The price of anarchy of serial, average and incremental cost sharing

We compute the price of anarchy (PoA) of three familiar demand games, i.e., the smallest ratio of the equilibrium to efficient surplus, over all convex preferences quasi-linear in money. For any convex cost, the PoA is at least in the average and serial games, where n is the number of users. It is zero in the incremental game for piecewise linear cost functions. With quadratic costs, the PoA of the serial game is , and for the average and incremental games. This generalizes if the marginal cost is convex or concave, and its elasticity is bounded.      

Characterization of additive cost sharing methods

The set of additive cost sharing methods when individual demands are integer valued and cost shares are non negative is characterized by its extreme points and by a network flow representation. The extreme methods allocate costs incrementally along a chain of demand vectors independent of the cost function. The result generalizes Wang’s characterization in that we do not assume the Dummy axiom.     

A geometric approach to the price of anarchy in nonatomic congestion 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 and on nonatomic congestion games. This novel proof also facilitates two new types of theoretical 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 stronger bounds on the inefficiency of equilibria for situations in which the equilibrium costs are within reasonable limits of the fixed costs. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of nonatomic congestion games, which provide the framework in which we describe this work.     

Group strategyproof cost sharing: The role of indifferences

Every agent reports his willingness to pay for one unit of a good. A mechanism allocates goods and cost shares to some agents. We characterize the group strategyproof (GSP) mechanisms under two alternative continuity conditions interpreted as tie-breaking rules. With the maximalist rule (MAX) an indifferent agent is always served. With the minimalist rule (MIN) an indifferent agent does not get a unit of the good.GSP and MAX characterize the cross-monotonic mechanisms. These mechanisms are appropriate whenever symmetry is required. On the other hand, GSP and MIN characterize the sequential mechanisms. These mechanisms are appropriate whenever there is scarcity of the good.Our results are independent of an underlying cost function; they unify and strengthen earlier results for particular classes of cost functions.     

On games with identical equilibrium payoffs

Summary. This paper compares the sets of Nash, coalition- proof Nash and strong Nash equilibrium payoffs of normal form games which are closely related. We propose sufficient conditions for equivalent or closely related games to have identical sets of equilibrium payoffs. Received: April 23, 1999; revised version: November 23, 1999     

Strong monotonicity in surplus sharing

Summary. We consider three new axioms for surplus sharing problems. The first is strong monotonicity which says that workers should be rewarded for increases in productivity and the second says that productive workers should receive some compensation. The third requires that the surplus sharing rule should be well defined (and continuous on) the set of threshold functions. We show that none of the standard "equitable" mechanisms satisfy any of these axioms and then present a constructive characterization of mechanisms which do. Using this we construct several new mechanisms. These are the Almost Flat mechanism, the Spread Aumann-Shapley mechanism, and the Spread Serial mechanism, which have many desirable properties.Received: 15 September 2002, Revised: 11 March 2003, JEL Classification Numbers: C71, H4.I would like to thank Hervé Moulin, Rich Mclean, Yves Sprumont and an anonymous referee for helpful comments.     

Stackelberg thresholds in network routing games or the value of altruism

It is well known that the Nash equilibrium in network routing games can have strictly higher cost than the optimum cost. In Stackelberg routing games, where a fraction of flow is centrally-controlled, a natural problem is to route the centrally-controlled flow such that the overall cost of the resulting equilibrium is minimized.We consider the scenario where the network administrator wants to know the minimum amount of centrally-controlled flow such that the cost of the resulting equilibrium solution is strictly less than the cost of the Nash equilibrium. We call this threshold the Stackelberg threshold and prove that for networks of parallel links with linear latency functions, it is equal to the minimum of the Nash flows on links carrying more optimum flow than Nash flow.Our approach also provides a simpler proof of characterization of the minimum fraction that must be centrally controlled to induce the optimum solution.     

The worst absolute surplus loss in the problem of commons: random priority versus average cost

A good is produced with increasing marginal cost. A group of agents want at most one unit of that good. The two classic methods that solve this problem are average cost and random priority. In the first method users request a unit ex ante and every agent who gets a unit pay average cost of the number of produced units. Under random priority users are ordered without bias and the mechanism successively offers the units at price equal to marginal cost. We compare these mechanisms by the worst absolute surplus loss and find that random priority unambiguously performs better than average cost for any cost function and any number of agents. Fixing the cost function, we show that the ratio of worst absolute surplus losses will be bounded by positive constants for any number of agents, hence the above advantage of random priority is not very large. I especially thank Herve Moulin for helpful comments and suggestions. I also appreciate the feedback from Justin Leroux and an anonymous referee.     

