# The Expected Utility Theory

• Tapan Biswas
Chapter

## Abstract

In the eyes of many people, gambling is not regarded as a desirable activity. Some religious institutions actually condemn it as a sin. In spite of this, at least two important branches of academic pursuit, namely probability theory and decision theory, owe their origins to a careful analysis of the activities and outcomes of gambling. In the seventeenth century, mathematicians like Blaise Pascal and Pierre de Fermat argued that, people usually select the gamble yielding the highest expected return. If the possible outcomes of a gamble (wins and losses) are given by the vector (A1, A2, ..., An) with an associated probability vector (p1, p2, ..., pn), the expected return from the gamble is given by ∑ipiAi. Pascal and Fermat argued that people should accept the gamble which offers the highest expected return. This implies that, in the selection of gambles, individual preferences do not matter. It is the difference in individual subjective perception of probabilities which causes someone to offer a gamble and someone else to accept it. The theory also implies that the variance of returns do not matter. If any rational individual is forced to choose between the following two gambles:
$$\begin{gathered} {L_1}:\,Win\,\ 10\,with\;probability\,{1 \mathord{\left/ {\vphantom {1 {2\,or\,lose\,\ 10\,with\,probability}}} \right. \kern-\nulldelimiterspace} {2\,or\,lose\,\ 10\,with\,probability}}\,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ {L_2}\,:\,Win\,\ 1000\,with\;probability\,{1 \mathord{\left/ {\vphantom {1 {2\,or\,lose\,\ 1000\,with\,probability}}} \right. \kern-\nulldelimiterspace} {2\,or\,lose\,\ 1000\,with\,probability}}\,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ \end{gathered}$$
she should be indifferent between the alternative prospects forced upon her.