Desktop version

Home arrow Education

Keynes and Ramsey (and Savage and Good)

Ramsey explicitly addressed in an unpublished note (see Ramsey, 1990b)3 Keynes's question understood as a question about the acquisition of new information about data of observation and experiment. Leonard

Jimmie Savage (1954, s. 6.2) and Irving John Good (1967; reprinted in Good, 1983, ch. 17) offered essentially the same account as Ramsey, apparently independently both of each other and of Ramsey.

Ramsey, Savage, and Good all argued that acquiring the new data and then choosing maximizes expected utility provided certain conditions are satisfied. The collection of data should be cost-free. The inquirer X should be convinced that he will update X's initial probability judgements using Bayes's theorem via conditionalization. The outcome of experimentation makes a difference as to which option maximizes expected utility. If the same option maximizes expected utility regardless of the outcome of experimentation and observation, the expected utility of acquiring the new data and then choosing is the same as the expected utility of choosing without the benefit of the data.

Let us see how this argument works in a simple case. Suppose agent X is told that an urn contains 90 black balls and 10 white balls (HB), or 10 black balls and 90 white balls (HW), or 50 black and 50 white balls (HN). X is also told that whether HB, HW, or HN is true depends on the outcome of random process that assigns equal chance of 1/3 to each alternative. As far as X is concerned the probabilities of the three hypotheses are equal. X is offered three options given in Table 3.1.

X should evaluate the three options equally according to expected utility relative to the initial state of information.

Let X now be offered the opportunity to observe the outcome of a random selection of a single ball before making a choice. The probability P(black/HB) = P(white/HW) = 0.9 and P(white/HB) = P(black/HW) = 0.1. P(black/HN) = P(white/HN) = 0.5. Since P(HB) = P(HW) = P(HN) 0.33, by Bayes's theorem P(HB/black) = P(HW/white) = 0.6. P(HN/black) = P(HN/ white) = 0.33+ and P(HB/white) = P(HW/black) = 0.07-.

Thus, adding the information that the ball drawn is black to X's body of full beliefs and updating via conditionalization and Bayes's theorem lead to a belief probability of 0.6 for HB. This equals the expected value of A. The expected value of B is 1/15 and for C is 0.1/3. So the result recommends A as the best option. The addition of the information that

Table 3.1 Random process and evaluation of hypotheses

Hb

Hn

hw

A

1

0

0

B

0

0

1

C

0

1

0

the ball drawn is white determines B as the best option with the same expected value of 0.6. Thus, on the supposition that X will maximize expected utility on obtaining the new information whatever it may be, X evaluates making the observation and choosing the option that then maximizes expected utility as itself carrying an expected utility of 0.6. This is better than the expected utility 0.33+ of choosing any one of A, B, or C without the benefit of the observation.

Can X improve X's predicament still further by observing another ball selected from the urn at random? (I shall suppose that the first ball is returned to the urn prior to this second selection.) If the first ball selected is black, the second can be black or white. If black, the posterior for HB will be boosted even higher (0.76). A will be recommended as before with even higher expectation. If white, the posterior for HB will be reduced to 0.21. The posterior for HW will increase to 0.21 and the posterior for HN will be 0.58. Option C is then optimal. The expected value of obtaining the new information will be 0.75. This is higher than 0.6.

It is demonstrable that collecting new data in this way can never be worse than refusing the new data as long as no extra cost is incurred and we ignore the risk of importing error in acquiring the data.

Appealing to expected utility when deciding whether to obtain more data is not quite the same as appealing to expected utility in choosing between terminal options. Keynes seems quite clear that weight of argument is not relative to any specific decision problem. Moreover, as long as a is not entailed by h, the weight of argument for a can always be increased by strengthening h relevantly. (See the condition " V (a/hh1) = V (a/h), unless h1 is irrelevant, in which case V (a/hh1) = V (a/h)" in Keynes's Treatise on Probability) (TP, p. 79).

Nonetheless, Ramsey offered an answer to Keynes's problem. When information is cost free, risk free and relevant to the decision under consideration, it pays to obtain it relative to the aims of the problem at hand.

In spite of its distinguished provenance and the indubitable validity of the argument under the assumptions upon which it is made, the Ramsey argument has severely limited applicability.

Keep in mind that the inquirer X is in a context where X is deciding whether to perform an experiment and then reach a decision or to take the decision without experiment or observation. In that setting, the inquirer does not, as yet, know whether the experiment and observation will be made and, if it is, what it will be. X may be in a position to make determinate probability judgements as to what the outcome of experiment will be conditional on running the experiment or not as the case may be. Ramsey (and Savage and Good) all presupposed that the probabilities would be determinate. If they are not, the import of the argument is quite different.

If we set aside the possibility (which Keynes insisted upon) that probabilities are indeterminate, the calculations upon which the Ramsey-Savage-Good argument is based in our example and in general presuppose that errors of observation are ignored.

