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Extra resources for Utility Maximization, Choice and Preference (Studies in Economic Theory)
3) holds for these probabilities. These probabilities are called the individuals subjective prior. To see this, let be a ﬁnite set of states of nature. We call A Â events. Also, let L be a set of “lotteries,” where a lottery is a function W ! X that associates with each state of nature ! / 2 X . Note that this concept of a lottery does not include a probability distribution over the states of nature. , expressing the decision maker’s personal assessment of the probability that ! will occur. We suppose that the individual chooses among lotteries without knowing the state of nature, 23 24 Chapter 2 after which “Nature” chooses the state !
T/. Suppose there are moves by Nature, by which we mean that at one or more nodes in the game tree, there is a lottery over the various branches emanating from that node, rather than a player choosing at that node. For every terminal node t 2 T , there is a unique path p t in the game tree from the root node to t. We say p t is compatible with strategy proﬁle s if, for every branch b on p t , if player i moves at b h (the head node of b), then si chooses action b at b h . s; t/ D 0. s; t/ to be the product of all the probabilities associated with the nodes of p t at which Nature moves along p t , or 1 if Nature makes no moves along p t .
Brown, an onlooker, asks Mrs. ” She nods yes. ” She nods yes again. In which of the two situations is Mrs. Black more likely to have at least one other ace in her hand? Calculate the exact probabilities in the two cases. 30 The Brain and Kidney Problem A mad scientist is showing you around his foul-smelling laboratory. He motions to an opaque, formalin-ﬁlled jar. “This jar contains either a brain Probability and Decision Theory or a kidney, each with probability 1/2,” he exclaims. Searching around his workbench, he ﬁnds a brain and adds it to the jar.