The expected value of the utility function applied to a random outcome, using the decision-maker's probabilities. For a finite lottery it is . Under risk neutrality and quasilinear utility, auction utility is valuation minus payment upon winning and zero upon losing, so its expectation is the winning probability times the payoff conditional on winning.
A utility function on monetary amounts need not be increasing merely because preferences obey the expected utility hypothesis. The outcome labels do not themselves supply a preference order. For example, values , , define consistent expected utility comparisons even though the amount is least preferred. Preferring more money is an additional assumption. It is distinct from risk aversion, which concerns the effect of spreading uncertainty for a specified increasing utility function.
A probability distribution over possible outcomes. A finite lottery assigning probabilities to outcomes has expected utility . Mixing lotteries mixes their outcome probabilities; the independence axiom for lottery preferences describes how this operation preserves preference comparisons.
One standard mixture formulation says that if , some satisfies . Under the other Von Neumann-Morgenstern utility theorem assumptions, this excludes lexicographic comparisons and permits numerical utility function scales. It concerns mixtures of lotteries, rather than continuity of a monetary utility function on a prespecified consumption domain.
A compound lottery first selects branch with probability and then outcome with conditional probability . Its reduced lottery assigns the total probability . The law of total expectation gives the same expected utility in either description:This rule assumes that only the final outcome matters. A preference for a particular randomization procedure requires including that procedure in the outcome description; it cannot be captured by changing the arithmetic of expected utility on a fixed outcome set.
Articles by others on the same topic
Expected utility is a fundamental concept in decision theory and economics that provides a framework for evaluating choices under uncertainty. It is based on the idea that individuals make decisions by considering the potential outcomes of their choices, each associated with its likelihood of occurring, and assigning a utility value to each outcome. Here's a breakdown of the main components of expected utility: 1. **Outcomes**: These are the different possible results of a decision or action.