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People have been consciously coping with decision problems for at least as long as history has been recorded. Some of the best advice modern theory can offer appears to have been known to the ancients. For example, the biblical Jacob, fearing his brother Esau’s might, divides his camp into two bands, saying, “If Esau come to the one company and smite it, then the other company which is left shall escape” (Genesis 32:8)— thereby holding a claim to having invented diversification.

Historical Background

Yet decision theory, and, in particular, decision under uncertainty, was not explicitly studied until the mid-seventeenth century, when probability theory was introduced. The person most associated with the concepts of probability and expectation, Blaise Pascal, also introduced decision theory in his famous “wager.” In this argument, designed to convince nonbelievers that they will be better off becoming believers, Pascal introduced several basic notions of decision theory: (1) the decision matrix, in which one’s acts are independent of nature’s choices, or the “states of the world”; (2) domination between acts, in which one act is better than another no matter which state obtains the goal; (3) expected utility maximization, according to which the choice between undominated acts should be made according to the mathematical expectation of the utility of the outcomes they yield; (4) subjective probability over the states, which is an application of the mathematical probability model as a way to capture one’s beliefs; and (5) nonunique probabilities, in which one’s beliefs are too vague to be captured by a single probability vector. But even after this dramatic inauguration, decision theory was largely neglected until the twentieth century, with the exception of Daniel Bernoulli’s (1738) explicit introduction of the expected utility hypothesis applied to monetary payoff and given probabilities.

By contrast, mathematicians and philosophers have been interested in the mathematical theory of probability throughout the past centuries. Jacob Bernoulli (1713) discovered the law of large numbers and also discussed different types of probability. Thomas Bayes (1763) introduced the idea of Bayesian updating of “prior” probabilities to “posterior” ones.

Axiomatic Foundations

Whether all uncertainty can be quantified probabilistically has remained a topic of dispute from the very early writings on probability to the present. Frank Knight (1921) famously argued that this is not the case, and he distinguished between situations of risk, in which probabilities can be assumed given, and situations of uncertainty, in which probabilities are neither given nor can they be inferred from past statistical data. A major proponent of the opposite view was Frank Ramsey (1931), who, in the spirit of logical positivism, suggested defining and measuring one’s subjective probability by one’s willingness to bet. He suggested that reasonable decision makers will behave as if they had a subjective probability that guided their decisions, even if objective probabilities are not part of the description of the decision problem. Bruno de Finetti (1931, 1937) offered an “axiomatization” of subjective probabilities in the context of maximization of expected monetary value. That is, he provided a set of conditions on presumably observable choices and showed that they are equivalent to the claim that the decision maker maximizes expected value relative to some probability vector, which is taken to be that person’s subjective probability. The conditions are called “axioms” partly because they are presented as intuitive, if not compelling, and partly because they are in line with the logical positivist dictum of defining theoretical concepts (“subjective probability”) with observations (pairwise choices between bets).

A similar axiomatic derivation of the concept of “utility” was offered by John von Neumann and Oskar Morgenstern as a by-product of their introduction of game theory in Theory of Games and Economic Behavior (1944). They considered a presumably observable preference relation between pairs of “lotteries,” namely random variables with known distributions, and they showed that a set of axioms on the relation is equivalent to the claim that this relation can be represented by a utility function such that, confronted with any two choices, the decision maker would opt for the one that has a higher expected utility.

von Neumann and Morgenstern provided a definition of utility, coupled with the expected utility paradigm, based on a primitive notion of probability, whereas de Finetti did the opposite: He defined subjective probability based on a primitive notion of “utility.” However, when neither utility nor probability is well defined, it is not obvious that the theory of expected utility maximization relative to a subjective probability is very convincing, whether interpreted descriptively or normatively. This problem was rectified by Leonard J. Savage (1954), who showed that both utility and subjective probability can be derived, with the expected utility maximization rule, from basic axioms on “acts,” which are not defined numerically and presuppose neither probabilities nor utilities.

The axiomatic results of von Neumann and Morgenstern and Savage had a tremendous impact on research in decision theory, game theory, and in the applications of these in the social sciences. The mainstream view among theorists is that expected utility maximization, with respect to a subjective probability, is the only rational way of behavior. Moreover, many also believe that it is the only reasonable model to be used in applications in which a formal model attempts to describe reality. However, the theory has been challenged, mostly, but not solely, from a descriptive viewpoint.

Challenges

One of the earliest, and perhaps the most radical, objection to the theory was raised by Herbert Simon in 1957. He coined the term bounded rationality and argued that people do not optimize; rather, they “satisfice”; that is, as long as their performance is above a certain “aspiration level,” they stick to their previous choice. Only when their performance is below that threshold do they experiment with other choices. Simon thus challenged the very paradigm of optimization, and while his theory is seldom incorporated into formal decision models, it has had a remarkable impact on the thinking of many decision theorists, who have developed models that are classified as bounded rationality, even if their departure from the basic paradigm is much less dramatic than that of satisficing behavior.

Expected utility maximization was also attacked based on concrete examples in which it turned out to provide a poor prediction of people’s choices. Maurice Allais (1953) provided a “paradox” in the context of decision under risk (with known probabilities). In this example, many people violate a key axiom of von Neumann and Morgenstern (the independence axiom), and therefore behave in a way that cannot be captured by expected utility maximization (for any utility function). Allais’s example showed that people tend to put more weight on certainty than the standard theory predicted. In other words, people tend to behave in a way that is nonlinear in probabilities. Daniel Ellsberg (1961) proposed examples (also dubbed “paradoxes”), in which many people violate one of Savage’s basic axioms (the “sure thing principle”). In Ellsberg’s examples, many people behave in a way that cannot be described by subjective probability. Specifically, people tend to prefer situations with known probabilities to situations with unknown probabilities. This phenomenon is referred to as uncertainty aversion, or ambiguity aversion (following Knight’s and Ellsberg’s terms, respectively). Expected utility theory was generalized to deal with uncertainty aversion by David Schmeidler (1989) and Itzhak Gilboa and David Schmeidler (1989), among others.

Starting in the late 1960s, Daniel Kahneman and Amos Tversky launched a systematic experimental study of decision theoretic axioms. In carefully designed experiments, they have shown that practically any axiom of decision theory is violated in some examples. Moreover, they uncovered several implicit assumptions of the decision theory, which were also too idealized to describe actual choices. For example, they documented the “framing effect,” which shows that different representations of the same problem may result in different choices. They also suggested “prospect theory” (Kahneman and Tversky 1979) as an alternative to expected utility maximization for behavior under risk.

One key idea in prospect theory is that people respond to given probabilities in a way that is nonlinear in the probability, especially near the extreme values of 0 and 1. Another idea, with potentially far-reaching implications to research in political science, is that people react differently to gains as compared to losses. That is, the (monetary) bottom line is not all that matters to the decision maker: It also matters whether this bottom line is perceived as a gain or a loss relative to a “reference point” that decision makers have in their minds.

Conclusion

Formal decision theory has been extremely powerful in providing important insights into the behavior of agents in social and political environments. Formal models help analyze real-life situations and reveal analogies that might otherwise be difficult to identify. At the same time, formal models have been justifiably criticized on various grounds. Some of these criticisms have to do with assumptions of decision theory per se, such as the existence of probabilistic beliefs, and some have to do with assumptions of related fields, such as the concept of equilibrium in games. It is important not to discard the powerful insights that formal analysis might generate on account of some assumptions that need to be refined or replaced. It is to be hoped that future research will improve understanding of political phenomena using formal models, while taking each assumption thereof with a grain of salt.

Bibliography:

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