Even-chance lotteries in social choice theory

This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting.     

On some suggestions for having non-binary social choice functions

The various paradoxes of social choice uncovered by Arrow [1], Sen [10] and others have led some writers to question the basic assumption of a binary social choice function underlying most of these paradoxes. Schwartz [8], for example, proves an important theorem which may be considered to be a generalization of the famous paradox of Arrow, and then lays the blame for this paradox on the assumption of a binary social choice function. He then proceeds to define a type of choice functions which, like binary choice functions, define the best elements in sets of more than two alternatives on the basis of binary comparisons, but which, as he claims, have an advantage over binary choice functions, in so far as they always ensure the existence of best elements for sets of more than two alternatives irrespective of the results of binary comparisons. The purpose of this paper is to show that even a considerable weakening of the assumption of a binary social choice function does not go very far towards solving some of the paradoxes under consideration, and that if replacing the requirement of a binary social choice function by a Schwartz type social choice function solves these paradoxes, it does so only by violating the universally acceptable value judgment that in choosing from a set of alternatives, society should never choose an alternative which is Pareto inoptimal in that set (i.e., the socially best alternatives in a set should always be Pareto optimal). This argument is substantiated with the help of an extended version of Sen's [10] paradox of a Paretian liberal, and thus a by-product of our analysis is a generalization of the theorem of Sen [10]. The argument itself, however, is more general and applies also to the impossibility result proved by Schwartz [8].We are extremely grateful to Amartya Sen for his helpful comments.

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‘q-Pareto-Scalar’ Two-Stage Extremization Model and its Reducibility to One-Stage Model

A two-stage sequential choice model is studied, the first stage being defined by q-Pareto multicriterial choice rule, and the second stage being defined by scalar extremization model. In this model, at the first stage the q-Pareto rule choses alternatives which are not only undominated in terms of Pareto comparison, but also includes into choice the alternatives which are dominated by no more than q alternatives. Since the choice set of the first-stage usually contains too many elements, obtained set is used as a presentation for the second stage constructed by a scalar extremization model. The properties of the model are studied as well as its representability to one-stage scalar extremization model.     

Rational choice on non-finite sets by means of expansion-contraction axioms

The rationalization of a choice function, in terms of assumptions that involve expansion or contraction properties of the feasible set, over non-finite sets is analyzed. Schwartz's results (1976), stated in the finite case, are extended to this more general framework. Moreover, a characterization result when continuity conditions are imposed on the choice function, as well as on the binary relation that rationalizes it, is presented.     

Set and revealed preference axioms for multi-valued choice

Theory and Decision - We consider choice correspondences that assign a subset to every choice set of alternatives, where the total set of alternatives is an arbitrary finite or infinite set. We...     

The independence condition in the theory of social choice

Arrow's theorem is really a theorem about the independence condition. In order to show the very crucial role that this condition plays, the theorem is proved in a refined version, where the use of the Pareto condition is almost avoided.A distinction is made between group preference functions and group decision functions, yielding respectively preference relations and optimal subsets as values. Arrow's theorem is about the first kind, but some ambiguities and mistakes in his book are explained if we assume that he was really thinking of decision functions. The trouble then is that it is not clear how to formulate the independence condition for decision functions. Therefore the next step is to analyse Arrow's argument for accepting the independence condition.The most frequent ambiguity depends on an interpretation of A as the set of all conceivable alternatives, while the variable subset B is the set of all feasible or available alternatives. He then argues that preferences between alternatives that are not feasible shall not influence the choice from the set of available alternatives. But even if this principle is accepted, it only forces us to require independence with respect to some specific set B and not to every B simultaneously. Therefore the independence condition cannot be accepted on these grounds.Another argument is about an election where one of the candidates dies. On one interpretation this argument can be taken to support an independence requirement which leads to a contradiction. On another interpretation it is a condition about connexions between choices from different sets.The so-called problem of binary choice is found to be different from the independence problem and it plays no essential role in Arrow's impossibility result. Other impossibility results by Sen, Batra and Pattanaik and by Schwartz are of a different character.In the last section, several weaker independence conditions are presented. Their relations to Arrow's condition are stated and the arguments supporting them are discussed.     

Hidden information acquisition and static choice

This note explores the consequence of hidden information acquisition for static choice theory. We show that any choice function in the observable problem can be consistent with some well-behaved choice function in a metaproblem with unobservable costly information acquisition. This illustrates how choices may not satisfy consistency conditions because a decision maker's decision process (in this case, information acquisition) depends on her feasible set. It also illustrates the importance of modeling the source of violations of consistency conditions, rather than simply weakening axioms on preferences.     

Choosers as extension axioms

We consider the extension of a (strict) preference over a set to its power set. Elements of the power set are non-resolute outcomes. The final outcome is determined by an “(external) chooser” which is a resolute choice function. The individual whose preference is under consideration confronts a set of resolute choice functions which reflects the possible behaviors of the chooser. Every such set naturally induces an extension axiom (i.e., a rule that determines how an individual with a given preference over alternatives is required to rank certain sets). Our model allows to revisit various extension axioms of the literature. Interestingly, the Gärdenfors (1976) and Kelly (1977) principles are singled-out as the only two extension axioms compatible with the non-resolute outcome interpretation.     

The expected likelihood of transitivity: A survey

May developed an algebraic choice model to describe pairwise comparisons from an empirical study. A probabilistic choice variation of May's model has also been developed. This study presents a survey of work that considers the expected likelihood that a subject using the probabilistic model will have transitive responses for pairwise choices on a set of three alternatives. Of particular interest is the impact that various factors that influence the probabilistic choice model have on the expected likelihood of transitivity. These factors include the degree of accuracy with which the subject perceives the attributes of the alternatives, the number of attributes of comparison, and the consistency with which alternatives are ranked across attributes.This research was supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

Metapreferences and the reasons for stability in social choice: Thoughts on broadening and clarifying the debate

The standard model of collective choice looks at aggregation procedures which take individual preferences as existing for a specified set of alternatives. In this paper we propose that actors also have preferences for rules of choice or characteristics of choice processes (e.g., the perceived fairness of procedures or the popularity of outcomes) rather than simply for alternatives (outcomes) themselves. We argue that the positing of the existence of meta-preferences can illuminate a number of areas of choice theory. Here we focus on one such area: the problem of “too much” stability in majority rule decision making - a stability which belies the standard theoretical results on the generic instability of majority rule processes. We also show that discussion of the prevalence of stability in collective decision making needs to be clarified because there are at least six distinct types of stability which are sometimes confounded in the literature.     

Positionalist voting functions

Positionalist voting functions are those social choice functions where the positions of the alternatives in the voter's preference orders crucially influence the social ordering of the alternatives. An important subclass consists of those voting functions where numbers are assigned to the alternatives in the preference orders and the social ordering is computed from these numbers. Such voting functions are called representable. Various well-known conditions for voting functions are introduced and it is investigated which representable voting functions satisfy these conditions. It is shown that no representable voting function satisfies the Condorcet criterion. This condition and Arrow's independence condition, which are typical non-positionalist conditions, are shown to be incompatible. The Borda function, which is a well-known positionalist voting function, is studied extensively, conditions uniquely characterizing it are given and some modifications of the function are investigated.My thanks are due to professor Bengt Hansson for encouragement and several helpful suggestions.     

The aggregation of preferences: can we ignore the past?

The article shows that a Paretian social welfare function can be history independent and time consistent only if a stringent set of conditions is verified. Individual utilities must be additive. The social welfare function must be a linear combination of these utilities. Social preferences are stationary only if, in addition, all individuals have the same constant discount rate. The results are implemented in two frameworks: deterministic dynamic choice and dynamic choice under uncertainty. The applications highlight that the conditions are unlikely to be met by individual preferences, and that they severely restrict social preferences.     

Liberalism,non-binary choice and Pareto principle

It is shown in this paper that a very mild form of Pareto principle is compatible with a set of restrictive conditions. Deriving a choice set identical with the set of alternatives in the case of paradox of voting amounts to begging the problem. If we restrict that the choice set should be a proper sub-set of the original set, the paradox will be revived. In the realistic sense liberalism may well be treated as an outcome of the choice rather than as a basic value judgement. Choice of Rules of the Game ought to be the first step and then only society can seek the optimal situation under those Rules.I am very grateful to P. K. Pattanaik for helpful discussions and valuable comments on the first draft. I am also grateful to Prof. Amartya Sen whose lectures at the Delhi School of Economics introduced me to the theory of social choice.     

On the Analysis of Negative Freedom

This paper presents a non-preference-based approach to the analysis of negative freedom. It is argued that a proper understanding of (different conceptions of) negative freedom necessitates an examination of the consequences of changes in the set of feasible alternatives. For this reason the paper does not focus on freedom rankings of opportunity sets but on freedom rankings of opportunity situations, i.e., pairs consisting of a feasible set and an opportunity set. Three different freedom rankings of opportunity situations are axiomatically characterised. Each of the three rankings forms a generalisation of the purely cardinality-based freedom ranking of opportunity sets presented by Pattanaik and Xu (1990).     

Expected Utility Consistent Extensions of Preferences

We consider the problem of extending a (complete) order over a set to its power set. The extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expected utility consistent extensions whilst it allows to emphasize various subtleties, the effects of which seem to be underestimated – particularly in the literature on strategy-proof social choice correspondences.      

The uncovered set in spatial voting games

In a majority rule voting game, the uncovered set is the set of alternatives each of which can defeat every other alternative in the space either directly or indirectly at one remove. Research has suggested that outcomes under most reasonable agenda processes (both sincere and sophisticated) will be confined to the uncovered set. Most research on the uncovered set has been done in the context of voting games with a finite number of alternatives and relatively little is known about the properties of the uncovered set in spatial voting games.We examine the geometry of the uncovered set in spatial voting games and the geometry of two important subsets of the uncovered set, the Banks set and the Schattschneider set. In particular, we find both general upper and lower limits on the size of the uncovered set, and we give the exact bounds of the uncovered set for situations with three voters. For situations with three voters, we show that the Banks set is identical to the uncovered set.     

The Political Economy of Pension Reform in Eastern Europe

The retirement schemes in several East European countries underwent fundamental change in recent years, defying conventional wisdom in welfare state research. This article takes a new look at the determinants of paradigm choice in the area of old-age security, comparing the Polish, Hungarian and Czech experience from an actor-centred institutionalist perspective. The author points out that structural factors — notably the financial situation of retirement schemes and the level of external debt — largely determined the set of main pension reform actors, as well as their relative strength. The resulting actor constellations produced the basic paradigm choice, based on the actors' respective cognitive maps and consequent perception of pension reform alternatives. Tactical moves and the strategic potentialities of the chosen paradigm were also relevant.     

Choice Functions with States of Mind

An agent is rational by dominance then states of mind (DSM) if he makes his decisions by, first, preselecting the alternatives that are not dominated by any other alternative in the choice set, and second, by choosing the preselected alternative corresponding to his state of mind. This state of mind is dependent on the previous choices he made. An agent is rational by states of mind then dominance (SMD) if he makes his decisions by, first, preselecting the alternatives corresponding to his state of mind and second, choosing the alternative that is not dominated. We give axiomatizations for SMD-rational or DSM-rational choice functions.