Contracts and externalities: How things fall apart

A single principal interacts with several agents, offering them contracts. The crucial assumption of this paper is that the outside-option payoffs of the agents depend positively on how many uncontracted or “free” agents there are. We study how such a principal, unwelcome though he may be, approaches the problem of contract provision to agents when coordination failure among the latter group is explicitly ruled out. Two variants are considered. When the principal cannot re-approach agents, there is a unique equilibrium, in which contract provision is split up into two phases. In phase 1, simultaneous offers at good (though varying) terms are made to a number of agents. In phase 2, offers must be made sequentially, and their values are “discontinuously” lower: they are close to the very lowest of all the outside options. When the principal can repeatedly approach the same agent, there is a multiplicity of equilibria. In some of these, the agents have the power to force delay. They can hold off the principal's overtures temporarily, but they must succumb in finite time. In both models, despite being able to coordinate their actions, agents cannot resist an “invasion” by the principal and hold to their best payoff. It is in this sense that “things [eventually] fall apart”.     

Evolutionary stability and lexicographic preferences

We explore the interaction between evolutionary stability and lexicographic preferences. To do so, we define a limit Nash equilibrium for a lexicographic game as the limit of Nash equilibria of nearby games with continuous preferences. Nash equilibria of lexicographic games are limit Nash equilibria, but not conversely. Modified evolutionarily stable strategies (Binmore and Samuelson, 1992. J. Econ. Theory 57, 278–305) are limit Nash equilibria. Modified evolutionary stability differs from “lexicographic evolutionarily stability” (defined by extending the common characterization of evolutionary stability to lexicographic preferences) in the order in which limits in the payoff space and the space of invasion barriers are taken.     

Escaping Poverty: Risk-Taking and Endogenous Inequality in a Model of Equilibrium Growth

Recent macroeconomic models of income distribution generate equilibria characterized as poverty traps. These models specify a production indivisibility such that, due to problems of asymmetric information in credit or capital markets, poor agents are never able to acquire the resources necessary to overcome the indivisibility. In the context of an equilibrium growth model, this paper demonstrates that faced with such constraints, poor, risk-averse agents have incentives to voluntarily take on risk in the hopes of exiting poverty. Each period they remain in poverty, they sacrifice a small amount of current consumption to pool resources for this risky activity. Risk-taking, economic mobility, and the distribution of income are generated endogenously. Furthermore, beginning from identical endowments, “initial” inequality also emerges endogenously. It is shown that voluntary risk-taking eliminates many potential steady-state equilibria of this and other models—those that exhibit individual poverty traps. All agents (or dynasties) expect to escape at some future date, perhaps after an extended spell in poverty.     

A Noncooperative Analysis of Hotelling's Location Game

We study the location equilibrium in Hotelling's model of spatial competition. As d'Aspremontet al.have shown, with quadratic consumer transportation cost the two sellers will seek to move as far away from each other as possible. We show that the location game possesses an infinity of mixed strategy Nash equilibria. In these equilibria coordination failure invalidates the principle of “maximum differentiation” and firms may even locate at the same point.Journal of Economic LiteratureClassification Numbers: C72, D43, L11.     

Non-Additive Beliefs and Strategic Equilibria

This paper studies n-player games where players' beliefs about their opponents' behaviour are modelled as non-additive probabilities. The concept of an “equilibrium under uncertainty” which is introduced in this paper extends the equilibrium notion of Dow and Werlang (1994, J. Econom. Theory64, 305–324) to n-player games in strategic form. Existence of such an equilibrium is demonstrated under usual conditions. For low degrees of ambiguity, equilibria under uncertainty approximate Nash equilibria. At the other extreme, with a low degree of confidence, maximin equilibria appear. Finally, robustness against a lack of confidence may be viewed as a refinement for Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D81.     

Learning to Learn, Pattern Recognition, and Nash Equilibrium

The paper studies a large class of bounded-rationality, probabilistic learning models on strategic-form games. The main assumption is that players “recognize” cyclic patterns in the observed history of play. The main result is convergence with probability one to a fixed pattern of pure strategy Nash equilibria, in a large class of “simple games” in which the pure equilibria are nicely spread along the lattice of the game. We also prove that a necessary condition for convergence of behavior to a mixed strategy Nash equilibrium is that the players consider arbitrarily long histories when forming their predictions.Journal of Economic LiteratureClassification Numbers: C72, D83.     

Equilibria in systems of social interactions

In this paper, we establish existence and uniqueness results for equilibria in systems with an infinite number of agents and with local and global social interactions. We also examine the structure of the equilibrium distribution and derive a “Markov”property for the equilibrium distribution of a class of spatially homogeneous systems.     

