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.     

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.     

What motivates membership in non-renewable resource cartels?: The case of OPEC

In this paper, we analyze the question of membership in a non-renewable resource cartel, with specific application to OPEC. One would expect the benefits of cartel membership to be positively related to the size of remaining reserves, while domestic petroleum consumption should be negatively related to membership if countries care about consumer interests. Our econometric analysis indicates that larger reserves and lower consumption are positively associated with OPEC membership. On the other hand, membership does not appear to be systematically related to countries’ religious makeup. Our regressions correctly predict membership for the vast majority of oil-producing countries.     

An experimental study of commitment in Stackelberg games with observation costs

We report on experiments examining the value of commitment in Stackelberg games where the follower chooses whether to pay some cost to perfectly observe the leader's action. Várdy [Games Econ. Behav. (2004)] shows that in the unique pure-strategy subgame perfect equilibrium of this game, the value of commitment is lost completely; however, there exists a mixed-strategy subgame perfect equilibrium where the value of commitment is fully preserved. In the data, the value of commitment is largely preserved when the cost of looking is small, while it is lost when the cost is large. Nevertheless, for small observation costs, equilibrium behavior is clearly rejected. Instead, subjects persistently play non-equilibrium strategies in which the probability of the follower choosing to observe the leader's action is a decreasing function of the observation cost.     

The value of commitment in Stackelberg games with observation costs

We study Stackelberg games in which the follower faces a cost for observing the leader's action. We show that, irrespective of the size of the cost, the leader's value of commitment is lost completely in all pure-strategy equilibria. However, there also exists a mixed-strategy equilibrium that fully preserves the first-mover advantage. In this type of equilibrium, the probability that the follower looks at the leader's action is independent of the cost of looking.     

Cournot games with biconcave demand

Biconcavity is a simple condition on inverse demand that corresponds to the ordinary concept of concavity after simultaneous parameterized transformations of price and quantity. The notion is employed here in the framework of the homogeneous-good Cournot model with potentially heterogeneous firms. The analysis leads to unified conditions, respectively, for the existence of a pure-strategy equilibrium via nonincreasing best-response selections, for existence via quasiconcavity, and for the uniqueness of the equilibrium. The usefulness of the generalizations is illustrated in cases where inverse demand is either “nearly linear” or isoelastic. It is also shown that commonly made assumptions regarding large outputs are often redundant.     

Mergers in asymmetric Stackelberg markets

It is well known that the profitability of horizontal mergers with quantity competition is scarce. However, in an asymmetric Stackelberg market we obtain that some mergers are profitable. Our main result is that mergers among followers become profitable when the followers are inefficient enough. In this case, leaders reduce their output when followers merge and this reduction renders the merger profitable. This merger increases price and welfare is reduced.      

A new look for Stackelberg-Cournot equilibria in oligopolistic markets

Summary. This note deals with Cournot type oligopolies in which the market clearing price occasionally may be non-unique. A Stackelberg leading producer is present. Given that setting we explore continuity properties of the followers' reaction and provide sufficient conditions for existence of equilibrium. Received: June 20, 2000; revised version: April 24, 2001     

A cake for Cournot

A cake division mechanism is presented that is equivalent in terms of the size of the pieces of the cake n players’ receive in equilibrium to the quantities that n firms in a Cournot oligopoly supply in equilibrium. This mechanism extends to equivalence between cake division and Nash’s ‘divide the dollar’ game.     

Stackelberg equilibrium with many leaders and followers. The case of zero fixed costs

I study a version of the Stackelberg game with many identical firms in which leaders and followers use a continuous cost function with no fixed cost. Using lattice theoretical methods I provide a set of conditions that guarantee that the game has an equilibrium in pure strategies. With convex costs the model shows the same properties as a quasi-competitive Cournot model. The same happens with concave costs, but only when the number of followers is small. When this number is large the leaders preempt entry. I study the comparative statics and the limit behavior of the equilibrium and I show how the main determinants of market structure interact. More competition between the leaders always displaces the followers. Instead, how a stronger threat of entry affects the equilibrium depends on the technology. With strictly convex costs it is the followers that eventually displace the leaders.     

Endogenous Stackelberg equilibria with incomplete information

In this paper, the endogenous order of quantity decision is studied in a duopoly model with incomplete information. One firm knows the state of the demand curve while the other firm remains uninformed. Firms have to commit to a quantity in one out of two periods. While, a priori, simultaneous-move Cournot equilibria are possible, only Stackelberg equilibria, with either the informed or the uninformed firm moving first, emerge endogenously.     

