“Until the Bitter End: On Prospect Theory in a Dynamic Context,” S. Ebert & P. Strack (2012)

Let’s kick off job market season with an interesting paper by Sebastian Ebert, a post-doc at Bonn, and Philipp Strack, who is on the job market from Bonn (though this doesn’t appear to be his main job market paper). The paper concerns the implications of Tversky and Kahneman’s prospect theory is its 1992 form. This form of utility is nothing obscure: the 1992 paper has over 5,000 citations, and the original prospect theory paper has substantially more. Roughly, cumulative prospect theory (CPT) says that agents have utility which is concave above a reference point, convex below it, with big losses and gains that occur with small probability weighed particularly heavily. Such loss aversion is thought to explain, for example, the simultaneous existence of insurance and gambling, or the difference in willingness to pay for objects you possess versus objects you don’t possess.

As Machina, among others, pointed out a couple decades ago, once you leave expected utility, you are definitely going to be writing down preferences that generate strange behavior at least somewhere. This is a direct result of Savage’s theorem. If you are not an EU-maximizer, then you are violating at least one of Savage’s axioms, and those axioms in their totality are proven to avoid many types of behavior that we find normatively unappealing such as falling for the sunk cost fallacy. Ebert and Strack write down a really general version of CPT, even more general than the rough definition I gave above. They then note that loss aversion means I can always construct a right-skewed gamble with negative expected payout that the loss averse agent will accept. Why? Agents like big gains that occur with small probability. Right-skew the gamble so that a big gain occurs with a tiny amount of probability, and otherwise the agent loses a tiny amount. An agent with CPT preferences will accept this gamble. Such a gamble exists at any wealth level, no matter what the reference point. Likewise, there is a left-skewed, positive expected payoff gamble that is rejected at any wealth level.

If you take a theory-free definition of risk aversion to mean “Risk-averse agents never accept gambles with zero expected payoff” and “Risk-loving agents always accept a risk with zero expected payoff”, then the theorem in the previous paragraph means that CPT agents are neither risk-averse, nor risk-loving, at any wealth level. This is interesting because a naive description of the loss averse utility function is that CPT agents are “risk-averse above the reference point, and risk-loving below it”. But the fact that small probability events are given more weight, in Ebert and Strack’s words, dominates whatever curvature the utility function possesses when it comes to some types of gambles.

So what does this mean, then? Let’s take CPT agents into a dynamic framework, and let them be naive about their time inconsistency (since they are non EU-maximizers, they will be time inconsistent). Bring them to a casino where a random variable moves with negative drift. Give them an endowment of money and any reference point. The CPT agent gambles at any time t as long as she has some strategy which (naively) increases her CPT utility. By the skewness result above, we know she can, at the very least, gamble a very small amount, plan to stop if I lose, and plan to keep gambling if I win. There is always such a bet. If I do lose, then tomorrow I will bet again, since there is a gamble with positive expected utility gain no matter my wealth level. Since the process has negative drift, I will continue gambling until I go bankrupt. This result isn’t relying on any strange properties of continuous time or infinite state spaces; the authors construct an example on a 37-number roulette wheel using the original parameterization of Kahneman and Tversky which has the CPT agent bet all the way to bankruptcy.

What do we learn? Two things. First, a lot of what is supposedly explained by prospect theory may, in fact, be explained by the skewness preference which the heavy weighting on low probability events in CPT, a fact mentioned by a number of papers the authors cite. Second, not to go all Burke on you, but when dealing with qualitative models, we have good reason to stick to the orthodoxy in many cases. The logical consequences of orthodox models will generally have been explored in great depth. The logical consequences of alternatives will not have been explored in the same way. All of our models of dynamic utility are problematic: expected utility falls in the Rabin critique, ambiguity aversion implies sunk cost fallacies, and prospect theory is vulnerable in the ways described here. But any theory which has been used for a long time will have its flaws shown more visibly than newer, alternative theories. We shouldn’t mistake the lack of visible flaws for their lack more generally.

SSRN Feb. 2012 working paper (no IDEAS version).

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