Probability and Statistical Inference. Robert Bartoszynski
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Название: Probability and Statistical Inference

Автор: Robert Bartoszynski

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119243823

isbn:

СКАЧАТЬ lottery images to lottery images, then

equation

      which means that

equation equation

      which gives

equation

      The only problem lies in finding an event images with subjective probability images. Empirically, an event images has subjective probability images if, for any objects images and images, the person is indifferent between lotteries images and images. Such an event was found experimentally (Davidson et al., 1957). It is related to a toss of a die with three of the faces marked with the nonsense combination images, and the other three with the nonsense combination images (these combinations evoked the least number of associations).

      Let us remark at this point that the system of Savage involves determining first an event with probability images, then the utilities, and then the subjective probabilities. Luce and Krantz (1971) suggested an axiom system (leading to an appropriate scheme) that allows simultaneous determination of utilities and probabilities. The reader interested in these topics is referred to the monograph by Krantz et al. (1971).

      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 images is negative. Consider the bet (lottery) images, (i.e., a lottery in which X pays the sum images if images occurs and pays the sum images if images does not occur). We have here (identifying, for simplicity, the amounts of money with their utilities)

equation

      so that SEU is positive for a large enough images if images. Thus, following the principle of maximizing SEU, X should accept this lottery over the status quo (no bet), but he will lose in any case—the amount images or the amount images.

      Suppose now that images. Consider the bet images whose SEU is images. Since images, making images large enough, the bet appears favorable to X, yet he is bound to lose the amount images on every trial.

      If images or if the additivity axiom is not satisfied, one can also design bets that will formally be favorable for X (SEU will be positive) but that X will be bound to lose. Determination of these bets is left to the reader.

      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