Saint Petersburg Paradox, Expected Utility Principle
AMS 221
April 12, 2016
Saint Petersburg Paradox
What is the fair price of a bet? We assume that the fair value is the expected value. For this to be
operational, we need a notion of utility.
Consider the game where a fair coin is tossed until the first heads is obtained. If this process takes n tosses,
then your reward is 2n−1 . How much are you willing to pay for this bet? What should the fair bet price be?
We’ll need to calculate the expected value to decide.
The expected value of the gain is:
∞
X
1
2n−1 n → ∞
2
n=1
So you should be willing to pay an arbitrarily large sum of money to buy this bet. This is a paradox.
What is going wrong? We can’t change the probability of the fair coin landing heads-up, so we’ll need to
mess with the reward. For a large n, the reward is 2n , for n + 1, it is 2n+1 , so an additional step doubles
the reward in dollars (or utils, or whatever). This is regardless of the size of n.
Bernoulli’s take on this problem is that the utility gain of money is inversely proportional to current wealth.
That is:
c∆
u(z + ∆) − u(z) =
,
z
where z is the current level of wealth, ∆ is the change in wealth, and c is a constant greater than 0. In the
limit, the derivative of the utility is a constant divided by c:
u0 (z) =
c
⇒ u(z) = c log z − log z0
z
Remember, the expected value is generally written as
∞
X
u(n)p(n)
n=1
So Bernoulli kind of solved the problem. Well, he solved this problem, but he didn’t solve every problem.
Something that could solve the problem is bounded utility, which was proposed by Lindley.
Bernoulli’s approach solves the Saint Petersburg Paradox, but does not solve all possible paradoxes of this
kind. For that, we need a bounded utility function. That is:
∞
X
u(n)p(n) ≤ c
n=1
∞
X
p(n) = c
n=1
if u(n) ≤ c, ∀n. Ultimately, you should consider a utility function that increases to some asymptotic value.
Expected Utility Principle
• Define a set of outcomes or rewards.
1
• Define a set of states of nature Θ.
• Define an action as a function a from Θ to Z.
• Define the set of possible actions as A.
• The utility of a reward is measured by a real-valued function u(z).
• The states of nature happen according to a probability distribution π(Θ).
R
We can think of actions as probability distributions on z by setting pa (z) = θ:a(θ)=z π(θ)dθ. We then refer
to actions as lotteries.
The expected utility of an action a is
Z
U (a) =
u(a(θ))π(θ)dθ
Θ
Assuming that Z is finite, we have that the expected utility is
X
U (a) =
p(z)u(z)
z∈Z
We say that a∗ is optimal if a∗ = arg max Uπ (a). This is the expected utility principle.
Is it possible to deduce the expected utility principle from a set of axioms?
• Denote ≺ a binary preference operator. Thus, a ≺ a0 means that “action a0 is preferred to action a.
• Denote an operator that indicates indifference. Thus, a a0 means that action a and action a0 are
equivalent.
• Denote an operator for preference or indifference. Thus, a a0 means a0 is preferred to or the same
as a
We assume:
• Completeness: for any two actions, a, a0 ∈ A, one and only one of these is true: a ≺ a0 or a a0 , or neither
• Transitivity: For any a, a0 , and a00 ∈ A, a a0 , and a0 a00 ⇒ a a00 .
• Notation: a, a0 ∈ A, α ∈ [0, 1], and a00 = αa + (1 − α)a0 is the action that assigns probability
αpa (z) + (1 − α)pa0 (z) to z ∈ Z.
And now the axiomatic deduction:
1. ≺ is complete and transitive
2. Independence: a, a0 , a00 ∈ A, α ∈ [0, 1]. If a a0 ⇒ (1 − α)a00 + αa (1 − α)a00 + αa0 .
3. a, a0 , a00 ∈ A with a a0 a00 . There exist α, β ∈ [0, 1] such that
αa + (1 − α)a00 a0 βa + (1 − β)a00
Theorem - Von Neumann-Morgenstern Theory of Utility
1, 2, and 3 are true if and only if there exists a function a such that, for every pair of actions a and a0 ,
X
X
a a0 ⇐⇒
pa (z)u(z) >
Pa0 (z)u(z)
z∈Z
z∈Z
which can also be written as
0
Z
a a ⇐⇒
Z
u(a(θ))π(θ)dθ >
Θ
Θ
2
u(a0 (θ))π(θ)dθ