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.     

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.     

For an integrative theory of social behaviour: Theorising with and beyond rational choice theory

Most sociologists are not content with merely relating macrosocial phenomena to preceding macrosocial causes in their causal explanations of social phenomena. Instead they are seeking to provide (non‐reductive) microfoundations with which they can corroborate and make understandable the connection between macrosocial phenomena. In order to do so a theory (or theories) of human action is required. One such theory, rational choice theory (RCT), has long been viewed with strong suspicion in sociology. I show such suspicion to be partially justified. RCT cannot be a general theory of social behaviour. Nonetheless, there are important insights in various versions of RCT that should not be discarded. In order to improve upon RCT and move toward a more unified or integrative theory of action social‐psychological research has to be taken note of. I demonstrate how dual‐process theories and the research on heuristics can help sociologists move beyond RCT without contradicting some of its more basic insights.     

Independent social choice correspondences

A fixed agenda social choice correspondence on outcome set X maps each profile of individual preferences into a nonempty subset of X. If satisfies an analogue of Arrow's independence of irrelevant alternatives condition, then either the range of contains exactly two alternatives, or else there is at most one individual whose preferences have any bearing on . This is the case even if is not defined for any proper subset of X.     

Inequality and social choice

This paper is concerned with the welfare and inequality rankings of income distributions whose Lorenz curves cross. A simple inequality measure is proposed which gives unambiguous results in many cases.     

Weakly implementable social choice rules

Previous investigations have shown that a social choice function which is partially implementable must be characterized by pervasive veto power. This paper investigates how much additional latitude in the design of social choice functions, and how much relief from this vetoers result, can be achieved by examining multi-valued social choice rules and relaxing the requirement of partial implementability to a requirement that we call weak partial implementability. We find that the power structures which characterize partially implementable social choice functions, including the veto properties, also characterize weakly partially implementable social choice rules. The conclusion is that invoking multi-valuedness and implementation of appealing social choice rules in strong Nash equilibria. Our results apparently exhaust the possibilities for implementation in strong Nash equilibrium. If any implementation possibility results are to be achieved, they can apparently come only by weakening the equilibrium requirement.     

Economic choice in generalized expected utility theory

Generalized expected utility models have enjoyed considerable success in explaining observed choices under uncertainty. However, there has been only limited progress in deriving comparative static results. This paper presents a general framework which permits the incorporation of a wide range of generalized expected utility models, but is sufficiently powerful to permit the derivation of comparative static results. The central idea is to represent preferences by the expected utility of a transformed probability distribution.     

Regret theory with general choice sets

The regret theory of choice under uncertainty proposed by Loomes and Sugden has performed well in explaining and predicting violations of Expected Utility theory. The original version of the model was confined to pairwise choices, which limited its usefulness as an economic theory of choice. Axioms for a more general form of regret theory have been proposed by Loomes and Sugden. In this article, it is shown that a simple nonmanipulability requirement is sufficient to characterize the functional form for regret theory with general choice sets. The stochastic dominance and comparative static properties of the model are outlined. A number of special cases are derived in which regret theory is equivalent to other well-known theories of choice under uncertainty.I would like to thank Graham Loomes, Robert Sugden, Peter Wakker, and Nancy Wallace for helpful comments and criticism.     

A note on Chichilnisky's social choice paradox

One of the main results in topological social choice states the non-existence of a continuous, anonymous, and unanimous aggregation rule on spheres. This note provides a proof based upon simple methods such as integration.     

An application of social choice theory to U.S.D.A. forest service decision making

To meet the requirements of the National Environmental Policy Act of 1969, the U.S.D.A. Forest Service has changed the way forest plans are developed. The focus of this paper is to address the group-decision problem using social choice theory, specifically the voting models of Condorcet and Borda. The elements of a social choice problem are voters, alternatives, preferences, and aggregation. A case study from the Shoshone National Forest is used to demonstrate the use of the voting models from social choice theory. The solutions derived from the analysis are strategy and coalitional strategy proof implying that behaviors intended to influence the outcome, such as vote trading, would be unsuccessful.     

Decision-making under ignorance with implications for social choice

A new investigation is launched into the problem of decision-making in the face of complete ignorance, and linked to the problem of social choice. In the first section the author introduces a set of properties which might characterize a criterion for decision-making under complete ignorance. Two of these properties are novel: independence of non-discriminating states, and weak pessimism. The second section provides a new characterization of the so-called principle of insufficient reason. In the third part, lexicographic maximin and maximax criteria are characterized. Finally, the author's results are linked to the problem of social choice.     

The reasonable man — a social choice approach

Most legal systems rely heavily on the notion of the reasonable man. Here an attempt is made to analyze the reasonable man in a social choice model. The main argument is that if the reasonable man satisfies a certain set of axioms, he essentially coincides with one of the individuals, both in terms of his preferences and in terms of his expectations.     

The general relevance of the impossibility teorem in smooth social choice

A complete classification theorem for voting processes on a smooth choice spaceW of dimensionw is presented. Any voting process is classified by two integersv ^* () andw(), in terms of the existence or otherwise of the optima set, IO(), and the cycle set IC().In dimension belowv ^* () the cycle set is always empty, and in dimension abovew() the optima set is nearly always empty while the cycle set is open dense and path con nected. In the latter case agenda manipulation results in any outcome.For admissible (compact, convex) choice spaces, the two sets are related by the general equilibrium result that IO() union IC() is non-empty. This in turn implies existence of optima in low dimensions. The equilibrium theorem is used to examine voting games with an infinite electorate, and the nature ofstructure induced equilibria, induced by jurisdictional restrictions.This material is based on work supported by a Nuffield Foundation grant.     

Modernizing social security: Changing responsibilities and individual choice

Social security needs to be reformed to make it sustainable. We argue that it is necessary to consider not only system characteristics (such as the level and duration of benefits, insured population), but also the division of responsibilities. Social security is not solely a state matter; firms, trade unions, industrial sectors and the individual play a role as well. As social risks are to some extent endogenous, insurance ought to be taken care of at the level at which the risk can be most directly influenced in order to improve efficiency. This may partly take place within the public system, for example through risk rating. Alternatives would include shifting from collective to more individual arrangements, prominent examples being multipillar systems and savings accounts. Multipillar systems mainly imply a new division of responsibilities, taking the endogeneity of social risks and the implications of modern life cycles into account. At the same time they offer more individual choice. Savings accounts reduce moral hazard without affecting individual choice. Either way, in reforming social security a balance needs to be found between competing goals or principles: between equity and efficiency, between solidarity and freedom of choice, between the risks of moral hazard and adverse selection. This choice is difficult because social security is not only about "security". It is an important part of the way we organize our society and embodies many of society's values.     

On admissible strategies and manipulation of social choice procedures

A collective choice mechanism can be viewed as a game in normal form; in this article it is shown, for very attractive rules and for sets with any number of alternatives, how individuals involved in a collective decision problem can construct the preferences they choose to express. An example is given with a version of plurality rule. Manipulability results are deduced from such a characterization.     

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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A quantum leap for social theory

I discuss the revolutionary implications of Alexander Wendt's Quantum Mind and Social Science for social theory. These center on the significance given to society as a possibility space in which observable social reality amounts to quantum decoherent states in the system. I discuss how Wendt radicalises social constructivism and goes considerably beyond Bourdieu's conceptions of society as a field. A signature feature in Wendt's theory, the state as a hologram, comports well with the implied view of the social order in much public opinion and market research. In this respect, Wendt's theory is very attuned to the spirit of our times.