Columns are different things | |||
Option | 1 | 2 | 3 |
A | uA(1) | uA(2) | uA(3) |
B | uB(1) | uB(2) | uB(3) |
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 (... 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.''
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
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
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?"
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?"
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EU(Test1)=(.10)(-10) + (.10)(-100) = -11
EU(Test2)=(.25)(-10) + (0)(-100) = -2.5
but Test 2 leads to more errors
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EU(Test1)=(.10)(-10) + (.10)(-100) = -11
EU(Test2)=(.25)(-10) + (0)(-100) = -2.5
but Test 2 leads to more errors
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
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2: It is implied by certain axioms or principles:
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 |
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
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."
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
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.
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.