Errors of observation would be legitimately ignored if the inquirer X were absolutely certain that no such error could arise. However, to suppose that the inquirer rules out the logical possibility that forming the belief that a black (white) is drawn in response to observation when the ball drawn is white (black) is to suppose that X is more confident of the testimony of the senses than X normally should be. It would not be sound practice to assume in advance of making observations that the observations will be 100 per cent reliable.

Perhaps risk of error should be ignored not because importing false belief is not seriously possible but because it is not important. According to a vulgar form of pragmatism to which Peirce did not subscribe, the inquirer X should not attach any particular value to avoiding false belief unless it impacts on the consequences of X's actions relative to the practical goals X is committed to realizing. X might acknowledge the serious or epistemic possibility of errors of observation and continue to ignore them because they have no impact on X's expectations as to what the practical consequences of X's decisions will be.

Vulgar pragmatists insist that practical considerations always override cognitive goals. So unless risk of error is relevant to promoting or undermining the realization of practical goals, vulgar pragmatists could judge themselves justified in ignoring the possibility of error.

But risk of error can be relevant to the realization of practical goals. Thus, in the case where the issue is to make an observation of the color of one ball drawn from the urn in the case where p (HB) = 0.33, if the probability of error is greater than 46 per cent the expected value of making an observation will be less than one-third and, hence, will be disadvantageous. In those cases where the Ramsey argument leads to the result that acquiring new information via observation is neither advantageous nor disadvantageous, taking risk of error into account can make no difference. But where the expected value of the new information is positive, taking risk of error into account can undermine the Ramsey argument even when practical considerations alone are considered.

Taking cognitive values including risk of error seriously could deter the making of observations in cases where there is neither practical advantage nor disadvantage otherwise. Thus, if X had already observed a large number of draws from the urn and they were overwhelmingly black, making an additional observation would not make any difference to X's decision to choose option A. But a new observation might incur the risk of a false belief that the ball drawn is black when it is white or white when black. If this risk is slightly greater than the value of the information gained, making the observation would become disadvantageous. Thus, the import of the Ramsey argument is further undermined by insisting on the autonomy of cognitive values.

In the previous section, it was argued that pragmatic justifications for specific inductive inferences should prohibit the overriding of cognitive goals by practical ones. This consideration ought to suffice for the recognition of cognitive values as autonomous dimensions of value and the rejection of vulgar pragmatism and the utilitarianism that so often spawns it.

Ramsey also assumed that the agent X is convinced that, upon obtaining the data, X will update by conditionalizing on the data to form new probability judgements. But even if X is making probability judgements coherently, rationality does not mandate that X update probability judgements in this way any more than it mandates changing probabilities by Jeffrey Conditionalization.4 Rational agents should change probability judgements on acquiring new data via (temporal credal) conditionalization only if they do not revise their confirmational commitments—that is, their commitments as to what probability judgements should be relative to diverse potential states of full belief or evidence (see Levi, 1974; 1980). So even if we restrict discussion to ideally rational agents, X must predict that X will retain X's current confirma- tional commitment upon acquiring new information and that such confirmational commitment meets a condition of confirmational condi- tionalization as a requirement of synchronic rationality. Ideally rational X need not do this.

There is another non-trivial presupposition ingredient in the Ramsey argument. X must assume prior to acquiring new information that, after obtaining the new information, X will choose for the best. At the time t0 when X is contemplating the acquisition of new information, X may be in a position to decide whether or not to do so. But X is not in control of whether X will maximize in the future once the new information is acquired. X can predict only at the initial stage t0 whether X will do so or not. And uncertainty may infect this prediction as well.

The reservations registered concerning Ramsey's argument ought not to be taken as a dismissal of its insight. Many of the assumptions tacitly made by advocates of the argument are often reasonably adopted.

The most difficult one, in my judgement, concerns the risk of error. When that risk is sufficiently small, its impact is negligible. A good case can be made for the desirability of acquiring new information when the reservations concerning the assumptions of the Ramsey argument can be overlooked.

Nonetheless, the Ramsey argument does not provide an explication of the notion of weight of argument from evidence h to hypothesis a where the argument is the judgement of probability that a given h. Keynes did suggest that perhaps such an assessment of weight would be useful in determining whether the current total evidence is sufficient for terminating investigation and taking whatever practical decision is at issue. But the threshold level might differ depending upon what the practical decision problem is. Keynes was interested in what sort of measure of weight of argument is suitable for the purpose no matter what the threshold might be. The choice of a threshold might depend on the practical goals of the decision problem. Keynes seemed to have been interested in what the threshold is a threshold of. The assessment of weight of argument is in this sense independent of the specific goals of the practical decision problem. It is clear that the Ramsey argument cannot answer the question raised.

 
Source
< Prev   CONTENTS   Source   Next >

Related topics