Stackelberg Mixed Duopoly with a Foreign Competitor

We investigate Stackelberg mixed duopoly models where a state‐owned public firm and a foreign private firm compete. We examine a desirable role (either leader or follower) of the public firm. We also consider endogenous roles by adopting the observable delay game of Hamilton and Slutsky (1990). We find that, in contrast to Pal (1998) discussing a case of domestic competitors, the public firm should be the leader and that it becomes the leader in the endogenous role game. We also find that in contrast to Ono (1990) eliminating a foreign firm does not improve domestic welfare in mixed oligopolies.     

A dynamic Stackelberg model with production-adjustment costs

We investigate the (dynamic) stability of a stackelberg oligopoly model of a market of a homogeneous good, with output competition, one Stackelberg leader and a number of identical followers. We assume that each firm incurs quadratic production-adjustment costs if it changes its output. We present a simple necessary and sufficient condition for stability of the model. Using the condition, we compare the stability of this model with the stability of two related Cournot models in which all firms present are followers. It turns out that the Stackelberg model is more stable than these two Cournot models.     

Anonymity in nonatomic games

This paper purpose is twofold. First, it offers a critical review of the proofs of existence of pure strategy Nash Equilibria in nonatomic games. In particular, it focuses on the alternative ways of formalizing the critical assumption of anonymity. Second, the paper proves the existence of pure strategy Nash Equilibria by relaxing anonymity and allowing instead for “limited anonymity” (i.e. players’ decisions depend on the average strategy of a finite number of players’ subsets and not on the average strategy of the whole set of players). (JEL: C72, C79)     

Foreign market entry,upstream market power,and endogenous mode of downstream competition

In a differentiated duopoly model of trade and FDI featuring both horizontal and vertical product differentiation, we examine whether globalization and trade policy measures can generate welfare gains by leading firms to change their mode of competition. We show that when a high-quality foreign variety is manufactured under large frictions due to upstream monopoly power, a foreign firm can become a Bertrand competitor against a Cournot local rival in equilibrium, especially when the relative product quality of the foreign variety is sufficiently high and trade costs are sufficiently low (implying higher input price distortions due to double marginalization). Our results suggest that such strategic asymmetry is welfare improving and that the availability of FDI as an alternative to trade can make welfare-enhancing strategic asymmetry even more likely, especially when both input trade costs and fixed investment costs are sufficiently low and trade costs in final goods are sufficiently large.     

中国外汇储备投资组合选择——基于外汇储备循环路径的内生性分析

本文从中美两国经济的本质性差异出发,通过刻画中国外汇储备对外投资的"循环路径",构建了包括央行、金融市场和实体经济的斯塔克尔伯格及古诺模型,进而模拟出中国外汇储备对外投资对本国经济的间接贡献、合意的外汇储备投资组合,以及最优外汇储备投资规模。研究结果表明,中国外汇储备投资于美国风险资产的规模将影响外汇储备间接转化为美国对中国FDI的比例。同时,中国央行外汇储备规模及投资策略对危机时期的反应不足。改变外汇储备投资收益的主要方法包括降低居民的相对风险回避系数,通过政策引导促进居民消费,以及大力发展中国金融市场,降低对美国金融市场的依赖程度。     

Output commitment through product bundling: Experimental evidence

We analyze the impact of product bundling in experimental markets. One firm has monopoly power in a first market but competes with another firm à la Cournot in a second market. We compare treatments where the multi-product firm (i) always bundles, (ii) never bundles, and (iii) chooses whether to bundle or not. We also contrast the simultaneous and the sequential order of moves in the duopoly market. Our data indicate support for the theory of product bundling: with bundling and simultaneous moves, the multi-product firm offers the predicted number of units. When the multi-product firm is the Stackelberg leader, the predicted equilibrium is better attained with bundling, especially when it chooses to bundle, even though in theory bundling should not make a difference here. In sum, bundling works as a commitment device that enables the transfer of market power from one market to another.     

Strategic managerial incentives in a two-period Cournot duopoly

This paper examines the nature of optimal managerial incentives in the context of a duopoly marked by competition between the firm's managers in a dynamic production environment. If the marginal cost of production falls moderately over time or remains unchanged, there exists an equilibrium where one owner requires her manager to maximize profit, whereas the rival-owner requires her manager to maximize sales revenue. The profit-maximizing manager turns his firm into a Stackelberg-leader, while the sales-revenue-maximizing manager turns his firm into a Stackelberg-follower. Further, the profit-maximizing manager may generate a larger firm profit relative to the sales-revenue-maximizing manager.     

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     

Fault tolerance in large games

A Nash equilibrium is an optimal strategy for each player under the assumption that others play according to their respective Nash strategies, but it provides no guarantees in the presence of irrational players or coalitions of colluding players. In fact, no such guarantees exist in general. However, in this paper we show that large games are innately fault tolerant. We quantify the ways in which two subclasses of large games – λ-continuous games and anonymous games – are resilient against Byzantine faults (i.e. irrational behavior), coalitions, and asynchronous play. We also show that general large games have some non-trivial resilience against faults.