Utility

Utility Function

The measurement of utility has a long history in economics, starting with the measurement of preferences on an ordinal scale. Since von Neumann and Morgenstern (1947) developed the axiomatic foundation of utility, it can be measured on an interval scale. It is derived from personal preferences for either a sure thing or a gamble (Figure 1a). The decision maker is presented with a choice between obtaining either outcome o for sure or a gamble that returns with probability p a better outcome (o∗) and with complementary probability (1-p) a worse outcome (o∗). The gamble is written as [p o∗, (1-p) o∗] or simply [p o∗, o∗] (the square denotes a decision, the circle a chance node).

With o∗ and o∗ as best and worst outcomes and all outcomes oi in-between, the indifference probability πi, where the decision maker is indifferent between the sure thing and the gamble, can be determined as oi ∼ [πi o∗, o∗]. Pairs (o∗, πi) of these indifferences constitute the utility function (Figure 2a). Above the function the gamble is preferred, below the function the sure thing is preferred. For outcomes with the following preference order o∗ ϕo1 ϕo2ϕ … oi …ϕo∗ the corresponding indifference probabilities with 1.0 = π∗ > p > p >… >p, >… > π∗ = 0 (as well as their linear transformations) are the utilities of the corresponding outcomes. Whenever the preference oi ϕoj holds, the utility measurement of oi exceeds oj and πi > πj.

Axiomatic Foundation

The perceived utility of outcomes is the basis for expressing preferences. These preferences are analysed in order to measure true internal utilities. If the expressed preferences with respect to outcomes of alternatives (objects, commodities, events, strategies) and gambles meet the following six axioms, utility can be measured on an interval scale.

Dominance

A rational decision maker should never accept a dominated gamble whose best outcome matches [Page 1064]the outcome of the sure thing and should always accept a gamble whose worst outcome matches the outcome of the sure thing.

Figure 1. The choice between a sure thing and a gamble.

Figure 2. Utility functions.