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.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

A formulation of the determinism hypothesis

The author tries to formulate what a determinist believes to be true. The formulation is based on some concepts defined in a systems-theoretical manner, mainly on the concept of an experiment over the sets A ^m (a set of m-tuples of input values) and B ⁿ (a set of n-tuples of output values) in the time interval (t ₁, ..., t _k ) (symbolically E[t ₁,..., t _k , A ^m , B ⁿ ]), on the concept of a behavior of the system S ^m,n (=(A ^m , B ⁿ )) on the basis of the experiment E[t ₁, ..., t _k , A ^m , B ⁿ ] and, indeed, on the concept of deterministic behavior .... The resulting formulation of the deterministic hypothesis shows that this hypothesis expresses a belief that we always could find some hidden parameters.     

Echelons in Incomplete Relations

An efficient method of value assessment of a set of exchangeable alternatives A = {a ₁,a ₂, ,a _n} is presented. It particularly applies to situations where certain preferences may be easily evaluated or are already known, while other binary comparisons may not at once be available. Further applications are to ranking partial tournaments and the emergence and the characterisation of organisational hierarchy. By sequentially performing transitively efficient assessments of uncompared pairs, an initial weakly acyclical preference structure in A is transformed into an ordering of A in echelons. We call these nicely surveyable preference structures echelon orders. Theoretical properties of echelon orders are investigated, including a characterisation and a numerical representation.     

Inferring a linear ordering over a power set

An observer attempts to infer the unobserved ranking of two ideal objects, A and B, from observed rankings in which these objects are `accompanied' by `noise' components, C and D. In the first ranking, A is accompanied by C and B is accompanied by D, while in the second ranking, A is accompanied by D and B is accompanied by C. In both rankings, noisy-A is ranked above noisy-B. The observer infers that ideal-A is ranked above ideal-B. This commonly used inference rule is formalized for the case in which A,B,C,D are sets. Let X be a finite set and let be a linear ordering on 2^X. The following condition is imposed on . For every quadruple (A,B,C,D)Y, where Y is some domain in (2^X)⁴, if and , then . The implications and interpretation of this condition for various domains Y are discussed.     

Expected discounted utility

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann (Ann Math Stat 34(1):199–205, 1963, p. 201).

     

When an Event Makes a Difference

For (S, Σ) a measurable space, let and be convex, weak^* closed sets of probability measures on Σ. We show that if ∪ satisfies the Lyapunov property , then there exists a set A ∈ Σ such that min_μ1∈ μ₁(A) > max_μ2 ∈ (A). We give applications to Maxmin Expected Utility (MEU) and to the core of a lower probability.     

A model to support the assessment of subjective probabilities

Conclusions The purpose of this article has been to provide a new method for the determination of subjective probabilities. The kind of support which is suggested for the assessment process has several interesting properties. Constructed in accordance with the Savage theory the procedure considers a preference structure on the space of the possible actions A or on a suitably chosen space A ^* as its basis. This structure will be attained by pairwise comparison of the elements a _i A (or a _i ^*A ^*). It may be incomplete and inconsistent. A linear programming approach will use this information to derive numerical probabilities which are in accordance with the stated preference judgments. Therefore, it avoids the difficulties which arise when the judge has to assign the numerical values to the possible states of nature. On the other hand, this method presents the advantage of taking into consideration all relevant a priori information because of its systematic judgment process. Another important element is given by the use of coefficients which express the judge's confidence in the stated comparisons. Further relevance is added to the procedure by the relatively small consumption of time and work to report the judgments even in complex situations, and by the renunciation of any methodological knowledge on the part of the judges.This study was financed by a research grant of the Deutsche Forschungsgemeinschaft (DFG).     

