decision theory the theory of rational decision, often called ‘rational choice theory’ in political science and other social sciences. The basic idea (probably Pascal’s) was published at the end of Arnaud’s Port-Royal Logic (1662): ‘To judge what one must do to obtain a good or avoid an evil one must consider not only the good and the evil in itself but also the probability of its happening or not happening, and view geometrically the proportion that all these things have together.’ Where goods and evils are monetary, Daniel Bernoulli (1738) spelled the idea out in terms of expected utilities as figures of merit for actions, holding that ‘in the absence of the unusual, the utility resulting from a fixed small increase in wealth will be inversely proportional to the quantity of goods previously possessed.’ This was meant to solve the St. Petersburg paradox: Peter tosses a coi. . . until it should land ‘heads’ [on toss n]. . . . He agrees to give Paul one ducat if he gets ‘heads’ on the very first throw [and] with each additional throw the number of ducats he must pay is doubled. . . . Although the standard calculation shows that the value of Paul’s expectation [of gain] is infinitely great [i.e., the sum of all possible gains $ probabilities, 2n/2 $ ½n], it ha. . . to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats. In this case Paul’s expectation of utility is indeed finite on Bernoulli’s assumption of inverse proportionality; but as Karl Menger observed (1934), Bernoulli’s solution fails if payoffs are so large that utilities are inversely proportional to probabilities; then only boundedness of utility scales resolves the paradox.
Bernoulli’s idea of diminishing marginal utility of wealth survived in the neoclassical texts of W. S. Jevons (1871), Alfred Marshall (1890), and A. C. Pigou (1920), where personal utility judgment was understood to cause preference. But in the 1930s, operationalistic arguments of John Hicks and R. G. D. Allen persuaded economists that on the contrary, (1) utility is no cause but a description, in which (2) the numbers indicate preference order but not intensity. In their Theory of Games and Economic Behavior (1946), John von Neumann and Oskar Morgenstern undid (2) by pushing (1) further: ordinal preferences among risky prospects were now seen to be describable on ‘interval’ scales of subjective utility (like the Fahrenheit and Celsius scales for temperature), so that once utilities, e.g., 0 and 1, are assigned to any prospect and any preferred one, utilities of all prospects are determined by overall preferences among gambles, i.e., probability distributions over prospects. Thus, the utility midpoint between two prospects is marked by the distribution assigning probability ½ to each.
In fact, Ramsey had done that and more in a little-noticed essay (‘Truth and Probability,’ 1931) teasing subjective probabilities as well as utilities out of ordinal preferences among gambles. In a form independently invented by L. J. Savage (Foundations of Statistics, 1954), this approach is now widely accepted as a basis for rational decision analysis. The 1968 book of that title by Howard Raiffa became a theoretical centerpiece of M.B.A. curricula, whose graduates diffused it through industry, government, and the military in a simplified format for defensible decision making, namely, ‘cost–benefit analyses,’ substituting expected numbers of dollars, deaths, etc., for preference-based expected utilities.
