Risk, ambiguity and the Savage axioms.

*(English)*Zbl 1280.91045The author describes the following well-known experiment: An urn contains 90 balls 30 of which are red. The others are yellow or black in unknown proportion. The partitipants have to choose between two gambles:

(A) they win 100$ drawing a red ball and 0$ drawing a yellow or a black one, or

(B) they win 100$ drawing a yellow ball and 0$ drawing a red or a black one.

Here, the majority of participants prefer gamble (A) to (B).

Subsequently, these gambles are modified in such a way that black in both cases means success, too:

(C) they win 100$ drawing a red or black ball and 0$ drawing a yellow one, or

(D) they win 100$ drawing a yellow or black ball and 0$ drawing a red one.

Now the vast majority chooses gamble (D). This is in apparent contradiction to the earlier decision for gamble (A), since the black ball in (C) as well as in (D) now means more money, so it makes no difference (hence the designation as a paradox).

Ellsberg explains this result by the distinction between risk and uncertainty (ambiguity): ‘risk’ means that the probabilities are known (examples are: classical experiments such as dice, roulette, etc.) whereas ‘uncertainty’ means unknown probabilities. The typical participant suspects “cautiously” that the proportion of the yellow and black balls could turn to her disadvantage and both times decides for the known risk (chance of winning 1/3 in the first round, 2/3 in the second).

Contents: I. Are there uncertainties that are not risks? II. Uncertainties that are not risks. III. Why are some uncertainties not risks?

(A) they win 100$ drawing a red ball and 0$ drawing a yellow or a black one, or

(B) they win 100$ drawing a yellow ball and 0$ drawing a red or a black one.

Here, the majority of participants prefer gamble (A) to (B).

Subsequently, these gambles are modified in such a way that black in both cases means success, too:

(C) they win 100$ drawing a red or black ball and 0$ drawing a yellow one, or

(D) they win 100$ drawing a yellow or black ball and 0$ drawing a red one.

Now the vast majority chooses gamble (D). This is in apparent contradiction to the earlier decision for gamble (A), since the black ball in (C) as well as in (D) now means more money, so it makes no difference (hence the designation as a paradox).

Ellsberg explains this result by the distinction between risk and uncertainty (ambiguity): ‘risk’ means that the probabilities are known (examples are: classical experiments such as dice, roulette, etc.) whereas ‘uncertainty’ means unknown probabilities. The typical participant suspects “cautiously” that the proportion of the yellow and black balls could turn to her disadvantage and both times decides for the known risk (chance of winning 1/3 in the first round, 2/3 in the second).

Contents: I. Are there uncertainties that are not risks? II. Uncertainties that are not risks. III. Why are some uncertainties not risks?

##### MSC:

91B06 | Decision theory |

91B16 | Utility theory |

62A01 | Foundations and philosophical topics in statistics |

91A90 | Experimental studies |

91A60 | Probabilistic games; gambling |