This week we have been looking at Expected Utility Theory (EUT) and its alternatives. Many people have challenged the empirical and theoretical basis for EUT. For instance, Matthew Rabin (Econometrica 2000) claims that EUT has implausible implications. For instance, if an individual would prefer $0 to playing a lottery with a 50% chance of losing $100 and 50% chance of winning $110, this implies that this same person would refuse a play lottery that had a 50% chance of losing $800 and a 50% chance of winning $2090. The person also would refuse to play a lottery with a 50% of losing $1000 and a 50% chance of winning an infinite amount of money.

**What are the alternatives to EUT?**

An article by Chris Starmer (JEL 2000) outlines some of the alternative theories to EUT. Earlier this week we discussed Prospect Theory so I will not review that theory now. Here are some other alternatives.

UCSD professor Mark Machina is well known for his work on **generalized expected utility** analysis. In Machina (Econometrica 1982), he extends creates the following expected utility framework:

- V(
**q**)= Σ U(x_{i})p_{i} - where U(x
_{i})=∂[V(q)]/∂p_{i}**;**

**q**is the vector of probabilities (p_{1}, p_{2},…,p_{n}) corresponding to outcomes (x_{1}, x_{2},…,x_{n})

Machina showed that “…standard expected utility results (e.g.: risk aversion *iff* concavity of U(.)) also hold for the probability derivatives U(x_{i};q)= ∂V(**q**)/∂p_{i} of smooth non-expected utility preference functions V(.), so that U(.;**q**) can be thought of as the ‘local utility function’ of V(.) about **q**. For example, the property ‘concavity of U(.;**q**) at every **q**‘ is equivalent to global risk aversion of V(.).” Hypothesis II of the Machina paper has indifference curves which “fan out” in the Machina triangle and imply that individuals “become more risk averse as the prospects they face get better.”

**Regret Theory** was developed by David Bell (1985) and Loomes and Sugden (1986). Loomes and Sugden create the following utility function:

- V(
**q**)=Σ_{i}p_{i}[u(x_{i})+D(u(x_{i})-U)]

The function *D(.)* is a measure of disappointment. *U* is the prior expectation of the utility from the prospect. This theory takes into account that an agent may prefer to take $100 for sure rather than a lottery where 50% of the time one wins $200 and 50% of the time one wins $50, because the agent will fell much disappointment from receiving $50 if U=$100.

**Rank-Dependent Expected Utility **uses decision weighting–like those used in Prospect Theory–to give weights to different probabilities π(p). This model was first proposed by Quiggin (1982). Weights are attached “to any consequence of a prospect depends not only on the true probability of that consequence but also on its ranking relative to the other outcomes of the prospect…With consequences indexed…such that x_{1} is worst and x_{n} best, we can state rank-dependent expected utility theory as the hypothesis that agents maximize the decision weighted form with weights for i = 1,…, n – 1 given by

- w
_{i}=π(p_{i}+…+p_{n}) – π(p_{i+1}+…+p_{n}) - w
_{n}=π(p_{n})

Thus, individuals are using decision weights to compare the probabilities of doing better than an outcomes x_{i} and x_{i+1}. This procedure allows for extreme outcomes to have either very high or very low weights. However, “a small change in the value of some outcome of a prospect can have a dramatic effect on its decision weight if the change affects the rank order of the consequence; but a change in the value of an outcome, no

matter how large the change, can have no affect on the decision weight if it does not alter its rank.”

**Empirical Testing**

Which of these theories is best? Hey and Orme (Econometrica 1994) try to test these various theories empirically to see which best fits the data. Numerous theories are tested and there is no clear cut winner, but the paper is a very interesting read.

- Chris Starmer (2000) “Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk” Journal of Economic Literature, Vol. 38, No. 2., pp. 332-382.
- Hey and Orme (1994) “Investigating Generalizations of Expected Utility Theory Using Experimental Data”
*Econometrica*, Vol. 62, No. 6., pp. 1291-1326.