Inefficient Markov perfect equilibria in multilateral bargaining

We study a complete-information alternating-offer bargaining game in which one “active” player bargains with each of a number of other “passive” players one at a time. In contrast to most existing models, the order of reaching agreements is endogenously determined, hence the active player can “play off” some passive players against others by m oving back and forth bargaining with the passive players. We show that this model has a finite number of Markov Perfect Equilibria, some of which exhibiting wasteful delays. Moreover, the maximum number of delay periods that can be supported in Markov Perfect Equilibria increases in the order of the square of the number of players. We also show that these results are robust to a relaxing of the Markov requirements and to more general surplus functions. Received: November 19, 2001; revised version: August 20, 2002 RID="*" ID="*"This paper grew out of my dissertation submitted to Stanford University. I am deeply indebted to my advisor, Paul Milgrom, for his insights and guidance. I would also like to thank Douglas Bernheim, Sushil Bikhchandani, Harold Demsetz, Bryan Ellickson, Avner Greif, Peter Hammond, David Levine, Bentley Macleod, Joe Ostroy, John Pencavel, Jean-Laurent Rosenthal, David Starrett, Robert Wilson, Bill Zame and especially John Riley and Jeff Zwiebel for their helpful comments. I am grateful to an anonymous referee for extremely constructive suggestions.     

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     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

Arrow-Debreu equilibria with asymptotically heterogeneous expectations exist

Summary. In infinite horizon economies only local equivalence of beliefs is needed to ensure the existence of an Arrow–Debreu equilibrium. In fact, agents can even disagree completely in the long run in the sense that asymptotically, their beliefs are singular. Received: November 3, 2000; revised version: February 13, 2002     

A model of intergenerational transfers

Summary. Extending some existing literature, this paper formalizes the idea that intergenerational transfers occur because people care about the “characteristics” (i.e quantity and quality) of their offspring, rather than their children's welfare per se or consumption. The model analyzes this transfer motive in an infinite Markovian game framework, and it proves the existence of a stationary Markov Perfect equilibrium. Further, the analysis shows that under certain conditions, the proposed transfer motive will diminish, as the average income of an economy is sufficiently high. Thus, it suggests that as incomes continue to rise beyond a certain level, the (extended) life-cycle hypothesis will likely be a better and better approximation for explaining most people's saving behavior. This result also provides an explanation for the decline of the saving rates in the U.S. and other developed countries. Received: December 28, 1998; revised version: February 17, 2000     

Decentralizing Edgeworth equilibria in economies with many commodities

Summary. This paper proves core-equivalence theorems for exchange economies without ordered preferences, defined on locally convex Riesz commodity spaces such that the price space is a lattice. Properness assumptions are borrowed from some recent equilibrium existence results. Received: January 15, 1998; revised version: August 19, 1998     

A note on asymmetric and mixed strategy equilibria in the search-theoretic model of fiat money

Summary. The simple search-theoretic model of fiat money has three symmetric Nash equilibria: all agents accept money with probability 1; all agents accept money with probability 0; and all agents accept money with probability y in (0,1). Here I construct an asymmetric pure strategy equilibrium, payoff-equivalent to the symmetric mixed strategy equilibrium, where a fraction N in (0,1) of agents always accept money and 1-N never accept money. Counter to what has been conjectured previously, I find N > y. I also introduce evolutionary dynamics, and show that the economy converges to monetary exchange iff the initial proportion of agents accepting money exceeds N. Received: September 10, 1997; revised version: April 24, 1998     

Inefficient ex-post equilibria in efficient auctions

Summary. I consider a single-object English auction with two asymmetric bidders and show that it has a continuum of inefficient undominated ex-post equilibria. The result extends for the generalized VCG mechanism, Dasgupta-Maskin auction and, generally, for every auction that has an efficient ex-post equilibrium. Received: November 5, 2001; revised version: June 10, 2002 RID="*" ID="*"I am grateful to Vijay Krishna, Sergei Izmalkov and anonymous referee for many important comments.     

Incentive compatible core and competitive equilibria in differential information economies

Summary. If the allocations of a differential information economy are defined as incentive compatible state-contingent lotteries over consumption goods, competitive equilibrium allocations exist and belong to the (ex ante incentive) core. Furthermore, any competitive equilibrium allocation can be viewed as an element of the core of the n-fold replicated economy, for every n. The converse holds under the further assumption of independent private values but not in general, as shown by a counter-example. Received: August 9, 1999; revised version: September 12, 1999     

Investment coordination and demand complementarities

Summary. This paper establishes necessary conditions for demand complementarity to imply investment coordination failure and explores the welfare implications of coordinated investment. Our main results caution against demand complementarities as a motive for investment coordination. We find that: 1) generally, a strict notion of complementarity (Hicks) is necessary for the existence of an investment coordination problem and 2) that when the problem does exist, coordination lowers social welfare without countervailing sectoral asymmetries. Received: June 19, 1996; revised version: December 5, 1997     

