Utility theory
U: O-> R (utility maps from outcomes to a real number)
represents preferences over outcomes
~ means indifference
We need a way to talk about how preferences interact with
uncertainty:
A lottery is a random selection of one of a set of outcomes
according to specified probabilities
Axioms
• Completeness:  induces an ordering over
all pairs of the outcome space (allowing
ties)
• Transitivity: if o1 o2 and o2 o3, then o1 o3
• substitutability : if o1 ~ o2 then
[p:o1, p3:o3 … pn:o4] ~ [p:o2, p3:o3 … pn:o4]
• monotonicity: if o1 o2 and p > q then
[p:o1, (1-p):o2] > [q:o1, (1-q):o2]
Axioms
• continuity: if o1 >o2 and o2 >o3 then there
exists p such that
o2 ~ [p:o1, (1-p):o3]
• decomposibility: if (for all oi ) the probability
of lottery 1 selecting oi is the same as the
probability of lottery 2 selecting oi, then the
lotteries are equivalent.
Risk attitudes
• Which would you prefer?
– A lottery ticket that pays out $10 with probability
.5 and $0 otherwise, or
– A lottery ticket that pays out $3 with probability 1
• How about:
– A lottery ticket (A) that pays out $100,000,000
with probability .5 and $0 otherwise, or
– A lottery ticket (B) that pays out $30,000,000
with probability 1
Risk attitudes
• Usually, people do not simply go by expected value
• An agent is risk-neutral if she only cares about the
expected value of the lottery ticket
• An agent is risk-averse if she always prefers the
expected value of the lottery ticket to the lottery ticket
– Most people are like this
• An agent is risk-seeking if she always prefers the
lottery ticket to the expected value of the lottery ticket
• We capture the attitude about risk by mapping the
value to a utility.
Decreasing marginal utility
• Typically, at some point, having an extra dollar does
not make people much happier (decreasing marginal
utility). Marginal utility: utility of x given you already
have y.
utility
buy a nicer car (utility = 3)
Risk averse
The first you get is valued
the most.
buy a car (utility = 2)
buy a bike (utility = 1)
$200 $1500
How would you describe this curve?
$5000
money
Maximizing expected utility
utility
buy a nicer car (utility = 3)
buy a car (utility = 2)
buy a bike (utility = 1)
$200 $1500
$5000
money
• Lottery 1: get $1500 with probability 1
– gives expected utility 2
• Lottery 2: get $5000 with probability .4, $200 otherwise
– gives expected utility .4*3 + .6*1 = 1.8
– (expected amount of money = .4*$5000 + .6*$200 = $2120 > $1500)
Any lottery is going to be less than the weighted known value
Maximizing expected utility
utility
buy a nicer car (utility = 3)
buy a car (utility = 2)
buy a bike (utility = 1)
$200 $1500
$5000
money
• Any weighting of 200 and 5000 – represents a point on the red line above
• Even if you got the exact same amount as a no-risk value, you would prefer
the no risk value. Can you see why?
• Lottery 1: get $X with probability 1
• Lottery 2: get $5000 with probability p, $200 otherwise
• Is there ever a case where (in this scenario) you would prefer the lottery?
Different possible risk attitudes
under expected utility maximization
utility
•
•
•
•
money
Green has decreasing marginal utility → risk-averse
Blue has constant marginal utility → risk-neutral
Red has increasing marginal utility → risk-seeking
Grey’s marginal utility is sometimes increasing,
sometimes decreasing → neither risk-averse
(everywhere) nor risk-seeking (everywhere)
What is utility, anyway?
• Function u: O →  (O is the set of “outcomes”
that lotteries randomize over)
• What are its units?
– It doesn’t really matter
– If you replace your utility function by u’(o) = a +
b*u(o) (an affine transformation), your behavior will
be unchanged
• Why would you want to maximize expected
utility rather than expected value?
– This is a question about preferences over lotteries
Compound lotteries – a lottery which
gives out tickets to another lottery
• For two lottery tickets L and L’, let p*L + (1-p)L’ be the
“compound” lottery ticket where you get lottery ticket L with
probability p, and L’ with probability 1-p
p*L+(1-p)L’
p=50%
p*L+(1-p)L’
1-p=50%
50%
o1
=
L’
L
25%
o2
25%
75%
o3
o2
25%
o4
25%
50%
o1
o2
12.5%
o3
12.5%
o4
Goals of axioms
• Researchers are trying to show that (in
certain circumstances) we ONLY need to
worry about a single dimension utility
function (rather than something in multiple
dimensions)
Sufficient conditions for expected utility
• Idea: A decision maker chooses between prospects by
comparing expected utility (weighted sums of utility values
multiplied by probabilities)
• Expected utility theorem. (von Neumann and Morgenstern
1944) if an agent can choose between the lotteries, the utility of
an arbitrary lottery can be calculated as a linear combination of
the utility of its parts
• Suppose we have a relationship  over outcomes which
satisfies the axioms of completeness, transitivity,
substitutability, decomposability, monotinicity, and continuity
then there exists a function u: O →  so that L ≥ L’ if and only if
L gives a higher expected value of u than L’ as defined by
– U(o1)  U(o2) iff o1  o2 [utility is an accurate measure of goodness]
– U[p1:o1…pn:on] = p1U(o1) [ the total utility is based on utility of individual
outcomes]