Utility theory

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Notes for CS 8803 - Game Theory and Computer Science. Spring 2008

A good motivating example is the St. Petersburg Paradox. There is a set E of events. A preference relation \geq_R is a binary relation that is complete, transitive, and reflexive. We assume we have a preference relation \geq_R on the set of events. We denote A \geq_R B \wedge B \geq_R A by A = RB. If A \geq_R B and not B \geq_R A, then we write A > RB. We also consider randomized events. For r\in [0,1] and A,B \in E, a lottery,

rA+(1-r)B \in E

means the event where A is chosen with probability r and B with the remaining probability. We assume that every lotter is an event in E as well.

Roughly speaking, an event is what you get, and should capture your total happiness, including things that happened before. This may include chance events. Economically, it is typically used to model preferences over bundles of goods/money or lotteries over these.

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von Neumann-Morgenstern utility

A von Neumann-Morgenstern utility function u:E \rightarrow \reals for \geq_R is one that satisfies A \geq_R B \Leftrightarrow u(A)\geq u(B) for all A,B \in E. Moreover, one assigns the value ru(A) + (1 − r)u(B) to the chance event that one receives A with probability r and B with probability 1 − r. Why should we assume that you have such a utility function? One can actually give a set of preference axioms that imply that you have such a preference.

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Axiomatic formulation

This example illustrates how axioms can be used to motivate a concept.

The requirements on preferences over lotteries are as follows:

  1. rA + (1 − r)B = (1 − r)B + rA
  2. rA + (1 − r)(sB + (1 − s)C) = rA + (1 − r)sB + (1 − r)(1 − s)C.
  3. rA + (1 − r)A = A.
  4. If A = RB, then for any r \in [0,1], C \in E,~~ rA+(1-r)C =_R rB+(1-r)C.
  5. If A > RB, then for any r \in (0,1], C \in E,~~ rA+(1-r)C >_R rB + (1-r)C.
  6. Suppose A > RB > RC. Then there exists some r \in [0,1] such that,
rA + (1 − r)C = RB.

The last requirement is called continuity.


The theorem is the following:

Theorem. Suppose we have a preference relation \geq_R that satisfies axioms 1-6 above. Then there exists a von-Neumann Morgenstern utility function u:E \rightarrow \reals for the preferences. Moreover, u is unique up to linear transformation. In other words, if there is another von-Neumann morgenstern utility function u':E \rightarrow \reals

for \geq_R, then there must exist reals a > 0,b, such that u(A) = au'(A) + b, for all A \in E.

The proof of this is relatively straightforward, but quite tedious.



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Logarithmic utility for money

In this section, I will motivate a particular utility function. The one that assigns u(x) = logx. 

Would you take a gamble that paid +$120 with probability 1/2, and -$100, with probability 1/2?

The real answer to this question may depend on how much money you have. If you have less than $100, for example, it may be a mistake. If you have $200, then it represents a gamble between totals $320 or $100. In this case, the question can be phrased as: 

Would you take a gamble (on your entire wealth) that paid +60% or -40% with equal probability?

This may seem like a good idea, because (1.6x + 0.6x) / 2 = 1.1x, i.e., you have a 10% expected earning. However, what would happen if you repeatedly accepted such gambles on your entire wealth? After t times, you most likely would have about 1.6t / 20.6t / 2 = 0.96t / 2 money. This is not the expected value, but what we can say is that with very very high probability, you would have exponentially less money than you started with, asymptotically in t. If you played forever, with probability 1 your wealth would approach 0.

This follows from the Chernoff/Hoeffding bounds. In particular, let s be the number of times where the money went up minus the number of times where the money went down. (So it went up t / 2 + s / 2 times and down t / 2 − s / 2 times.) Then your wealth is:

1.6t / 2 + s / 20.6t / 2 − s / 2 = 0.96t / 2(1.6 / .6)s / 2.

However, Hoeffding bounds say that with probability \geq 1-e^{-k^2/2}, s\leq k\sqrt{t} so,

your wealth is \leq 0.96^{t/2}(2.7)^{k\sqrt{t}/2}.

What is a good rule for determining whether such a gamble is good? Well, given a probability distribution over multiplicative gains ri with probabilities p_i \geq 0, i=1,2,\ldots,n, after t days, a similar calculation gives a gain:

\prod_{i=1}^n r_i^{t p_i}=\left(\prod r_i^{p_i}\right)^t = e^{t \sum p_i \log r_i}.

If we simply assign a logarithmic utility function to wealth and accept a gamble iff it leads to a non-negative change in expected utility, then we will decline a gamble if and only if it leads to bankruptcy with probability 1 asymptotically in t.


The production of this material was supported in part by NSF award SES-0734780.

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