1
Utility can be concave in money
The first thousand is worth more to you than the ten-thousandth. When each extra unit adds less, the utility curve bends, and averaging over outcomes then gives less than the utility of the average.
Lesson · Decision and Rationality · Risk
Risk Aversion and the Value of Information
Expected utility scores utilities, not sums of money, and when utility rises more slowly than wealth a sure amount can beat a fair gamble on the same average. The same machinery prices a perfect forecast as the expected best payoff minus the best expected payoff, which is the most any forecast can be worth.
u(wealth) rises, but ever more slowly
fair bet: average money the same, EU lower
refusing a fair bet can be consistent
VPI = expected best − best expected
no forecast is worth more than the VPI
Utility is not money
Core ideas
What this lesson establishes
Two ideas share one mechanism: what a payoff is worth is a separate fact from how large it is, and what knowing the state is worth comes out of the same table.
2
A fair bet is fair in money only
A coin flip for plus or minus 96 around 100 has the same average wealth as keeping 100. In utility, with a concave curve, it has a lower average, so the sure thing wins.
3
Information is worth the gain from acting on the state
Learning the state before you choose lets you take the best act in each state. The value of perfect information is the expected best payoff minus the best expected payoff.
The material
A curve that bends, and a forecast with a price
First why a fair gamble can be refused, then how much a forecast is worth.
A fair bet is one whose average gain is zero. A person who has expected utility as their rule, and scores money at face value, would be indifferent to a fair bet, so refusing it would be a mistake. That is a consequence of scoring money at face value and not of expected utility itself, which scores utilities.
Definition · Risk aversion
A chooser is risk averse when she prefers a sure amount to a gamble with the same average amount. With expected utility, this holds exactly when the utility of wealth is concave: it rises, but each extra unit adds less.
Take utility equal to the square root of wealth. Wealth 100 has utility 10, wealth 196 has utility 14 and wealth 4 has utility 2. A coin flip between 196 and 4 averages 100, the same as keeping 100 for sure. Its expected utility is 14·(1/2) + 2·(1/2) = 8, which is below the 10 of keeping. The fair-gamble preset enters exactly these utilities, and the rule prefers to keep. The flip tile shows how far the coin would have to be bent: gambling wins only when heads has probability above 2/3.
There is nothing irrational in that preference unless the utility of money is a straight line, and nothing in expected utility requires that. What expected utility does require is that one set of utilities explains all the choices, a demand “The Allais Paradox and the Sure-Thing Principle” puts to the test.
Example · Insurance
Wealth is 100 and a fire, with probability 1/4, would cost 75. Cover costs 19. Insured, you hold 81 whatever happens, utility 9. Uninsured you hold 100, utility 10, or 25, utility 5. Insuring scores 9 and risking scores 10·(3/4) + 5·(1/4) = 35/4 = 8.75, so insuring wins.
That example is worth a second look: the cover costs 19 and the expected loss is 18.75, so in money the policy is slightly unfavourable. It is still worth buying, because the lost 75 comes out of a smaller pot, where each unit is worth more to you. A risk-neutral chooser, scoring money at face value, would decline. The insurance preset enters the utilities and the lab recommends insuring, and its last tile says the cover is worth buying whenever the chance of fire is above 1/5.
Definition · Value of perfect information
Let the best expected payoff be the highest expected utility of any act now. Let the expected best be the average, over states, of the best payoff available in each state. Their difference is the value of perfect information.
On the umbrella with rain at 1/3, take is best in rain at 2 and leave is best when dry at 3. The expected best is 2·(1/3) + 3·(2/3) = 8/3. The best expected is 4/3, from take. So a forecast that never misses is worth 4/3 in the same units as the payoffs.
The figure is a ceiling. A forecast that is sometimes wrong gives less than a perfect one, so no forecast is worth more than 4/3 here, and paying more for one is a mistake whatever it promises. If one act is best in every state, nothing can be gained by knowing the state and the value is zero.
Interactive
A decision table under every rule
the same payoffs, eight ways of choosing, each computed exactly
Method
Pricing a risk and a forecast
Convert to utilities first. Everything after that is the arithmetic of the last lesson.
1
Turn outcomes into utilities
Apply the utility curve to each wealth level before averaging. Averaging the money and converting afterwards gives a different, and wrong, answer.
2
Score each act by expected utility
Weight utilities by the state probabilities and add. Compare the sure act with the gamble.
3
Find the best payoff in each state
Look down each column for its highest entry and average these by the state probabilities. That is the expected best.
4
Subtract the best expected
Take the larger expected utility from the previous step away from the expected best. The remainder is the most any forecast could be worth.
Concave utility and the price of knowing
Score 0 / 0Choose the answer you can justify.
What a perfect forecast of rain is worth
Take pays 2 in rain and 1 when dry; leave pays −3 and 3. Rain has probability 1/3.
best in rain: take, 2 best when dry: leave, 3
expected best = 2·(1/3) + 3·(2/3) = 8/3
EU(take) = 4/3 EU(leave) = 1
best expected = 4/3
value of perfect information = 8/3 - 4/3 = 4/3
Without the forecast you take the umbrella and score 4/3. With it, you take the umbrella in rain and leave it when dry and score 8/3. The forecast is worth the difference, in the units of the payoffs, and an honest seller cannot charge more for a perfect one.
Common mistakes
1
Calling the refusal of a fair bet irrationalFair means the average money is zero, and expected utility averages utility, not money. With the square root of wealth the coin flip between 196 and 4 scores 8 and keeping 100 scores 10, so refusing is what the rule recommends. It would be a mistake only if the utility of money were a straight line, and nothing requires that.
2
Averaging the money and then converting to utilityThe utility of the average is not the average of the utilities. Convert each outcome first. Doing it the other way round erases the curvature that the whole example turns on.
3
Paying for a forecast because it is accurateA forecast is worth what acting on it gains, and no more than the perfect-information figure. If one act wins in every state, an accurate forecast is worth nothing, since you would choose the same act whatever it said.
Check your understanding
✓
Finish when you can show a sure thing beating a fair gamble and price a forecast.
Given a utility curve and a gamble, you should be able to compute both expected utilities and say which wins. Given a table with probabilities, you should be able to compute the expected best, the best expected, and the value of perfect information.
Note: The curve used here, the square root, is one choice among many concave curves, and the lab does not choose it for you. A different curve gives different numbers and the same shape of result. Every utility in this lesson is a number somebody chose.