## Friday, November 21, 2008

## Prospect Theory

One of my courses this quarter was dedicated to decision theory and behavioral economics. Decision Theory has its roots in the axiomatisation proposed by Von Neumann and Morgenstern and is based on 'objective' probabilities, such as a fair coin flip in their Theory of Games and Economic Behavior. Savage proposed an alternative in his Foundations of Statistics that incorporated the notion of subjective probabilities based on individuals having their own perception of the likelihood of events. Anscombe and Aumann later synthesised these two theories to devise a theory that included both subjective and objective probabilities. Nevertheless, the goal for all of this the Expected Utilty Representation of a Utility Function, i.e. representing a utility such that it can be decomposed into the probability of obtaining specific utility amounts and that this can be done in an additively separable manner (this is less complex than it sounds).

However, the problem is that humans often don't behave consistent with the axioms of Von Neumann and Morgenstern, Savage, or Anscombe and Aumann. Allais showed this in his 1952 paper. Kahneman and Tversky (K&T), two psychologists who shifted into economic choice theory, also did work on this in 1979 and subsequently. Previously, I had 'studied' their theory, called 'prospect theory' previously from textbooks, but had not read their original 1979 paper. It is part of our compulsory reading list for this course and so I recently read it. K&T used many surveys to try to come up with an idea of how people responded to risk, to uncertainty over outcomes. In orthodox economic theory an amount of money should be 'state neutral', i.e. you should treat 500 currency units (CUs) in the same manner regardless of whether you are going to gain that money or lose it. However, their evidence indicates that people treat losses and gains differently. People are 'risk-averse' over gains, but 'risk-loving' over losses. I will go over this in a moment.

Take the following example. You are given the choice between R450 for certain or a lottery which involves a 50% chance of winning R1000 and a 50% of getting nothing. The expected value of the lottery (the sum of the probability multiplied by the amount to be won) is R500, which is strictly greater than R450. A 'rational' person should therefore choose the lottery. However, people seem to prefer the situation in which they gain the money for certain. People are therefore 'risk-averse' over gains.

Now take another example. You are offered the following choice. You either lose R3000 for certain, or you face a lottery in which you lose R4000 with 80% probability or lose nothing, R0, with probability 20%. The expected loss of the lottery is R3200. A 'rational' person would thus choose the certain loss. However, empirically, people choose the second option in which they face a lottery rather than the certain loss. People are therefore 'risk-loving' over losses.

These conclusions are in direct conflict with orthodox decision theory. Now, I think that VNM, Savage and A&A provide graceful mathematical depictions of potential behavior. However, they postulate humans with decision capabilities and computational abilities the likes of which I believe are truly miraculous. Savage, for example, talks about being able to partition sets of events into smaller sets of equiprobable events in order for them to be able to rank different sets of events, this would require a super-machine, or a person who is entirely unlike me in their decision capabilities.

K&T's theories should also be seen as largely descriptive. They have taken data and constructed a model that would represent behavior that is consistent with the behave that they observe when people make decisions. This area of decision theory has seen several developments in recent times with Gilboa being the champion of its development, I don't know if I can comprehend Gilboa's math, but hey I'll try and I'll let you know one day... For now it's good to know though that economic theory is doing its best to construct models that use assumptions that are closer to the actual behavior of people, rather than abstract and incorrect postulates of (in)human conduct.

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