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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Practical reason or metapreferences? an undogmatic defense of kantian morality

This article presents the thesis that a critique of decisions is not necessarily (except in the trivial sense) a critique of preferences. This thesis runs contrary to the fundamental assumption in economic theory that a critique of decisions will always simultaneously be a critique of (subjective) preferences, since decision behavior is after all a manifestation of preferences. If this thesis is right, then the paradigm of so-called instrumental rationality is in serious trouble, not for external reasons but because of imminent inconsistencies. The thesis is developed in five parts: I. A preliminary remark to the economic theory of rationality in general. II. The cooperation problem as a challenge to the economic theory of rationality. III. An account of the most interesting attempt to save the theory. IV. A critique of that attempt. V. And the conclusion: practical reason is concerned with actions and not with preferences.     

Ordinal utility models of decision making under uncertainty

This paper studies two models of rational behavior under uncertainty whose predictions are invariant under ordinal transformations of utility. The quantile utility model assumes that the agent maximizes some quantile of the distribution of utility. The utility mass model assumes maximization of the probability of obtaining an outcome whose utility is higher than some fixed critical value. Both models satisfy weak stochastic dominance. Lexicographic refinements satisfy strong dominance.The study of these utility models suggests a significant generalization of traditional ideas of riskiness and risk preference. We define one action to be riskier than another if the utility distribution of the latter crosses that of the former from below. The single crossing property is equivalent to a minmax spread of a random variable. With relative risk defined by the single crossing criterion, the risk preference of a quantile utility maximizer increases with the utility distribution quantile that he maximizes. The risk preference of a utility mass maximizer increases with his critical utility value.     

On Seidenfeld‘s Criticism of Sophisticated Violations of the Independence Axiom 1

An agent who violates independence can avoid dynamic inconsistency in sequential choice if he is sophisticated enough to make use of backward induction in planning. However, Seidenfeld has demonstrated that such a sophisticated agent with dependent preferences is bound to violate the principle of dynamic substitution, according to which admissibility of a plan is preserved under substitution of indifferent options at various choice nodes in the decision tree. Since Seidenfeld considers dynamic substitution to be a coherence condition on dynamic choice, he concludes that sophistication cannot save a violator of independence from incoherence. In response to McClennens objection that relying on dynamic substitution when independence is at stake must be question-begging, Seidenfeld undertakes to prove that dynamic substitution follows from the principle of backward induction alone, provided we assume that the agents admissible choices from different sets of feasible plans are all based on a fixed underlying preference ordering of plans. This paper shows that Seidenfelds proof fails: depending on the interpretation, it is either invalid or based on an unacceptable assumption.     

Rational preference: Decision theory as a theory of practical rationality

In general, the technical apparatus of decision theory is well developed. It has loads of theorems, and they can be proved from axioms. Many of the theorems are interesting, and useful both from a philosophical and a practical perspective. But decision theory does not have a well agreed upon interpretation. Its technical terms, in particular, utility and preference do not have a single clear and uncontroversial meaning.How to interpret these terms depends, of course, on what purposes in pursuit of which one wants to put decision theory to use. One might want to use it as a model of economic decision-making, in order to predict the behavior of corporations or of the stock market. In that case, it might be useful to interpret the technical term utility as meaning money profit. Decision theory would then be an empirical theory. I want to look into the question of what utility could mean, if we want decision theory to function as a theory of practical rationality. I want to know whether it makes good sense to think of practical rationality as fully or even partly accounted for by decision theory. I shall lay my cards on the table: I hope it does make good sense to think of it that way. For, I think, if Humeans are right about practical rationality, then decision theory must play a very large part in their account. And I think Humeanism has very strong attractions.     

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.     