Random belief equilibrium in normal form games

In defining random belief equilibrium (RBE) in finite, normal form games we assume a player's beliefs about others' strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile, the focus. At an RBE: (1) Each chooses a best response relative to her beliefs. (2) Each player's expected choice coincides with the focus of the other players' belief distributions. RBE provides a statistical framework for estimation which we apply to data from three experimental games. We also characterize the limit-RBE as players' beliefs converge to certainty. When atoms in the belief distributions vanish in the limit, not all limit-RBE (called robust equilibria) are trembling hand perfect Nash equilibria and not all perfect equilibria are robust.     

Generic determinacy of Nash equilibrium in network-formation games

This paper proves the generic determinacy of Nash equilibrium in network-formation games: for a generic assignment of utilities to networks, the set of probability distributions on networks induced by Nash equilibria is finite.     

Nash equilibrium in games with incomplete preferences

Summary. This paper investigates Nash equilibrium under the possibility that preferences may be incomplete. I characterize the Nash-equilibrium-set of such a game as the union of the Nash-equilibrium-sets of certain derived games with complete preferences. These games with complete preferences can be derived from the original game by a simple linear procedure, provided that preferences admit a concave vector-representation. These theorems extend some results on finite games by Shapley and Aumann. The applicability of the theoretical results is illustrated with examples from oligopolistic theory, where firms are modelled to aim at maximizing both profits and sales (and thus have multiple objectives). Mixed strategy and trembling hand perfect equilibria are also discussed.Received: 22 September 2003, Revised: 24 June 2004, JEL Classification Numbers: D11, C72, D43.I would like to thank Jean-Pierre Benôit, Juan Dubra, Alejandrio Jofre, Debraj Ray, Kim-Sau Chung and the seminar participants at NYU and at the Universidad de Chile for their comments. I am most grateful to Efe Ok, for his comments, criticism, suggestions and questions.     

Regularity of pure strategy equilibrium points in a class of bargaining games

Summary. We develop an index theory for the Stationary Subgame Perfect (SSP) equilibrium set in a class of n-player sequential bargaining games with probabilistic recognition rules. For games with oligarchic voting rules (a class that includes unanimity rule), we establish conditions on individual utilities that ensure that for almost all discount factors, the number of SSP equilibria is odd and the equilibrium correspondence lower-hemicontinuous. For games with general, monotonic voting rules, we show generic (in discount factors) determinacy of SSP equilibria under the restriction that the agreement space is of dimension one. For non-oligarchic voting rules and agreement spaces of higher finite dimension, we establish generic determinacy for the subset of SSP equilibria in pure strategies. The analysis also extends to the case of fixed delay costs. Lastly, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.Received: 13 May 2004, Revised: 1 March 2005, JEL Classification Numbers: C62, C72, C78.I thank John Duggan and participants of the 2003 annual meeting of the American Political Science Association, Philadelphia, PA, the Political Economy Seminar at Northwestern University, and the Economic Theory seminar at the University of Rochester for helpful comments.     

Existence of equilibrium in common agency games with adverse selection

We establish the existence of subgame perfect equilibria in general menu games, known to be sufficient to analyze common agency problems. Our main result states that every menu game satisfying enough continuity properties has a subgame perfect equilibrium. Despite the continuity assumptions that we make, discontinuities naturally arise due to the absence, in general, of continuous optimal choices for the agent. Our approach, then, is based on (and generalizes) the existence theorem of [Simon, L., Zame, W., 1990. Discontinuous games and endogenous sharing rules. Econometrica 58 (4), 861–872] designed for discontinuous games.     

Catalog competition and Nash equilibrium in nonlinear pricing games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each seller competes for a buyer of unknown type by offering the buyer a catalog of products and prices. We call this game a catalog game. Our main objective is to show that catalog games have Nash equilibria. The Nash existence problem for catalog games is particularly contentious due to payoff discontinuities caused by tie-breaking. We make three contributions. First, we establish under very mild conditions on primitives that no matter what the tie-breaking rule, catalog games are uniformly payoff secure, and therefore have mixed extensions which are payoff secure. Second, we show that if the tie-breaking rule awards the sale to firms which value it most (i.e., breaks ties in favor of firms which stand to make the highest profit), then firm profits are reciprocally upper semicontinuous (i.e., the mixed catalog game is reciprocally upper semincontinuous). This in turn implies that the mixed catalog game satisfies Reny’s condition of better-reply security—a condition sufficient for existence (Reny in Econometrica 67:1029–1056, 1999). Third, we show by example that if the tie-breaking rule does not award the sale to firms which value it most (for example, if ties are broken randomly with equal probability), then the catalog game has no Nash equilibrium. This paper was written while the second author was Visiting Professor, Centre d’Economie de la Sorbonne, Universite Paris 1, Pantheon-Sorbonne. The second author thanks CES and Paris 1, and in particular, Bernard Cornet and Cuong Le Van for their support and hospitality. The second author also thanks the C&BA and EFLS at the University of Alabama for financial support. Both authors are grateful to Monique Florenzano and to participants in the April 2006 Paris 1 NSF/NBER Decentralization Conference for many helpful comments on an earlier version of the paper. Finally, both authors are especially grateful to an anonymous referee whose thoughtful comments led to substantial improvements in the paper. Monteiro acknowleges the financial support of Capes-Cofecub 468/04.     

