text: We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either games and, thus, leads -- via our first algorithm -- to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd [2016].
  @cite_30: Network oligopoly models have been recently proposed as an alternative to general equilibrium models for analysing conflict situations between spatially separated firms. However, the oligopoly models are, from an algorithmic standpoint, less well understood than their equilibrium counterparts. In this paper we propose four iterative schemes for computing Cournot-Nash equilibria on congested transportation networks. The algorithms will be analysed theoretically, and computational results will be provided (i) for an oligopoly test problem, and (ii) for a small-scale network oligopoly problem.
  @cite_38: 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.
  @cite_47: 
  @cite_43: We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is at most m, where m denotes the number of coalitions. For general networks, we present an instance showing that the price of anarchy is unbounded, even in the case of two coalitions. If latencies are restricted to polynomials with nonnegative coefficients and bounded degree, we prove upper bounds on the price of anarchy for general networks, which improve upon the current best ones except for affine latencies. In light of the negative results even for two coalitions, we analyze the effectiveness of Stackelberg strategies as a means to improve the quality of Nash equilibria. In this setting, an α fraction of the entire demand is first routed centrally by a Stackelberg leader according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by the coalitions (followers). For a single coalitional follower and parallel arcs, we develop an efficiently computable Stackelberg strategy that reduces the price of anarchy to one. For general networks and a single coalitional follower, we show that a simple strategy, called SCALE, reduces the price of anarchy to 1+α. Finally, we investigate SCALE for multiple coalitional followers, general networks, and affine latencies. We present the first known upper bound on the price of anarchy in this case. Our bound smoothly varies between 1.5 for α=0 and full efficiency for α=1.
  @cite_15: 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. >
  @cite_13: A noncooperative game is formulated on a transportation network with congestion. The players are associated with origin-destination pairs, and are facing demand functions at their respective destination nodes. A Nash-Cournot equilibrium is defined and conditions for existence and uniqueness of this solution are provided. The asymptotic behavior of the Nash-Cournot equilibrium is then shown to yield (under appropriate assumptions) a total flow vector corresponding to a Wardrop equilibrium.
summary: Atomic Splittable Congestion Games. Atomic splittable network congestion games with player-independent cost functions have been studied (seemingly independent) by @cite_15 and Haurie and Marcotte @cite_13 and Marcotte @cite_30 . Both lines of research mentioned that Rosens' existence result for concave games on compact strategy spaces implies the existence of pure Nash equilibria via Kakutani's fixed-point theorem. @cite_38 presented the first upper bounds on the price of anarchy in atomic splittable congestion games. These were later improved by Harks @cite_43 and finally shown to be tight by Schoppmann and Roughgarden @cite_47 .