Aggregation of Probability Judgments
讨论加总多个概率判断时出现的Arrow式悖论，通过两个决策场景说明不当加总可能产生错误结果，并给出相关公式与定理。
THIS PAPER DISCUSSES the problem of aggregating probability judgements and shows that Arrow-type paradoxes arise in this context, just as in the context of aggregating preferences. The need to aggregate probability judgements may arise in several different sorts of situation. Two examples which concern decision making under risk are: (i) when an individual, prior to making a decision, consults a number of experts who differ in their assessments of the probabilities of alternative states of nature; (ii) when the individuals constituting a society have to make a joint decision on the basis of identical utility functions but, again, differing assessments of the probabilities of alternative states of nature.2 We illustrate (i) above by means of a simple example: An individual consults two legal experts, who give him their probability judgements concerning the success or failure (there are just two possible outcomes) of proposed litigation. Scenario 1: The individual knows that litigation will succeed if judge J presides and barrister B defends, but will not succeed otherwise. From the experts' probability judgements the individual is able to infer that one expert has received a message that judge J will preside, and the other a message that barrister B will defend. (Either message, or both, may be false.) Given these messages, the individual updates in Bayesian fashion the prior probabilities he assigns to success and failure. Scenario 2: Success or failure depends on interpretation of the law. Pooling the information on which the experts' probability judgements are based is, we suppose, impracticable. (Perhaps they meet to discuss the case and cannot agree.) Aggregation of probability judgements in situations such as Scenario 1 (and according to axioms which we discuss in the next section) is clearly a wrong procedure which may lead to totally erroneous results. In situations such as Scenario 2 such aggregation may however have a useful role. Formulae for aggregating probability judgements have been discussed by, among others, Wagner (1982), Bordley (1982), Genest, Weerahandi, and Zidek (1984), and Fishburn and Rubinstein (1984). In this paper we generalize some results contained in the latter two papers, before going on to consider paradoxes. An early, Arrow-type impossibility result, based on rather strong assumptions, is due to Dalkey (1972). In Section 2 notation and axioms are introduced, and some results given. In Section 3 three propositions are stated and proved.