Читать книгу Probability and Statistical Inference - Robert Bartoszynski - Страница 65

Let us finally consider briefly the third interpretation of probability, namely as a degree of certainty, or belief, about the occurrence of an event. Most often, this probability is associated not so much with an event as with the truth of a proposition asserting the occurrence of this event.

The material of this section assumes some degree of familiarity with the concept of expectation, formally defined only in later chapters. For the sake of completeness, in the simple form needed here, this concept is defined below. In the presentation, we follow more or less the historical development, refining gradually the conceptual structures introduced. The basic concept here is that of a lottery, defined by an event, say , and two objects, say and . Such a lottery, written simply , will mean that the participant (X) in the lottery receives object if the event occurs, and receives object if the event occurs.

The second concept is that of expectation associated with the lottery , defined as

(2.13)

where and are measures of how much the objects and are “worth” to the participant. When and are sums of money (or prices of objects and ), and we put , the quantity (2.13) is sometimes called expected value. In cases where and are values that person X attaches to and (at a given moment), these values do not necessarily coincide with prices. We then refer to and as utilities of and , and the quantity (2.13) is called expected utility (EU). Finally, when in the latter case, the probability is the subjective assessment of likelihood of the event by X, the quantity (2.13) is called subjective expected utility (SEU).

First, it has been shown by Ramsey (1926) that the degree of certainty about the occurrence of an event (of a given person) can be measured. Consider an event , and the following choice suggested to X (whose subjective probability we want to determine). X is namely given a choice between the following two options:

1 Sure option: receive some fixed amount , which is the same as lottery , for any event .

2 A lottery option. Receive some fixed amount, say $100, if occurs, and receive nothing if does not occur, which is lottery . One should expect that if is very small, X will probably prefer the lottery. On the other hand, if is close to , X may prefer the sure option.

Therefore, there should exist an amount such that X will be indifferent between the sure option with and the lottery option. With the amount of money as a representation of its value (or utility), the expected return from the lottery equals

which, in turn, equals . Consequently, we have . Obviously, under the stated assumption that utility of money is proportional to the dollar amount, the choice of is not relevant here, and the same value for would be obtained if we choose another “base value” in the lottery option (this can be tested empirically).

This scheme of measurement may provide an assessment of the values of the (subjective) probabilities of a given person, for a class of events. It is of considerable interest that the same scheme was suggested in 1944 by von Neumann and Morgenstern (1944) as a tool for measuring utilities. They assumed that probabilities are known (i.e., the person whose utility is being assessed knows the objective probabilities of events, and his subjective and objective probabilities coincide). If a person is now indifferent between the lottery as above, and the sure option of receiving an object, say , then the utility of object must equal the expected value of the lottery, which is . This allows one to measure utilities on the scale that has a zero set on nothing (status quo) and “unit” as the utility of $100. The scheme of von Neumann and Morgenstern was later improved by some authors, culminating with the theorem of Blackwell and Girshick (1954).

Still the disadvantages of both approaches were due to the fact that to determine utilities, one needed to assume knowledge of probabilities by the subject, while conversely, to determine subjective probabilities, one needed to assume knowledge of utilities. The discovery that one can determine both utilities and subjective probabilities of the same person is due to Savage (1954). We present here the basic idea of the experiment rather than formal axioms (to avoid obscuring the issue by technicalities).

Let denote events, and let denote some objects, whose probabilities and utilities are to be determined (keep in mind that both and refer to a particular person X, the subject of the experiment). We now accept the main postulate of the theory, that of the two lotteries, X will prefer the one that has higher SEU.

Suppose that we find an event with subjective probability , so that . If X prefers lottery to lottery , then

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 . That is, if X is indifferent between lotteries and , we have

which gives

The only problem lies in finding an event with subjective probability . Empirically, an event has subjective probability if, for any objects and , the person is indifferent between lotteries and . 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 , and the other three with the nonsense combination (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 , 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 is negative. Consider the bet (lottery) , (i.e., a lottery in which X pays the sum if occurs and pays the sum if does not occur). We have here (identifying, for simplicity, the amounts of money with their utilities)

so that SEU is positive for a large enough if . 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 or the amount .

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

If 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.