0Expected Utility: Theorems
The theory of expected utility was developed by the founders of game theory, namely John von Neumann and Oskar Morgenstern, in their 1944 book Theory of Games and Economic Behavior.
In a rather unconventional way, we shall first (in this section) state the main result of the theory (which we split into two theorems) and then (in the following section) explain the assumptions (or axioms) behind that result. The reader who is not interested in understanding the conceptual foundations of expected utility theory, but wants to understand what the theory says and how it can be used, can study this section and skip the next. Let O be a set of basic outcomes. Note that a basic outcome need not be a sum of money: it could be the state of an individual’s health, or whether the individual under consideration receives an award, or whether it will rain on the day of her planned outdoor party, etc. Let L (O) be the set of simple lotteries (or probability distributions) over O. We will assume throughout that O is a finite set: O = {o1,o2,..., om} (m ≥ 1). Thus, an element of L (O) is of the form o1 o2 ... om p1 p2 ... pm with 0 ≤ pi ≤ 1, for all i = 1,2,...,m, and p1 + p2 +...+ pm = 1. We will use the symbol L (with or without subscript) to denote an element of L (O), that is, a simple lottery. Lotteries are used to represent situations of uncertainty.
For example, if m = 4 and the individual faces the lottery L = o1 o2 o3 o4 2 5 0 1 5 2 5 , then she knows that, eventually, the outcome will be one and only one of o1,o2,o3,o4, but does not know which one; furthermore, she can quantify her uncertainty by assigning probabilities to these outcomes. We interpret these probabilities either as objectively obtained from relevant (past) data or as subjective estimates by the individual.
For example, an individual who is considering whether or not to insure her bicycle against theft for the following 12 months knows that there are two relevant basic outcomes: either the bicycle will be stolen, or it will not be stolen. Furthermore, she can look up data on past bicycle thefts in her area and use the proportion of stolen bicycles as an “objective” estimate of the probability that her bicycle will be stolen.
Alternatively, she can use a more subjective estimate: she might use a lower probability of theft than suggested by the data because she knows herself to be very conscientious and – unlike other people – always to lock her bicycle when left unattended.
The assignment of zero probability to a particular basic outcome is taken to be an expression of belief, not impossibility: the individual is confident that the outcome will not arise, but she cannot rule out that outcome on logical grounds or by appealing to the laws of nature.
