There are two cylinders I and II in front of you, both of which contain 100 red balls and black balls. You are told that the number of red balls in the second cylinder is 50, and the number of red balls in the first cylinder is unknown. If a red ball or a black ball is taken out from cylinder I and cylinder II respectively, it will be marked as red I, black I, red II and black II respectively. Now take a ball from these two tanks, and let you guess the color of the ball before taking it out. If your guess is correct, then you will get 100 dollars. If your guess is wrong, then you will get nothing. In order to determine your subjective preference order, you need to answer the following questions:
(1) Do you prefer the looks of red I and black I, or are you unbiased about their looks?
(2) Do you prefer to bet red or black?
(3) Do you prefer to bet on Red One or Red Two?
(4) Do you prefer black I or black II?
Ellsberg found that most people's answers to questions 1 and 2 were impartial. But the answer to question 3 is more inclined to the appearance of red, and the answer to question 4 is more inclined to the appearance of black.
He thinks, according to Savage's theory, if you bet on Red Two, then as an observer, you will experimentally infer that you think the appearance of Red Two is more likely than Red One ... At the same time, if you bet on Black Two, you can infer that you think Black Two is more likely than Black One, but according to the knowledge of probability, we know that this is impossible, because if Black Two is more likely than Black One, then Red One must be more likely than Red Two. Therefore, in the case of uncertainty, the subjective probability can not be assigned, nor can any probability measure be determined.
Another example given by ellsberg directly aims at the principle of certainty, which is expressed as follows:
There are 30 red balls and 60 black balls and yellow balls in a jar, and the proportion is unknown. Now take a ball out of the jar and let people choose four behaviors in the following two situations.
Behavior I is betting on red balls. When a red ball is taken out, it can get 100 dollars, while balls of other colors get nothing.
Behavior 2 is betting on black balls. A black ball can get 100 dollars, and balls of other colors can get nothing.
Behavior III is betting on red or yellow balls. When the red ball and the yellow ball are taken out, they can get $65,438+000 respectively, while when the black ball is taken out, they can get nothing.
Behavior Ⅳ is betting on black or yellow balls. When the black ball and the yellow ball are taken out, they can get $65,438+000 respectively, while the red ball gets nothing.