Probability Logic for Type Spaces

Using a formal propositional language with operators “individual i assigns probability at least α” for countably many α, we devise an axiom system which is sound and complete with respect to the class of type spaces in the sense of Harsanyi (1967–1968, Management Science, 14 159–182). A crucial inference rule requires that degrees of belief be compatible for any two sets of assertions which are equivalent in a suitably defined natural sense. The completeness proof relies on a theorem of the alternative from convex analysis, and uses the method of filtration by finite sub-languages. Journal of Economic Literature Classification Numbers: D80, D82.     

Substantive Rationality and Backward Induction

S. Modica and A. Rustichini (1994, Theory and Decision37, 107–124) provided a logic for reasoning about knowledge where agents may be unaware of certain propositions. However, their original approach had the unpleasant property that nontrivial unawareness was incompatible with partitional information structures. More recently, Modica and Rustichini (1999, Games Econ. Behav.27, 265–298) have provided an approach that allows for nontrivial unawareness in partitional information structures. Here it is shown that their approach can be viewed as a special case of a general approach to unawareness considered by R. Fagin and J. Y. Halpern (1988, Artificial Intelligence34, 39–76). Journal of Economic Literature Classification Numbers: D80, D83.     

The Role of Absolute Continuity in “Merging of Opinions” and “Rational Learning”

D. Blackwell and L. Dubins (1962, Ann. Math. Statist.38, 882–886) showed that opinions merge when priors are absolutely continuous. E. Kalai and E. Lehrer (1993, Econometrica61, 1019–1045) use this result to show that players in a repeated game eventually play like a Nash equilibrium. We provide an alternative proof of merging of opinions that clarifies the role of absolute continuity while casting doubt on the relevance of the result. Persistent disagreement, the opposite of merging, allows the construction of a sequence of mutually favorable “bets.” By a law of large numbers, both agents are certain they will win these bets on average. This certain disagreement violates absolute continuity. Journal of Economic Literature Classification Numbers: C11, C69, C72, D83.     

Optimality and Robustness of the English Auction

In Milgrom and Weber's (1982, Econometrica50, 1089–1122) “general symmetric model,” under a few additional regularity conditions, the English auction maximizes the seller's expected profit within the class of all posterior-implementable trading procedures and fails to do so among all interim incentive-compatible procedures in which “losers do not pay.” These results suggest that appropriate notions of robustness and simplicity which imply the optimality of the English auction for a risk-neutral seller must impose “bargaining-like” features on the set of feasible trading mechanisms. Journal of Economic Literature Classification Numbers: D44, D82.     

Private Value Allocations in Large Economies with Differential Information

We consider a Radner-type (e.g., Radner, 1968, Econometrica36, 31–58) pure exchange economy with differential information and a continuum of agents. We show that under appropriate assumptions the set of Aumann–Shapley private value allocations in such an economy coincides with the set of Radner competitive equilibrium allocations. Journal of Economic Literature Classification Numbers: C71, D51, D82.     

Benefit and Distance Functions

We explore the relationship between R. W. Shephard's input distance function (“Cost and Production Functions,” Princeton Univ. Press, Princeton, 1953) and D. G. Luenberger's benefit function (J. Math. Econ.21(1992a), 461–481). We point out that the latter can be recognized in a production context as a directional input distance function which can exhaustively characterize technologies in both price and input space. D. McFadden's (Cost, revenue, and profit functions,in“Production Economics: A Dual Approach to Theory and Applications, “North-Holland/Elsevier, New York, 1978) composition rules for input sets and input distance functions are then extended to the directional input distance function.Journal of Economic LiteratureClassification Numbers : D21, D24, D29.     

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.     

A Proof of Calibration via Blackwell's Approachability Theorem

Over the past few years many proofs of the existence of calibration have been discovered. Each of the following provides a different algorithm and proof of convergence: D. Foster and R. Vohra (1991, Technical Report, University of Chicago), (1998, Biometrika85, 379–390), S. Hart (1995, personal communication), D. Fudenberg and D. Levine (1999, Games Econ. Behavior29, 104–130), and S. Hart and A. Mas-Colell (1997, Technical Report, Hebrew University). Does the literature really need one more? Probably not. But the algorithm proposed here has two virtues. First, it only randomizes between two forecasts that are very close to each other (either p or p + ε). In other words, the randomization only hides the last digit of the forecast. Second, it follows directly from Blackwell's approachability theorem, which shortens the proof substantially. Journal of Economic Literature Classification Numbers: C70, C73, C53.     

