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Thinking and Deciding

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EXPECTED-UTILITY THEORY 241<br />

are still sufficiently great that .01·u(being saved)+.99·u(the small inconvenience of<br />

living a Christian life if God does not exist) is greater than .01 · u(being damned)+<br />

.99 · u(a normal life).<br />

Another way of looking at this is to say that the utility difference between being<br />

saved <strong>and</strong> being damned is so great that it is worthwhile to live the Christian life,<br />

even if the probability of God’s existence is very small. We can use the difference<br />

between outcomes to determine expected utility. When we compare two options,<br />

we can compare them within each column in a utility table. We then multiply the<br />

difference between the two utilities by the probability of the state. The following<br />

table shows this in a general form, with two options, A <strong>and</strong> B.<br />

Here, the expected utility of option A is<br />

<strong>and</strong> the utility of option B is<br />

State of the world<br />

Option 1 2 3<br />

A uA (1) uA (2) uA (3)<br />

B uB (1) uB (2) uB (3)<br />

EUA = p(1)uA (1) + p(2)uA (2) + p(3)uA (3)<br />

EUB = p(1)uB (1) + p(2)uB (2) + p(3)uB (3)<br />

so the difference between the two utilities is<br />

EUA −EUB = p(1)[uA (1)−uB (1)]+p(2)[uA (2)−uB (2)]+p(3)[uA (3)−uB (3)]<br />

(10.3)<br />

It is the difference that determines which option is better. If the difference is positive,<br />

option A is better, <strong>and</strong>, if it is negative, B is better.<br />

You can see here why it does not matter which outcome is selected as the zero<br />

point. If we add a constant to everything in the same column, that constant gets<br />

multiplied by the probability <strong>and</strong> then added to the utilities of both options, <strong>and</strong> the<br />

difference between them is unchanged. For example, if we add C to both uA (3)<br />

<strong>and</strong> uB (3), then the rightmost term in equation 10.3 becomes p(3)[(uA (3) + C) −<br />

(uB (3) + C)], <strong>and</strong> the Cs cancel out to give back the same term that was there<br />

before. Naively, we tend to think of some outcomes as being favorable <strong>and</strong> others as<br />

unfavorable in themselves; however, outcomes are all relative. Thus, a well-known<br />

decision theorist, asked, “How’s your wife?” answered, “Compared to what?”<br />

The units do not matter either, provided that the same units are used throughout<br />

the whole table. Multiplying everything in the table by the same constant (<strong>and</strong> thus

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