The expected utility hypothesis stems from Daniel Bernoulli's (1738) solution to the famous St. Petersburg Paradox. The Paradox challenges the old idea that people value random ventures according to its expected return. The Paradox posed the following situation: a fair coin will be tossed until a head appears; if the first head appears on the nth toss, then the payoff is 2ⁿ ducats. How much should one pay to play this game? The paradox, of course, is that the expected return is infinite, namely:

E(w) = å _i=1¥ (1/2ⁿ)·2ⁿ = (1/2)·2 + (1/4)2² + (1/8)2³ + .... = 1 + 1 + 1 + ..... = ¥

Yet while the expected payoff is infinite, one would not suppose, at least intuitively, that real-world people would be willing to pay an infinite amount of money to play this.

Daniel Bernoulli's solution involved two ideas that have since revolutionized economics: firstly, that people's utility from wealth, u(w), is not linearly related to wealth (w) but rather increases at a decreasing rate - the famous idea of diminishing marginal utility, u¢ (Y) > 0 and u¢ ¢ (Y) < 0; (ii) that a person's valuation of a risky venture is not the expected return of that venture, but rather the expected utility from that venture. In the St. Petersburg case, the value of the game to an agent (assuming initial wealth is zero) is:

E(u) = å _i=1¥ (1/2ⁿ)·u(2ⁿ) = (1/2)·u(2) + (1/4)·u(2²) + (1/8)·u(2³) + .... < ¥

which Bernoulli conjectured is finite because of the principle of diminishing marginal utility. Consequently, people would only be willing to pay a finite amount of money to play this, even though its expected return is infinite. In general, by Bernoulli's logic, the valuation of any risky venture takes the expected utility form:

E(u | p, X) = å _{xÎ X} p(x)u(x)

where X is the set of possible outcomes, p(x) is the probability of a particular outcome x Î X and u: X ® R is a utility function over outcomes.