Fairness,Public Good,and Emotional Aspects of Punishment Behavior

We report an experiment on two treatments of an ultimatum minigame. In one treatment, responders reactions are hidden to proposers. We observe high rejection rates reflecting responders intrinsic resistance to unfairness. In the second treatment, proposers are informed, allowing for dynamic effects over eight rounds of play. The higher rejection rates can be attributed to responders provision of a public good: Punishment creates a group reputation for being tough and effectively educate proposers. Since rejection rates with informed proposers drop to the level of the treatment with non-informed proposers, the hypothesis of responders enjoyment of overt punishment is not supported.     

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 systems concept of society: Beyond individualism and holism

Three rival views of the nature of society are sketched: individualism, holism, and systemism. The ontological and methodological components of these doctrines are formulated and analyzed. Individualism is found wanting for making no room for social relations or emergent properties; holism, for refusing to analyze both of them and for losing sight of the individual.A systems view is then sketched, and it is essentially this: A society is a system of interrelated individuals sharing an environment. This commonsensical idea is formalized as follows: A society is representable as an ordered triple Composition of , Environment of , Structure of , where the structure of is the collection of relations (in particular connections) among components of . Included in the structure of any are the relations of work and of managing which are regarded as typical of human society in contrast to animal societies.Other concepts formalized in the paper are those of subsystem (in particular social subsystem), resultant property, and emergent or gestalt property. The notion of subsystem is used to build the notion of an F-sector of a society, defined as the set of all social subsystems performing a certain function F (e.g. the set of all schools). In turn, an F-institution is defined as the family of all F-sectors. Being abstractions, institutions should not be attributed a life and a mind of their own. But, since an institution is analyzable in terms of concrete totalities (namely social subsystems), it does not comply with the individualist requirement either.It is also shown that the systems view is inherent in any mathematical model in social science, since any such schema is essentially a set of individuals endowed with a certain structure. And it is stressed that the systems view combines the desirable features of both individualism and holism.     

Axiomatization of a class of share functions for n-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity.In this paper we generalize the axiom of additivity by introducing a positive real valued function on the class of cooperative games in characteristic function form. The so-called axiom of -additivity generalizes the classical axiom of additivity by putting the weight (v) on the value of the gamev . We show that any additive function determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and -additivity on the subclass of games on which is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.     

On Harrod's first refining principle

Harrod introduced a refinement to crude Utilitarianism with the aim of reconciling it with common sense ethics. It is shown (a) that this refinement (later known as Rule Utilitarianism) does not maximise utility (b) the principle which truly maximizes utility, marginal private benefit equals marginal social cost, requires that a number of forbidden acts like lying be performed. Hence Harrod's claim that his refined Utilitarianism is the foundation of moral institutions cannot be sustained. Some more modern forms of Utilitarianism are reinterpreted in this paper as utility maximizing decision rules. While they produce more utility than Harrod's rule, they require breaking the moral rules some of the time, just like the marginal rule mentioned above. However, Harrod's rule is useful in warning the members of a group, considered as a single moral agent, of the externalities that lie beyond the immediate consequences of the collective action.     

Inductive-nomological explanations and psychological laws

This paper explores some major conceptions of non-deductive explanations, with special reference to the ability of these conceptions to account for some important types of psychological explanations. The central focus is on a kind of tendency generalization and the nature of the explanations that are made possible by generalizations of this sort. It is argued that although these generalizations cannot plausibly be cast in the Hempelian form P(G, F) = r, they are lawlike, and the explanations they ground can have a degree of explanatory power commensurate with exhibiting the phenomenon to be explained as nomically expectable.     

The Optimality of the Expert and Majority Rules Under Exponentially Distributed Competence

We study the uncertain dichotomous choice model. In this model a set of decision makers is required to select one of two alternatives, say support or reject a certain proposal. Applications of this model are relevant to many areas, such as political science, economics, business and management. The purpose of this paper is to estimate and compare the probabilities that different decision rules may be optimal. We consider the expert rule, the majority rule and a few in-between rules. The information on the decisional skills is incomplete, and these skills arise from an exponential distribution. It turns out that the probability that the expert rule is optimal far exceeds the probability that the majority rule is optimal, especially as the number of the decision makers becomes large.