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I need help calculating the max expected utility. I want to LEARN this stuff, so if I can be so picky please give some explanation with answers (just an answer won't do me any good).

Let P(x) = 0.2 for a Boolean Random Variable X.

Assume that you have to make a decision D=1 or 2, leading to different utilities, both functions of X:

U(D=1,x)= 400, and U(D=1, ~x)= 2

U(D=2,x)= 20, and U(D=2, ~x)= 100

Compute the Expected Utilities and state what choice you would make.

First attempt at a solution.

If I choose D=1, then...

0.2 * (400) + 0.8 * (2) = 81.6

If I choose D=2 then...

0.2 * (20) + 0.8 * (100) = 84

So I would choose D=2. But I have little confidence in that though because this is the first time I've done anything like this.

CamHart
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    Looks good to me! You're doing the exact right thing. You are seeking to maximize what you expect to get. Even though $U(D = 1, x) = 400$ looks nice, it is unlikely you will get it. – ml0105 Dec 01 '14 at 18:46

1 Answers1

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This looks right. It often helps to create a table (decision matrix) for this sort problem where you put the decisions along one axis then the outcomes along another and the utility payoffs in the corresponding table entries. For example for this problem you might write

|         | D=1   | D=2  |
| --------|-------| -----|
|  X,P=0.2| 400   |  20  |
| ~X,P=0.8|   2   | 100  |

Then you'll have that the expected utility of choosing decision 1 is the sum of the entries in the $D=1$ column weighted by their respective probabilities. The expected utility of choosing decision 2 is computed similarly by looking at column 2. Hence as you have shown $$ u(D=1)=0.2(400)+0.8(2)=81.6 $$ $$ u(D=2)=0.2(20)+0.8(100)=84. $$ So to maximize expected utility you should pick 2. This representation is particularly useful when you're trying to analyze other decision strategies. For example you can apply the minimax principle to this problem simply by picking the column whose minimal entry is the largest out of the minimal entries of the other columns. In this case, you can see that's decision 2. You can also easily check if one decision dominates the other by comparing columns side by side.

CalebKoch
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