CE VALUE ADDITIVITY PROPERTY

for all lotteries L and all scalar amounts delta. In words, if all payoffs in a lottery are enhanced or reduced by an amount delta, then the value of the lottery is adjusted by the same amount. Diagramatically, in the finite case,

 THEOREM

 A utility function U(x) has the delta-property if and only if U(x) is either linear or exponential in form.
(Proved by Ron Howard in a paper entitled "Risk Preference" 1970)

 Note: The linear utility function is equivalent to making decisions based on EMV, that is, Expected Monetary Value, or expected values. If one uses EMV as the basis for decisions, one is said to be "risk neutral" as this is the limiting case of exponential utility as the risk tolerance value tends towards infinity. A decision maker with a finite risk tolerance is said to be "risk averse", and the risk aversion coefficient is given by the reciprocal of the risk tolerance. Thus high risk aversion corresponds to low risk tolerance levels, and low risk aversion corresponds to high risk tolerance levels.

 Another way of stating the delta property is in terms of value additivity: the certain equivalent value of the joint receipt of two independently distributed lotteries L₁ and L₂ is equal to the CE value of L₁ plus the CE value of L₂. Value additivity is clearly required in situations requiring approval of a traditional accounting department, and has an intuitive appeal which suggests its elevation to the status of an axiom in more general contexts. Additionally, it enables one to extend the traditional notions of EVPI (Expected Value of Perfect Information) and EVSI (Exptected Value of Sample Information) to the risk averse case in an unambiguous way, as shown later in this presentation.

 CE VALUE ADDITIVITY PROPERTY

 The delta-property is equivalent to value additivity, that is, if L₁ and L₂ are two independently distributed gambles, then