<p>this paper <a href="http://else.econ.ucl.ac.uk/papers/uploaded/243.pdf">http://else.econ.ucl.ac.uk/papers/uploaded/243.pdf</a> (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 <a href="https://ideas.repec.org/p/lec/leecon/13-24.html">https://ideas.repec.org/p/lec/leecon/13-24.html</a></p>

CC BY-SA 3.0
2014-11-19T23:00:59.097Z
2014-11-19T23:00:59.097Z
user157623
<p>Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-Colell, Whinston and Green's Microeconomic Theory.</p>

<p>A person prefers a trip to Paris to watching a television program about Paris to nothing.</p>

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

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

<p>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.</p>

CC BY-SA 3.0
2014-11-19T21:55:53.567Z
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
CC BY-SA 3.0
2014-11-19T21:59:41.723Z
@jmbejara Okay, but this critique must also apply to the Allais paradox or anything with gambles.
Pburg
CC BY-SA 3.0
2014-11-19T22:19:59.507Z
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
CC BY-SA 3.0
2014-11-19T22:25:25.490Z
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
CC BY-SA 3.0
2014-11-19T23:08:15.747Z
Ok. Sorry. I see what you're saying. That's interesting. I've opened another question to help further this train of thought. http://economics.stackexchange.com/questions/134/can-the-machina-paradox-by-expanding-the-choice-set
jmbejara
CC BY-SA 3.0
2014-11-19T20:34:16.157Z
2014-11-19T20:34:16.157Z
Pburg
<p>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 : </p>

<blockquote>
  <p>Gamble 1: Win $1 million 99% of the time, win a penny 1% of the time.</p>
  
  <p>Gamble 2: Win $1 million 99% of the time, win nothing 1% of the time. </p>
</blockquote>

<p>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.</p>

CC BY-SA 4.0
2018-11-26T07:15:43.903Z
Any idea how I can finish the proof of EUT with three outcomes?
OGC
CC BY-SA 3.0
2014-11-20T05:47:07.900Z
2014-11-20T05:47:07.900Z
Martin Van der Linden
<p>Let me mention another quite well-known one: The calibration theorem by <a href="http://onlinelibrary.wiley.com/doi/10.1111/1468-0262.00158/abstract" rel="nofollow noreferrer">Rabin (2000)</a> and <a href="http://www.jstor.org/stable/2696549" rel="nofollow noreferrer">Rabin and Thaler (2002)</a>. The idea is that over small stakes individuals must be essentially risk-averse, but in reality they are not. </p>

<p>Only assuming a weakly concave and strictly increasing utility function, Rabin shows that risk aversion on small stakes implies obviously unrealistic risk aversion over large stakes. In other words, under expected-utility theory, a resistance to accept small stake gambles with positive expected value leads to absurd conclusions about individuals' behavior in large stake gambles. </p>

<p>For example, an individual rejecting a coin flip with a gain of USD 125 and a loss of USD 100 would not accept a gain USD $\infty$ and lose USD 600 gamble.</p>

<p>The papers are worth reading, but keep in mind the rebuttals, e.g., by Cox and Sadiraj (2006) or Palacios-Huerta and Serrano (2006).</p>

CC BY-SA 3.0
2016-12-29T23:37:36.033Z
2016-12-29T23:37:36.033Z
Bayesian
<p>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.</p>

CC BY-SA 3.0
2017-01-26T12:52:18.923Z
This answer could be greatly improved by linking to some of the relevant experiments.
Giskard
CC BY-SA 3.0
2017-01-27T21:12:33.077Z
There are many relevant articles in behavioral economics - and many of them by the two authors. I think it is best to post one answer for each paradox such that people can discuss one issue at a time in the comments and not all at once.
Bayesian
CC BY-SA 3.0
2015-04-15T06:09:50.327Z
2015-04-15T06:09:50.327Z
Sander Heinsalu
<p>Picking up <a href="https://economics.stackexchange.com/a/5145/11590">my comment under this answer</a>.</p>

<p>One striking issue relevant to decisions not captured by expected utility is the framing effect discussed by <a href="http://psych.hanover.edu/classes/cognition/papers/tversky81.pdf" rel="nofollow noreferrer">Tversky and Kahneman (1981)</a> and others.
In their experimental study, they let two different (but with the same characteristics) groups choose between two options. Both groups actually face the same choices, but the wording is different. One group chooses between A and B, and one group between C and D. It is always one safe and one risky choice. While 72 percent picked the save option A vs B, 78 percent picked the risky option D vs C, although in expected-utility terms $A=C$ and $B=D$. So this observation is not compatible with expected utility.</p>

<p>A disease is expected to kill 600 people if no action is taken.</p>

<p>You have two options (program $A$ and $B$) to fight the disease:</p>

<p>If $A$ is adopted, 200 people will be saved.</p>

<p>If $B$ is adopted, all 600 are saved with probability 1/3 and with probability 2/3 no people are saved.</p>

<p>Another group of people faced the choice between program $C$ and $D$</p>

<p>If $C$ is adopted, 400 people will die.</p>

<p>If $D$ is adopted, nobody dies with probability 1/3 and with probability 2/3 all people die.</p>

CC BY-SA 3.0
2017-01-27T21:28:21.933Z
2017-01-27T21:28:21.933Z
2017-04-13T12:51:38.253Z
Bayesian
<p>This is <a href="https://cogsci.stackexchange.com/questions/6426/list-of-experiments-contradicting-the-expected-utility-model">a question I asked on the cognitive science beta</a> which never got any answer there. 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. </p>

