A μ-σ-Risk Aversion Paradox and Wealth Dependent Utility

We report a surprising property of --preferences: the assumption of nonincreasing relative risk aversion implies the optimal portfolio being riskless. We discuss a solution of that paradox using wealth dependent utility functions in detail. Using the revealed preference theory we show that (general, i.e. not necessary -) wealth dependent utility functions can be characterized by Wald's axiom.     

A Comparison of Some Distance-Based Choice Rules in Ranking Environments

We discuss the relationships between positional rules (such as plurality and approval voting as well as the Borda count), Dodgsons, Kemenys and Litvaks methods of reaching consensus. The discrepancies between methods are seen as results of different intuitive conceptions of consensus goal states and ways of measuring distances therefrom. Saaris geometric methodology is resorted to in the analysis of the consensus reaching methods.     

Comparative accuracy of value solutions in non-sidepayment games with empty core

This article reports a test of the predictive accuracy of solution concepts in cooperative non-sidepayment n-person games with empty core. Six solutions were tested. Three of these were value solutions (i.e., -transfer value, -transfer nucleolus, and -transfer disruption value) and three were equilibrium solutions (deterrence set, stable set, and imputation set). The test was based on a laboratory experiment utilizing 5-person, 2-choice normal form games with empty core; other related data sets were also analyzed. Goodness-of-fit results based on discrepancy scores show that the three value solutions are about equally accurate in predicting outcomes, and that all three are substantially more accurate than the other solutions tested.     

A paradox in decision theory and some experimental results: The relative nature of decisions

A simple two-choice single outcome valued decision under risk is presented which should show up the limitations in the classical approach of von Neumann, its extensions and its alternatives. An empirical testing of this hypothesis strongly supports this criticism. A rationale for explaining the apparent irrational decision is put forward and the case is made for understanding the relative nature of decision choices especially when multi-criteria are involved.     

Expected utility, μ-σ preferences,and linear distribution classes: A further result

This article is an extension of Meyer and Sinn's results on the representation of arbitrary von Neumann-Morgenstern functions in - space when the probability distributions to be compared belong to a linear distribution class. It shows that, when absolute risk aversion decreases, stays constant, or increases not too fast, an increase in , given , increases the indifference curve slope: increased riskiness increases the required marginal compensation for risk when risk is measured by the standard deviation of wealth or income.I gratefully acknowledge useful comments by Wolfgang Buchholz, Kai Konrad, Jack Meyer, Hans Schneeweiß, Dominique Thon, and Bengt Arne Wickström.     

Rational choice and economic behavior

This paper discusses several concepts that can be used to provide a foundation for a unified, theory of rational, economic behavior. First, decision-making is defined to be a process that takes place with reference to both subjective and objective time, that distinguishes between plans and actions, between information and states and that explicitly incorporates the collection and processing of information. This conception of decision making is then related to several important aspects of behavioral economics, the dependence of values on experience, the use of behavioral rules, the occurrence of multiple goals and environmental feedback.Our conclusions are (1) the non-transitivity of observed or revealed preferences is a characteristic of learning and hence is to be expected of rational decision-makers; (2) the learning of values through experience suggests the sensibleness of short time horizons and the making of choices according to flexible utility; (3) certain rules of thumb used to allow for risk are closely related to principles of Safety-First and can also be based directly on the hypothesis that the feeling of risk (the probability of disaster) is identified with extreme departures from recently executed decisions. (4) The maximization of a hierarchy of goals, or of a lexicographical utility function, is closely related to the search for feasibility and the practice of satisficing. (5) When the dim perception of environmental feedback and the effect of learning on values are acknowledged the intertemporal optimality of planned decision trajectories is seen to be a characteristic of subjective not objective time. This explains why decision making is so often best characterized by rolling plans. In short, we find that economic man - like any other - is an existential being whose plans are based on hopes and fears and whose every act involves a leap of faith.This paper is based on a talk presented at the Conference, New Beginnings in Economics, Akron, Ohio, March 15, 1969. Work on this paper was supported by a grant from the National Science Foundation.     

Correlated strategies as Institutions

Two institutions that are often implicit or overlooked in noncooperative games are the assumption of Nash behavior to solve a game, and the ability to correlate strategies. We consider two behavioral paradoxes; one in which maximin behavior rules out all Nash equilibria (Chicken), and another in which minimax supergame behavior leads to an inefficient outcome in comparison to the unique stage game equilibrium (asymmetric Deadlock). Nash outcomes are achieved in both paradoxes by allowing for correlated strategies, even when individual behavior remains minimax or maximin. However, the interpretation of correlation as a public institution differs for each case.     

