# Experiments contradicting the expected utility model

This is a question I asked on the cognitive science beta, but which never got any answer. I do not know what the policy should be for question migration/reposting (maybe worth discussing in the meta?), but I hoped it might get more answers (i.e. at least one ;)) here.

I am looking for a list of experiments which cannot be accounted for by the expected utility model. By the expected utility model, I mean the model of individual preferences over vectors of uncertain events (e.g. $\Big(P(rain) = 0.4, P(sunshine) = 0.6\Big)$ and $\Big(P(rain) = 0.6, P(sunshine) = 0.4\Big)$) which satisfies a list of axioms proposed by Von Neuman and Morgernstern, namely

• Completeness
• Transitivity
• Continuity
• Independence

A rigorous formulation of these axioms can be found on page 8 of Axiomatic Foundations of Expected Utility and Subjective Probability, by Edi Karni, from the Handbook of Economics of risk and uncertainty..

Alternatively, by Von-Neuman and Morgenstern's representation theorem (page 9 of the same reference), these axioms are know to be equivalent to the fact that the preferences of the agent can be represented by a utility function of the form (in the discrete case):

$U(L) = \sum_{all~possible~events "e"} P(e)u(e)$

where $P(e)$ is again the probability that $e$ occurs and $u(e)$ is the utility of getting event $e$ for sure.

The violations of these axioms I am most interested in are the ones related to the Independence axiom (violations of completeness, transitivity and continuity would probably deserve a separate question. See this question for an example of intransitivity.).

I am looking for situations which cannot be accounted for by the expected utility model. Some well-known examples are the the Allais and Ellsberg paradoxes (although there is still a debate regarding Ellsberg paradox). On the other hand, I do not see the Saint-Peterborough paradox as contradicting expected utility theory, because it can be accounted for by the theory if one assumes an appropriate degree of risk aversion. But you are much welcome to argue against that.

I hope this question can serve as a repository of famous experiments contradicting expected utility theory, so feel free to add many.

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## 4 Answers

this paper http://else.econ.ucl.ac.uk/papers/uploaded/243.pdf (Choi 2007) has a nice state of the art experiment that deals with rationality and expected utility is a special case of it. In general only 17% of consumers are compatible with rationality ergo the remaining part cannot be expected utility maximizers. Quah has a nice paper on the revealed preference theory of expected utility (among other models), he uses Choi dataset to test expected utility hypothesis that is going to be rejected more times than rationality https://ideas.repec.org/p/lec/leecon/13-24.html

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Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-Colell, Whinston and Green's Microeconomic Theory.

A person prefers a trip to Paris to watching a television program about Paris to nothing.

Gamble 1: Win a trip to Paris 99% of the time, the television program 1% of the time.

Gamble 2: Win a trip to Paris 99% of the time, nothing 1% of the time.

It's reasonable to suppose that given the preferences over items, the second gamble might be preferred to the first. Someone who lost the trip to Paris might be so disappointed that they wouldn't be able to stand watching a program about how great it is.

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I think one problem here is that the case that you're describing is a case of state-dependent utility. That doesn't invalidate the expected utility model. You just need to be more exhaustive when you write out all potential consumption bundles. – jmbejara Nov 19 '14 at 21:55
@jmbejara Okay, but this critique must also apply to the Allais paradox or anything with gambles. – Pburg Nov 19 '14 at 21:59
No, that's not correct. In your example you asserted that the person had lost a trip to Paris. So, the person is in a different state of being. The Allais paradox or the Ellsberg paradox do not assume that the person is in a different state of being. – jmbejara Nov 19 '14 at 22:19
The person hasn't lost anything, they are evaluating gambles ex ante. They anticipate that regret. There could be a similar dynamic to the Allais paradox, where I would feel awful if I turned down a sure $\$1$million for a high chance of$\$5$ million but lost. – Pburg Nov 19 '14 at 22:25
Ok. Sorry. I see what you're saying. That's interesting. I've opened another question to help further this train of thought. economics.stackexchange.com/questions/134/… – jmbejara Nov 19 '14 at 23:08

Following @Pburg answer and the subsequent discussion in the comments, I wanted to post an alternative Machina Paradox I thought of. Although it might be less pervasive in real life, it seems stronger to me in the sense that it does not rely on some kind of complementarity between the "different" components of each outcome. Consider the following alternative :

Gamble 1: Win $1 million 99% of the time, win a penny 1% of the time. Gamble 2: Win$1 million 99% of the time, win nothing 1% of the time.

I suspect that most people prefer winning \$1 million for sure to winning a penny for sure to winning nothing for sure, while some people nevertheless prefer gamble 2 to gamble 1.

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Kahneman and Tversky's experiments and many in behavioural economics contradict the existence of a utility function (preferences not complete and transitive), therefore also contradict expected utility.

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