SAVAGE, LEONARD JAMES

(b. Detroit, Michigan, 20 November 1917; d. New Haven, Connecticut, 1 November 1971), statistics, mathematics, philosophy. For the original article on Savage see DSB vol.18, Supplement II.

Savage was a founder of twentieth-century Bayesian statistics. Savage’s ideas about probability and utility inspired theorists in many fields. Besides advancing his own discipline, statistics, he made significant contributions to philosophy.

Savage’s representation theorem for probability and utility is a cornerstone of the theory of subjective probability and utility. It shows that preferences satisfying basic principles of rationality may be represented as maximizing expected utility. Such representation theorems are crucial components of decision theory. Under one interpretation, they ground definitions of probability and utility. Under another interpretation, they demonstrate that probability and utility may be inferred from preferences. Under either interpretation, they form foundations for principles of rational decision making.

Because of the importance of Savage’s ideas, scholars continue to analyze and refine them. James M. Joyce’s book, The Foundations of Causal Decision Theory (1999),

Figure 1 .Allais’s Paradox.

extends Savage’s theory, improving and making more versatile Savage’s representation theorem. Joyce’s representation theorem offers a choice between causal and evidential decision theory. Its account of conditional probability may make a decision’s evaluation depend on either the decision’s effect on the future or the evidence it furnishes about the future. Given the first interpretation of conditional probability, the representation theorem grounds a causal decision theory and, given the second interpretation, it grounds an evidential decision theory.

Savage assumes that for any outcome, there is an act that yields that outcome in every case. Some consequences of an act obtain only if the act’s outcome is not constant. Savage’s assumption therefore puts aside those consequences. In particular, it prevents treating risk as a consequence of an act. Risk arises only from acts that do not yield the same outcome in every case. Joyce’s representation theorem gains realism by replacing Savage’s assumption of constant acts with structural axioms requiring coherent extensions of preferences.

Furthermore, Joyce attains Savage’s goal of streamlining the expected utility principle so that it uses summary “small worlds” instead of “grand worlds” complete in every detail. To accomplish the streamlining, Joyce takes the objects of probabilities and utilities to be propositions and makes calculations of an option’s expected utility invariant with respect to partitions of states.

Savage’s theory incorporates the sure-thing principle, which he presents in The Foundations of Statistics(2nd ed., 1972). That principle states that a rational agent’s ranking of a pair of gambles having the same consequence in a state or set of states S agrees with the agent’s ranking of any other pair of gambles the same as the first pair except for having some other common consequence in S. This principle is the target of a number of criticisms. The best known is Allais’s paradox, presented by Maurice Allais in an essay, “Foundations of a Positive Theory of Choice Involving Risk and a Criticism of the Postulates and Axioms of the American School” (1979).

Allais argues that some violations of the sure-thing principle are not irrational. His example presents four gambles involving a 100-ticket lottery. The table (Figure 1) lists in units of $100,000 the prizes the gambles yield for each ticket number. Changing A’s and B’s common consequence for tickets 12–100 from 5 to 0 yields C and D respectively. Hence the sure-thing principle prohibits preferring A to B and simultaneously preferring D to C. Yet most people have those preferences, which seem coherent. To guarantee a large gain, a typical person prefers gamble A, which offers $500,000 for all 100 tickets, to gamble B, which yields $0 for ticket number 1. Also, being attracted by the biggest prize, a typical person prefers gamble D, which offers $2,500,000 for ten tickets, to gamble C, which offers $500,000 for eleven tickets.

The sure-thing principle assumes that if two options have identical consequences in some cases, then a rational agent prefers the first option to the second option if and only if in the remaining cases he prefers the consequences of the first option to the consequences of the second option. The gambles A and B have the same consequences in 89 cases. So a preference between them indicates a preference concerning their profiles of consequences in the remaining 11 cases. Similarly, the gambles C and D have the same consequences in 89 cases. So a preference between them indicates a preference concerning their profiles of consequences in the remaining 11 cases. According to the sure-thing principle, preferring A to B and preferring D to C each express a preference between the same two 11-case profiles of consequences. However, they express opposite preferences between those profiles. To prevent inconsistency, the sure-thing principle prohibits the typical preferences concerning the gambles.