Five legitimate definitions of correlated equilibrium in games with incomplete information

Aumann's (1987) theorem shows that correlated equilibrium is an expression of Bayesian rationality. We extend this result to games with incomplete information.First, we rely on Harsanyi's (1967) model and represent the underlying multiperson decision problem as a fixed game with imperfect information. We survey four definitions of correlated equilibrium which have appeared in the literature. We show that these definitions are not equivalent to each other. We prove that one of them fits Aumann's framework; the agents normal form correlated equilibrium is an expression of Bayesian rationality in games with incomplete information.We also follow a universal Bayesian approach based on Mertens and Zamir's (1985) construction of the universal beliefs space. Hierarchies of beliefs over independent variables (states of nature) and dependent variables (actions) are then constructed simultaneously. We establish that the universal set of Bayesian solutions satisfies another extension of Aumann's theorem.We get the following corollary: once the types of the players are not fixed by the model, the various definitions of correlated equilibrium previously considered are equivalent.     

On Playing Fair: Professor Binmore on Game Theory and the Social Contract

This paper critically reviews Ken Binmores non- utilitarian and game theoretic solution to the Arrow problem. Binmores solution belongs to the same family as Rawls maximin criterion and requires the use of Nash bargaining theory, empathetic preferences, and results in evolutionary game theory. Harsanyi has earlier presented a solution that relies on utilitarianism, which requires some exogenous valuation criterion and is therefore incompatible with liberalism. Binmores rigorous demonstration of the maximin principle for the first time presents a real alternative to a utilitarian solution.     

Disparate goods and Rawls' difference principle: A social choice theoretic treatment

Rawls' Difference Principle asserts that a basic economic structure is just if it makes the worst off people as well off as is feasible. How well off someone is is to be measured by an index of primary social goods. It is this index that gives content to the principle, and Rawls gives no adequate directions for constructing it. In this essay a version of the difference principle is proposed that fits much of what Rawls says, but that makes use of no index. Instead of invoking an index of primary social goods, the principle formulated here invokes a partial ordering of prospects for opportunities.     

A strategic foundation for the cooperator's advantage

Orbell and Dawes develop a non-game theoretic heuristic that yields a cooperator's advantage by allowing players to project their own cooperate-defect choices onto potential partners (1991, p. 515). With appropriate parameter values their heuristic yields a cooperative environment, but the cooperation depends, simply, on optimism about others' behavior (1991, p. 526). In earlier work, Dawes (1989) established a statistical foundation for such optimism. In this paper, I adapt some of the concerns of Dawes (1989) and develop a game theoretic model based on a modification of the Harsanyi structure of games with incomplete information (1967–1968). I show that the commonly made conjecture that strategic play is incompatible with cooperation and the cooperator's advantage is false.     

Cost-benefit versus expected utility acceptance rules

A rule for the acceptance of scientific hypotheses called the principle of cost-benefit dominance is shown to be more effective and efficient than the well-known principle of the maximization of expected (epistemic) utility. Harvey's defense of his theory of the circulation of blood in animals is examined as a historical paradigm case of a successful defense of a scientific hypothesis and as an implicit application of the cost-benefit dominance rule advocated here. Finally, various concepts of dominance are considered by means of which the effectiveness of our rule may be increased.The number of friends who have kindly given me suggestions and encouragement is almost embarrassingly large, but I would like to express my gratitude to Myles Brand, Cliff Hooker, David Hull, Scott Kleiner, Hugh Lehman, Werner Leinfellner, Andrew McLaughlin and Tom W. Settle.     

Testing Descriptive Utility Theories: Violations of Stochastic Dominance and Cumulative Independence

Choices between gambles show systematic violations of stochastic dominance. For example, most people choose ($6, .05; $91, .03; $99, .92) over ($6, .02; $8, .03; $99, .95), violating dominance. Choices also violate two cumulative independence conditions: (1) If S = (z, r; x, p; y, q) R = (z, r; x, p; y, q) then S = (x, r; y, p + q) R = (x, r + p; y, q). (2) If S = (x, p; y, q; z, r) R = (x, p; y, q; z, r) then S = (x, p + q; y, r) R = (x, p; y, q + r), where 0 < z < x < x < y < y < y < z.Violations contradict any utility theory satisfying transivity, outcome monotonicity, coalescing, and comonotonic independence. Because rank-and sign-dependent utility theories, including cumulative prospect theory (CPT), satisfy these properties, they cannot explain these results.However, the configural weight model of Birnbaum and McIntosh (1996) predicted the observed violations of stochastic dominance, cumulative independence, and branch independence. This model assumes the utility of a gamble is a weighted average of outcomes\' utilities, where each configural weight is a function of the rank order of the outcome\'s value among distinct values and that outcome\'s probability. The configural weight, TAX model with the same number of parameters as CPT fit the data of most individuals better than the model of CPT.     

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.     

