Ranking opportunity sets in a simple intertemporal framework

We extend the problem of ranking subsets (opportunity sets) of a set of alternatives to an intertemporal setting, whereby the agent makes a sequence of choice decisions over time. In particular, we show in a simple two-period model that an agent who satisfies five plausible axioms ranks opportunity sets in a lexicographic fashion. That is, the agent ranks opportunity sets based solely on the relative desirability of their first-period alternatives, and only if the first-period alternatives are thought equally desirable does the agent consider the second-period alternatives. We discuss this strong time-discounting result and the role of the axioms, and also show that the popular “independence” axioms employed in the existing “static” literature are inconsistent with some reasonable axioms regarding intertemporal choice.      

A unifying impossibility theorem

This paper identifies and illuminates a common impossibility principle underlying a number of impossibility theorems in social choice. We consider social choice correspondences assigning a choice set to each non-empty subset of social alternatives. Three simple axioms are imposed as follows: unanimity, independence of preferences over infeasible alternatives, and choice consistency with respect to choices out of all possible alternatives. With more than three social alternatives and the universal preference domain, any social choice correspondence that satisfies our axioms is serially dictatorial. A number of known impossibility theorems—including Arrow’s Impossibility Theorem, the Muller–Satterthwaite Theorem, and the impossibility theorem under strategic candidacy—follow as corollaries.     

Extending the Nash solution to choice problems with reference points

In 1985 Aumann axiomatized the Shapley NTU value by non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives. We show that, when replacing unanimity by “unanimity for the grand coalition” and translation covariance, these axioms characterize the Nash solution on the class of n-person choice problems with reference points. A classical bargaining problem consists of a convex feasible set that contains the disagreement point here called reference point. The feasible set of a choice problem does not necessarily contain the reference point and may not be convex. However, we assume that it satisfies some standard properties. Our result is robust so that the characterization is still valid for many subclasses of choice problems, among those is the class of classical bargaining problems. Moreover, we show that each of the employed axioms – including independence of irrelevant alternatives – may be logically independent of the remaining axioms.     

Characterization of generalized weak orders and revealed preference

Summary This short paper provides an alternative framework to axiomatize various binary preference relations such as semiorder, weak semiorder etc. A set of simple axioms is presented in terms of revealed-preferred and revealed-inferior alternatives which makes the connection between various binary preference relations transparent; and every single axiom is necessary and sufficient for the existence of a binary preference relation of a specified type.We thank Bhasker Dutta, Peter Fishburn, Prasanta Pattanaik, Robert Russell and Thomas Schwartz for helpful suggestions     

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.     

Risk–risk versus standard gamble procedures for measuring health state utilities

Contingent valuation (CV) has been widely used to measure the potential benefits derived from different policy decisions. However, doubt now exists about the validity of the CV method and alternative approaches to benefit valuation have been proposed. The paper reports on the results of a study which was designed to test the viability of two of the most prominent of the alternatives: the risk–risk (RR) and standard gamble (SG) approaches. If individual preferences are consistent with the axioms of von Neumann–Morgenstern expected utility theory (EUT) then the two methods should generate the same interval scales for any given set of health states. However, the results show that SG utilities are substantially higher than RR ones, thus casting doubt on these axioms. The paper discusses alternatives to EUT which might better expalin the discrepancies found. It also considers whether the results might be explained in terms of status-quo bias and/or by the relative difficulty of RR questions. The results presented may have important implications for other areas of applied research in which there exists uncertainty about outcomes.     

Temptation-Driven Preferences

"My own behaviour baffles me. For I find myself not doing what I really want to do but doing what I really loathe." Saint Paul
What behaviour can be explained using the hypothesis that the agent faces temptation but is otherwise a "standard rational agent"? In earlier work, Gul and Pesendorfer (2001) use a set betweenness axiom to restrict the set of preferences considered by Dekel, Lipman and Rustichini (2001) to those explainable via temptation. We argue that set betweenness rules out plausible and interesting forms of temptation including some which may be important in applications. We propose a pair of alternative axioms called DFC, desire for commitment , and AIC, approximate improvements are chosen . DFC characterizes temptation as situations in which given any set of alternatives, the agent prefers committing herself to some particular item from the set rather than leaving herself the flexibility of choosing later. AIC is based on the idea that if adding an option to a menu improves the menu, it is because that option is chosen under some circumstances. From this interpretation, the axiom concludes that if an improvement is worse (as a commitment) than some commitment from the menu, then the best commitment from the improved menu is strictly preferred to facing that menu. We show that these axioms characterize a natural generalization of the Gul–Pesendorfer representation.     

Parallel axiomatizations of majority and unanimity

The relative majority rule and the unanimity rule are characterized in the two alternatives case. The main axioms rely on the idea of determining the preference of a group by partitioning it into two subgroups and aggregating next their preferences.     

