Utility

Utility is:
• A numerical measure of goal achievement.
• A numerical measure of ``good''.
Utility isn't just:
• Pleasure.
• Money.
• Happiness.
• Outcomes that have ``usefulness'' for something else.

Aspects of utility theory:

• Expected-Utility Theory (uncertain states)
• Multiattribute Utility Theory (MAUT) (attributes)
• Utilitarianism (people)
• Intertemporal choice (times)
 Columns are different things Option 1 2 3 A uA(1) uA(2) uA(3) B uB(1) uB(2) uB(3)

Predicted utility, decision utility, experienced utility (Kahneman)

Predicted ------> Experienced
Do people know what will be good for them?

Predicted ------> Decision
Do people use their hedonic knowledge?

Decision --------> Experienced
Do people choose what will be good for them?

Redelmeier and Kahneman experiment:

Expected value theory

How should we combine utility and probability of gambles?

Expected value (... utility) of a monetary gamble:
\$4 for a heart, \$2 for a diamond, \$1 for anything else.
How much \$ should you `expect'?

If you did this many times, you could expect:
\$4 about 13 of every 52 times (so p(\$4)=13/52).
\$2 about 13 of every 52 times (so p(\$2)=13/52).
\$1 about 26 of every 52 times (so p(\$1)=26/52).
So...
(\$4 × 0.25) +
(\$2 × 0.25) +
(\$1 × 0.50) = \$2 each time on average.

In Mathematical symbols: EV = ∑ipivi

In English: ``To calculate the expected value of a gamble, multiply the value of each possible outcome by the probability of that outcome, and add up all the results.''

Pascal's wager

 State of the world Option God exists God does not exist Live Christian life Saved (very good) Small inconvenience Live otherwise Damned (very bad) Normal life

 Illustrative probabilities Option p(exist)=.1 p(not exist)=.9 Live Christian life 100 -1 Live otherwise -100 0

EU(Christian) = (.1)(100) + (.9)(-1) = 9.1
EU(otherwise) = (.1)(-100) + (.9)(0) = -10

Pascal's wager

 State of the world Option God exists God does not exist Live Christian life Saved (very good) Small inconvenience Live otherwise Damned (very bad) Normal life

 Illustrative probabilities Option p(exist)=.1 p(not exist)=.9 Live Christian life 100 -1 Live otherwise -100 0

EU(Christian) = (.1)(100) + (.9)(-1) = 9.1
EU(otherwise) = (.1)(-100) + (.9)(0) = -10

Application to sentencing

 Option p(innocent)=.1 p(guilty)=.9 Convict -100 0 Acquit 0 -5

EU(Acquit) - EU(Convict) = (.9)(-5) - (.1)(-100) = -4.5 + 10 = 5.5

but Convict would minimize error rate

Question: Is this the same as "For every false conviction we let 9 guilty people go free?"

Application to sentencing

 Option p(innocent)=.1 p(guilty)=.9 Convict -100 0 Acquit 0 -5

EU(Acquit) - EU(Convict) = (.9)(-5) - (.1)(-100) = -4.5 + 10 = 5.5

but Convict would minimize error rate

Question: Is this the same as "For every false conviction we let 9 guilty people go free?"

Application to testing

 TEST 1 Benign Malignant pos. .10 .40 neg. .40 .10
 TEST 2 Benign Malignant pos. .25 .50 neg. .25 0

U(miss)=-100, U(false alarm)=-10, others 0, so

EU(Test1)=(.10)(-10) + (.10)(-100) = -11
EU(Test2)=(.25)(-10) + (0)(-100) = -2.5

but Test 2 leads to more errors

Application to testing

 TEST 1 Benign Malignant pos. .10 .40 neg. .40 .10
 TEST 2 Benign Malignant pos. .25 .50 neg. .25 0

U(miss)=-100, U(false alarm)=-10, others 0, so

EU(Test1)=(.10)(-10) + (.10)(-100) = -11
EU(Test2)=(.25)(-10) + (0)(-100) = -2.5

but Test 2 leads to more errors

Properties of utility

• Utility is relative.
• Utility has no true zero.
• Utility is measured in arbitrary units.
 State of the world Option 1 2 3 A uA(1) uA(2) uA(3) B uB(1) uB(2) uB(3)

