Transcript Expected Utility

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 (1pr)
 This notion is very general and offers a
variety of possible payoff structures
 x and y may be monetary payoffs
x
pr
(x,y,pr)
(1-pr)
y
 x and y are both lotteries
pr~
pr
(1-pr~)
x1
x2
pr*
y1
(1-pr)
(1-pr*)
y2
pr*, pr~ = other probabilities
(x,y,pr) = ((x1,x2, pr*) (y1,y2,pr~)pr)
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
 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
 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
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 19th century
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
 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)
 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
 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~
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
 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