Consensus and common knowledge of an aggregate of decisions

McKelvey and Page [McKelvey, R., Page, T., 1986. Common knowledge, consensus and aggregate information. Econometrica 54, 109–127] generalized Aumann's [Aumann, R.J., 1976. Agreeing to disagree. Ann. Statist. 4, 1236–1239] agreement theorem to the case where agents have common knowledge of a statistic of their posterior probabilities of some event. They showed that if individuals have the same prior, and if the statistic satisfies a stochastic regularity condition, then common knowledge of it implies equality of all posteriors. We show a similar result in a more general setting where agents have common knowledge of a statistic of their individual decisions. Decisions can be posteriors as well as discrete actions such as buy or sell. We show that if the decision rule followed by individuals is balanced union consistent, and if the statistic of individual decisions is exhaustive, then common knowledge of it implies equality of all decisions. We give an example showing that neither Cave's [Cave, J., 1983. Learning to agree. Econ. Letters 12, 147–152] union consistency condition nor Parikh and Krasucki's [Parikh, R., Krasucki, P., 1990. Communication, consensus and knowledge. J. Econ. Theory 52, 178–189] convexity condition is sufficient to guarantee the result.     

The Choquet Bargaining Solutions

We axiomatically investigate the problem of rationalizing bargaining solutions by social welfare functions that are linear in every rank-ordered subset of Rⁿ₊ . Such functions, the so-called Choquet integrals, have been widely used in the theories of collective and individual choice. We refer to bargaining solutions that can be rationalized by Choquet integrals as Choquet bargaining solutions. Our main result is a complete characterization of Choquet bargaining solutions. As a corollary of our main result, we also obtain a characterization of the generalized Gini bargaining solutions introduced by Blackorby et al. (1994, Econometrica62, 1161–1178). Journal of Economic Literature Classification Numbers: D71, C78.     

Evolution in Finitely Repeated Coordination Games

I apply a version of Kandori et al. (1993, Econometrica, 61, 29–56) and Young's (1993, Econometrica, 61, 57–84) evolutionary dynamic to finitely repeated coordination games. The dynamic is modified by allowing mutations to affect only off path beliefs. I find that repetition within a match leads agents to sacrifice current payoffs in order to increase payoffs in later stages. As a consequence, evolution leads to (almost) efficiency. Journal of Economic Literature Classification Number: C79.     

Forecasting the Forecasts of Others in the Frequency Domain

This paper studies a class of models developed by R. M. Townsend (1983, J. Polit. Econ.91, 546–588) and T. J. Sargent (1991, J. Econom. Dynam. Control15, 245–273). These models feature dynamic signal extraction problems and an infinite regress in expectations. This paper uses frequency domain methods to compute an analytical solution to the fixed point problem posed by the infinite regress in expectations. The advantage of a frequency domain approach vis-à-vis a time domain approach derives from the fact that these models produce equilibria with non-fundamental moving average representations, in which market observations do not reveal the underlying shocks to agents' information sets. As a result, decision rules contain moving average components that are more easily handled in the frequency domain than in the time domain.     

Subgame-perfect implementation of bargaining solutions

We study house allocation problems introduced by L. Shapley and H. Scarf (1974, J. Math. Econ.1, 23–28). We prove that a mechanism (a social choice function) is individually rational, anonymous, strategy-proof, and nonbossy (but not necessarily Pareto efficient) if and only if it is either the core mechanism or the no-trade mechanism, where the no-trade mechanism is the one that selects the initial allocation for each profile of preferences. This result confirms the intuition that even if we are willing to accept inefficiency, there exists no interesting strategy-proof mechanism other than the core mechanism. Journal of Economic Literature Classification Numbers: C71, C78, D71, D78, D89.     

Rational Choice and the Condorcet Jury Theorem

In a recent paper, Austen-Smith and Banks (1996,Amer. Polit. Sci. Rev.90, 34–45), criticize the current literature on the Condorcet jury theorem as neglecting the behavioral underpinnings of decision-making. They leave open the question whether allowing mixed strategies would sustain the conclusions of the Condorcet jury theorem. In this paper, it is shown that these conclusions can hold in equilibrium. In other words, “a rational choice foundation for the claim that majorities invariably ‘do better’ than individuals” is derived. (Journal of Economic LiteratureClassification Number: D72.     

