# List of experiments contradicting the expected utility model

I proposed this question as an example question for the Area51 proposal "Mathematical modeling". User Artem Kaznatcheev suggested that it be asked here too, which I thought was a good idea.

So 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.. 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 deserve a separate question. See this question for an example of intransitivity.).

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.

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.

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

• Interesting question. When you say 'contradicting', I presume you mean cases where people's actual behaviour systematically departs from what UI theory would predict (like the Allais paradox), rather than actual logical paradoxes in UI (like the St. Petersburg paradox)? I'm not sure if 'paradox' is the appropriate term for these descriptive violations of normative theory. You might want to look here as a starting point (I'll try and provide a proper answer later). – Eoin May 14 '14 at 14:07
• Thanks for your comment. I am not sure what you mean by "UI". "Cases where people's actual behaviour systematically departs from what expected utility theory would predict" is precisely what I am looking for though. Equivalently, I could ask for situations which violate the Von-Neuman and Morgenstern independence axiom (or continuity, but that is maybe less interesting). I guess I used paradoxes because the most famous cases are called "paradoxes". Maybe I should reframe the question and the title... – Martin Van der Linden May 14 '14 at 14:55
• Also, it would be better if you expanded your question to be more self-contained for those that might not be familiar, by for instance giving a one sentence sketch of expected utility theory (because there are different levels of generality people use). I would also avoid "Most famous paradoxes" since that is a judgement call and experimental results are only 'paradoxical' if you make the mistake of assuming expected utility theory describes reality; just 'experiments contradicting EUT' seems more neutral. – Artem Kaznatcheev May 14 '14 at 17:14
• Sorry, I was being lazy, and also amazingly sloppy; I meant EU (for Expected Utility). I see there might be a clash of terminology here between those of us from psychology and economics backgrounds: your 'positive theory' seems to be our 'descriptive model'. Also +1 on @ArtemKaznatcheev's point about how the question is phrased. – Eoin May 14 '14 at 17:54
• I completely disagree with @NickStauner, there is absolutely no way in which EUT fits with Cross Validated, and it is perfectly on topic here because psychology was central to forming behavioral economics, and early behavioral economics formed largely as a way to explain failures of EUT. – Artem Kaznatcheev May 14 '14 at 21:13