Minimizing Regret: The General Case

In repeated games with differential information on one side, the labelling “general case” refers to games in which the action of the informed player is not known to the uninformed, who can only observe a signal which is the random outcome of his and his opponent's action. Here we consider the problem of minimizing regret (in the sense first formulated by J. Hannan (1956, in Contributions to the Theory of Games, Vol. III, Annals of Mathematics Studies, Vol. 39, Princeton, NJ: Princeton Univ. Press.) when the information available is of this type. We give a simple condition describing the approachable set. Journal of Economic Literature Classification Number: D81, D82, 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.     

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.     

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.     

Topological games at Princeton, a mathematical memoir

The games of the title are “Nash” (or Hex), “Milnor” (or Y), “Shapley” (or Projective Plane) and “Gale” (or Bridg-It) all of which were discovered (or re-discovered) in Princeton in 1948–1949. After giving the basic topological connections, I will discuss more recent ramifications related to computational complexity theory. A recurrent theme will be non-constructive proofs, or how we can know something can be done without having the slightest idea of how to do it.     

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.     

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.     

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.     

Result-oriented and process-oriented evaluations of indicative planning

This paper comments on the methodology that Estrin and Holmes (Estrin, Saul, and Holmes, Peter, “Indicative Planning in Developed Economies.” J. Comp. Econom. 14, 4:000-000, 1990) employ to evaluate indicative planning in developed countries. An evaluation that focuses on results suffers from inevitable informational problems, thus necessitating an examination of the processes of indicative planning.     

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.     

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.     

When Is Efficiency Separable from Distribution in the Provision of Club Goods?

Using the homogeneous of degree zero inverse congestion (“quality”) function employed most frequently in the literature, we characterise completely the families of utility functions which are then necessary and sufficient for break-even utilitarian welfare maximising provision of club goods with self-selection to be separable from distribution. Two types of separation—involving, alternatively (a) the price and quality; (b) the price, quality, facility size and total utilisation—are considered. When separation is possible, utility functions and optimal decisions take simple, intuitive forms. Our results extend the scope of separation results previously obtained only for pure public goods. Journal of Economic Literature Classification Numbers: D62, H23, H40.     

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.     

Games of Manipulation in Marriage Problems

We analyze the “equilibrium” outcomes of thepreference revelation gamesinduced byPareto efficientandindividually rationalsolutions in the context ofmarriage problems. We employ a Nash equilibrium refinement which allows deviations by a set of permissible coalitions, and show that the set of equilibrium outcomes coincides with a variant of the core that allows blocking by only permissible coalitions,Journal of Economic LiteratureClassification Numbers: C71, C78, D71, D78.     

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.     

Quasi-weightings, Trading, and Desirability Relations in Simple Games

We investigate the problem of simultaneously characterizing various classes of simple games in three different ways: (i) the existence of “quasi-weightings” in which smaller numerical values are assigned to losing coalitions than are assigned to winning ones, (ii) the robustness of “winningness” with respect to trades involving groups of players, and (iii) the acyclicity of certain strict “orderings” that intuitively correspond to the idea of one player being more desirable than another.Journal of Economic LiteratureClassification Number: C71.     

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.     

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.     

A new solution to the problem of adjudicating conflicting claims

Summary. For the problem of adjudicating conflicting claims, we consider the requirement that each agent should receive at least 1/n his claim truncated at the amount to divide, where n is the number of claimants (Moreno-Ternero and Villar, 2004a). We identify two families of rules satisfying this bound. We then formulate the requirement that for each problem, the awards vector should be obtainable in two equivalent ways, (i) directly or (ii) in two steps, first assigning to each claimant his lower bound and then applying the rule to the appropriately revised problem. We show that there is only one rule satisfying this requirement. We name it the “ rule”, as it is obtained by a recursion. We then undertake a systematic investigation of the properties of the rule.Received: 7 September 2004, Revised: 21 April 2005, JEL Classification Numbers: C79, D63, D74.William Thomson: Correspondence toWe thank Christopher Chambers, Youngsub Chun, Tarík Kara, Çağatay Kayí, Juan Moreno-Ternero, Toyotaka Sakai, and Michel Truchon for their comments. We also thank a referee for useful comments. This paper was presented at Columbia University, CORE, Bilkent University, Sabancí University, Université Laval, and Caltech, and at the Mallorca Review of Economic Design meeting. Thomson acknowledges support from the NSF under grant SES-0214691.