April 8 Outline

Expected Utility Theory and Prospect Theory

April 8, 1999

Why is expected utility the "right" thing to do?

Not at all clear why one should choose the option with highest expected utility if you’re only going to play once

Basics of Axiomatic EUT

Start with a set of raw consequences {a,b,c,…}. For example: apple, banana, cookie

Consider choices among all lottery mixtures of the raw consequences

Lottery mixture (A,p;B) = "You get A with prob. p; otherwise you get B"

Given certain properties (axioms) hold among your choices between lotteries, then it’s as if you choose by picking the lottery with highest expected utility

Transitivity

If A is preferred to B, and B is preferred to C, then A is preferred to C

In symbols:

If A >*B, and B >*C, then A>*C

Intuition: Implies a consistent ordering of the lotteries (and raw consequences too)

Solvability/Continuity

If A >* B >* C, then there’s some probability p between 0 and 1 such that

B ~ (A,p;C)

Can mix best and worst options in a lottery so that you’re indifferent between the lottery and the middle option

Intuition: can trade-off value and probability

Independence/Cancellation

If A>*B, then (A,p;C) >* (B,p;C), for any possible C

Intuition: If you like apples better than bananas, you’ll choose any lottery that pays off in apples over an equivalent lottery that pays off in bananas

Invariance / Compound Lotteries

((A,p;B),q;B) ~ (A,pq;B)

Multi-stage lotteries treated as same as equivalent single-stage lotteries

Intuition: should follow rules of probability theory

Representation Theorem

If these axioms hold, then there exists (mathematically) a utility function u(.), defined for each of the raw consequences, and choices between lotteries are based on choosing the one with highest expected utility

Comments

Utility is extracted from choices

Utility represents the way value and probability interact

Utility need not correspond to subjective value (but fortunately it does most of the time)

No circularity / tautology. Choices are rational if they obey the axioms.

Utility provides a common scale for comparing different outcomes

Who cares? What good is it?

Can be used to predict or guide behavior

Decision analysis

Can test individual axioms, and build better models of behavior by cutting/pasting/tweaking different axioms

Subjective expected utility

Descriptive Problems with EUT

Asian disease expected to kill 600 people

Program A: 200 people will be saved [72%]

Program B: 1/3 chance 600 will be saved, 2/3 chance no one will be saved [28%]

Program C: 400 people will die [22%]

Program D: 1/3 chance nobody will die, 2/3 chance 600 people will die [78%]

Descriptive Problems with EUT

A: sure gain of $240 [82%]

B: 25% chance to gain $1000; 75% to gain nothing [18%]

C: sure loss of $750 [27%]

D: 75% chance to lose $1000; 25% to lose nothing [73%]

Prospect theory value function

Three key features

Evaluate gains and losses relative to a reference point

Risk-averse for gains, risk-seeking for losses

Loss aversion: losses loom larger than gains