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.     

Evolution in Games with Endogenous Mistake Probabilities

J. Bergin and B. Lipman (Econometrica64 (1996), 943-956) show that the selection effect from the random mutations in the adaptive population dynamics in M. Kandori, G. Mailath, and R. Rob (Econometrica61 (1993), 29-56) and P. Young (Econometrica61 (1993), 57-84) is due to restrictions on how these mutation rates vary across population states. We here model mutation rates as endogenously determined mistake probabilities, by assuming that players with some effort can control the probability of implementing the intended strategy. This is shown to corroborate the results in Kandori-Mailath-Rob (1993) and, under certain regularity conditions, those in Young (1993). Journal of Economic Literature Classification Number: C72.     

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.     

Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

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.     

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.     

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.     

Evolutionary Learning in Signalling Games

We study equilibrium selection by evolutionary learning in monotone two-type signalling games. The learning process we study extends that introduced by Young (1993, Econometrica61, 57–84) to deal with incomplete information and sequential moves; it thus involves stochastic trembles. For vanishing trembles the process gives rise to strong selection among sequential equilibria: if the game has separating equilibria, then in the long run only play according to the so-called Riley equilibrium will be observed frequently. Journal of Economic Literature Classification Number: C72.     

Non-Walrasian Labor Markets and Real Business Cycles

The standard real business cycle literature mainly focuses on Walrasian models designed to fit the U.S. institutional framework. Differences between the United States and Europe, mostly evident in the labor market, suggest that a purely Walrasian model may be inappropriate for the study of European business cycles. I present a stochastic version of the dynamic general equilibrium model of Daveri and Maffezzoli (2000, “A Numerical Approach to Fiscal Policy, Unemployment and Growth in Europe,” Econometrics and Applied Economics Working Paper 2000-4, IEP, Università Bocconi), where unemployment is generated by monopolistic unions, and calibrate it to reproduce several long-run features of the Italian and U.S. economies. This framework is then compared with an indivisible labor model built on Hansen (1985, Journal of Monetary Economics16, 309–328) and Rogerson and Wright (1988, Journal of Monetary Economics22, 501–515). I focus on the impulse response functions, the standard business cycle statistics, and the ability to reproduce the cyclical components of the main macroeconomic variables. The main results are as follows: (i) the impulse response functions of the monopoly union (MU) model show a higher degree of overall persistence; (ii) the business cycle statistics are similar; (iii) the MU model enjoys a statistically significant advantage in reproducing the Italian business cycle, but not that of the United States. Journal of Economics Literature Classification Numbers: E32, E24, J23, J51.     

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.     

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.     

Growing through Cycles in an Infinitely Lived Agent Economy

This paper extends the analysis of K. Matsuyama (Econometrica67 (1999), 335–347) to the case of an infinitely lived representative agent economy. The economy grows endogenously through endogenous fluctuations, perpetually moving back and forth between two phases. In one phase, there is no innovation, the market structure is competitive, and the economy grows solely by capital accumulation. In the other phase, new goods are introduced and the market structure is monopolistic. In the long run, both investment and innovation grow at the same rate, but the economy alternates between the periods of high investment and the periods of higher innovation. Journal of Economic Literature Classification Numbers: E32, O11.     

Commuters route choice behaviour

The paper reports laboratory experiments with a two route choice scenario. In each session 18 subjects had to choose between a main road M and a side road S. The capacity of M was larger. Feedback was given in treatment I only on the subjects' own travel time and in treatment II on travel time for M and S. The main results are as follows:

• Mean numbers on M and S are near to pure equilibrium.

• Fluctuations persist until the end of the sessions.

• The total number of changes is significantly greater in treatment I.

• Subjects' road changes and payoffs are negatively correlated.

• A direct response mode results in more changes for bad payoffs whereas a contrary response mode shows opposite reactions.

• Simulations of an extended payoff sum learning model fits the main results of the statistical evaluation of the data.

Data-dependent selection of the lag truncation parameter in unit root tests of the Phillips-Perron type

The Andrews (Econometrica, 1991, 59, 817–858) plug-in method of heteroscedastic and autocovariance consistent covariance matrix estimation is used to construct estimators of the long-run variance parameter for use in Phillips-Perron unit root tests. This allows the lag truncation parameter to be data dependent. Monte Carlo size and power estimates are obtained suggesting that this apparently natural approach does not provide significant improvements in test performance.     

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.     

Projecting soviet energy requirements using a vintage capital model

A Solow-type vintage capital model with capital utilization is used to estimate the long-term average embedded rate of improvement in energy efficiency of Soviet fixed-capital commissionings: 2.6% per year. The results are combined with planned Soviet capital stock growth to forecast cap Soviet energy requirements through 1990. The implied Soviet capital utilization goals call for greater energy savings than have been achieved in the past. The results also identify an upper bound for rates of hidden inflation contained in Soviet capital commissionings data. It appears an estimate such as S. Cohn's 1% (Soviet Stud. 33, 2:269–299, 1981) is far more likely than A. Nove's 6 to 7% (Soviet Stud. 33, 1:142–145, 198 1).     

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.     

On Learning to Cooperate

This paper shows that experimentation can have the effect of inducing outcomes that are “far” from Nash in a learning model similar to that studied by D. Fudenberg and D. K. Levine (1993, Econometrica61, 547-573). The game under consideration is a version of a finitely repeated prisoner's dilemma, and the average length of players' lifetimes is arbitrarily large, but fixed. If the number of stages in the prisoner's dilemma is sufficiently large, then experimentation induces cooperation within the society on an arbitrarily large scale, even though the beliefs of almost all players are eventually within an arbitrary η>0 of being correct at all of the information sets in the game. Journal of Economic Literature Classification Numbers: C72.