Theorists disagree about the rationality of the typical preferences concerning Allais’s gambles and hence about the sure-thing principle. Moreover, those who defend the rationality of the typical preferences, disagree about the best way of justifying those preferences.

The sure-thing principle is also known as the independence axiom because it makes preferences concerning profiles of consequences independent of the context in which the profiles occur. In response to Allais’s paradox, Mark Machina’s article, “‘Expected Utility’ Theory without the Independence Axiom” (1982), relaxes the sure-thing principle. Machina holds that this principle is too demanding to be a requirement of rationality.

Tamara Horowitz’s essay, “The Backtracking Fallacy” (2006), analyzes Savage’s argument for the sure-thing principle. The argument treats probabilities that depend on evidence and therefore probabilities that depend on what is known. Horowitz distinguishes supposing a proposition true and supposing it to be known true. She contends that Savage’s argument inadvertently slips from one form of supposition to the other and, as a result, fails to establish the sure-thing principle. However, the sure-thing principle, in contrast with the argument for it, avoids equivocation, and some guarded formulations of it may succeed.

Many scholars gratefully acknowledge their debt to Savage. The International Society for Bayesian Analysis annually presents the Leonard J. Savage Dissertation Award to two outstanding doctoral dissertations in Bayesian econometrics and statistics, fields that Savage loved and brilliantly advanced.

SUPPLEMENTARY BIBLIOGRAPHY

For a bibliography of Savage’s published works, see The Writings of Leonard Jimmie Savage: A Memorial Selection (Washington, DC American Statistical Association and the Institute of Mathematical Statistics, 1981).

WORKS BY SAVAGE

With Edwin Hewitt. “Symmetric Measures on Cartesian Products.” Transactions of the American Mathematical Society 80 (1955): 470–501.

With Lester Dubins. How to Gamble if You Must: Inequalities for Stochastic Processes. New York: McGraw-Hill, 1965.

“Implications of Personal Probability for Induction.” Journal of Philosophy 64 (1967): 593–607.

“Elicitation of Personal Probabilities and Expectations.” Journal of the American Statistical Association 66 (1971): 783–801.

The Foundations of Statistics. New York: Wiley, 1954. 2nd ed. New York: Dover, 1972.

OTHER SOURCES

Allais, Maurice. “Foundations of a Positive Theory of Choice Involving Risk and a Criticism of the Postulates and Axioms of the American School.” In Expected Utility Hypotheses and the Allais Paradox, edited by Maurice Allais and Ole Hagen. Dordrecht, Netherlands: Reidel, 1979. The original French version of this essay was written in 1952 in response to Savage’s early formulations of his theory.

Gärdenfors, Peter, and Nils-Eric Sahlin, eds. Decision, Probability, and Utility: Selected Readings. Cambridge, U.K., and New York: Cambridge University Press, 1988. Besides containing selections from Savage’s work, this collection contains many essays appraising Savage’s ideas.

Horowitz, Tamara. The Epistemology of A Priori Knowledge. Edited by Joseph L. Camp Jr. Oxford, and New York: Oxford University Press, 2006. The principal essay in this collection, “The Backtracking Fallacy,” is an accessible and thorough treatment of Savage’s sure-thing principle.

Joyce, James M. The Foundations of Causal Decision Theory. New York, and Cambridge, U.K.: Cambridge University Press, 1999. Presents Savage’s decision theory and extends it to provide foundations for causal decision theory.

Lindley, Dennis V. “L. J. Savage: His Work in Probability and Statistics.” Annals of Statistics 8 (1980): 1–24.

Machina, Mark. “‘Expected Utility’ Theory without the Independence Axiom.” Econometrica 50 (1982): 277–323. Revises Savage’s theory in response to Allais’s objections. Weirich, Paul. “Expected Utility and Risk.” British Journal for the Philosophy of Science 37 (1986): 419–442. Analyzes Allais’s objections to Savage’s theory.

Paul Weirich