Stackelberg leadership in a dynamic duopoly with stochastic capital accumulation

I propose a dynamic duopoly model where firms enter simultaneously but compete hierarchically á la Stackelberg at each instant over time. They accumulate capacity through costly investment, with capital accumulation dynamics being affected by an additive shock the mean and variance of which are known. The main findings are the following. First, the Stackelberg game is uncontrollable by the leader; hence, it is time consistent. Second, the leader invests more than the follower; as a result, in the steady state, the leader’s capacity and profits are larger than the follower’s. Therefore, the present analysis does not confirm Gibrat’s Law, since the individual growth rate is determined by the timing of moves.JEL Classification: C61, C73, D43, D92, L13Financial support within the project The post-entry performance of firms: technology, growth and survival lead by Enrico Santarelli, co-financed by the University of Bologna and MIUR, is gratefully acknowledged. I thank Uwe Cantner, Roberto Cellini, Roberto Golinelli, Helen Louri, Enrico Santarelli, Antonello Scurcu, Peter Thompson, two anonymous referees and the audience at the final workshop of the project (Bologna, November 22-23, 2002) for useful comments and discussion. The usual disclaimer applies.     

Subgame Monotonicity in Extensive Form Evolutionary Games

A theory of subgame monotonicity is developed for finite extensive form games to model evolutionary processes that respect the subgame structure of the game. It is shown how these evolutionary processes can be decomposed into their evolution on smaller games and also recombined to simplify the analysis of the original game. When applied to the replicator dynamic, a generalization of the Wright manifold from population genetics is a natural consequence of monotonicity. Subgame monotonicity and the Wright manifold are also shown to arise naturally in imitation models based on the extensive form. Journal of Economic Literature Classification Numbers: C70, C72.     

Tests of Fairness Models Based on Equity Considerations in a Three-Person Ultimatum Game

Two recent models incorporating fairness considerations into the economics literature based on agents' concerns about the distribution of payoffs between themselves and others (Fehr-Schmidt, 1999,Quarterly Journal of Economics. 114 (3), 769–816; Bolton-Ockenfels, 2000,American Economic Review. 90, 166–193) are investigated using a new three-person ultimatum game: One person allocates a sum of money to two others, one of which is randomly chosen to accept or reject the offer. Rejection gives both the responder and the proposer zero income and a positive consolation prize for the non-responder. The data show essentially no reductions in rejection rates, holding offers constant, with and without consolation prizes, contrary to both models' predictions.     

Ex ante contracting with endogenously determined communication plans

Summary. In this paper we introduce a new model of ex ante contracting for economies with asymmetric information to examine endogenously determined communication plans for information sharing in the interim stage. In contrast to the models used in previous research, in the present model agents negotiate not only on a contract of state contingent allocations but also on a communication plan, a set of rules describing how agents will reveal part of their private information at the interim stage to execute the trade contracts. We prove a result about the nested structure of the set of allocations implementable by various communication plans and establish the existence of core strategies for this cooperative game under various regularity conditions. Received: 4 March 1998; revised version: 17 September 1999     

Finitely repeated games with social preferences

A well-known result from the theory of finitely repeated games states that if the stage game has a unique equilibrium, then there is a unique subgame perfect equilibrium in the finitely repeated game in which the equilibrium of the stage game is being played in every period. Here I show that this result does in general not hold anymore if players have social preferences of the form frequently assumed in the recent literature, for example in the inequity aversion models of Fehr and Schmidt (Quartely Journal of Economics 114:817–868, 1999) or Bolton and Ockenfels (American Economic Review 100:166–193, 2000). In fact, repeating the unique stage game equilibrium may not be a subgame perfect equilibrium at all. This finding should have relevance for all experiments with repeated interaction, whether with fixed, random or perfect stranger matching.     

