Showing papers in "Theory and Decision in 1972"

Even-chance lotteries in social choice theory

Abstract: 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.

On some suggestions for having non-binary social choice functions

Abstract: The various paradoxes of social choice uncovered by Arrow [1], Sen [10] and others1 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,2 and then lays the blame for this paradox on the assumption of a binary social choice function.3 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.4 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].

What if utility functions do not exist

Abstract: Often the preferences of decision-makers are sufficiently inconsistent so as to preclude the existence of a utility function in the classical sense. Several alternatives for dealing with this situation are discussed. One alternative, that of modifying classical demands on utility functions, is emphasized and described in the context of the theory of measurement developed in recent years by behavioral scientists. The measurement theory approach is illustrated by discussing the concept of the dimension of a partial order. Even if we cannot assign numerical utility or worth values which reflect preferences in the classical utility function sense, from the measurement theory point of view we can still learn a lot about the preferences by finding several measures of worth so that a given alternative x is preferred to an alternative y if and only if x is ranked higher than y on each of the worth scales. If such measures can be found, it follows that the preferences define a partial order, and the smallest number of such scales needed is called the dimension of the partial order. If one-dimensional preferences (those amenable to classical utility assignments) cannot be found, then the next best thing is to search for partially ordered preferences with as small a dimension as possible. Several conditions under which a partial order is two-dimensional are described.

On the Onassis problem

Abstract: A decision problem is by convention characterized by its outcome matrix and by a subsequent utility evaluation. In trying to set up an outcome matrix based on wealth values it may occur that due to ambiguities inherent in the decision problem it is not clear which standard of value (or numeraire) should be used in order to measure wealth. A typical example of this kind is Stutzel's so called Onassis Paradox. We show that problems of this kind can be solved within the conventional framework of decision theory. The analysis proceeds in two steps. First, state-dependent utility functions are derived; second, a model for evaluating these utility functions is presented.

Democracy and interdependent preferences

Abstract: In a democracy, the views and wishes of the citizens are to count. The problem here is how this dictum is to be understood. I suggest that a proper analysis requires that each person's preferences be seen in the context of his beliefs concerning the preferences of the others and of the preferences he would have if he held different beliefs. Preferences founded on erroneous beliefs call for special consideration. So also do certain reluctantly held preferences. I propose a decision-policy involving the identification of social equilibria of preference.

Liberalism, non-binary choice and Pareto principle

Abstract: 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.

The mythology of methodology

Abstract: The purpose of this paper is to interject an alternate mood and basis for the doing of the philosophy of science. The mood is one of ‘feeling’ and the basis is one of Jungian psychology. The paper argues that there are many bases for the doing of the philosophy of science and that the logic of science is only one of these. To dismiss all applications of psychology out of hand with the charge (or label) of psychologism is as bad and as much a disease as psychologism itself.

Conditional expected, extensive utility

Abstract: Luce and Krantz (1971) presented an axiom system for conditional expected utility. In this theory a conditional decision is a function whose domain is a non-null subevent and whose range is a subset of a set of consequences. Given a family of conditional decisions that is closed under unions of decisions whose domains are disjoint and under restrictions to non-null subevents, the second major primitive is an ordering of the family. Axioms were given that are adequate to construct a numerical utility function over decisions and a probability function over events for which the conditional expectation of the utility is order preserving. Several of the axioms are quite complex and seem a bit artificial, and the proof is very long. Here the structure is modified by adding to the set of outcomes a concatenation operation, and the representation theorem is modified by requiring that the utility function be additive over this binary operation as well as exhibiting the expected utility property. The advantages of this pair of changes are, first, it exploits the obvious fact that the union of consequences is itself a consequence; second, it reduces the mathematical burden carried by the set theoretic structure of conditional decisions and, as a result, the axioms can be made much easier to understand; and third, it permits a considerably shorter proof because one can draw more readily on known results. The major drawback of this approach is, of course, that it is inconsistent with the evidence that utility is not additive over consequences - at least, not over increasing amounts of a single good (diminishing marginal utility).