Extending an order on a Set to the power set: Some remarks on Kannai and Peleg's approach

Kannai and Peleg have shown that given an ordering over a set, it is impossible to induce an ordering over the power set satisfying certain plausible axioms. We prove an impossibility and also a possibility result in this context with closely related sets of axioms, and argue that the dividing line between impossibility and possibility here is rather thin. Also, we distinguish three possible intuitive interpretations for the formal framework of Kannai and Peleg, and argue that the acceptability of specific formal axioms may crucialy depend on the particular interpretation that one chooses to adopt.     

A general impossibility result on strategy-proof social choice hyperfunctions

A social choice hyperfunction picks a non-empty set of alternatives at each admissible preference profile over sets of alternatives. We analyze the manipulability of social choice hyperfunctions. We identify a domain D^λ of lexicographic orderings which exhibits an impossibility of the Gibbard–Satterthwaite type. Moreover, this impossibility is inherited by all well-known superdomains of D^λ. As most of the standard extension axioms induce superdomains of D^λ while social choice correspondences are particular social choice hyperfunctions, we are able to generalize many impossibility results in the literature.     

Finite-dimensional utilities

Summary. Sufficient axioms are identified for the existence of a finite- dimensional quasilinear utility function whose lexicographically ordered vectors preserve a decision maker's preference order on a mixture set . It is shown that those axioms are also necessary for the linear lexicographic representation when the underlying set is a mixture space. Received: August 20, 1998; revised version: December 14, 1998     

To Majority through the Search for Unanimity

The relative majority rule is characterized, when there are only two alternatives, in terms of axioms of unanimity, reducibility, ontoness (which expresses citizen sovereignty), and no veto power (a weakening of almost unanimity). The strongest axiom is reducibility, which embodies the preference aggregation procedure that successively synthesizes two divergent preferences until divergence disappears and, therefore, unanimity can be applied.     

A characterization of consistent collective choice rules

We characterize a class of collective choice rules such that collective preference relations are consistent. Consistency is a weakening of transitivity and a strengthening of acyclicity requiring that there be no cycles with at least one strict preference, which excludes the possibility of a “money pump.” The properties of collective choice rules used in our characterization are unrestricted domain, strong Pareto, anonymity and neutrality. If there are at most as many individuals as there are alternatives, the axioms provide an alternative characterization of the Pareto rule. If there are more individuals than alternatives, however, further rules become available.     

Utility of gambling I: entropy modified linear weighted utility

Behavioral axioms about preference orderings among gambles and their joint receipt lead to numerical representations consisting of a subjective utility term plus a term depending upon the events and the subjective weights. The results here are for uncertain alternatives, in much the same sense as Savage’s usage. Several open problems are described. Results for the risky case are in a second article.      

An axiomatic approach to choice under uncertainty with catastrophic risks

This paper analyses decision under uncertainty with catastrophic risks, and is motivated by problems emerging from global environmental risks. These are typically low-probability events with major irreversible consequences. For such risks, the Von Neumann–Morgenstern (NM) axioms for decision making under uncertainty are not appropriate, since they are shown here to be insensitive to low-probability events. The paper introduces an alternative set of axioms requiring sensitivity to both low- and large-probability events. Through a new representation theorem in functional analysis, the results characterize all the operators whose maximization leads to the fulfillment of these axioms. They involve a convex combination of expected utility and a criterion based on the desire to avoid low probability and potentially catastrophic events. It is shown that the new axioms help resolve the Allais paradox. Open questions about risk aversion, games under uncertainty and calculus of variations are discussed.     

A simple characterization of majority rule

Summary. Assuming an odd number of voters, E. S. Maskin recently provided a characterization of majority rule based on full transitivity. This paper characterizes majority rule with a set of axioms that includes two of Maskin's, dispenses with another, and contains weak versions of his other two axioms. It allows the number of voters to be odd or even. Received: December 23, 1998; revised version: May 10, 1999     

Some remarks on ranking opportunity sets and Arrow impossibility theorems: correspondence results

An earlier work by Dutta and Sen provides characterizations of a set of decision rules for the ranking of opportunity sets. This paper begins by demonstrating the redundancy of one of the axioms in the said characterizations and goes on to analyze in detail one of the theorems, the Generalized Utilitarian rules theorem, which is incorrect. Basically, we find that one of the axioms that the authors provide is not in all cases satisfied for the Utilitarian rules. In this paper, we discuss this issue before proposing an alternative characterization which preserves the spirit of the original authors.     

Extending Nash's Axioms to Nonconvex Problems

Nash's axioms are extended to a domain of nonconvex problems to characterise in a simple way the Extended Nash Bargaining Solution proposed by Conley and Wilkie (1996,Games Econ. Behav.13, 26–38). Independence of irrelevant alternatives is weakened on the new domain to take into account the possibility that the solution of the convexified problem constitutes a “relevant” alternative.Journal of Economic LiteratureClassification Number: C71, C78.     

A theory of anticipated utility

A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.     

A uniqueness result for extending orders; with application to collective choice as inconsistency resolution

Two axioms about extending a weak ordering of a set Ω to its power set 2^Ω uniquely determine a lexicographic weak ordering of 2^Ω. The solution is applied to collective choice to resolve majority cycle inconsistencies, and also contrasted with Condorcet.