**APPENDIX A: EXPECTED UTILITY THEOREM**

Expected utility theory posits a set of rational choice axioms. If these assumptions are satisfied, then the expected utility theorem shows the existence of a utility function with the elegant property that the preferences over risky outcomes can be represented by the expected utility of these outcomes.

A binary investment opportunity, or a binary lottery, consists of three numbers: two outcomes and a probability of the first outcome. We denote this as *L* = {p,*x*,*y*}, where p is the probability of outcome *x* (so (*1-*p) is the probability of *y*).

We can also combine lotteries. This is called a compound lottery. Thus, the compound lottery *L* = {p,*L*1,*L*2} lets you win the lottery *L*1 with probability p, and *L*2 with probability (1-p). When we combine lotteries *L*1 and *L*2 to create *L*, we denote this as

Axioms

1. **Complete ordering axiom.** For any pair of lotteries *L*1 and *L*2, one of the following is true: *L*1 is strictly preferred to *L*2 , *L*2 is strictly preferred to *L*1, or the investor is indifferent between the two lotteries.

2. **Continuity axiom.** For three lotteries, assume that *L*1 is strictly preferred to *L*2, and *L*2 is strictly preferred to *L*3*.* Then there exists some probability p so that the investor is indifferent between *L*2 for sure and

.

3. **Independence axiom.** Suppose the investor is indifferent between lotteries *L*1 and *L*2. Let *L*3 be another lottery. Then, the investor is indifferent between

and

Similarly, if *L*1 is strictly preferred to *L*2, then the first combination is strictly preferred to the second.

4. **Unequal probability axiom.** Assume that *L*1 is strictly preferred to *L*2. If *L*i = {p ,* L*1, *L*2} and *L*k = {r, *L*1, *L*2 } then *L*i* * is strictly preferred to *L*k if and only if p > r.

The set of possible risky investment opportunities is assumed to be finite. Denote the most preferred (best) as *L*B and the least preferred (worst) as *L*W.

Expected Utility Theorem

If these axioms are satisfied, then there exists a utility function, U, such that for any two lotteries *L*1 and *L*2, *L*1 is preferred to *L*2 if and only if the expected utility from *L*1 is greater than that from *L*2,

Proof:

Set *U*(*L*B) = 1 and *U*(*L*w) = 0. Now, consider a lottery *L*1. According to the continuity axiom, there exists a probability, say p1, such that the investor is indifferent between *L*1 and the compound lottery opportunity (p1,*L*B,*L*w).

Set

Similarly, for any other lottery *L*2, we set *U*(*L*2) = p2, where the investor is indifferent between *L*2 and (p2 ,*L*B ,* L*w ).

We want to make sure that the way we are assigning utility numbers preserves the investor's preferences. From Axiom 1, the investor is either indifferent between *L**1* and *L*2 or strictly prefers *L*1 or strictly prefers *L*2. By Axiom 4, this will be the case only if respectively p1 = p2 , p1 > p2 , or p1 < p2, and so *U* preserves the preferences.

Finally, we need to show that we can compare any two compound lotteries simply by taking expected utilities. This requires us to show that if *L* = (p,* L*1,* L*2) is a compound lottery, then

To show this, we need the independence axiom. First, substitute (p1,* L**B*,* L**w*) for *L*1, and (p2,* L*B,* L*w* *) for *L*2. Then, the investor is indifferent between

and

which equals

which is

or

By construction U(*L*B* *)=1 and U(*L*w)=0. Therefore, this equals

which establishes the expected utility theorem.

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