Expected Utility And Certainty Equivalence
1. A prospect is a set of outcomes, each one of which has a probability attached. For example, the prospect of flipping a coin can be described as, Pr(heads) = .5 and Pr(tails) = .5
2. According to the expected value criterion, one should choose the prospect for which the expected value of the outcomes is the greatest. Expected value is calculated according to the formula
E(v) = Pr(i)V(i), where Pr(i) is the probability of outcome i and Vi is the value of outcome i
3. If outcomes are expressed in money, utility may not be proportional to money. Instead, the concept of diminishing marginal utility suggests that as the value of the outcome expressed in money increases, the increase in utility resulting from one more dollar decreases.
4. Although it is impossible to specify any one form of a utility function is universally valid for everyone, the following functional form is convenient for illustrating problems related to decision making under uncertainty:
U(V) = 100((V/Vmax)^1/k))
where n^k means n to the power k, as in Excel notation
U is utility on a scale with U =0 when V=0 and U = 100 when V = Vmax
V is the value of an outcome
Vmax is the maximum possible value of the outcome
k is a coefficient of risk aversion:
k = 1 implies risk neutrality
k > 1 implies risk aversion
k < 1 implies positive risk preference
The following graph shows a graph of the utility function for Vmax = 1000 and various values of k
5. The inverse utility function is derived by solving the utility function for V. It gives the value V which has a given utility U:
V(U) = Vmax (U/100)^k
6. If a prospect consists of several outcomes, each having a value Vi, a utility Ui, and a probability Pri, then the expected value for the scenario is Ev = Pri * Vi and...
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