On the foundations of mean-variance analyses

Let (, ) and (, ) be mean-standard deviation pairs of two probability distributions on the real line. Mean-variance analyses presume that the preferred distribution depends solely on these pairs, with primary preference given to larger mean and smaller variance. This presumption, in conjunction with the assumption that one distribution is better than a second distribution if the mass of the first is completely to the right of the mass of the second, implies that (, ) is preferred to (, ) if and only if either > or ( = and < ), provided that the set of distributions is sufficiently rich. The latter provision fails if the outcomes of all distributions lie in a finite interval, but then it is still possible to arrive at more liberal dominance conclusions between (, ) and (, ).This research was supported by the Office of Naval Research.     

Why I am not an objective Bayesian; some reflections prompted by Rosenkrantz

Summary The objective Bayesian program has as its fundamental tenet (in addition to the three Bayesian postulates) the requirement that, from a given knowledge base a particular probability function is uniquely appropriate. This amounts to fixing initial probabilities, based on relatively little information, because Bayes' theorem (conditionalization) then determines the posterior probabilities when the belief state is altered by enlarging the knowledge base. Moreover, in order to reconstruct orthodox statistical procedures within a Bayesian framework, only privileged ignorance probability functions will work.To serve all these ends objective Bayesianism seeks additional principles for specifying ignorance and partial information probabilities. H. Jeffreys' method of invariance (or Jaynes' modification thereof) is used to solve the former problem, and E. Jaynes' rule of maximizing entropy (subject to invariance for continuous distributions) has recently been thought to solve the latter. I have argued that neither policy is acceptable to a Bayesian since each is inconsistent with conditionalization. Invariance fails to give a consistent representation to the state of ignorance professed. The difficulties here parallel familiar weaknesses in the old Laplacean principle of insufficient reason. Maximizing entropy is unsatisfactory because the partial information it works with fails to capture the effect of uncertainty about related nuisance factors. The result is a probability function that represents a state richer in empirical content than the belief state targeted for representation. Alternatively, by conditionalizing on information about a nuisance parameter one may move from a distribution of lower to higher entropy, despite the obvious increase in information available.Each of these two complaints appear to me to be symptoms of the program's inability to formulate rules for picking privileged probability distributions that serve to represent ignorance or near ignorance. Certainly the methods advocated by Jeffreys, Jaynes and Rosenkrantz are mathematically convenient idealizations wherein specified distributions are elevated to the roles of ignorance and partial information distributions. But the cost that goes with the idealization is a violation of conditionalization, and if that is the ante that we must put up to back objective Bayesianism then I propose we look for a different candidate to earn our support.³¹     

How can an expert system help in choosing the optimal decision?

We introduce a rationality principle for a preference relation on an arbitrary set of lotteries. Such a principle is a necessary and sufficient condition for the existence of an expected utility agreeing with . The same principle also guarantees a rational extension of the preference relation to any larger set of lotteries. When the extended relation is unique with respect to the alternatives under consideration, the decision maker does not need a numerical evaluation in order to make a choice. Such a rationality condition needs little information in order to be applied, and its verification amounts to solving a linear system.The present research is supported by the Research Contract of CNR (Research National Council) 1989 and 1990 Decision Models under uncertainty and risk, for expert systems with incomplete and revisable information.     

Separating marginal utility and probabilistic risk aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a probabilistic risk attitude towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaari's axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (aversion) of preferences with respect to probability mixtures, and convexity (pessimism) of the probability transformation — are equivalent.     

Flexibility,endogenous risk,and the protection premium

We introduce two types of protection premia. The unconstrained protection premium, ^u, is the individual's willingness to pay for certain protection efficiency given flexibility to adjust optimally the investment in protection. The constrained protection premium, ^c, measures willingness to pay for certain protection efficiency given no flexibility to adjust the investment in protection. ^u depends on tastes and wealth as well as protection technology whereas ^c depends only on technology. We show that ^c cannot exceed ^u and develop necessary conditions for ^c=^u. Optimal protection for an individual with decision flexibility may be larger or smaller than that desired under no flexibility.Journal Paper No. J-15504 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 3048.     

Predictive superiority of the beta-characteristic function in cooperative non-sidepayment N-person games

This article reports an experimental study of decision-making outcomes in cooperative non-sidepayment games. The objective of this test was to determine which characteristic function, V (S) or V (S), provides the most accurate basis for payoff predictions from solution concepts. The experiment tested three solution concepts (core, stable set, imputation set) in the context of 5-person, 2-strategy non-sidepayment games. Predictions from each of the three solution concepts were computed on the basis of both V (S) and V (S), making a total of six predictive theories under test. Consistent with earlier studies (Michener et al., 1984a; Michener et al., 1985), two basic findings emerged. First, the data show that for each of the solutions tested, the prediction from any solution concept computed from V(S) was more accurate than the prediction from the same solution concept computed from V (S). Second, the -core was the most accurate of the six theories tested. Overall, these results support the view that V (S) is superior to V (S) as a basis for payoff predictions in cooperative non-sidepayment games.