Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

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, and the quantity (2.13) is called expected utility (EU). Finally, when in the latter case, the probability images

is the subjective assessment of likelihood of the event images

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 images , 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 images such that X will be indifferent between the sure option with images 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 images . Consequently, we have images . Obviously, under the stated assumption that utility of money is proportional to the dollar amount, the choice of images is not relevant here, and the same value for images 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 images , then the utility images of object images must equal the expected value of the lottery, which is images . 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 images denote events, and let images denote some objects, whose probabilities images and utilities images are to be determined (keep in mind that both images and images 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 images with subjective probability images , so that images . If X prefers lottery images to lottery images , 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 images . That is, if X is indifferent between lotteries images and images , we have

which gives

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

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