Under what circumstances can preferences under uncertainty be modeled by expected utility? The classic von Neumann-Morgenstern theorem constructs a utility function that, when multiplied by corresponding probabilities, yields an ordering that is consistent with the ordering of the preferences of the agent.
Being able to represent choice under uncertainty via expectations is a powerful tool. As just one example, utility functions tend to be convex to capture diminishing marginal utility. Hence, by Jenson's inequality, the notion of risk aversion (the utility of a certain outcome of 1 dollar is higher than the utility of a 50% chance of 2 dollars and 50% chance of zero dollar) pops out of such models "for free". More generally, the rich mathematical and optimization frameworks that accompany expectations is made available for economic modeling.
The von Neumann-Morgenstern theorem itself relies on two core axioms of economics, a third axiom that is required to prove the existence of a utility function, and finally a fourth that is required to prove the existence of a utility function such that the expectation of the utility in each possible state corresponds to the actual preference ordering of the agent. It is this fourth axiom that is the lynchpin for the von Neumann-Morgenstern proof.
The late Daniel Kahneman, Nobel prize winner and critic of "mainstream" economics, took issue with this fourth axiom. (As an aside, his pop-science book "Thinking Fast and Slow" took (unjustified) issue with far more in economics than just this fourth axiom. He repeatedly conflates the concept of rationality and of the representation of preferences via expectations, leading the casual reader to believe that he had essentially undermined all of economics). As a simple example, the "Allais Paradox" posits two gambles that would appear to contradict the fourth axiom. In response to this shortcoming, he created an entirely new framework he termed prospect theory.
So does Kahneman's work prove the death-knell for expected utility? I don't believe so. There is a trick for resolving Allais' Paradox while retaining the expected utility framework: switching the "real-world" probabilities into an equivalent measure. A simple re-weighting of the probabilities generates expected utility that is consistent with the actual preference ordering, and it is not difficult to construct a general re-weighting that accomplishes this for any "gamble" in which at least one state has a higher utility than the alternative. Put in such a way, the fourth axiom is actually not necessary for expectations to be an appropriate representation of preferences. By adding the single line "under some probability measure" to the von Neumann-Morgenstern theorem, the entire analytical toolkit of expected utility is back at our disposal.
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Very neat. But, might you give a worked example of how to use an equivalent measure to solve Allais' Paradox?
Sure, if you use a simple utility function (say, 1-e^{-x}), where x is millions of dollars, then the "sure bet" in choice 1 has utility 1-exp(-1)=.63 while the "gamble" ([.01, .89, .1]) under the real-world measure has utility (1-exp(-1))*.89+(1-exp(-5))*.1=.66. By adjusting the probabilities to (as an example) [.1, .89, .01] then the under this new measure the utility is (1-exp(-1))*.89+(1-exp(-5))*.01=.57, which is less than .63.
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