls, we show that even for two-player linear congestion games on series-parallel networks, it is NP-hard to decide whether the optimal solution is the unique pure Nash equilibrium or there is another pure Nash equilibrium of total cost at least $6/5$ times the optimal cost.
In this paper we propose a new methodology for determining approximate Nash equilibria of noncooperative bimatrix games, and based on that, we provide an efficient algorithm that computes $0.3393$-approximate equilibria, the best approximation to date. The methodology is based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players' payoffs from the best payoff each player could achieve given the strategy chosen by the other. We then seek to minimize such a function using descent procedures. Because it is unlikely to be able to find global minima in polynomial time, given the recently proven intractability of the problem, we concentrate on the computation of stationary points and prove that they can be approximated arbitrarily closely in polynomial time and that they have the above-mentioned approximation property. Our result provides the best $\epsilon$ to date for polynomially computable $\epsilon$-approximate Nash equilibria of bimatrix games. Furthermore, our methodology for computing approximate Nash equilibria has not been used by others.
In this paper, we focus on the core stability of vertex cover games, which arise from vertex cover problems on graphs. Based on duality theory of linear programming, we prove that a balanced vertex cover game has a stable core if and only if every edge belongs to a maximum matching in the underlying graph. We also prove that for a totally balanced vertex cover game, the core largeness, extendability, and exactness are all equivalent, which implies core stability. Furthermore, we show that core stability and the three related properties can be determined efficiently.
We revisit the problem of incentive-compatible interdomain routing, examining the quite realistic special case in which the utilities of autonomous systems (ASes) are linear functions of the traffic in the incident links and the traffic leaving each AS. We show that incentive-compatibility toward maximizing total welfare is achievable efficiently, and in the uncapacitated case, by an algorithm that can be easily implemented by the border gateway protocol (BGP), the standard protocol for interdomain routing.
Let $G=(V,E)$ be a graph modeling a network in which each edge is owned by a selfish agent, which establishes the cost for traversing its edge (i.e., assigns a weight to its edge) by pursuing only its personal utility. In such a setting, we aim at designing approximate truthful mechanisms for several NP-hard traversal problems on $G$, such as the graphical traveling salesman problem, the rural postman problem, and the mixed Chinese postman problem, each of which in general requires an edge of $G$ to be used several times. Thus, in game-theoretic terms, these are one-parameter problems, but with a peculiarity: the workload of each agent is a natural number. In this paper we refine the classical notion of monotonicity of an algorithm so as to capture exactly this property, and we then provide a general mechanism-design technique that guarantees this monotonicity and that allows one to compute efficiently the corresponding payments. In this way, we show that the former two problems and the latter one admit a $3/2$- and a $2$-approximate truthful mechanism, respectively. Thus, for the first two problems we match the best known approximation ratios holding for their corresponding centralized versions, while for the third one we are only a $4/3$-factor away from it.
Congestion games are a well-studied model for resource sharing among uncoordinated selfish players. Usually, one assumes that the resources in a congestion game do not have any preferences regarding the players that can access them. In typical load-balancing applications, however, different jobs can have different priorities, and jobs with higher priorities get, for example, larger shares of processor time. We extend the classical notion of congestion game and introduce a model in which each resource can assign priorities to the players, and players with higher priorities can displace players with lower priorities. Not only does our model extend classical congestion games, it can also be seen as a model of two-sided markets with ties. Hence it unifies previous results for these two classical models.
We prove that singleton congestion games with priorities are potential games. Furthermore, we show that every player-specific singleton congestion game with priorities possesses a pure Nash equilibrium that can be found in polynomial time. Finally, we extend our results to matroid congestion games, in which the strategy spaces of the players are matroids over the resources.
We present a game-theoretic study of hybrid communication networks in which mobile devices can connect to a base station, The maximal number of allowed hops might be bounded in order to guarantee small latency. We introduce hybrid connectivity games to study the impact of selfishness on this kind of infrastructure.
Mobile devices are represented by selfish players, each of which aims at establishing an uplink path to the base station minimizing its individual cost. Our model assumes that intermediate nodes on an uplink path are reimbursed for transmitting the packets of other devices. Depending on the model, the reimbursements can be paid either by a benevolent network operator or by the senders of the packets using micropayments via a clearing agency that possibly collects a small percentage as commission.
Our main findings are these: If there is no constraint on the number of allowed hops on the path to the base station, then the existence of equilibria is guaranteed regardless of whether the network operator or the senders pay for forwarding packets. If there is an upper bound on the number of allowed hops on the uplink path, then the existence of equilibria depends on who pays for forwarding packets. If the network operator pays, then the existence of equilibria is guaranteed only if at most one intermediate node is allowed. If the senders pay for forwarding their packets, then equilibria are guaranteed to exist given any upper bound on the number of allowed hops.
Our equilibrium analysis gives a first game-theoretic motivation for the implementation of micropayment schemes in which senders pay for forwarding their packets. We further support this evidence by giving an upper bound on the “price of anarchy” for this kind of hybrid connectivity game that is independent of the number of nodes but depends only on the number of hops and the power gradient.
We create Atropos, a new two-player game based on Sperner's lemma. Our game has simple rules and several desirable properties. First, Atropos is always certain to have a winner. Second, the game is impartial, meaning that both players always can make the same moves. Third, like many other interesting games such as Hex and geography, we prove that deciding whether one can win a game of Atropos is a PSPACE-complete problem. We provide a web-based version of the game, playable at http://cs-people.bu.edu/paithan/atropos/. In addition, we propose other games, also based on fixed-point theorems.
We introduce a restriction of the stable roommates problem in which room-mate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, they can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of an algorithm due to R. W. Irving, D. Michail, K. Mehlhorn, K. Paluch, and K. Telikepalli. “Rank-Maximal Matchings.” to a nonbipartite setting. Also, we describe several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs.