1
Risk and ambiguity are different
Risk is a gamble with known probabilities, like red here. Ambiguity is a gamble whose probabilities are not known, like black. The prizes are the same; what differs is how much you know.
Lesson · Decision and Rationality · Risk
Ambiguity and the Ellsberg Urn
An urn holds 30 red balls and 60 that are black or yellow in an unknown mix. People prefer to bet on red rather than black, and to bet on black or yellow rather than red or yellow. Each choice needs a different estimate of the black share, so no one estimate supports both.
urn: 30 red, 60 black or yellow, mix unknown
I: bet red, or bet black
II: red or yellow, or black or yellow
red in I needs p(black) < 1/3
black or yellow in II needs p(black) > 1/3
both at once fit no single p
Known odds against unknown odds
Core ideas
What this lesson establishes
The puzzle does not ask whether you like risk. It asks whether you prefer to know the odds.
2
A single guess settles every bet
If you hold one credence p for the black share, each bet has a probability, and expected utility ranks all four bets from it. Choices that need two different values of p cannot come from one guess.
3
Worst-case rules can avoid the guess
A rule that scores a bet by its worst case over the possible mixes needs no single p, and it reproduces the popular pattern. It pays for that by leaving expected utility.
The material
The urn, the two bets, and the proportion that decides them
First the setup, then the algebra that shows no proportion fits, then the one rule that does.
Daniel Ellsberg described an urn of 90 balls in 1961. Thirty are red. The other 60 are black or yellow, in a proportion nobody has told you. One ball will be drawn. A bet pays 100 if its colour comes up and nothing otherwise, so on a scale where winning is worth 1 and losing 0, a bet is worth its probability.
In the first decision you choose between a bet on red and a bet on black. In the second you choose between a bet on red or yellow and a bet on black or yellow. Most people choose red in the first and black or yellow in the second. In both the pick is the bet whose chance is known: red is exactly 30 in 90, and black or yellow is exactly 60 in 90.
Let p be your credence that a drawn ball is black. Yellow is then 2/3 − p, since red is 1/3 and the three add to 1. Score each bet by its probability of winning.
bet chance of winning
red 1/3
black p
red or yellow 1/3 + (2/3 − p) = 1 − p
black or yellow p + (2/3 − p) = 2/3
Red beats black exactly when 1/3 > p. Black or yellow beats red or yellow exactly when 2/3 > 1 − p, which is p > 1/3. The popular pair needs p below a third and above a third. No credence satisfies both, and the lab shows it: the first preset puts black at 2/9 and red wins; the second puts black at 4/9 and black or yellow wins; no one table holds both.
Theorem · The pair has no credence
A strict preference for red over black needs p < 1/3. A strict preference for black or yellow over red or yellow needs p > 1/3. No single p does both, so the pattern is inconsistent with a probability of black, and with expected utility, whatever the utilities.
It is tempting to call this risk aversion. It is not. Both red and black pay the same prize on the same terms, so any utility for the prize multiplies both scores by the same positive factor and leaves their order alone. A concave utility cannot prefer red to black unless the probability of red is higher, and nothing in the urn says it is. The preference is for a known probability.
Example · Black at a quarter
Suppose you believe p = 1/4. Then red at 1/3 beats black at 1/4 in the first decision, and red or yellow at 3/4 beats black or yellow at 2/3 in the second. A believer in p = 1/4 bets on red twice. The popular pattern bets on red once and on black or yellow once.
One rule does reproduce the pattern. Let the possible mixes be 0, 30 or 60 black balls, and score each bet by its worst case over the three. Red wins one third in every mix, so its worst case is 1/3; black wins 0 in the first mix, so its worst case is 0. Red or yellow can fall to 1/3, while black or yellow wins 2/3 in every mix. The worst-case rule picks red in the first decision and black or yellow in the second.
The third preset sets this up, with one state per mix and each mix given probability 1/3. Under expected utility red and black tie, and red or yellow ties black or yellow; averaging over the mixes has erased the difference between known and unknown. Switch the rule to maximin and black or yellow comes out alone on top.
Which view to hold is the question. A defender of the pattern says ignorance is not the same as having a credence of one third, and a chooser who has no basis for any p has no reason to average. A critic answers that a credence is what betting behaviour reveals, and that a chooser who will not assign one to black has declined to treat the bets as bets. The lab computes each rule and takes no side.
Interactive
A decision table under every rule
the same payoffs, eight ways of choosing, each computed exactly
Method
Testing the Ellsberg pattern against any credence
Four steps, and the unknown stays a symbol until the last.
1
Name the unknown
Let p be the probability of black. Yellow is whatever is left after red at 1/3 and black, so 2/3 − p.
2
Score the four bets
Add the probabilities of every colour a bet wins on. A bet's expected utility is that sum, with winning scored 1.
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State each choice as an inequality
Red over black is 1/3 > p. Black or yellow over red or yellow is 2/3 > 1 − p. Solve each for p.
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Look for a p that satisfies both
If the solution sets do not overlap, the pair of choices fits no credence, and the pattern is not expected utility's.
A single credence against the pattern
Score 0 / 0Choose the answer you can justify.
The two decisions as inequalities in p
Let p be the black share. Red is 1/3 and yellow is 2/3 − p.
I: red wins 1/3, black wins p
red preferred when 1/3 > p
II: red or yellow wins 1 - p
black or yellow wins 2/3
black or yellow preferred when 2/3 > 1 - p
that is p > 1/3
both strict: p < 1/3 and p > 1/3
At p = 1/3 the bets tie in both decisions, so the credence in the middle is indifferent twice. The popular pattern is strict twice, in opposite directions, and for no credence is that possible.
Common mistakes
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Treating ambiguity aversion as risk aversionRisk aversion is a fact about how prizes are valued. Red and black offer the same prize on the same terms, so a curve for the prize multiplies both scores by the same factor and cannot separate them; change the payoff 1 to 5 in the lab and the order is unchanged. What separates them is whether the probability is known.
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Assuming ignorance means a credence of 1/3Giving each colour an equal share is a choice, not a consequence of ignorance. It turns the unknown into a known and then removes the preference. Someone who refuses to do so is not confused; the open question is whether they are consistent.
3
Reading the inconsistency as a fact about psychologyThe result is a statement about the pair of choices: no probability of black supports both. It does not say how many people choose this way or why, which is an empirical matter.
Check your understanding
✓
Finish when you can show that two choices fit no single credence.
Given the urn, you should be able to write each bet's chance in terms of one unknown, state each choice as an inequality, find the value at which the bets tie, and show that the popular pair needs the unknown to be on both sides of it.
Note: The urn is idealised: the red count is known and the rest is stipulated as unknown. The lab scores the table you enter and has no view on whether ignorance of a proportion should be treated as a probability. That is the question the lesson leaves open.