The idea is the same as in choice under certainty. There we defined the utility of every basket on the diagonal, $u(x,x)=x$, and gave any other basket the utility of the diagonal basket it is indifferent to. Under expected utility the ruler is no longer the diagonal but the family of lotteries between the best and the worst prize.
This is the von Neumann–Morgenstern construction turned into a question, and it is the standard way of measuring utility in health economics.
Example. Three prizes. Ann's choices between sure prizes reveal the ranking: San Pellegrino $\succ$ KitKat $\succ$ banana.
Same trick, new unknown. In the standard model beliefs are probabilities. Ann's probability that it rains in Chapel Hill tomorrow is not written anywhere: we extract it from her choices, one indifference at a time.
Notice that $u$ cancels: all we used is $u(b)\gt u(w)$. So we do not need to know Ann's utility to extract her beliefs. Any two prizes she can rank will do.
Example. Prizes as before, San Pellegrino $\succ$ banana, and $A$ = rain in Chapel Hill tomorrow.
In the standard model, individuals update their (probabilities) beliefs using:
Later, we shall see how beliefs (prior probabilities) are “formed” in the standard model…
Let's consider two alternative hypotheses regarding some coin:
Ann and Bart believe that only these two hypotheses are possible but they ascribe different likelihoods. Ann's prior probabilities are $\Pr^A[U]=\Pr^A[B]=1/2$ and Bart's are $\Pr^B[U]=1/3$ and $\Pr^B[B]=2/3$.
Let $p_t(h^t)$ be Ann's belief that the hypothesis $U$ is true given that the particular history of coin tosses, $h^t$, was observed. We have that: $$p_0(\emptyset)=1/2,$$ $$p_1(T)=\Pr^A[U\mid T]=\dfrac{\Pr^A[U\cap T]}{\Pr^A[T]}=\dfrac{p_0\cdot \tfrac12}{p_0\cdot \tfrac12 +(1-p_0)\cdot \tfrac14},$$ $$p_t(h^{t-1},T)=\dfrac{p_{t-1}(h^{t-1})\cdot \tfrac12}{p_{t-1}(h^{t-1})\cdot \tfrac12 +(1-p_{t-1}(h^{t-1}))\cdot \tfrac14}.$$
In sum: given some coin toss outcome $H$ or $T$ and knowing the prior probability of the hypothesis $U$ being true, we are able to compute the posterior (updated) probability that the hypothesis $U$ is true.
Result: If two individuals assign positive probabilities to the same set of hypotheses, it does not matter if their priors differ. In the limit, as more information becomes available, their posteriors will merge. They will think alike.
Heuristics are rules of thumb that deliver satisfactory outcomes without requiring high computational costs.
Common heuristics used to evaluate probabilities:
After a plane crash makes the news, people overestimate the risk of flying: a crash is vivid and easy to recall, the car accidents of the same week are not.
Are there more English words that start with K, or that have K as the third letter? Most people say “start with K”, because those are easier to think of. There are about twice as many with K in third place.
Linda is 31, single, outspoken, and was a philosophy major deeply concerned with social justice. Which is more likely?
(a) Linda is a bank teller.
(b) Linda is a bank teller and active in the feminist movement.
Most people pick (b). But (b) is a special case of (a): every Linda in (b) is also in (a), so (b) cannot be more likely.
In a city, 85% of the cabs are Green and 15% are Blue. A cab was involved in a hit-and-run at night. A witness says it was Blue. Tested under the same conditions, the witness identifies the color correctly 80% of the time. How likely is it that the cab was Blue?
Most people answer about $0.8$. Bayes' rule gives $$\begin{aligned}&\Pr[\text{Blue}\mid\text{says Blue}]\\ &=\dfrac{0.15\cdot 0.8}{0.15\cdot 0.8+0.85\cdot 0.2}=\dfrac{0.12}{0.29}\approx 0.41.\end{aligned}$$ The base rate, $15\%$ Blue, was ignored.
Gambler's fallacy. After five reds in a row at roulette, black feels “due”. The wheel has no memory: black is still $18/38$.
Hot hand. A basketball player who hit three shots in a row is “hot”, so the next shot feels more likely to go in than her season average.
Both come from the same belief: a short run must balance out (fallacy) or must continue (hot hand), when a short run is just what randomness looks like.
Ask a class to predict their exam score, to rank themselves in the class, and to give a range they are 90% sure contains the class average. Most predict too high, most rank themselves above the median, and most ranges miss.
The planning fallacy. Professor Parreiras predicts a quick walk from Gardner to Fetzer Hall and arrives late to class.
The better-than-average effect: most drivers rate themselves above the median driver, which cannot be true of most drivers.
Ask people for a range they are 90% sure contains the length of the Nile. Fewer than half of the ranges contain it.
The hard-easy effect: on hard tasks people overestimate their score but rank themselves too low; on easy tasks the reverse. Ask who is below average at juggling and most hands go up, which cannot be true either.
Two students read the same file about a car accident, one as the injured party and one as the driver, and are asked what a fair settlement would be. Each expects the judge to side with them. The further apart their answers, the more often they fail to settle, and both lose.
Depressive realism. Students press a button and a light sometimes comes on; afterwards they rate how much control they had over it. Non-depressed students overrate their control. Depressed students, whose flinch is weaker, get it right: sadder but wiser.
Supporters and opponents of the death penalty read the same two studies, one finding that it deters crime and one finding that it does not. Each side rates the study on its side as the better one, and both leave more convinced than they came.
A good grade is deserved, a bad one is an unfair exam. Investors credit their skill for gains and blame the market for losses, so after a run of gains they trade more, and their returns fall.
People asked on a hot day believe more in global warming than people asked on a cold day. Thirsty people rate a drought as more likely.
Cold-weather gear ordered on an unusually cold day is returned more often once the weather turns. Shopping when hungry fills the cart with more than you will eat.
A lucky shirt on game day, knocking on wood, not saying the name of the disease.
Students who skipped the reading feel more likely to be called on in class. Saying “I have never had a flat tire” feels like asking for one.
People refuse a glass of juice a sterilized cockroach was dipped in, and will not wear a freshly laundered sweater that once belonged to a murderer.