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     

On the inconsistent behavior in voting for incumbents and for term limitation

This paper considers the seemingly inconsistent behavior of individuals who simultaneously vote for incumbents and for limitations on their terms in office. We argue that such behavior may occur even if voters pursue their self-interests in both candidate and term-limitation elections. First, we formulate elections for Congressional candidates as a two-person game, where each candidate maximizes votes by proposing a distribution of benefits to voters. Then we discuss the term limitation at the state level, where voters in each district compare, over time, the average benefits obtained from two alternative series of campaign games: one with a longer tenure associated with no term limit and the other with a shorter tenure created by the introduction of a term limit. In elections of candidates for Congress, the incumbent is successful because he can generate more aggregate benefits for voters. We show, however, that at some critical point of the tenure, his behavior will be less beneficial to his core constituents. In term-limitation elections, those voters tend to support a term limit. In some cases, they represent a majority in the state, and term limits are enacted. Received: February 23, 1999; revised version: January 24, 2000     

The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000     

Long run equilibria in an asymmetric oligopoly

Summary. Consider an oligopolistic industry composed of two groups (or populations) of firms, the low cost firms and the high cost firms. The firms produce a homogeneous good. I study the finite population evolutionarily stable strategy defined by Schaffer (1988), and the long run equilibrium in the stochastic evolutionary dynamics based on imitation and experimentation of strategies by firms in each group. I will show the following results. 1) The finite population evolutionarily stable strategy (ESS) output is equal to the competitive (or Walrasian) output in each group of the firms. 2) Under the assumption that the marginal cost is increasing, the ESS state is the long run equilibrium in the stochastic evolutionary dynamics in the limit as the output grid step, which will be defined in the paper, approaches to zero. Received: September 19, 1997; revised: June 18, 1998     

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     

Production equilibria in vector lattices

The general purpose of this paper is to prove quasiequilibrium existence theorems for production economies with general consumption sets in an infinite dimensional commodity space, without assuming any monotonicity of preferences or free-disposal in production. The commodity space is a vector lattice commodity space whose topological dual is a sublattice of its order dual. We formulate two kinds of properness concepts for agents' preferences and production sets, which reduce to more classical ones when the commodity space is locally convex and the consumption sets coincide with the positive cone. Assuming properness allows for extension theorems of quasiequilibrium prices obtained for the economy restricted to some order ideal of the commodity space. As an application, the existence of quasiequilibrium in the whole economy is proved without any assumption of monotonicity of preferences or free-disposal in production. Received: March 9, 1999; revised version: September 4, 2000     

Recursive structure and equilibria in games with private monitoring

Summary. In each stage of a repeated game with private monitoring, the players receive payoffs and privately observe signals which depend on the players' actions and the state of world. I show that, contrary to a widely held belief, such games admit a recursive structure. More precisely, I construct a representation of the original sequential problem as a sequence of static games with incomplete information. This establishes the ground for a characterization of strategies and, hence, of behavior in interactive-decision settings where private information is present. Finally, the representation is used to give a recursive characterization of the equilibrium payoff set, by means of a multi-player generalization of dynamic programming. Received: February 11, 2002; revised version: July 22, 2002 RID="*" ID="*" I am very grateful to In-Koo Cho, Larry Epstein, Denis Gromb, Stephen Morris, Paolo Siconolfi, Lones Smith and Max Stinchcombe for several insights and suggestions. A referee's comments helped improving the exposition. Finally, I wish to thank the participants to the seminars at MEDS, NYU, Columbia University, Caltech, UCLA, University of Rochester, University of Texas-Austin, Northwestern Summer Microeconomics Conference 98, Summer in Tel Aviv 98, and NASM98.     

Informational efficiency properties of rational expectations equilibria in non-convex economies

Summary. A simple example shows that although non-convexities might prevent the existence of a fully revealing rational expectations equilibrium, they need not prevent the existence of a non-informative one. Indeed, the economy in this example does not possess any fully revealing equilibria, but does have a continuum of non-informative ones. Received: February 9, 1999; revised version: October 20, 1999     

The equilibrium set of two-player games with complementarities is a sublattice

Summary. I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space. Received: May 31, 2001; revised version: October 4, 2002     

Sensitivity analysis for applied general equilibrium models in the presence of multiple Walrasian equilibria

Summary. Pagan and Shannon's (1985) widely used approach employs local linearizations of a system of non-linear equations to obtain asymptotic distributions for the endogenous parameters (such as prices) from distributions over the exogenous parameters (such as estimates of taste, technology, or policy variables, for example). However, this approach ignores both the possibility of multiple equilibria as well as the problem (related to that of multiplicity) that critical points might be contained in the confidence interval of an exogenous parameter. We generalize Pagan and Shannon's approach to account for multiple equilibria by assuming that the choice of equilibrium is described by a random selection. We develop an asymptotic theory regarding equilibrium prices, which establishes that their probability density function is multimodal and that it converges to a weighted sum of normal density functions. An important insight is that if a model allows multiple equilibria, say , but multiplicity is ignored, then the computed solution for the i-th equilibrium generally no longer coincides with the expected value of that i-th equilibrium. The error can be large and correspond to several standard deviations of the mean's estimate. Received: December 7, 1999; revised version: December 4, 2000