We study a class of games in which a finite number of agents each controls a quantity of flow to be routed through a network, and are able to split their own flow between multiple paths through the network. Recent work on this model has contrasted the social cost of Nash equilibria with the best possible social cost. Here we show that additional costs are incurred in situations where a selfish leader'' agent allocates his flow, and then commits to that choice so that other agents are compelled to minimise their own cost based on the first agent's choice. We find that even in simple networks, the leader can often improve his own cost at the expense of increased social cost. Focusing on the 2-player case, we give upper and lower bounds on the worst-case additional cost incurred.
In this paper we initiate the study of how collusion alters the quality of solutions obtained in competitive games. The price of anarchy aims to measure the cost of the lack of coordination by comparing the quality of a Nash equilibrium to that of a centrally designed optimal solution. This notion assumes that players act not only selfishly, but also independently. We propose a framework for modeling groups of colluding players, in which members of a coalition cooperate so as to selfishly maximize their collective welfare. Clearly, such coalitions can improve the social welfare of the participants, but they can also harm the welfare of those outside the coalition. One might hope that the improvement for the coalition participants outweighs the negative effects on the others. This would imply that increased cooperation can only improved the overall solution quality of stable outcomes. However, increases in coordination can actually lead to significant decreases in total social welfare. In light of this, we propose the price of collusion as a measure of the possible negative effect of collusion, specifying the factor by which solution quality can deteriorate in the presence of coalitions. We give examples to show that the price of collusion can be arbitrarily high even in convex games. Our main results show that in the context of load-balancing games, the price of collusion depends upon the disparity in market power among the game participants. We show that in some symmetric nonatomic games (where all users have access to the same set of strategies) increased cooperation always improves the solution quality, and in the discrete analogs of such games, the price of collusion is bounded by two.    The essence of the routing problem in real networks is that the traffic demand from a source to destination must be satisfied by choosing a single path between source and destination. The splittable version of this problem is when demand can be satisfied by many paths, namely a flow from source to destination. The unsplittable, or discrete version of the problem is more realistic yet is more complex from the algorithmic point of view; in some settings optimizing such unsplittable traffic flow is computationally intractable.In this paper, we assume this more realistic unsplittable model, and investigate the "price of anarchy", or deterioration of network performance measured in total traffic latency under the selfish user behavior. We show that for linear edge latency functions the price of anarchy is exactly @math 2.5 for unweighted demand. These results are easily extended to (weighted or unweighted) atomic "congestion games", where paths are replaced by general subsets. We also show that for polynomials of degree d edge latency functions the price of anarchy is dδ(d). Our results hold also for mixed strategies.Previous results of Roughgarden and Tardos showed that for linear edge latency functions the price of anarchy is exactly 4 3 under the assumption that each user controls only a negligible fraction of the overall traffic (this result also holds for the splittable case). Note that under the assumption of negligible traffic pure and mixed strategies are equivalent and also splittable and unsplittable models are equivalent.   We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times---the total latency---is minimized.In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance.In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4 3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total latency of the routes chosen by unregulated selfish network users may be arbitrarily larger than the minimum possible total latency; however, we prove that it is no more than the total latency incurred by optimally routing twice as much traffic. The authors consider a communication network shared by several selfish users. Each user seeks to optimize its own performance by controlling the routing of its given flow demand, giving rise to a noncooperative game. They investigate the Nash equilibrium of such systems. For a two-node multiple links system, uniqueness of the Nash equilibrium is proven under reasonable convexity conditions. It is shown that this Nash equilibrium point possesses interesting monotonicity properties. For general networks, these convexity conditions are not sufficient for guaranteeing uniqueness, and a counterexample is presented. Nonetheless, uniqueness of the Nash equilibrium for general topologies is established under various assumptions. >  We study the problem of traffic routing in noncooperative networks. In such networks, users may follow selfish strategies to optimize their own performance measure and therefore, their behavior does not have to lead to optimal performance of the entire network. In this article we investigate the worst-case coordination ratio, which is a game-theoretic measure aiming to reflect the price of selfish routing. Following a line of previous work, we focus on the most basic networks consisting of parallel links with linear latency functions. Our main result is that the worst-case coordination ratio on m parallel links of possibly different speeds is Θ(log m log log log m). In fact, we are able to give an exact description of the worst-case coordination ratio, depending on the number of links and ratio of speed of the fastest link over the speed of the slowest link. For example, for the special case in which all m parallel links have the same speed, we can prove that the worst-case coordination ratio is Γ(−1) (m) p Θ(1), with Γ denoting the Gamma (factorial) function. Our bounds entirely resolve an open problem posed recently by Koutsoupias and Papadimitriou [1999].  