<p>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. <span class="math-container">$\Big(P(rain) = 0.4, P(sunshine) = 0.6\Big)$</span> and <span class="math-container">$\Big(P(rain) = 0.6, P(sunshine) = 0.4\Big)$</span>)  which satisfies a list of axioms proposed by Von Neuman and Morgernstern, namely</p>

<ul>
<li>Completeness</li>
<li>Transitivity</li>
<li>Continuity</li>
<li>Independence </li>
</ul>

<p>A rigorous formulation of these axioms can be found on page 8 of <a href="http://www.econ2.jhu.edu/people/karni/chapter.pdf" rel="nofollow noreferrer">Axiomatic Foundations of Expected Utility and Subjective Probability, by Edi Karni, from the Handbook of Economics of risk and uncertainty.</a>. </p>

<p>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):</p>

<p><span class="math-container">$U(L) = \sum_{all~possible~events "e"} P(e)u(e)$</span></p>

<p>where <span class="math-container">$P(e)$</span> is again the probability that <span class="math-container">$e$</span> occurs and <span class="math-container">$u(e)$</span> is the utility of getting event <span class="math-container">$e$</span> for sure. </p>

<p><strong>The violations of these axioms I am most interested in are the ones related to the Independence axiom</strong> (violations of completeness, transitivity and continuity would probably deserve a separate question. See <a href="https://cogsci.stackexchange.com/questions/5150/are-people-more-likely-to-pick-the-odd-one-out/5162#5162">this question</a> for an example of intransitivity.).</p>

<p>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 <a href="http://www.vanderbilt.edu/econ/conference/rational-choice/Bradley.pdf" rel="nofollow noreferrer">still a debate regarding Ellsberg paradox</a>). 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.</p>

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

<expected-utility><decision-theory><uncertainty>
Experiments contradicting the expected utility model
Martin Van der Linden
# Experiments contradicting the expected utility model

(Asked by: username_0 on 2014-11-19)

This is [a question I asked on the cognitive science beta](https://cogsci.stackexchange.com/questions/6426/list-of-experiments-contradicting-the-expected-utility-model) which never got any answer there. 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.](http://www.econ2.jhu.edu/people/karni/chapter.pdf).

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](https://cogsci.stackexchange.com/questions/5150/are-people-more-likely-to-pick-the-odd-one-out/5162#5162) 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](http://www.vanderbilt.edu/econ/conference/rational-choice/Bradley.pdf)). 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.

**Tags:** expected-utility, decision-theory, uncertainty


---

## 6 answers

### Answer by username_1 on 2014-11-19. Score: 13

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>


---

### Answer by username_2 on 2014-11-19. Score: 10

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.

> **Comments:**
> **username_3**: 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.  
> **username_2**: @username_3 Okay, but this critique must also apply to the Allais paradox or anything with gambles.  
> **username_3**: 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.  
> **username_2**: 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.  
> **username_3**: Ok. Sorry. I see what you're saying. That's interesting. I've opened another question to help further this train of thought. http://economics.stackexchange.com/questions/134/can-the-machina-paradox-by-expanding-the-choice-set  

---

### Answer by username_0 on 2014-11-20. Score: 5

Following @username_2 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.

> **Comments:**
> **username_4**: Any idea how I can finish the proof of EUT with three outcomes?  

---

### Answer by username_5 on 2017-01-27. Score: 5

Picking up [my comment under this answer](https://economics.stackexchange.com/a/5145/11590).

One striking issue relevant to decisions not captured by expected utility is the framing effect discussed by [Tversky and Kahneman (1981)](http://psych.hanover.edu/classes/cognition/papers/tversky81.pdf) and others.
In their experimental study, they let two different (but with the same characteristics) groups choose between two options. Both groups actually face the same choices, but the wording is different. One group chooses between A and B, and one group between C and D. It is always one safe and one risky choice. While 72 percent picked the save option A vs B, 78 percent picked the risky option D vs C, although in expected-utility terms $A=C$ and $B=D$. So this observation is not compatible with expected utility.

A disease is expected to kill 600 people if no action is taken.

You have two options (program $A$ and $B$) to fight the disease:

If $A$ is adopted, 200 people will be saved.

If $B$ is adopted, all 600 are saved with probability 1/3 and with probability 2/3 no people are saved.

Another group of people faced the choice between program $C$ and $D$

If $C$ is adopted, 400 people will die.

If $D$ is adopted, nobody dies with probability 1/3 and with probability 2/3 all people die.


---

### Answer by username_5 on 2016-12-29. Score: 4

Let me mention another quite well-known one: The calibration theorem by [Rabin (2000)](http://onlinelibrary.wiley.com/doi/10.1111/1468-0262.00158/abstract) and [Rabin and Thaler (2002)](http://www.jstor.org/stable/2696549). The idea is that over small stakes individuals must be essentially risk-averse, but in reality they are not.

Only assuming a weakly concave and strictly increasing utility function, Rabin shows that risk aversion on small stakes implies obviously unrealistic risk aversion over large stakes. In other words, under expected-utility theory, a resistance to accept small stake gambles with positive expected value leads to absurd conclusions about individuals' behavior in large stake gambles.

For example, an individual rejecting a coin flip with a gain of USD 125 and a loss of USD 100 would not accept a gain USD $\infty$ and lose USD 600 gamble.

The papers are worth reading, but keep in mind the rebuttals, e.g., by Cox and Sadiraj (2006) or Palacios-Huerta and Serrano (2006).


---

### Answer by username_6 on 2015-04-15. Score: 4

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.

> **Comments:**
> **username_7**: This answer could be greatly improved by linking to some of the relevant experiments.  
> **username_5**: There are many relevant articles in behavioral economics - and many of them by the two authors. I think it is best to post one answer for each paradox such that people can discuss one issue at a time in the comments and not all at once.  

---