A differentiable homotopy to compute Nash equilibria of n-person games

Summary. The literature on the computation of Nash equilibria in n-person games is dominated by simplicial methods. This paper is the first to introduce a globally convergent algorithm that fully exploits the differentiability present in the problem. It presents an everywhere differentiable homotopy to do the computations. The homotopy path can therefore be followed by several numerical techniques. Moreover, instead of computing some Nash equilibrium, the algorithm is constructed in such a way that it computes the Nash equilibrium selected by the tracing procedure of Harsanyi and Selten. As a by-product of our proofs it follows that for a generic game the tracing procedure defines a unique feasible path. The numerical performance of the algorithm is illustrated by means of several examples. Received: December 21, 1999; revised version: December 27, 2000     

A New Bargaining Set of an N-Person Game and Endogenous Coalition Formation

The two most fundamental questions in cooperative game theory are: When a game is played, what coalitions will be formed and what payoff vectors will be chosen? Few solution concepts or theories in the current literature provide satisfactory answers to both questions; answers are especially lacking for the first one. In this paper we introduce a new bargaining set, which is the first solution concept in cooperative game theory that provides answers to both fundamental questions endogenously. Journal of Economic Literature Classification Number: C71.     

Oligopoly equilibria in exchange economies: a limit theorem

In a pure exchange economy, agents have the possibility of behaving strategically by putting only a part of their initial endowments on the market. An oligopoly equilibrium is defined to be a Nash equilibrium of the game in which agents choose simultaneously quantities to be put on the market. It is proved that under standard hypotheses, the oligopoly equilibrium leads to the competitive equilibrium when the economy is replicated an infinite number of times. Received: May 26, 1999; revised version: April 3, 2000     

Von Neumann-Morgenstern solution social choice functions: An impossibility theorem

Recent literature has made significant progress in characterizing those social choice functions that can arise, or be “implemented,” as the equilibria of an underlying noncooperative game. This paper studies the implementability of social choice functions via cooperative games. Specifically, we show that if a social choice function arises, in each environment, as a Von Neumann-Morgenstern solution of an underlying cooperative game, whose dominance structure is monotonic and neutral, then the social choice function is essentially oligarchic, in exactly the same sense that “core” selecting choice functions are oligarchic.     

Multilateral bargaining: conditional and unconditional offers

Summary. We present a game in which n persons split a cake, where a distinction between conditional and unconditional offers is made. This distinction sheds light on the contrasting results obtained in the previous literature of multilateral bargaining. By allowing the proposer to make both conditional and unconditional offers, we show that the game has a unique subgame perfect Nash equilibrium outcome. Received: March 14, 2000; revised version: March 13, 2001     

Political Competition, Rent Seeking and the Choice of Environmental Policy Instruments: Comment

In this comment, I analyze Damania's political economy model (Environmental and Resource Economics 13: 415–433, 1999), correcting some flaws and clarifying some ambiguities. I arguethat the political parties are identical at the outset of the game. Onlyafter the parties have chosen the instrument (standards or taxation) and thestrictness of environmental policy do the environmentalists and thepolluting firms know which party to support in the election campaign. Inequilibrium, both parties choose the same platform, so that both have anequal probability of winning the election.     

Learning with noiseless information and payoff-relevant signals

Summary. In models of active learning or experimentation, agents modify their actions to affect the distribution of a signal that provides information about future payoffs. A standard result in the experimentation literature is that agents experiment, if at all, to increase information. This finding is a direct consequence of Blackwell's theorem: one experiment is more informative than another if and only if all expected utility maximizers prefer to observe the first. Blackwell's theorem presupposes, however, that the observed signal only conveys information and does not directly affect future payoffs. Often, however, signals are directly payoff relevant, a phenomenon that we call signal dependence. For example, if a firm is uncertain about its demand and uses today's sales as a signal of tomorrow's demand, then that signal may also directly affect tomorrow's profit if the good is durable or if consumers form consumption habits. Datta, Mirman and Schlee [9] and Bertocchi and Spagat [4] show that, if the signal is payoff relevant, experimentation may indeed reduce information. Here we show that, despite the inapplicability of Blackwell's Theorem, agents always experiment to increase information if the information structure is noiseless: given the true value of the unknown parameter, the signal realization is deterministic. We then apply our framework to analyze Lazear's [16] model of retail clearance sales, a model with both signal dependence and noiseless information. Received: February 19, 1999; revised version: August 11, 1999     

