Bayesian decision theory – a critique26 February, 2013 at 18:35 | Posted in Economics | Leave a comment
A farmer is offered a choice between, on the one hand, getting a horse if it is raining tomorrow and a cow if it is not raining and, on the other hand, a cow if it is raining and a horse if it is not. He prefers getting a horse to getting a cow; this is a ‘pure preference’. But which of the offered alternatives does he prefer? Assume that he professes to be indifferent as between them. How shall we then understand his attitude?
To this question there is an answer, first proposed by F. P. Ramsey, which has later come to play a great role in so-called Bayesian decision theory …
Ramsey thought that an attitude of indifference here means that the person rates the two events, ‘rain’ and ‘not rain’, as equally probable. Accepting this, one can then proceed as follows:
Assume that our farmer is next presented with this option: On the one hand a horse if it is raining and a sheep if it is not raining and, on the other hand, a cow if it is raining and a hog if it is not raining. Again he says he is indifferent. This, on Ramsey’s view, means that the value to him of a cow is as much less the value of the horse as the value of a sheep is less than that of a hog. With this the way is open to a metrization of value and the introduction of utility functions. This done, one can use attiyudes of indifference in other, more complex, conditional options for defining arbitrary degrees of (subjective) probability. The product of the value of a good an dthe probability of its materialization is called expected utility. Attitudes of preference in options aim at maximizing this quantity.
Ramsey’s method is elegant and ingenious. Nevertheless, it seems to rest on a mistake. It ignores the distinction between two senses of ‘indifference’.
The farmer who, when presented with the first of the above two options, professes an attitude of indifference can do so for one or two reasons. Either he ‘simply has no idea’ about the chances of rainfall for tomorrow and therefore cannot make up his mind about which alternative is more to his advantage.
This does not mean that he thinks rain and not-rain equally likely; he simply suspends judgement. Or, he considers them equally likely and therefore judges the two alternatives to be equally advantageous. He could, for example, support his attitude with the argument that if he repeatedly opted for one of the alternatives, no matter which one, on average half the number of times he would ‘probably’ get a horse, which is to his advantage, and half the number of times a cow, which is to his disadvantage. So, therefore, he is indifferent as between the alternatives. It is, in other words not his judgement of indifference which gives meaning to the probabilities for him; but it is his prior estimate of the probabilities which determines his attitude of indifference.