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.
A large body of recent work (initiated mainly by Roughgarden and Tardos @cite_2 @cite_19 ) has studied from a game-theory perspective, how selfishness can degrade the overall performance of a system that has multiple (selfish) users. Much of this work has focused on situations where users have access to shared resources, and the cost of using a resource increases as the resource attracts more usage. Our focus here is on the parallel links'' network topology, also referred to as scheduling jobs to a set of load-dependent machines, which is one of the most commonly studied models (e.g. @cite_10 @cite_11 @cite_16 @cite_21 @cite_6 @cite_20 ). Papers such as @cite_1 @cite_4 @cite_16 have studied the price of anarchy for these games in the unsplittable flow'' setting, where each user may only use a single resource. In contrast we study the splittable flow'' setting of @cite_15 . This version (finitely many players, splittable flow) was shown in @cite_15 @cite_8 to possess unique pure Nash equilibria (see Definition ). @cite_14 study the cost of selfish behaviour in this model, and compare it with the cost of selfish behaviour in the Wardrop model (i.e. infinitely many infinitesimal users).
@cite_14 @cite_4 @cite_8 @cite_21 @cite_1 @cite_6 @cite_19 @cite_2 @cite_15 @cite_16 @cite_10 @cite_20 @cite_11
2145512846 1554620983 2072113937 2107983838
It is well known that in a network with arbitrary (convex) latency functions that are a function of edge traffic, the worst-case ratio, over all inputs, of the system delay caused due to selfish behavior versus the system delay of the optimal centralized solution may be unbounded even if the system consists of only two parallel links. This ratio is called the price of anarchy (PoA). In this paper, we investigate ways by which one can reduce the performance degradation due to selfish behavior. We investigate two primary methods (a) Stackelberg routing strategies, where a central authority, e.g., network manager, controls a fixed fraction of the flow, and can route this flow in any desired way so as to influence the flow of selfish users; and (b) network tolls, where tolls are imposed on the edges to modify the latencies of the edges, and thereby influence the induced Nash equilibrium. We obtain results demonstrating the effectiveness of both Stackelberg strategies and tolls in controlling the price of anarchy. For Stackelberg strategies, we obtain the first results for nonatomic routing in graphs more general than parallel-link graphs, and strengthen existing results for parallel-link graphs, (i) In series-parallel graphs, we show that Stackelberg routing reduces the PoA to a constant (depending on the fraction of flow controlled). (ii) For general graphs, we obtain latency-class specific bounds on the PoA with Stackelberg routing, which give a continuous trade-off between the fraction of flow controlled and the price of anarchy, (iii) In parallel-link graphs, we show that for any given class L of latency functions, Stackelberg routing reduces the PoA to at most α + (1 - α) · ρ(L), where α is the fraction of flow controlled and ρ(L) is the PoA of class L (when α = 0). For network tolls, motivated by the known strong results for nonatomic games, we consider the more general setting of atomic splittable routing games. We show that tolls inducing an optimal flow always exist, even for general asymmetric games with heterogeneous users, and can be computed efficiently by solving a convex program. Furthermore, we give a complete characterization of flows that can be induced via tolls. These are the first results on the effectiveness of tolls for atomic splittable games. We present a short geometric proof for the price of anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks. This novel proof also facilitates two new types of results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive improved bounds on the inefficiency of Nash equilibria for situations in which the equilibrium travel times are within reasonable limits of the free-flow travel times. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of congestion games, which provides the framework in which we describe this work. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. We consider routing games where the performance of each user is dictated by the worst (bottleneck) element it employs. We are given a network, finitely many (selfish) users, each associated with a positive flow demand, and a load-dependent performance function for each network element; the social (i.e., system) objective is to optimize the performance of the worst element in the network (i.e., the network bottleneck). Although we show that such "bottleneck" routing games appear in a variety of practical scenarios, they have not been considered yet. Accordingly, we study their properties, considering two routing scenarios, namely when a user can split its traffic over more than one path (splittable bottleneck game) and when it cannot (unsplittable bottleneck game). First, we prove that, for both splittable and unsplittable bottleneck games, there is a (not necessarily unique) Nash equilibrium. Then, we consider the rate of convergence to a Nash equilibrium in each game. Finally, we investigate the efficiency of the Nash equilibria in both games with respect to the social optimum; specifically, while for both games we show that the price of anarchy is unbounded, we identify for each game conditions under which Nash equilibria are socially optimal.