Mechanism design with partial state verifiability

We study implementation in environments where agents have limited ability to imitate others. Agents are randomly and privately endowed with type-dependent sets of messages. So sending a message becomes a partial proof regarding type. For environments where agents can send any combination of available messages, we develop an Extended Revelation Principle and characterize the incentive constraints which implementable allocations must satisfy. When not all message combinations are feasible, static mechanisms no longer suffice. If a ‘punishment’ allocation exists for each agent, then implementable allocations can be characterized as equilibria of a “Revelation Game,” in which agents first select from the menus of allocation rules, then the mediator requests each agent to send some verifying messages. When a punishment allocation fails to exist for some agent, dynamic games in which agents gradually reveal their evidence implement a larger set of outcomes. The latter result provides a foundation for a theory of debate.     

Games with Imperfectly Observable Commitment

[1]claims that, in models of commitment, “the first-mover advantage is eliminated when there is aslightamount of noise associated with the observation of the first-mover's selection.” We show that the validity of this claim depends crucially on the restriction to pure strategy equilibria. The game analyzed by Bagwell always has a mixed equilibrium that is close to the Stackelberg equilibrium when the noise is small. Furthermore, an equilibrium selection theory that combines elements from the theory of[7]with elements from the theory of [6], actually selects this “noisy Stackelberg equilibrium.”Journal of Economic LiteratureClassification Number: C72.     

Concatenate coordination and mutual coordination

We tell of the evolving meaning of the term coordination as used by economists. The paper is based on systematic electronic searches (on “coord,” etc.) of major works and leading journals. The term coordination first emerged in professional economics around 1880, to describe the directed productive concatenation of factors or activities within a firm. Also, transportation economists used the term to describe the concatenation of routes and trips of a transportation system. These usages represent what we term concatenate coordination. The next major development came in the 1930s from several LSE economists (Hayek, Plant, Hutt, and Coase), who extended that concept beyond the eye of any actual coordinator. That is, they wrote of the concatenate coordination of a system of polycentric or spontaneous activities. These various applications of concatenate coordination prevailed until the next major development, namely, Thomas Schelling and game models. Here coordination referred to a mutual meshing of actions. Game theorists developed crisp ideas of coordination games (like “battle of the sexes”), coordination equilibria, convention, and path dependence. This “coordination” was not a refashioning, but rather a distinct concept, one we distinguish as mutual coordination. As game models became more familiar to economists, it was mutual coordination that economists increasingly had in mind when they spoke of “coordination.” Economists switched, so to speak, to a new semantic equilibrium. Now, mutual coordination overshadows the older notion of concatenate coordination. The two senses of coordination are conceptually distinct and correspond neatly to the two dictionary definitions of the verb to coordinate. Both are crucial to economics. We suggest that distinguishing between the two senses can help to clarify “coordination” talk. Also, compared to talk of “efficiency” and “optimality,” concatenate coordination allows for a richer, more humanistic, and more openly aesthetic discussion of social affairs. The narrative is backed up by Excel worksheets that report on systematic content searches of the writings of economics using the worldwide web and, using JSTOR, of Quarterly Journal of Economics, Economic Journal, Journal of Political Economy, American Economic Review, and Economica.     

Green consumers and public policy: On socially contingent moral motivation

“Green” consumers appear to accept individual responsibility for the provision of public goods. The propensity to take such responsibility may depend on beliefs about others’ behavior, even for consumers motivated by internalized moral norms, not by social sanctions. This effect can produce multiple equilibria with either high or low demand for “green” products. Permanent increases in green consumption may be achieved by imposing temporary taxes or subsidies, or through advertising that influences beliefs about others’ behavior or about external effects. If a tax is interpreted as taking responsibility away from the individual, however, taxes can reduce the influence of moral motivation.     

On the Generic Finiteness of Equilibrium Outcomes in Plurality Games

In this paper we prove that for generic (noncooperative) voting games under plurality rule the set of equilibria that induce a mixed distribution over the outcomes (i.e., with two or more candidates elected with positive probability) is finite and, furthermore, each of these equilibria is regular. From that we deduce the finiteness of the set of equilibrium distributions over outcomes. Furthermore we offer an example (S. Govindan and A. McLennan, 1997, “On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms,” mimeo) that shows the impossibility of extending such results to a general framework, even just to voting games. Journal of Economic Literature Classification Numbers: C72, D72.     

Imitation processes with small mutations

This note characterizes the impact of adding rare stochastic mutations to an “imitation dynamic,” meaning a process with the properties that absent strategies remain absent, and non-homogeneous states are transient. The resulting system will spend almost all of its time at the absorbing states of the no-mutation process. The work of Freidlin and Wentzell [Random Perturbations of Dynamical Systems, Springer, New York, 1984] and its extensions provide a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply. This note provides a simpler and more intuitive algorithm. Loosely speaking, in a process with K strategies, it is sufficient to find the invariant distribution of a K×K Markov matrix on the K homogeneous states, where the probability of a transit from “all play i” to “all play j” is the probability of a transition from the state “all agents but 1 play i, 1 plays j” to the state “all play j”.     