EU = ∑iPiUi

Here, the expected utility of option A is
 EUA = p(1)uA(1) + p(2)uA(2) + p(3)uA(3)
and the utility of option B is
 EUB = p(1)uB(1) + p(2)uB(2) + p(3)uB(3)
so the difference between the two utilities is
 EUA - EUB = p(1)[uA(1)-uB(1)] + p(2)[uA(2)-uB(2)] + p(3)[uA(3)-uB(3)]

Why is Expected-Utility Theory normative?

1: No other policy maximizes utility in the long run.

2: It is implied by certain axioms or principles:

• Weak ordering
• connectedness: A > B, or A = B, or A < B
• transitivity: A > B and B > C implies A > C
• Sure-thing principle:
Our preference for option A as compared to option B should be independent of anything which is the same regardless of which option we chose.
e.g., bananas or apples > oranges or apples implies bananas or pears > oranges or pears

The sure-thing principle

 State Option Win Lose Lottery 1 Europe V Lottery 2 Caribbean V

 30 balls 60 balls red black yellow Option X \$100 \$0 \$0 Option Y \$0 \$100 \$0 Option V \$100 \$0 \$100 Option W \$0 \$100 \$100

Ambiguity and omission bias (Ritov & Baron, 1990)

In the state you live in, there had been several epidemics of a certain kind of flu, which can be fatal to children under 3. ... 10 out of 10,000 children will die.

A vaccine for this kind of flu has been developed and tested. The vaccine eliminates the probability of getting the flu. The vaccine, however, might cause side effects that are also sometimes fatal.

Would you vaccinate your child [support a law requiring vaccination] if the overall death rate for vaccinated children were:

__ 0 in 10,000
__ 1 in 10,000
...
__ 10 in 10,000

Ambiguity manipulation

Basic case. "The children who die from the side effects of the vaccination are not necessarily the same ones who would die from the flu."

Risk group for vaccine. "Suppose it were discovered that 100 out of every 10,000 children were susceptible to death from the side effects of the vaccine (if any such deaths occur). Children who were not susceptible do not experience any adverse effects. The test to determine who is susceptible is not generally available and cannot be given."

Benjamin Franklin on vaccination

"In 1736 I lost one of my sons, a fine boy of four years old, by the small-pox, taken in the common way. I long regretted bitterly, and still regret that I had not given it to him by inoculation. This I mention for the sake of parents who omit that operation, on the supposition that they should never forgive themselves if a child died under it; my example showing that the regret may be the same either way, and that, therefore, the safer should be chosen."

The Utility of Money

What is the most you would pay for a chance to win \$20 if a coin comes up heads?
Peter tosses a coin and continues to do so until it should land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he does it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation. (Bernoulli, 1738/1954, p. 31)
```EV = (1/2)*1 + (1/4)*2 + (1/8)*4 + (1/16)*8 + .... =
=   1/2   +   1/2   +   1/2   +   1/2    + .... = infinity
```

Bentham on declining marginal utility

Of two people having unequal fortunes, he who has most wealth must by a legislator be regarded as having most happiness. But the quantity of happiness will not go on increasing in anything near the same proportion as the quantity of wealth: ten thousand times the quantity of wealth will not bring with it ten thousand times the quantity of happiness.

It will even be matter of doubt, whether ten thousand times the wealth will in general bring with it twice the happiness. The effect of wealth in the production of happiness goes on diminishing, as the quantity by which the wealth of one man exceeds that of another goes on increasing: In other words, the quantity of happiness produced by a particle of wealth (each particle being of the same magnitude) will be less at every particle; the second will produce less than the first, the third than the second, and so on.

Risk aversion

Graph illustrating the fact that the expected utility of a fair bet is less than the utility of not betting when the uility function is concave, because of "declining marginal utility" of money.

This is just one descriptive theory of risk aversion.
It is not necessarily the whole story.
It has implications for taxation policy.

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