INCENTIVE ISSUES IN R&D CONSORTIA: INSIGHTS FROM APPLIED GAME THEORY

This article sketches how insights from applied game theory can be applied to Research and Development (R&D) consortia using a case study on an international plant breeding consortium. The insights jointly comprise a new “logic of collective action in R&D,” which is inspired by Olson’s Logic of Collective Action but goes beyond it. We analyze R&D consortia as institutions that respond to a variety of incentive problems which are obstacles to realizing the benefits of cooperation that arise due to the public goods nature of outputs, complementarities of inputs, and economies of scale and scope. Additionally, we sketch a “big‐picture” consortium game, which abstracts from specific incentive issues. (JEL B41, D02, H41, O31, O32)     

On the computation of stable sets and strictly perfect equilibria

Summary. In this paper a procedure is described that computes for a given bimatrix game all stable sets in the sense of Kohlberg and Mertens (1986). Further the procedure is refined to find the strictly perfect equilibria (if any) of such a game. Recieved: December 29, 1999; revised version: February 17, 2000     

Axiomatizations of two types of Shapley values for games on union closed systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication.     

The number of commodities required to represent a market game

An n-person game with side payments that can arise from an economic market is called a market game. It is proved here that any n-person market game can be obtained from a market with at most n ? 1 commodities. Moreover, no smaller number will suffice in some cases (e.g., the unanimity game). This result is then applied to settle a question regarding market games with a continuum of traders.     

Endogenous leadership and sustainability of enhanced cooperation in a repeated interactions model of tax competition: Endogenous leadership in tax competition

In this study, the endogenous timing of moves is analyzed in an infinitely repeated game setting of capital tax competition between a subgroup (a tax union) of countries agreeing on partial tax harmonization and outside countries. It is shown that in a subgame perfect equilibrium of the infinitely repeated tax competition game, they simultaneously set capital taxes in every stage game when a tax union comprises similar countries with respect to productivity, whereas they may set capital taxes sequentially in every stage game when a tax union comprises dissimilar countries. This finding is significantly different from Ogawa (2013), although we also assume that capital is owned by the country's residents, as in Ogawa's model. This is because a disadvantaged member country of the tax union would suffer from larger losses when a tax union comprising dissimilar countries, and thus the tax union will choose the strategy of moving Late for the sake of sustaining tax harmonization to avoid such losses.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

Noncooperative Equilibrium and Chamberlinian Monopolistic Competition

This paper addresses the existence of equilibrium for a nonatomic Bertrand game in a Chamberlinian environment. We reformulate O. Hart′s model (Rev. Econ. Stud. 52, 1985, 529–546) as a nonatomic game and show that under dispersion of tastes and continuity of the tastes density, there exists a pure-strategies ε-equilibrium where prices exceed marginal cost. Unlike Hart′s model there is no need to impose uniformity (or even independence) on the distribution of the m-tuple of differential commodities that consumers care about. Moreover, demand curves are allowed to vary across firms and in equilibrium firms may earn profits. Journal of Economic Literature Classification Numbers: B21, D43, L13.     

Reciprocal relationships and mechanism design

We study an incomplete information game in which players can coordinate their actions by contracting among themselves. We model this relationship as a reciprocal contracting procedure where each player has the ability to make commitments contingent on the other players' commitments. We differ from the rest of the literature on reciprocal contracting by assuming that punishments cannot be enforced in the event that cooperation breaks down. We fully characterize the outcomes that can be supported as perfect Bayesian equilibrium outcomes in such an environment. We use our characterization to show that the set of supportable outcomes with reciprocal contracting is larger than the set of outcomes available in a centralized mechanism design environment in which the mechanism designer is constrained by his inability to enforce punishments against non‐participants. The difference stems from the players' ability in our contracting game to convey partial information about their types at the time they offer contracts. We discuss the implications of our analysis for modelling collusion between multiple agents interacting with the same principal.