A Bridge between Classical and Subjective Decision Theories

Simone Cerreia-Vioglio, Fabio Maccheroni, and Massimo Marinacci (Department of Decision Sciences) with Luigi Montrucchio (Università di Torino), in their Classical Subjective Expected Utility (Proceedings of the National Academy of Sciences, forthcoming), build a framework that combines the classical decision theory approach à la Wald and the subjective approach à la Savage.

Decision problems are part of our daily life. For instance a relatively simple problem could be to decide whether to buy or not to buy a new laptop given that our current one is becoming obsolete. More delicate and sophisticated problems are governments' decisions about the economic policies to implement, given the available data on economic variables, such as unemployment rate, gross domestic product, rates of inflation, and the like. Decision theory aims to model decision problems and provide criteria that may help decision makers to identify the best course of action.

Abraham Wald is considered the father of classical statistical decision theory. In his seminal book, published in 1950, he studies problems in which decision makers choose among actions whose outcome depend on some verifiable states of the environment. He posits a set of "objective" probability distributions that may generate the states, one of them being the true distribution that actually generates them. Within such set, he considers the least expected utility of each possible action. He then suggests to choose the action that features the highest of such least expected utilities. In the decision theoretic jargon, the decision maker should maximinimize.

The main assumption of the Waldean framework is that the decision maker is able to posit a class of probability distributions for the states, an assumption that is often made in applications (especially in empirical works that rely on time series).

In contrast, Leonard Savage in his famous 1954 book develops a purely subjective approach in which the decision makers maximize the expected utility of actions, computed with respect to a subjective probability on states that quantifies their beliefs. He thus provides the behavioral foundations of Bayesian decision theory. Unlike Wald, he does not postulate any class of objective distributions on the states.

These two decision theoretic approaches were often portrayed as irriducible alternatives. Cerreia-Vioglio, Maccheroni, Marinacci and Montrucchio show that, instead, they may nicely complement each other. They propose a model that is able to incorporate a collection of probability distributions, as posited by Wald, within an otherwise Savagean subjective setting. In this way, they provide a decision theoretic framework that takes such distributions as a primitive and, at the same time, leaves room for subjective judgments.