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.     

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.     

Payoff Kinks in Preferences over Lotteries

This paper identifies two distinct types of payoff kinks that can be exhibited by preference functions over monetary lotteries—locally separable vs. locally nonseparable—and illustrates their relationship to the payoff and probability derivatives of such functions. Expected utility and Fréchet differentiable preference functions are found to be incapable of exhibiting locally nonseparable payoff kinks; rank-dependent preference functions are incapable of avoiding them.     

The impact of risk attitude,uncertainty and disequilibria on optimal production and inventory

Summary The empirical facts seem to indicate that in real economies the effect of uncertainty tends to decrease production. The limitations of empirical investigations presented should be stressed: they were performed mainly on an aggregate level, they mainly refer to Austrian manufacturing, they rely heavily on questionnaires. Above all empirical investigation will never be able to decide normative questions or to explain the behavior in the general equilibrium. Nevertheless in the short run, given all the rigidities and disequilibria which exist, uncertainty tends to lower optimal production even in absence of risk aversion. Risk aversion becomes important for large, for once-for-all decisions, but it is not the only channel through which uncertainty changes decisions.Technological concavity created by concave marginal revenues or by convex marginal costs, marginal costs of uncertainty in disequilibria model or asymmetric costs of revisions of the preliminary decision are able to bias the decision downward in a real world economy without invokingPaper presented to the 2nd Conference on the Foundations of Risk and Utility (FUR), Venezia, 1984.     

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.     

Group decision making under uncertainty a note on the aggregation of “ordinal probabilities”

This paper is a first attempt to study the problem of aggregation of individual ordinal probabilistic beliefs in an Arrowian framework. We exhibit some properties an aggregation rule must fulfil; in particular we prove the existence of a quasi-dictator.     

Friedman's ‘instrumentalism’ and constructive empiricism in economics

This reassessment of the long debate about Friedman's thesis on the pointlessness of testing assumptions in economics shows that Friedman's three famous examples, on which a large part of the credit given to this thesis is based, far from substantiating it, can be used to establish radically opposite conclusions. Furthermore, it is shown that this so-called instrumentalist thesis, when applied by Friedman to economics, is of a quite different nature and raises much more serious problems than the standard instrumentalist thesis devised by some methodologists of physics. To disentangle these ambiguities concerning realism and instrumentalism applied to physics or to economics, this paper refers to Van Fraassen's constructive empiricism, which is helpful in reformulating, in a more satisfactory way, the essentials of Friedman's considerations about empiricism and anti-realism.     

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.     

Minimax,information and ultrapessimism

Discussing the foundations of the minimax principle, Savage (1954) argued that it is utterly untenable for statistics because it is ultrapessimistic when applied to negative income, but claimed that such objection is not relevant when the principle is applied to regret. In this paper I rebut the latter claim. I first present an example where ultrapessimism, as Savage understood it, applies to minimax regret but not to minimax negative income. Then, for a sequential decision problems with two terminal acts and a finite number of states of nature, I give necessary and sufficient conditions for a decision rule to be ultrapessimistic, and show that for every payoff table with at least three states, be it in regret form or not, there exist an experiment such that the minimax rule is ultrapessimistic. I conclude with some more general remarks on information and the value of experimentation for a minimax agent.     

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.     

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.     

Substitution of indifferent options at choice nodes and admissibility: a reply to Rabinowicz

Tiebreak rules are necessary for revealing indifference in non- sequential decisions. I focus on a preference relation that satisfies Ordering and fails Independence in the following way. Lotteries a and b are indifferent but the compound lottery 0.5f, 0.5b is strictly preferred to the compound lottery 0.5f, 0.5a. Using tiebreak rules the following is shown here: In sequential decisions when backward induction is applied, a preference like the one just described must alter the preference relation between a and b at certain choice nodes, i.e., indifference between a and b is not stable. Using this result, I answer a question posed by Rabinowicz (1997) concerning admissibility in sequential decisions when indifferent options are substituted at choice nodes.     

A note on Abraham Lincoln in probabilityland

Abraham Lincoln's dictum that you may fool all of the people some of the time; and some of the people all of the time; but you can't fool all of the people all of the time, is interpreted in terms of a simple binomial model, and potential ambiguities in Lincoln's assertion are clarified.     

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).

     

Decomposing Hindsight Bias

Traditional accounts of hindsight bias inadequately distinguish primary hindsight bias from both secondary and tertiary hindsight bias. A subject exhibits primary bias when she assigns a higher ex ante probability estimate to actual outcomes, secondary bias when she believes that she herself would have made the same estimate of the prior probability of an event before receiving outcome information as she made after receiving it, and tertiary bias when she believes that third parties lacking outcome information were unreasonable if they did not make the same prior probability judgments that subjects now possessing such information make.In our experiments, we find that when people can readily calculate the actual ex ante probability of an outcome, they don't reassess that probability when told what outcomes actually occurred. They reassess only in situations in which they are unable to assess prior probabilities or when given information that the outcome was not simply a result of sampling or chance but the result of an imperceptible feature of the initial situation. Observed primary bias may therefore often be rational.     

Metatickles and the dynamics of deliberation

The traditional or orthodox decision rule of maximizing conditional expected utility has recently come under attack by critics who advance alternative causal decision theories. The traditional theory has, however, been defended. And these defenses have in turn been criticized. Here, I examine two objections to such defenses and advance a theory about the dynamics of deliberation (a diachronic theory about the process of deliberation) within the framework of which both objections to the defenses of the traditional theory fail.     

Majority vote of even and odd experts in a polychotomous choice situation

The performance of majority vote by an odd number of voters in a dichotomous situation is a much-studied subject. In this paper, we study the performance of group consensus in a situation of polychotomous choice. The differences in majority vote behaviour between even and odd numbers of expert panellists are examined, and the effects of adding new members are derived. Unlike the dichotomous model, optimality in the present context may not be uniquely defined, so the probabilities of the consensus being correct or erroneous are both considered as the number of experts is increased.     

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.     

Comparative Risk Sensitivity with Reference-Dependent Preferences

Experimental evidence suggests that individuals are risk averse over gains and risk seeking over losses (i.e., they have S-shaped utility functions in an expected utility setting) and that they are loss averse. Furthermore, the evidence leads to a single definition of S-shaped utility, but it has led to several alternative specifications of loss aversion. This paper characterizes the relations more S-shaped than and more loss averse than for a utility function, and in so doing arrives at a new definition of loss aversion based on average instead of marginal utility.     

Consensus By Identifying Extremists

Given a finite state space and common priors, common knowledge of the identity of an agent with the minimal (or maximal) expectation of a random variable implies consensus, i.e., common knowledge of common expectations. This extremist statistic induces consensus when repeatedly announced, and yet, with n agents, requires at most log₂ n bits to broadcast.