Social choice and group decision form the native ground of interpersonal comparison of personal utilities. Thus, John C. Harsanyi (1955) proved that if (1) individual and social preferences all satisfy the von Neumann-Morgenstern axioms, and (2) society is indifferent between two prospects whenever all individuals are, and (3) society prefers one prospect to another whenever someone does and nobody has the opposite preference, then social utilities are expressible as sums of individual utilities on interval scales obtained by stretching or compressing the individual scales by amounts determined by the social preferences. Arguably, the theorem shows how to derive interpersonal comparisons of individual preference intensities from social preference orderings that are thought to treat individual preferences on a par. Somewhat earlier, Kenneth Arrow had written that ‘interpersonal comparison of utilities has no meaning and, in fact, there is no meaning relevant to welfare economics in the measurability of individual utility’ (Social Choice and Individual Values, 1951) – a position later abandoned (P. Laslett and W. G. Runciman, eds., Philosophy, Politics and Society, 1967). Arrow’s ‘impossibility theorem’ is illustrated by cyclic preferences (observed by Condorcet in 1785) among candidates A, B, C of voters 1, 2, 3, who rank them ABC, BCA, CAB, respectively, in decreasing order of preference, so that majority rule yields intransitive preferences for the group of three, of whom two (1, 3) prefer A to B and two (1, 2) prefer B to C but two (2, 3) prefer C to A. In general, the theorem denies existence of technically democratic schemes for forming social preferences from citizens’ preferences. A clause tendentiously called ‘independence of irrelevant alternatives’ in the definition of ‘democratic’ rules out appeal to preferences among non-candidates as a way to form social preferences among candidates, thus ruling out the preferences among gambles used in Harsanyi’s theorem. (See John Broome, Weighing Goods, 1991, for further information and references.) Savage derived the agent’s probabilities for states as well as utilities for consequences from preferences among abstract acts, represented by deterministic assignments of consequences to states. An act’s place in the preference ordering is then reflected by its expected utility, a probability-weighted average of the utilities of its consequences in the various states. Savage’s states and consequences formed distinct sets, with every assignment of consequences to states constituting an act. While Ramsey had also taken acts to be functions from states to consequences, he took consequences to be propositions (sets of states), and assigned utilities to states, not consequences. A further step in that direction represents acts, too, by propositions (see Ethan Bolker, Functions Resembling Quotients of Measures, University Microfilms, 1965; and Richard Jeffrey, The Logic of Decision, 1965, 1990). Bolker’s representation theorem states conditions under which preferences between truth of propositions determine probabilities and utilities nearly enough to make the position of a proposition in one’s preference ranking reflect its ‘desirability,’ i.e., one’s expectation of utility conditionally on it. Alongside such basic properties as transitivity and connexity, a workhorse among Savage’s assumptions was the ‘sure-thing principle’:
Preferences among acts having the same consequences in certain states are unaffected by arbitrary changes in those consequences. This implies that agents see states as probabilistically independent of acts, and therefore implies that an act cannot be preferred to one that dominates it in the sense that the dominant act’s consequences in each state have utilities at least as great as the other’s. Unlike the sure thing principle, the principle ‘Choose so as to maximize CEU (conditional expectation of utility)’ rationalizes action aiming to enhance probabilities of preferred states of nature, as in quitting cigarettes to increase life expectancy. But as Nozick pointed out in 1969, there are problems in which choiceworthiness goes by dominance rather than CEU, as when the smoker (like R. A. Fisher in 1959) believes that the statistical association between smoking and lung cancer is due to a genetic allele, possessors of which are more likely than others to smoke and to contract lung cancer, although among them smokers are not especially likely to contract lung cancer. In such (‘Newcomb’) problems choices are ineffectual signs of conditions that agents would promote or prevent if they could. Causal decision theories modify the CEU formula to obtain figures of merit distinguishing causal efficacy from evidentiary significance – e.g., replacing conditional probabilities by probabilities of counterfactual conditionals; or forming a weighted average of CEU’s under all hypotheses about causes, with agents’ unconditional probabilities of hypotheses as weights; etc.
Mathematical statisticians leery of subjective probability have cultivated Abraham Wald’s Theory of Statistical Decision Functions (1950), treating statistical estimation, experimental design, and hypothesis testing as zero-sum ‘games against nature.’ For an account of the opposite assimilation, of game theory to probabilistic decision theory, see Skyrms, Dynamics of Rational Deliberation (1990).
The ‘preference logics’ of Sören Halldén, The Logic of ‘Better’ (1957), and G. H. von Wright, The Logic of Preference (1963), sidestep probability. Thus, Halldén holds that when truth of p is preferred to truth of q, falsity of q must be preferred to falsity of p, and von Wright (with Aristotle) holds that ‘this is more choiceworthy than that if this is choiceworthy without that, but that is not choiceworthy without this’ (Topics III, 118a). Both principles fail in the absence of special probabilistic assumptions, e.g., equiprobability of p with q. Received wisdom counts decision theory clearly false as a description of human behavior, seeing its proper status as normative. But some, notably Davidson, see the theory as constitutive of the very concept of preference, so that, e.g., preferences can no more be intransitive than propositions can be at once true and false. See also EMPIRICAL DECISION THEORY, GAME THEORY, RATIONALITY, SOCIAL CHOICE THEORY. R.J.