Decision Theory

6State independence

Savage’s theory provides an axiomatization that yields unique probabilities and utilities, and provides a foundation for Bayesian decision theory. A key element in this enterprise is the definition of constant outcomes: that is, outcomes that have the same value to the decision maker irrespective of the states of the world. Go back to the Samarkand story of Chapter 4: you are about to ship home a valuable rug you just bought for $9000. At which price would you be indifferent between buying a shipping insurance for the full value of the rug, or taking the risk? Compared to where we were in Chapter 4, we can now answer this question with or without an externally given probability for the rug to be lost. Implicitly, however, all the approaches we have available so far assume that the only relevant state of the world for this decision is whether the rug will be lost. For example, we assume we can consider the value of the sum of money paid for the rug, or the value of the sum necessary to buy insurance, as fixed quantities that do not change with the state of the world. But what if the value of the rug to us depended on how much we can resell it for in New York? Or what if we had to pay for the insurance in the currency of Uzbekistan? Both of these considerations may introduce additional elements of uncertainty that make it hard to know the utility of the relevant outcomes without specifying the states of a “bigger world” that includes the exchange rates and the ...