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.     

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.     

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.     

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.     

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.     

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 optimality in collective institutional choice

Conditions are investigated under which democratic choice of the division of land between collective and “private” use and of the distribution of collective income between “needs” and “work” payments will produce a Pareto-optimal land allocation and optimal collective labor incentives. Sen's optimal rule for the degree of “needs” distribution is found to result from self-interested voting on this parameter when the distribution of labor inputs is unskewed or when votes are weighted by labor contributions. This in turn increases the optimality of land allocation, which can be further improved by a simple rental scheme. J. Comp. Econ., Dec. 1981, 5(4), pp. 392–403. Brown University, Providence, Rhode Island.     

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.     

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.     

University education in future studies— a mark III critique: Part II

In 1970 an initial survey was made of futures courses at university level based on forty courses that could be located in the United States and Canada [Technol. Forecast. Soc. Change 2, 133–148 (1970)]. This was replicated during 1970–1972 on approximately 200 North American courses and reported changes in the field at the Third World Future Research Conference in Bucharest, September 1972 (Technol. Forecast. Soc. Change, 4, 387–407 [1973]). A third interation was funded by the World Institute Council and published in abstracted form with their permission here. The conclusions to be drawn from some six years of analysis of such courses, and drawn especially from the specifics of the 300 courses in the third iteration, can be lumped under five main headings: (1) All disciplines are to some degree finally alerted to the future implications of both their research and their teaching. (2) There remains still much “froth and nonsense” in futures courses. How to control this and encourage lively creativity is a complex problem-especially since the intellectual image of futurism today is not that high! (3) There appears to be no reliable sociocultural change theory backing future studies and there is precious little solid theory backing future studies and there is precious little solid theory in the field itself-offset by some progress in methodology. (4) Despite these strictures, if all future-oriented courses in American/Canadian universities were lumped together (future Studies per se; technology forecasting/assessment; policy sciences; peace studies; demography; environmental/ecology courses; general systems theory/system analysis/system dynamics; mathematical modelling and game theory; eschatology; utopian literature and science fiction; even black studies and women's courses) the total might approach 5000 and it is still growing, as is popular and academic understanding of the necessity “to plan ahead” in this period of human malaise. (5) Informal education of varied worth, from free swinging communes to middle management cram courses conducted by think-tanks, may turn out to be more useful in future-oriented advanced education than that which takes place in formal “walled” traditional institutions.     

Subjective states: A more robust model

We study the demand for flexibility and what it reveals about subjective uncertainty. As in Kreps [D. Kreps, 1979. A representation theorem for ‘preference for flexibility’. Econometrica 47, 565–577], Nehring [K. Nehring, 1996. Preference for flexibility and freedom of choice in a Savage framework. UC Davis Working Paper; K. Nehring, 1999. Preference for flexibility in a Savage framework. Econometrica 67, 101–119] and Dekel et al. [E. Dekel, B. Lipman, A. Rustichini, 2001. Representing preferences with a unique subjective state space. Econometrica 69, 891–934], the latter is represented by a subjective state space consisting of possible future preferences over actions to be chosen ex post. One contribution is to provide axiomatic foundations for a range of alternative hypotheses about the nature of these ex post preferences. Secondly, we establish a sense in which the subjective state space is uniquely pinned down by the ex ante ranking of (random) menus. Finally, we demonstrate the tractability of our representation by showing that it can model the two comparative notions “2 desires more flexibility than 1” and “2 is more averse to flexibility-risk than is 1.”     

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.     

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.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

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.     

Will East Germany Become a New Mezzogiorno?

Despite massive regional policy efforts, GDP per capita in Southern Italy has only briefly converged on Northern Italian levels in the 1960's. Failure of convergence since then is associated with a policy switch from investment toward income maintenance, with reduced wage sensitivity to regional labor market conditions and with increases in rent-seeking opportunities and corruption. East Germany's early experience of rapid wage and income, but not productivity, convergence raised fears that aMezzogiornoscenario could be repeated. Since then, however, investment and productivity have risen while wage setting has become more flexible. Given East Germany's greater “social capability” for growth, and provided that investment continues to be encouraged, the prospects for convergence are now more promising.J. Comp. Econom.,June 1997,24(3), pp. 241–264. Magdalen College, University of Oxford, Oxford OX1 4AU, United Kingdom; University College London, London WC1E 6BT, United Kingdom; and School of Oriental and African Studies, University of London, London WC1H 0XG, United Kingdom.     

The core of an m-sided assignment game

A generalization of Shapley-Shubik's (1972, Int. J. Game Theory, 1, 111–130) housing market in which there are m types of agents instead of two is considered. These games can have empty cores. A subclass of such games with nonempty cores is presented.     

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).     

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.