Incentive monotonicity and equilibrium selection in 2×2 matrix games

For the class of 2×2 matrix games with two strict Nash equilibria the paper introduces an equilibrium refinement called incentive monotonicity. It selects the risk-dominant equilibrium if interests are conflicting, while it remains silent in games with common interests. These results suggest that the equilibrium-selection problem might be more difficult in games with common interests, which is certainly the case if risk dominance and payoff dominance go in opposite directions.     

Incomplete risk sharing arrangements and the value of information

Summary. The paper constructs a theoretical framework in which the value of information in general equilibrium is determined by the interaction of two opposing mechanisms: first, more information about future random events leads to better individual decisions and, therefore, higher welfare. This is the ‘Blackwell effect’ where information has positive value. Second, more information in advance of trading limits the risk sharing opportunities in the economy and, therefore, reduces welfare. This is the ‘Hirshleifer effect’ where information has negative value. We demonstrate that in an economy with production information has positive value if the information refers to non-tradable risks; hence, such information does not destroy the Blackwell theorem. Information which refers to tradable risks may invalidate the Blackwell theorem if the consumers are highly risk averse. The critical level of relative risk aversion beyond which the value of information becomes negative is less than 0.5. Received: May 14, 2001; revised version: March 5, 2002     

Stability of the cooperative equilibrium in N-person prisoners' dilemma with sequential moves

Summary. Nishihara [3] showed that N-person prisoners' dilemma has a cooperative Nash equilibrium, if the players decide their actions sequentially in the order determined by Nature under a certain information structure, and if each player's payoffs satisfy a certain inequality. This paper examines the stability of this cooperative equilibrium against two matters: players' slight mistakes and deviations by coalitions. The main results are as follows: (i) if the inequality on each player's payoffs strictly holds, then the cooperative equilibrium is a strictly proper equilibrium; (ii) if N≤3, and if full cooperation is Pareto efficient in N-person prisoners' dilemma, then the cooperative equilibrium is a strong Nash equilibrium; (iii) the cooperative equilibrium is in general a coalition-proof Nash equilibrium. Received: June 23, 1997; revised version: December 2, 1997     

Adjusting the model to adjust the world: constructive mechanisms in postwar general equilibrium theory

Economic methodologists most often study the relations between models and reality while focusing on the issues of the model's epistemic relevance in terms of its relation to the ‘real world’ and representing the real world in a model. We complement the discussion by bringing the model's constructive mechanisms or self-implementing technologies in play. By this, we mean the elements of the economic model that are aimed at ‘implementing’ it by envisaging the ways to change the reality in order to bring it more in line with the model. We are thus concerned mainly not with the ways to change the model to ‘fit’ the reality, but rather with the model's own armature that is supposed to transform the world along theoretical lines. The case we study is Arrow–Debreu–McKenzie general equilibrium model. In particular, we show the following: gradient methods and stability could be regarded as constructive mechanisms of general equilibrium modeling in the context of market socialism debates; the obsession of general equilibrium theorists with these concepts can be partly explained by the fact that they hoped not to be faithful to reality, but rather to adjust it to fit the theoretical model; mechanism design theory initiated by the stability theorist Leonid Hurwicz could be seen as a successor of this position. We conclude by showing the relevance of this analysis for epistemic culture of much of contemporary economics and hence, claim that it is an important complement to the traditional philosophy of economic modeling.     

Continuity of the first price auction Nash equilibrium correspondence

Summary. Despite the complexity of the first price auction in the general asymmetric case, analytical results have started to emerge in the literature. Authors have also searched to gain insights by computing numerical estimates of the equilibria for some probability distributions of the valuations. This paper proves that the Nash equilibrium of the first price auction depends continuously, for the weak topology, on the valuation distributions and thus brings robustness to the numerical results as well as some theoretical results. As an example of application, we disprove a conjecture of comparative statics. Received: February 1, 1999; revised version: July 27, 2001