Decision-making under ignorance with implications for social choice

A new investigation is launched into the problem of decision-making in the face of complete ignorance, and linked to the problem of social choice. In the first section the author introduces a set of properties which might characterize a criterion for decision-making under complete ignorance. Two of these properties are novel: independence of non-discriminating states, and weak pessimism. The second section provides a new characterization of the so-called principle of insufficient reason. In the third part, lexicographic maximin and maximax criteria are characterized. Finally, the author's results are linked to the problem of social choice.     

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.     

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.     

Experiments with cooperative 2 × 2 games

Nash's solution of a two-person cooperative game prescribes a coordinated mixed strategy solution involving Pareto-optimal outcomes of the game. Testing this normative solution experimentally presents problems in as much as rather detailed explanations must be given to the subjects of the meaning of threat strategy, strategy mixture, expected payoff, etc. To the extent that it is desired to test the solution using naive subjects, the problem arises of imparting to them a minimal level of understanding about the issue involved in the game without actually suggesting the solution.Experiments were performed to test the properties of the solution of a cooperative two-person game as these are embodied in three of Nash's four axioms: Symmetry, Pareto-optimality, and Invariance with respect to positive linear transformations. Of these, the last was definitely discorroborated, suggesting that interpersonal comparison of utilities plays an important part in the negotiations.Some evidence was also found for a conjecture generated by previous experiments, namely that an externally imposed threat (penalty for non-cooperation) tends to bring the players closer together than the threats generated by the subjects themselves in the process of negotiation.     

Mean variance preferences and the heat equation

Chipman (1979) proves that for an expected utility maximizer choosing from a domain of normal distributions with mean and variance ² the induced preference functionV(, ) satisfies a differential equation known as the heat equation. The purpose of this note is to provide a generalization and simple proof of this result which does not depend on the normality assumption.     

Far-sighted equilibria in 2 × 2, non-cooperative,repeated games

Consider a two-person simultaneous-move game in strategic form. Suppose this game is played over and over at discrete points in time. Suppose, furthermore, that communication is not possible, but nevertheless we observe some regularity in the sequence of outcomes. The aim of this paper is to provide an explanation for the question why such regularity might persist for many (i.e., infinite) periods.Each player, when contemplating a deviation, considers a sequential-move game, roughly speaking of the following form: if I change my strategy this period, then in the next my opponent will take his strategy b and afterwards I can switch to my strategy a, but then I am worse off since at that outcome my opponent has no incentive to change anymore, whatever I do. Theoretically, however, there is no end to such reaction chains. In case that deviating by some player gives him less utility in the long run than before deviation, we say that the original regular sequence of outcomes is far-sighted stable for that player. It is a far-sighted equilibrium if it is far-sighted stable for both players.     

A deterministic event tree approach to uncertainty,randomness and probability in individual chance processes

Both Popper and Good have noted that a deterministic microscopic physical approach to probability requires subjective assumptions about the statistical distribution of initial conditions. However, they did not use such a fact for defining an a priori probability, but rather recurred to the standard observation of repetitive events. This observational probability may be hard to assess for real-life decision problems under uncertainty that very often are - strictly speaking - non-repetitive, one-time events. This may be a reason for the popularity of subjective probability in decision models. Unfortunately, such subjective probabilities often merely reflect attitudes towards risk, and not the underlying physical processes.In order to get as objective as possible a definition of probability for one-time events, this paper identifies the origin of randomness in individual chance processes. By focusing on the dynamics of the process, rather than on the (static) device, it is found that any process contains two components: observer-independent (= objective) and observer-dependent (= subjective). Randomness, if present, arises from the subjective definition of the rules of the game, and is not - as in Popper's propensity - a physical property of the chance device. In this way, the classical definition of probability is no longer a primitive notion based upon equally possible cases, but is derived from the underlying microscopic processes, plus a subjective, clearly identified, estimate of the branching ratios in an event tree. That is, equipossibility is not an intrinsic property of the system object/subject but is forced upon the system via the rules of the game/measurement.Also, the typically undefined concept of symmetry in games of chance is broken down into objective and subjective components. It is found that macroscopic symmetry may hold under microscopic asymmetry. A similar analysis of urn drawings shows no conceptual difference with other games of chance (contrary to Allais' opinion). Finally, the randomness in Lande's knife problem is not due to objective fortuity (as in Popper's view) but to the rules of the game (the theoretical difficulties arise from intermingling microscopic trajectories and macroscopic events).Dedicated to Professor Maurice Allais on the occasion of the Nobel Prize in Economics awarded December, 1988.     

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.     

On inference from inconsistent premisses

The main object of this paper is to provide the logical machinery needed for a viable basis for talking of the consequences, the content, or of equivalences between inconsistent sets of premisses.With reference to its maximal consistent subsets (m.c.s.), two kinds of consequences of a propositional set S are defined. A proposition P is a weak consequence (W-consequence) of S if it is a logical consequence of at least one m.c.s. of S, and P is an inevitable consequence (I-consequence) of S if it is a logical consequence of all the m.c.s. of S. The set of W-consequences of a set S it determines (up to logical equivalence) its m.c.s. (This enables us to define a normal form for every set such that any two sets having the same W-consequences have the same normal form.) The W-consequences and I-consequences will not do to define the content of a set S. The first is too broad, may include propositions mutually inconsistent, the second is too narrow. A via media between these concepts is accordingly defined: P is a P-consequence of S, where P is some preference criterion yielding some of the m.c.s. of S as preferred to others, and P is a consequence of all of the P-preferred m.c.s. of S. The bulk of the paper is devoted to discussion of various preference criteria, and also surveys the application of this machinery in diverse contexts - for example, in connection with the processing of mutually inconsistent reports.     

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.     

Singular causal explanations

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.Work on this paper was supported by a University of Maryland Faculty Research Award. Earlier versions were read at the University of Minnesota and at the 1971 Western Division meetings of the American Philosophical Association. I have profited from criticisms raised on these occasions. I am especially grateful for the comments of James Lesher, Peter Machamer, John Vollrath, and the students in my Macalester College seminar.     

Science as a rational enterprise

Scientists often disagree about whether a new theory is better than the current theory. From this some (e.g., Thomas Kuhn) have inferred that the values of science are changing and subjective, and hence that science is an irrational enterprise. As an alternative, this paper develops a rational model of the scientific enterprise according to which the scope and elegance of theories are important elements in the scientist's utility function. The varied speed of acceptance of new theories by scientists can be explained in terms of the optimal allocation of time among different scientific activities. The model thus accounts for the rationality of science in a way that is broadly consistent with the empirical evidence on the history and practice of science.     

Liberals,information and Webster's principles

Critiques two social choice principles employed by Webster's analysis of using information to resolve Sen's paradox of the Impossibility of a Paretian Liberal.     

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.     

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.     

Focal points in pure coordination games: An experimental investigation

This paper reports an experimental investigation of the hypothesis that in coordination games, players draw on shared concepts of salience to identify focal points on which they can coordinate. The experiment involves games in which equilibria can be distinguished from one another only in terms of the way strategies are labelled. The games are designed to test a number of specific hypotheses about the determinants of salience. These hypotheses are generally confirmed by the results of the experiment.     

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.     

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.     

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.     

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.