appendix - Appendix 1: Utility Theory Much of the theory...

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Unformatted text preview: Appendix 1: Utility Theory Much of the theory presented is based on utility theory at a fundamental level. This theory gives a justification for our assumptions (1) that the payoff functions are numerical valued and (2) that a randomized payoff may be replaced by its expectation. There are many expostions on this subject at various levels of sophistication. The basic theory was developed in the book of Von Neumann and Morgenstern (1947). Further developments are given in Savage (1954), Blackwell and Girshick (1954) and Luce and Raiffa (1957). More recent descriptions may be found in Owen (1982) and Shubik (1984), and a more complete exposition of the theory may be found in Fishburn (1988). Here is a brief description of the basics of linear utility theory. The method a rational person uses in choosing between two alternative actions, a 1 and a 2 , is quite complex. In general situations, the payoff for choosing an action is not necessarily numerical, but may instead represent complex entities such as you receive a ticket to a ball game tomorrow when there is a good chance of rain and your raincoat is torn or you lose five dollars on a bet to someone you dislike and the chances are that he is going to rub it in. Such entities we refer to as payoffs or prizes . The rational person in choosing between two actions evaluates the value of the various payoffs and balances it with the probabilities with which he thinks the payoffs will occur. He may do, and usually does, such an evaluation subconsciously. We give here a mathematical model by which such choices among actions are made. This model is based on the notion that a rational person can express his preferences among payoffs in a method consistent with certain axioms. The basic conclusion is that the value to him of a payoff may be expressed as a numerical function, called a utility , defined on the set of payoffs, and that the preference between lotteries giving him a probability distribution over the payoffs is based only on the expected value of the utility of the lottery. Let P denote the set of payoffs of the game. We use P , P 1 , P 2 , and so on to denote payoffs (that is, elements of P ). Definition. A preference relation on P , or simply preference on P , is a (weak) linear ordering, , on P ; that is, (a) (linearity) if P 1 and P 2 are in P , then either P 1 P 2 or P 2 P 1 (or both), and (b) (transitivity) if P 1 , P 2 and P 3 are in P , and if P 1 P 2 and P 2 P 3 , then P 1 P 3 . If P 1 P 2 and P 2 P 1 , then we say P 1 and P 2 are equivalent and write P 1 ' P 2 . We assume that our rational being can express his preferences over the set P in a way that is consistent with some preference relation. If P 1 P 2 and P 1 6' P 2 , we say that our rational person prefers P 2 to P 1 and write P 1 P 2 . If P 1 ' P 2 , we say that he is indifferent between P 1 and P 2 . The statement P 1 P 2 means either he either prefers P...
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appendix - Appendix 1: Utility Theory Much of the theory...

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