On the nonexistence of universal information structures

This paper extends the nonexistence result of Heifetz and Samet (Games Econ. Behav. 22 (1998) 260-273). They have shown that there exists no universal knowledge space to which every knowledge space can be mapped in a knowledge-preserving manner. We show that an analogous nonexistence result holds in the more general context of information structures. These structures can be viewed as generalizations of knowledge spaces that describe non-probabilistic beliefs.     

Choquet Rationality

Consider a decision problem under uncertainty for a decision maker with known (utility) payoffs over prizes. We say that an act is Choquet (Shafer, Bernoulli) rational if for some capacity (belief function, probability) over the set of states, it maximizes her “expected” utility. We show that an act may be Choquet rational without being Bernoulli rational, but it is Choquet rational if and only if it is Shafer rational. Journal of Economic Literature Classification Numbers: C72, D81.     

On the Existence of Stable Roommate Matchings

This paper identifies a condition called “no odd rings” that is sufficient for the existence of stable roommate matchings in the weak preferences case. It shows that the process of allowing randomly chosen blocking pairs to match converges to a stable roommate matching with probability one as long as there are no odd rings. This random-paths-to-stability result generalizes that of Roth and Vande Vate (1990, Econometrica58, 1475–1480) and may not hold if there are odd rings. The “no odd rings” condition can also be used to prove a number of other sufficient conditions that are more economically interpretable. Journal of Economic Literature Classification Numbers: C78, D71.     

Rent Seeking and Government Ownership of Firms: An Application to China''s Township–Village Enterprises

Using its control of regulated inputs, a government agency extracts rents from a manager who undertakes an investment. Such government rent-seeking activity leads to a typical hold-up problem. Government ownership serves as a second-best commitment mechanism, through which the government agency will restrain itself from the rent-seeking activity and may even offer the manager assistance in the form of tax breaks and subsidies. This mechanism works at a cost, however, as government ownership also compromises ex post managerial incentives and creates distortion in resource allocation. Nevertheless, government ownership Pareto dominates private ownership under certain conditions. These conditions correspond to a host of stylized empirical observations concerning local-government-owned firms, i.e., township–village enterprises, during China's transition to a market economy.J. Comp. Econ., June 2002 30(4), pp. 787–811. Department of Economics, University of Illinois at Urbana–Champaign, 328 David Kinley Hall, Urbana, Illinois 61801. © 2002 Association for Comparative Economic Studies. Published by Elsevier Science (USA). All rights reserved.Journal of Economic Literature Classification Numbers: D23, D72, L33.     

Debt Overhang and Barter in Russia

This paper develops a model in which costly barter is used by firms to protect working capital against outside creditors. Although creditors could agree to postpone debt payments and to avoid destroying the firm's working capital, if the firm cannot commit not to divert cash ex post, the outcome of renegotiation still provides ex ante incentives to use barter. We show that the greater is the debt overhang, the more likely is the use of barter, with and without the possibility of debt restructuring. Empirical evidence from Russian firm-level data is shown to be consistent with the model's predictions. J. Comp. Econ., December 2002, 30(4), pp. 635–656. New Economic School, CEFIR, CEPR, and WDI, Nakhimovsky pr. 47, Moscow 117418, Russia; Sloan School of Management, M.I.T., 50 Memorial Drive, Cambridge, Massachusetts 02142; and ROSES–CNRS and CEPR, Maison des Sciences économiques, 106–112 Bd de l'Hôpital, 75647 Paris Cedex 13, France. © 2002 Association for Comparative Economic Studies. Published by Elsevier Science (USA). All rights reserved.Journal of Economic Literature Classification Numbers: E41, G34, P31.     

Evolution and Mixed Strategies

Selten (1980, J. Theoret. Biol., 84, 93–101) showed that no mixed equilibria are evolutionarily stable when players can condition their strategies on their roles in a game. Alternatively, Harsanyi's (1973, Int. J. Game Theory, 2, 1–23) purification argument implies that all mixed equilibria are approximations of strict, and hence evolutionarily stable, equilibria of games with slightly perturbed payoffs. This paper reconciles these results: Approximations of mixed equilibria have high invasion barriers, and hence are likely to persist, when payoff perturbations are relatively important and role identification is relatively noisy. When payoff perturbations are unimportant and role identification is precise, approximations of mixed equilibria will have small invasion barriers and are unlikely to persist. Journal of Economic Literature Classification Numbers: C70, C78.