Abstract. In situations without central coordination, the price of anarchy relates the quality of any Nash equilibrium to the quality of a global optimum. Instead of assuming that all players choose their actions simultaneously, here we consider games where players choose their actions sequentially. The sequential price of anarchy, recently introduced by Paes Leme, Syrgkanis, and Tardos [11], then relates the quality of any subgame perfect equilibrium to the quality of a global optimum. The effect of sequential decision making on the quality of equilibria, however, depends on the specific game under consideration. Here we analyze the sequential price of anarchy for atomic congestion games with affine cost functions. We derive several lower and upper bounds, showing that sequential decisions mitigate the worst case outcomes known for the classical price of anarchy [5,2]. Next to tight bounds on the sequential price of anarchy, a methodological contribution of our work is, among other things, a "factor revealing" integer linear programming approach that we use to solve the case of three players. Model and NotationWe consider atomic congestion games with affine latency functions. The input of an instance I ∈ I consists of a finite set of resources R, a finite set of players N = {1, . . . , n}, and for each player i ∈ N a collection A i of possible actions A i ⊆ R. In other words, each players' action is to choose a subset of resources A i that are feasible for him. We say a resource r ∈ R is chosen by player i if r ∈ A i , where A i is the action chosen by player i. By A = (A i ) i∈N we denote a possible outcome, that is, a complete profile of actions chosen by all players i ∈ N .Each resource r ∈ R has a constant activation cost d r ≥ 0 and a variable cost or weight w r ≥ 0 that expresses the fact that the resource gets more congested the more players choose it. The total cost of resource r ∈ R, for some outcome A, is then f r (A) = d r + w r · n r (A), where n r (A) denotes the number of players choosing resource r in outcome A. Given outcome A, the negative utility of player i is the total cost of all resources chosen by that player cost i (A) = r∈Ai f r (A). Players aim to minimize their costs. For later convenience, the total constantResearch supported by CTIT (www.ctit.nl) and 3TU.AMI (www.3tu.nl), project "Mechanisms for Decentralized Service Systems".

Abstract. In the "The curse of simultaneity", Paes Leme at al. show that there are interesting classes of games for which sequential decision making and corresponding subgame perfect equilibria avoid worst case Nash equilibria, resulting in substantial improvements for the price of anarchy. This is called the sequential price of anarchy. A handful of papers have lately analysed it for various problems, yet one of the most interesting open problems was to pin down its value for linear atomic routing (also: network congestion) games, where the price of anarchy equals 5/2. The main contribution of this paper is the surprising result that the sequential price of anarchy is unbounded even for linear symmetric routing games, thereby showing that sequentiality can be arbitrarily worse than simultaneity for this important class of games. Complementing this unboundedness result we solve an open problem in the area by establishing that the (regular) price of anarchy for linear symmetric routing games equals 5/2. Additionally, we prove that in these games, even we two players, computing the outcome of a subgame perfect equilibrium is NP-hard. The latter explains, to some extent, the difficulty of analyzing subgame perfect equilibria.

Motivated by the organization of online service systems, we study models for throughput scheduling in a decentralized setting. In throughput scheduling, we have a set of jobs j with values wj, processing times pj, and release dates rj and deadlines and dj, to be processed nonpreemptively on a set of unrelated machines. The goal is to maximize the total value of jobs scheduled within their time window [rj, dj]. While several approximation algorithms with different performance guarantees exist for this and related models, we are interested in the situation where subsets of servers are governed by selfish players. We give a universal result that bounds the price of decentralization, in the sense that any local α-approximation algorithms, α ≥ 1, yield equilibria that are at most a factor (α + 1) away from the global optimum, and this bound is tight. For models with identical machines, we improve this bound to α √ e/(α √ e − 1) ≈ (α + 1/2), which is shown to be tight, too. We also address some variations of the problem.

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Network routing games, and more generally congestion games play a central role in algorithmic game theory, comparable to the role of the traveling salesman problem in combinatorial optimization. It is known that the price of anarchy is independent of the network topology for non-atomic congestion games. In other words, it is independent of the structure of the strategy spaces of the players, and for a ne cost functions it equals 4/3. In this paper, we show that the dependence of the price of anarchy on the network topology is considerably more intricate for atomic congestion games. More specifically, we consider congestion games with a ne cost functions where the strategy spaces of players are symmetric and equal to the set of bases of a k-uniform matroid. In this setting, we show that the price of anarchy is strictly larger than the price of anarchy for singleton strategy spaces where the latter is 4/3. As our main result we show that the price of anarchy can be bounded from above by 28/13 ¥ 2.15. This constitutes a substantial improvement over the price of anarchy bound 5/2, which is known to be tight for arbitrary network routing games with a ne cost functions. ACM Subject Classification IntroductionUnderstanding the impact of selfish behavior on the performance of a system is an important question in algorithmic game theory. One of the cornerstones of the substantial literature on this topic is the famous result of Roughgarden and Tardos [26]. They considered the tra c model of Wardrop [29] in a network with a ne flow-dependent congestion cost functions on the edges. Given a set of commodities, each specified by a source node, a target node, and a flow demand, a Wardrop equilibrium is a multicommodity flow with the property that every commodity uses only paths that minimize the cost. For this setting, Roughgarden and Tardos proved that the total cost of an equilibrium flow is not worse than 4/3 times that of a system optimum. This ratio was coined the price of anarchy by Koutsoupias and Papadimitriou [18] who introduced it as a measure of a system's performance degradation due to selfish behavior. A surprising consequence of the result of Roughgarden and Tardos is that the worst case price of anarchy in congested networks is attained for very simple single-commodity networks already considered a century ago by Pigou [22]. Pigou-style networks consist of only two nodes connected by two parallel links. In fact, Roughgarden [25] proved that for any set of cost functions, the price of anarchy is independent of the network topology as it is always