Rules of behavior and expected utility theory. Compatibility versus dependence

Conclusions We have seen that many decision rules which are intuitively and/or empirically supported and compatible with MEU, are compatible with it but not dependent on it.There are of course rules of behavior which are implied in MEU and also depend on it like this:If the hope of winning any of the prizes in a lottery motivates you to buy a ticket, and if you win half the amount of the highest prize, you should play double or nothing with your prize.Suppose you would prefer a one in a million chance of winning $2 million to a two in a million chance of winning $1 million, but your first choice is not available so you buy a ticket for $1 million. If you win, you should play 50–50 double or nothing with your prize. Generalize: put x in the place of $1 million and p in the place of 1/1 000000, and test yourself against this principle (Pf = preferred to): (p, 2x)Pf(2p, x) (0.5, 2x)Pf(1, x).See Friedmann and Savage (1968).In the real world lotteries are multiprize, i.e., composite games of elements like these. The same applies: If you would not have preferred the highest prize exchanged for a higher probability of some lower prize, then winning a lower prize would put you in the market for some simple bet like above. If you stand up to this test, you are a unique person because, as we know, such bets are not made.While in the process of finishing the final draft, I got hold of (Samuelson, 1983). He expresses grave doubts as to what he calls the dogma of Expected Utility maximizing. In a somewhat apologetic way, he preserves some formulations deriving behavior from EUM because, as he states in a general way, they do not depend on that particular dogma. More specifically: many models incompatible with EUM imply risk aversion, which would result also from maximizing the expectation of a concave utility function,In view of the authoritarian disposition of some of the strongest defenders of EUM and of Samuelson's (well deserved) authority and his leading role in the school of EUM theory, his open expression of doubt may well mark the beginning of the last chapter in the history of the rise and fall of the most powerful school that has so far been active in 20th century decision theory.     

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.     

A model of consumption-dependent temptation

The paper characterizes axiomatically a class of temptation-driven preferences. The key (but not the only) novelty of the paper is the idea that the alternative which tempts when an item x is consumed may not be the same as the alternative which tempts when another item y is consumed. For any single item to be ultimately consumed, the other items can be ranked by how much they tempt. An individual contemplates, as an alternative consumption, only the item that tempts most. The utility of a menu is then equal to the utility of the item that is consumed less a (psychological) cost of resisting temptation; this cost depends on both: the item to be consumed and the item that tempts. Unlike most of the existing literature, the axioms are imposed on deterministic menus, not on menus of lotteries.     

Preferences and decision errors in the winner’s curse

The problem of asymmetric information causes a winner’s curse in many environments. Given many unsuccessful attempts to eliminate it, we hypothesize that some people ‘prefer’ the lotteries underlying the winner’s curse. Study 1 shows that after removing the hypothesized cause of error, asymmetric information, half the subjects still prefer winner’s curse lotteries, implying past efforts to de-bias the winner’s curse may have been more successful than previously recognized since subjects prefer these lotteries. Study 2 shows risk-seeking preferences only partially explain lottery preferences, while non-monetary sources of utility may explain the rest. Study 2 suggests lottery preferences are not independent of context, and offers methods to reduce the winner’s curse.

The Becker-DeGroot-Marschak mechanism and nonexpected utility: A testable approach

The Becker-DeGroot-Marschak mechanism is widely used to elicit decision makers' selling prices of lotteries. This mechanism leads, however, to the preference reversal phenomenon, which seemed to indicate nontransitive preferences. To solve this puzzle, Karni and Safra (1987) introduced a new interpretation of this mechanism based on two-stage lotteries without the independence axiom. In this article, we suggest a set of empirically testable hypotheses based on their interpretation of the mechanism. One of these tests can be used to find the utility and the probability transformation functions of an anticipated utility maximizer.This article consists of an earlier paper with a similar title (University of Toronto WP #8809) and the paper Elicitation of Certainty Equivalents and the Becker-DeGroot-Marschak Mechanism.     

Comparative probability for conditional events: A new look through coherence

We study, from the standpoint of coherence, comparative probabilities on an arbitrary familyE of conditional events. Given a binary relation ·, coherence conditions on · are related to de Finetti's coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of ·. In this context, the numerical reference frame is that of de Finetti's coherent subjective conditional probability, which is not introduced (as in Kolmogoroff's approach) through a ratio between probability measures.Another relevant feature of our approach is that the family & need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.     