Is man doomed to progress?

This paper is dedicated to the empirical exploration of the welfare effect of expectations and progress per se. Using 10 waves of the Russian Longitudinal Monitoring Survey (RLMS), a panel household survey rich in subjective variables, the analysis suggests that for a given total stock of inter-temporal consumption, agents are more satisfied with an increasing time profile of consumption: they seem to have a strong “taste for improvement”. This contributes to qualify the “Easterlin paradox” that income growth does not make people happy.     

Consistent representative democracy

We study axioms which define “representative democracy” in an environment in which agents vote over a finite set of alternatives. We focus on a property that states that whether votes are aggregated directly or indirectly makes no difference. We call this property representative consistency. Representative consistency formalizes the idea that a voting rule should be immune to gerrymandering. We characterize the class of rules satisfying unanimity, anonymity, and representative consistency. We call these rules “partial priority rules.” A partial priority rule can be interpreted as a rule in which each agent can “veto” certain alternatives. We investigate the implications of imposing other axioms to the list specified above. We also study the partial priority rules in the context of specific economic models.     

Bounding the inefficiency of equilibria in nonatomic congestion games

Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a “best possible” outcome. We study a nonatomic version of the congestion games defined by Rosenthal [Int. J. Game Theory 2 (1973) 65], and identify games in which equilibria are approximately optimal in the sense that no other outcome achieves a significantly larger total payoff to the players—games in which optimization by individuals approximately optimizes the social good, in spite of the lack of coordination between players. Our results extend previous work on traffic routing games.     

Eliciting Preferences to Assign Positions and Compensation

We describe strategy-proof rules for economies where an agent is assigned a position (e.g., a job) plus some of a divisible good. For the 2-agent–2-position case we derive a robust characterization. For the multi-agent–position case, many “arbitrary” such rules exist, so we consider additional requirements. By also requiring coalitional strategy-proofness or nonbossiness, the range of a solution is restricted to the point that such rules are not more complex than those for the Shapley–Scarf housing model (no divisible good). Third, we show that essentially only constant solutions are immune to manipulations involving “bribes.” Finally, we demonstrate a conflict between efficiency and strategy-proofness. The results extend to models (without externalities) in which agents share positions. Journal of Economic Literature Classification Numbers: C72, D70.     

When is reputation bad?

In traditional reputation models, the ability to build a reputation is good for the long-run player. In [Ely, J., Valimaki, J., 2003. Bad reputation. NAJ Econ. 4, 2; http://www.najecon.org/v4.htm. Quart. J. Econ. 118 (2003) 785–814], Ely and Valimaki give an example in which reputation is unambiguously bad. This paper characterizes a class of games in which that insight holds. The key to bad reputation is that participation is optional for the short-run players, and that every action of the long-run player that makes the short-run players want to participate has a chance of being interpreted as a signal that the long-run player is “bad.” We allow a broad set of commitment types, allowing many types, including the “Stackelberg type” used to prove positive results on reputation. Although reputation need not be bad if the probability of the Stackelberg type is too high, the relative probability of the Stackelberg type can be high when all commitment types are unlikely.     

Selective Acceptance and Inefficiency in a Two-Issue Complete Information Bargaining Game

This paper explores the effects of a “selective acceptance” rule on the outcome of two-issue negotiations. The alternating-offer game introduced here allows for the possibility that settlement may be reached on one issue while negotiation continues about the other. This model captures features of laws that are generally believed to increase efficiency. The analysis shows that if one issue is indivisible, there are inefficient subgame perfect equilibria with no Pareto-improving alternative equilibria. With opposing valuations, rapid communication guarantees inefficiency. These are unique examples of this strong form of inefficiency in an alternating-offer bargaining game with complete (and perfect) information. Journal of Economic Literature Classification Numbers: C72, J30.     

Forward and backward dynamics in implicitly defined overlapping generations models

In dynamic economic models derived from optimization principles, the forward equilibrium dynamics may not be uniquely defined, while the backward dynamics is well defined. We derive properties of the global forward equilibrium paths based on properties of the backward dynamics. We propose the framework of iterated function systems (IFSs) to describe the set of forward equilibria and apply the IFS framework to a one- and a two-dimensional version of the overlapping generations (OLG)-model. We show that if the backward dynamics is chaotic and has a homoclinic orbit (a “snap-back repeller”), the set of forward equilibrium paths converges to a fractal attractor. Forward equilibria may be interpreted as sunspot equilibria, where a random sunspot sequence determines equilibrium selection at each date.