The Sequential Price of Anarchy for Atomic Congestion Games

Abstract

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".

Dynamic Resource Allocation Games

“…In the cost-sharing setting, the upper bound follows from the same argument as traditional games. For congestion games, it follows by applying a recent result by [10] to our setting. The details can be found in the full version.…”

“…The objectives of the players refer to the generated path and no costs are involved. Closest to our model is the model of [15], and its subsequent works [8,10]. They study RAGs in which players arrive and select strategies one by one, yet in one shot.…”

“…We show that the DPoA and DPoS in dynamic singleton cost-sharing games as well as dynamic singleton congestion games coincide with the PoA and PoS in the corresponding simultaneous class. As mentioned above, [15,8,10] study games in which players arrive one after the other. Since their games are sequential, they always have an SPE.…”

Quality of equilibria in resource allocation games

“…On the other hand, their results indicate the counterintuitive nature of sequential equilibria, which makes them very interesting from a theoretical perspective. Since then, a handful of papers were written on the sequential price of anarchy [6,32,51,57,58,63,80] for different classes of games. In this chapter we briefly list these results.…”

“…Moreover, we obtain bounds on the sequential price of anarchy for the special cases of affine singleton congestion games (Section 3.4), and affine symmetric singleton congestion games (Section 3.5). This chapter is based on 'The sequential price of anarchy for atomic congestion games' [58].…”