Generic Uniqueness of Equilibrium in Large Crowding Games

Milchtaich, Igal

doi:10.1287/moor.25.3.349.12220

Cited by 55 publications

(52 citation statements)

References 18 publications

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“…It can be shown (cf. [12]) that, in some precise sense, the equilibrium is generically unique. However, for present purposes, it suffices to establish the uniqueness of the equilibrium payoffs.…”

Section: The Setupmentioning

confidence: 99%

“…Note that the heterogeneity of the population is assumed to involve only the fixed utility (or cost) and not the cost of congestion (the variable cost). For a more general model, in which different players may be affected to a different degree by congestion, see [12]. The fixed-utility assignment f :…”

Section: The Setupmentioning

confidence: 99%

“…(6) 12 It is shown below that, under the condition 11 Aumann and Kurz [2] call v the "Harsanyi coalitional form" of the strategic game. Selten [21] uses the term "1/2-characteristic" to refer to essentially the same thing.…”

Section: Cooperationmentioning

confidence: 99%

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Social optimality and cooperation in nonatomic congestion games

Milchtaich

2004

Journal of Economic Theory

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Congestion externalities may result in non-optimal equilibria. For these to occur, it suffices that facilities differ in their fixed utilities or costs. As this paper shows, the only case in which equilibria are always socially optimal, regardless of the fixed components, in that in which the costs increase logarithmically with the size of the set of users. Therefore, achieving a socially optimal choice of facilities generally requires some form of external intervention or cooperation. For heterogeneous populations (in which the fixed utilities or costs vary across users as well as across facilities), this raises the question of utility or cost sharing. The sharing rule proposed in this paper is the Harsanyi transferable-utility value of the game-which is based on the users' marginal contributions to the bargaining power A. Neyman for suggesting the idea of investigating the value of such games. I also thank an anonymous referee and an associate editor for useful comments. several, or many, independent decision-makers interact by using the same facilities. In the nonatomic congestion games studied in this paper, each player only chooses one action or facility, and his utility strictly decreases as the size of the set of other players choosing the same action or facility increases. Different players do not necessarily achieve the same utility when making the same choice. In this respect, the population of players is heterogeneous. However, the manner in which utility decreases with increased congestion is the same for all. The following example (adapted from [4]) illustrates these assumptions. People in a professional meeting may prefer to go to different sessions, since the intrinsic quality they assign to each session varies. As more people crowd into a room, it becomes more difficult to see and hear. The actual quality each person assigns to a session therefore depends on both the intrinsic quality and the number of other people present. The delays experienced by clients in a computer network when many of them simultaneously try to access the same server are another example.In nonatomic congestion games of the kind considered here, the equilibrium payoffs are always unique. Moreover, the equilibria are Pareto efficient in the sense that it is not possible to modify an equilibrium in such a way that some players become better off without making some of the others worse off. However, the equilibria need not be socially optimal. That is, they may all be inferior to some non-equilibrium assignment of facilities in terms of the aggregate, or equivalently average, utility or cost. (Note that, unlike Pareto efficiency, the notion of social optimality involves interpersonal comparisons of utilities.) For the equilibria to maximize social welfare, a player's utility from choosing a facility should reflect the external effects of his choice on the other players. In other words, the cost or benefit for the individual must mirror the social cost or benefit. The first main result of this paper is that, under the assumption...

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Section: The Setupmentioning

confidence: 99%

Section: The Setupmentioning

confidence: 99%

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Social optimality and cooperation in nonatomic congestion games

Milchtaich

2004

Journal of Economic Theory

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“…We study nonatomic congestion games, in which infinitely many players interact and each player has a negligible ability to affect the others. Nonatomic congestion games have recently been studied in [28][29][30][31]. They are a subclass of the class of nonatomic games introduced by Schmeidler [32].…”

Section: Extension To Nonatomic Congestion Gamesmentioning

confidence: 99%

Truthful Mechanisms for Selfish Routing and Two-Parameter Agents

Thielen

Krumke

2009

Algorithmic Game Theory

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Abstract. We prove a general monotonicity result about Nash flows in directed networks and use it for the design of truthful mechanisms in the setting where each edge of the network is controlled by a different selfish agent, who incurs costs when her edge is used. The costs for each edge are assumed to be linear in the load on the edge. To compensate for these costs, the agents impose tolls for the usage of edges. When nonatomic selfish network users choose their paths through the network independently and each user tries to minimize a weighted sum of her latency and the toll she has to pay to the edges, a Nash flow is obtained. Our monotonicity result implies that the load on an edge in this setting can not increase when the toll on the edge is increased, so the assignment of load to the edges by a Nash flow yields a monotone algorithm. By a well-known result, the monotonicity of the algorithm then allows us to design truthful mechanisms based on the load assignment by Nash flows.Moreover, we consider a mechanism design setting with two-parameter agents, which is a generalization of the case of one-parameter agents considered by Archer and Tardos [1]. While the private data of an agent in the one-parameter case consists of a single nonnegative real number specifying the agent's cost per unit of load assigned to her, the private data of a two-parameter agent consists of a pair of nonnegative real numbers, where the first one specifies the cost of the agent per unit load as in the one-parameter case, and the second one specifies a fixed cost, which the agent incurs independently of the load assignment. We give a complete characterization of the set of output functions that can be turned into truthful mechanisms for two-parameter agents. Namely, we prove that an output function for the two-parameter setting can be turned into a truthful mechanism if and only if the load assigned to every agent is nonincreasing in the agent's bid for her per unit cost and, for almost all fixed bids for the agent's per unit cost, the load assigned to her is independent of the agent's bid for her fixed cost. When the load assigned to an agent is continuous in the agent's bid for her per unit cost, it must be completely independent of the agent's bid for her fixed cost. These results motivate our choice of linear cost functions without fixed costs for the edges in the selfish routing setting, but the results also seem to be interesting in the context of algorithmic mechanism design themselves.

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“…The stage game used in a repeated minority game is strongly related to congestion games introduced by Rosenthal (1973) and to crowding games studied by Milchtaich (1998Milchtaich ( , 2000.…”

Section: Introductionmentioning

confidence: 99%

Discounted and finitely repeated minority games with public signals

Renault¹,

Scarlatti²,

Scarsini³

2008

Mathematical Social Sciences

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We consider a repeated game where at each stage players simultaneously choose one of two rooms. The players who choose the less crowded room are rewarded with one euro. The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced, hence the game has imperfect public monitoring. An undiscounted version of this game was considered by Renault et al. (2005), who proved a folk theorem. Here we consider a discounted version and a finitely repeated version of the game, and we strengthen our previous result by showing that the set of equilibrium payoffs Hausdorff-converges to the feasible set as either the discount factor goes to one or the number of repetition goes to infinity. We show that the set of public equilibria for this game is strictly smaller than the set of private equilibria.

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Generic Uniqueness of Equilibrium in Large Crowding Games

Cited by 55 publications

References 18 publications

Social optimality and cooperation in nonatomic congestion games

Social optimality and cooperation in nonatomic congestion games

Truthful Mechanisms for Selfish Routing and Two-Parameter Agents

Discounted and finitely repeated minority games with public signals

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