A strong (Ross) characterization of multivariate risk aversion

Using the addition of uncorrelated noise as a natural definition of increasing risk for multivariate lotteries, I interpret risk aversion as the willingness to pay a (possibly random) vector premium in exchange for a reduction in multivariate risk. If no restriction is placed on the sign of any co-ordinate of the vector premium then (as was the case in Kihlstrom and Mirman's (1974) analysis) only pairs of expected utility maximizers with thesame ordinal preferences for outcomes can be ranked in terms of their aversion to increasing risk. However, if we restrict the premium to be a non-negative random variable then comparisons of aversion to increasing risk may be possible between expected utility maximizers withdistinct ordinal preferences for outcomes. The relationship between their utility functions is precisely the multi-dimensional analog of Ross's (1981)global condition forstrongly more risk averse.     

Third-generation prospect theory

We present a new theory of decision under uncertainty: third-generation prospect theory (PT³). This retains the predictive power of previous versions of prospect theory, but extends that theory by allowing reference points to be uncertain while decision weights are specified in a rank-dependent way. We show that PT³ preferences respect a state-conditional form of stochastic dominance. The theory predicts the observed tendency for willingness-to-accept valuations of lotteries to be greater than willingness-to-pay valuations. When PT³ is made operational by using simple functional forms with parameter values derived from existing experimental evidence, it predicts observed patterns of the preference reversal phenomenon.

Changes in preference

A basic framework for studies of changes in preference is introduced, and four types of changes in preference are identified.Revision by a sentence such as A is better thanB means that a preference forA overB is acquired. The result ofcontraction by A is better thanB is that the subject no longer holdsA to be better thanB. Inaddition andsubtraction, an alternative is added to, or subtracted from, respectively, the set of alternatives that are under consideration. Formal models of these four types of change are introduced, and they are shown to satisfy plausible postulates for rational changes in preferences.     

Stochastic Choice and Consistency in Decision Making Under Risk: An Experimental Study

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time.     

Rush and Procrastination Under Hyperbolic Discounting and Interdependent Activities

We analyze the decision of individuals with time-inconsistent preferences to invest in projects yielding either current costs and future benefits or current benefits and future costs. We show that competition between agents for the same project mitigates the tendency to procrastinate on the first type of activities (i.e. to undertake them too late) and to rush on the second one (i.e. to undertake them too early). Competition can therefore increase the expected welfare of each individual. On the contrary, complementarity of projects exacerbates the tendency to rush and to procrastinate and therefore it can decrease the expected welfare of each individual.     

Simulation methodology

Operational researchers, management scientists, and industrial engineers have been asked by Russell Ackoff to become systems scientists, yet he stated that Systems Science is not a science. (TIMS Interfaces, 2 (4), 41). A. C. Fabergé (Science 184, 1330) notes that the original intent of operational researchers was that they be scientists, trained to observe. Hugh J. Miser (Operations Research 22, 903), views operations research as a science, noting that its progress indeed is of a cyclic nature.The present paper delineates explicitly the attributes of simulation methodology. Simulation is shown to be both an art and a science; its methodology, properly used, is founded both on confirmed (validated) observation and scrutinised (verified) art work.The paper delineates the existing procedures by which computer-directed models can be cyclically scrutinised and confirmed and therefore deemed credible. The complexities of the phenomena observed by social scientists are amenable to human understanding by properly applied simulation; the methodology of the scientist of systems (the systemic scientist).

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Résumé Russell Ackoff propose à ceux qui s'occupent de recherches opérationnelle, industrielle, et de gestion, d'agir en systems scientists, et pourtant il affirme que systems science n'est pas une science (TIMS Interfaces 2 (4), 41). A. C. Fabergé (Science 184, 1330) remarque, qu'à l'origine, le but de ceux qui s'occupaient de recherche opérationnelle était d'agir en hommes de science instruits à observer. Hugh J. Miser (Operational Research 22, 903) considère la recherche opérationnelle comme science, notant que ses progrès sont en effet de nature cyclique.La présente étude délimite explicitement les attributs de la méthode de la simulation. Il est démontré que la simulation est à la fois un art et une science; sa méthode, lorsqu'utilisée correctement, repose sur l'observation validée et le modèle vérifié.L'étude délimite les moyens actuels dont nous disposons pour vérifier et valider cycliquement les modèles bâtis à l'aide d'ordinateurs, établissant ainsi leur crédibilité. La nature complexe des phénomènes étudiés par les sciences sociales peut être comprise à l'aide de la simulation: la méthode dont se servent les hommes de science qui étudient les systèmes (les scientistes systémiques).