1. Expected Utility  We discuss “expected utility” in the context of simple lotteries  A generic lottery is denoted (x, y, pr)  The lottery offers payoff (or consequence) x with probability pr  and offers payoff y with probability (1- pr)

2.  This notion is very general and offers a variety of possible payoff structures  x and y may be monetary payoffs pr (1-pr) x y (x,y,pr)

3.  x and y are both lotteries pr (1-pr) pr* (1-pr*) x1x1 y1y1 y2y2 pr*, pr~ = other probabilities (x,y,pr) = ((x 1,x 2, pr*) (y 1,y 2,pr~)pr) x2x2 pr~ (1-pr~)

4. PROPERTIES OF UTILITY  One way to begin an analysis of individuals choices that we say are characterized in a utility function is to state a basic set of postulates, or axioms, that characterize what we call “rational behavior o Completeness: If A and B are any two situations, the individual can always specify exactly one of the following possibilities: A is preferred to B B is preferred to A A and B are equally attractive here, people are not paralyzed by indecision --- this rules out that A is preferred to B, and, B is preferred to A

5.  There exists a preference relationship defined on lotteries, which is complete and transitive --- remember properties of utility?? completeness and transitivity  The preference relationship is continuous--- We can get U(x) = U(x,y,1) as a payment which is a degenerate lottery

6.  Transitivity: If an individual reports A is preferred to B and that B is preferred to C, then A is preferred to C --- choices are internally consistent  Continuity: If an individual reports A is preferred to B, the situations suitably close to A must also be preferred to B --- this helps us analyze relatively small changes in income and prices

7. UTILITY  Given the assumptions of completeness, transitivity, and continuity, it is possible to show formally that people are able to rank in order all possible situations from least desirable to most desirable  This ranking we call “utility” after the inventor, Jeremy Bentham, a 19 th century

8. Some more axioms  Agents are concerned with the net cumulative probability of each outcome like (x,y,pr=1) = x, (x,y,pr) = (x,y,(1-pr))l and (x,z,pr) = (x,y,pr +(1-pr)pr*) if z = (x,y,pr*) --- notice that (x,y,pr) = x iff (if and only if) pr = 1

9.  The utility of situation A and situation B would be denoted U(A,B)  Bentham suggested the utilitarian approach as “more is better”  Therefore, if a person prefers situation A to situation B, then U(A) is greater than U(B)

10.  Let the lotteries (x,y,pr) and (x,z,pr) be any two lotteries --- then y is preferred to z iff (x,y,pr) is preferred to (x,z,pr)  There exists a best (most preferred lottery), say B, as well as a worst lottery (least desired), L

11.  Let x, k, z be consequences or payoffs for which x is strictly preferred to k and k is strictly preferred to z --- then there exists a probability pr such that (x,z,pr) is approximately k  Let x be strictly preferred to y--- then (x,y,pr*) is preferred to (x,y,pr~) Iff pr* > pr~

12. Now the expected utility function  If all the axioms above are satisfied, then there exists a function, U, defined on the lottery space so that:  U(x,y,pr) = prU(x) + (1-pr)U(y)  John Von Neumann and Oscar Morgenstern actually developed these axioms and the expected utility framework

13.  So, by convention we refer to  U(x,y,pr) = prU(x) + (1-pr)U(y)  as the Von Neumann-Morgenstern framework or utility function --- or just VNM for short  There are several alternatives to this framework that are offered --- but these are in the realm of advanced financial economics