On having the opportunity

Abstract: There are least two general notions of having the opportunity to perform an action, one that concerns what is permitted on the basis of rules, regulations, or customs and another that concerns what is possible according to the physical or mental circumstances. The notion of having the opportunity within a set of circumstances is investigated and analysis in terms of physical possibility is proposed. A distinction is then made between a strong, person-dependent sense of ‘opportunity’ and a weak, person-independent sense. It is also observed that care must be taken to distinguish the proposed analyses from criteria for determining whether someone has an opportunity.

Singular causal explanations

Abstract: Singular causal explanations cite explicitly, or may be paraphrased to cite explicitly, a particular factor as the cause of another particular factor. During recent years there has emerged a consensus account of the nature of an important feature of such explanations, the distinction between a factor regarded correctly in a given context of inquiry as ‘the cause’ of a given result and those other causally relevant factors, sometimes called ‘mere conditions’, which are not regarded correctly in that context of inquiry as the cause of that result. In this paper that consensus account is characterized and developed. The developed version is then used to illuminate some recent discussions of singular causal explanations.

An incompleteness problem in Harsanyi's general theory of games and certain related theories of non-cooperative games

Abstract: The theory of games recently proposed by John C. Harsanyi in ‘A General Theory of Rational Behavior in Game Situations’, (Econometrica, Vol. 34, No. 3) has one anomalous feature, viz., that it generates for a special class of non-cooperative games solutions which are not equilibrium points. It is argued that this feature of the theory turns on an argument concerning the instability of weak equilibrium points, and that this argument, in turn, involves appeal to an unrestricted version of a postulate subsequently included in the theory in restricted form. It is then shown that if this line of reasoning is permitted, then one must, by parity of reasoning, permit another instability argument. But, if both of these instability arguments are permitted in the construction of the theory, the resultant theory must be incomplete, in the sense that there will be simple non-cooperative games for which such a theory cannot yield solutions. This result is then generalized and shown to be endemic to all theories which have made the equilibrium condition central to the treatment of non-cooperative games. Some suggestions are then offered concerning how this incompleteness problem can be resolved, and what one might expect concerning the postulate structure and implications of a theory of games which embodies the revisions necessitated by a resolution of this problem.

Notes on the so-called incompleteness problem and on the proposed alternative concept of rational behavior

Abstract: In reply to McClennen, the paper argues that his criticism is based on a mistaken assumption about the meaning of rationality postulates, to be called the ‘Implication Principle’. Once we realize that the Implication Principle has no validity, McClennen's criticisms of what he calls the ‘Reductio Argument’ and what he calls the ‘Incentive Argument’ fall to the ground. The rest of the paper criticizes the rationality concept McClennen proposes in lieu of that used by orthodox game theory. It is argued that McClennen's concept is inconsistent with the behavior of real-life intelligent egoists; it is incompatible with the way ‘payoffs’ are defined in game theory; and it would be highly dangerous as a practical guide to human behavior.

Prerequisites for quantification in the empirical sciences

Abstract: The present paper discusses the usual classification of qualitative, comparative and quantitative concepts in the language of empirical sciences. The quantification of empirical properties is introduced on the basis of quasi-serial order formed by means of implicitly defined qualitative and comparative concepts and the concept of relative coincidence. The paper introduces conditions of additivity of quantitative concepts relative to the fundamental concepts of quasi-serial order and the corresponding aggregative operations which are based on analogues of Tarski's postulates for the operation of addition.

A graph-theoretical approach to the aggregation of individual preference orderings

Abstract: A graph-theoretical approach as first outlined by Harary et al.1 is developed for solving the problem of aggregating a given set of individual preference orderings provided that no individual preferential judgment is inconsistent with the collective preference ordering obtained by aggregation.

Decision-theoretic aspects of risk-taking behaviour

Abstract: A crucial problem in economic decision (game) theory is to specify by an algorithm the standards of behaviour which determine the process of decision-making Recent research has indicated that the attitude to risk is an important argument for finding unique solutions for decision problems In this study different behaviour-constants of decision-making are reported which clarify that a logical equivalency exists between decision-making, conflict, achievement motive and risk-taking By using information theoretic arguments as a transform rule, it is suggested to measure risk-preference and risk-aversion in terms of redundancy and related measures