Expected Utility Theory

Expected utility theory (EUT) states how an EUT decision maker makes choices among options that have specified characteristics. Each option is in some sense viewed by the EUT decision maker as beneficial to the EUT decision maker, but the option also has risks associated with the benefits and the EUT decision maker must bear the adverse outcomes associated with these risks should they occur. In addition, both the benefits and the risks of the options are uncertain, hence EUT decision makers must consider a set of uncertainties of benefits and risks among options (or alternatives) as they make their way through the decisions they face. Compared with other types of decision makers who pursue different routes in coming to a choice among alternatives with trade-offs, the EUT decision maker makes his or her choice in one way: by comparing the weighted sums of the options that are open to him or her. The weighted sums of options are obtained by adding the utility values of each of the outcomes multiplied by each outcome's respective probability of occurrence across the set of outcomes open to the EUT decision maker.

The origins of the EUT can be traced back to 1738, when Daniel Bernoulli wrote what he described as a new theory of the measurement of risk. But what assumptions was Bernoulli coming up against that required a “new” formulation?

Floris Heukelom traces the history of the mathematics of rational behavior to 1654, when Chevalier de Méré instigated Blaise Pascal, and therewith Pierre Fermat, to consider gambling problems. Heukelom notes that from an examination of a large body of literature on Enlightenment mathematicians who were interested in probability, it seemed as if these mathematicians were not making a real distinction between the determination of what they considered to be an answer to the question “What should the rational solution to the problem be in situations of uncertainty?” and the question “What would a rational person actually do (or how would a rational person act) in those same situations of uncertainty?” For these mathematicians, the two questions were one and the same.

One such construction of a gamble is the St. Petersburg game that came under the scrutiny of Bernoulli. Chris Starmer notes the following about EUT as it was first proposed by Bernoulli to the St. Petersburg game. Starmer notes that Bernoulli proposed EUT in response to an apparent puzzle surrounding what price a reasonable [Page 475]person should be prepared to pay to enter a gamble. It was the conventional wisdom at the time that it would be reasonable to pay anything up to the expected value of a gamble. But Bernoulli proposed making a game out of flipping a coin repeatedly until a tail is produced, and let us make a game of this situation. The game rules are as follows: If one is willing to participate in the game, one will receive a payoff of, say, $2n, where n is the number of the throw producing the first tail (T). If one goes about looking for players for this game, one finds that people do not want to get involved in this game, where, in fact, the expected monetary payoff is infinite. In fact, and to the surprise of theoretical mathematicians, people are only prepared to pay a relatively small amount to even enter the game. Bernoulli argued that the “value” of such a gamble to an individual is not, in general, equal to its expected monetary value as theoretical mathematicians believe. Rather, Bernoulli argued and proposed a theory in which individuals place subjective values, or utilities, on monetary outcomes. Here, for Bernoulli, the value of a gamble is the expectation of these utilities.

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