004 blume 2 savage - Beliefs 1 Why Beliefs VnM Expected...

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Beliefs October 26, 2006 1 Why Beliefs? VnM Expected Utility: There is a state space S (which, for the moment only, is finite) and an objective probability distribution p on S . There is also a set O of outcomes. An act is a map f : S O . A preference order is a VnM order if acts are ranked by expected utility. That is, there is a payoff function u : O R such that f g iff s u ( f ( s ) ) p ( s ) s u ( g ( s ) ) p ( s ) (1) Why is this VnM? Where are the probability distributions on out- comes? To each act f we can associate a probability distribution p f on O , such that f g iff o u ( o ) p f ( o ) o u ( o ) p g ( o ). To see this, define p f ( o ) to be the probability that outcome o is realized with act f . That is, p f ( o ) = { s : f ( s )= o } p ( s ) 1
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So, s u ( f ( s ) ) p ( s ) = o { s : f ( s )= o } u ( f ( s ) ) p ( s ) = o { s : f ( s )= o } u ( o ) p ( s ) = o u ( o ) p f ( o ) This is just the familiar change of variables formula from basic stats. Where can probabilities come from? 1. Frequencies 2. Beliefs — expressed by asking 3. Beliefs — expressed through behavior Empirical frequencies are fine for bets at Las Vegas. But not for lots of things. Savage wrote in 1954 that “I, personally, consider it more probable that a Republican president will be elected in 1996 than that it will snow in Chicago sometime in the month of May, 1994. But even this late spring snow seems to me more probable than that Adolf Hitler is still alive.” 1. A Dem was elected in 1996. 2. I couldn’t find 1994 data, but Chicago had May snow in both 2001 and in 2002. 3. Hitler? Savage’s idea is to derive beliefs from preferences. (And presumably preferences can be elicited by presenting people with choices.) Eliciting a DM’s preferences over various acts — bets on events A and B — will tell us which event the DM thinks is more likely. That is, we will derive an ordering on events in which A B means that the DM believes A to be more likely than B . This order is called a qualitative probability or a comparative 2
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probability . He goes on to give conditions under which this order can be represented by a probability distribution p , and further conditions under which the preference ordering has a representation like that of equation (1) with respect to p . So “why beliefs”? Because in a well-structured theory of choice under uncertainty, subjective probability can play the same role that objective, which is to say frequentist, probability can play in the VnM choice theory. From preferences we derive a probability distribution p on S such that with an approprite payoff function u : O R also derived from preferences, the representation (1) holds. 2 Representing Beliefs This section is just a list of ways that beliefs can be represented: qualitative probability probability (Kolmogorov) Objective — frequentist Subjective plausibility measure possibility measure belief functions capacities sets of beliefs conditional probability systems There are reasons for eliciting belief that are distinct from decision theory, and the connection of some of these representations to decision theory is not known by me.
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