Название: Probability and Statistical Inference
Автор: Robert Bartoszynski
Издательство: John Wiley & Sons Limited
Жанр: Математика
isbn: 9781119243823
isbn:
which means that
A number of experiments on selected objects will allow us to estimate the utilities, potentially with an arbitrary accuracy (taking two particular objects as zero and a unit of the utility scale). In turn, if we know the utilities, we can determine the subjective probability of any event
which gives
The only problem lies in finding an event
Let us remark at this point that the system of Savage involves determining first an event with probability
A natural question arises: Are the three axioms of probability theory satisfied here (at least in their finite versions, without countable additivity)? On the one hand, this is the empirical question: The probabilities of various events can be determined numerically (for a given person), and then used to check whether the axioms hold. On the other hand, a superficial glance could lead one to conclude that there is no reason why person X's probabilities should obey any axioms: After all, subjective probabilities that do not satisfy probability axioms are not logically inconsistent.
However, there is a reason why a person's subjective probabilities should satisfy the axioms. For any axiom violated by the subjective probability of X (and X accepts the principle of SEU), one could design a bet that appears favorable to X (hence a bet that he will accept), but yet the bet is such that X is sure to lose.
Indeed, suppose first that the probability of some event
so that SEU is positive for a large enough
Suppose now that
If
Problems
1 2.7.1 Peter and Tom attend the same college. One day Tom buys a ticket for a rock concert. Tickets are already sold out and are in great demand. Peter, who does not have a ticket, agrees to play the following game with Tom. For a fee of $25, Peter will toss a coin three times and receive the ticket if all tosses show up heads. Otherwise, for an additional fee of $50, Peter will toss a coin two more times and receive the ticket if both tosses show up heads. If not, then for an additional fee of $100, Peter will toss a coin and receive the ticket if the toss shows up heads. Otherwise, all money will be given to Tom, and he also will keep the ticket. Assuming that the coin is fair, subjective probabilities of various outcomes coincide with objective probabilities, and that Peter's utility is linear in money, show that Peter's utility of the ticket exceeds $200.
2 2.7.2 Refer to Problem 2.7.1. Tom would agree on the following conditions: Peter pays him $50 and tosses a coin, winning the ticket if it comes up heads, and otherwise losing $50. In such a situation, should they both agree that Peter buys the ticket from Tom for $150?
3 2.7.3 Suppose that Tom is confronted СКАЧАТЬ