<p>I am not sure I understand the problems outlined in the other answers. Seems to me that if we assume students maximize their expected utility and lotteries A1 and B2 (or B1 and A2) are chosen by someone, we would have</p>

<p><span class="math-container">\begin{align*}
100\% \ U(B) &amp; &gt; 10\% \ U(A) + 89\% \ U(B) + 1\% \ U(F) \\
\\
11\% \ U(B) + 89\% \ U(F) &amp; &lt; 10\% \ U(A) + 90\% \ U(F)
\end{align*}</span>
which are contradictory. There is no need to assume anything about the function <span class="math-container">$U$</span> for these to be contradictory.</p>

<p>One runs into the usual problem of trying to elicit true preferences in a poll, but so does the original paradox.</p>

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2019-01-04T08:38:07.440Z
@EnergyNumbers Please comment.
Giskard
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2019-01-04T08:38:29.240Z
@KitsuneCavalry Please comment. (Also how does one do carbon copies in comments?)
Giskard
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2019-01-04T18:30:48.413Z
Idk if there is a feature for cc pinging. I did not get this ping b/c I have to comment here first in order to be replied to.
Kitsune Cavalry
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2019-01-04T18:31:55.510Z
Given OP's comments, it seems that getting a score of 0-17 is all failing, but then there is a jump in benefit for when you pass. The range of B grades we'd expect indifference until a jump to B+ or something like that. So it appears preferences are discontinuous, and you need continuous preferences to have Von-Neumann Morgenstern utility representation. I may not be thinking about this rigorously enough though.
Kitsune Cavalry
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2019-01-04T18:39:54.877Z
@KitsuneCavalry I thought A,B,C (perhaps even D) grades all pass?
Giskard
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2019-01-04T18:53:35.507Z
That is my impression too. I am assuming though that one prefers a higher "pass" than a lower one, at a discontinuity. But I think maybe my error is that I am confusing discontinuous utility for discontinuous preferences.
Kitsune Cavalry
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2019-01-04T22:32:41.787Z
@KitsuneCavalry I am not well versed in topology but I think that everything here is continuous, only the set of alternatives is not finite, and hence in some sense not [dense](https://en.wikipedia.org/wiki/Dense_set).
Giskard
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2019-01-04T22:39:50.823Z
I'm just tryna do the little epsilon balls in my heads to think of preference continuity but my brain is too tired atm lmao
Kitsune Cavalry
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2019-01-04T08:37:14.290Z
2019-01-04T08:37:14.290Z
Giskard
<p>No, it is incorrect to reformulate it like that. You've formulated something, but it's not the Allais paradox.</p>

<p>The point of the Allais paradox is that the rewards are (in the theory being tested) quantitative and of constant marginal utility. Grades are not quantitative, and are not claimed to have constant marginal utility.</p>

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2018-12-04T16:00:03.857Z
The idea was to use the Italian grading system, in which grades are in a 30-points scale, where pass is equal to 18.
plastico
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2018-12-04T16:02:29.813Z
So, again, a pass isn't quantitative (it's pass or fail). Is anyone claiming constant marginal utility here?
410 gone
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2018-12-04T16:24:16.917Z
But any score above the minimum score which is required to pass the exam has constant marginal utility. What if we consider that any score below 18 still has constant marginal utility because not passing the exam, having a score 5 or 17 is not the same? I means, if you take 17 it means that you will have to study less than someone who has taken 5 in order to pass the exam the next time you take it.
plastico
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2018-12-04T16:27:11.143Z
I suggest you plot what you think is the marginal utility, and edit it into your question. Also, add in all this extra information which might be relevant, about study effort and opportunities to retake the exam
410 gone
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2018-12-04T17:00:36.157Z
I think you are right
plastico
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2018-12-05T01:44:54.000Z
@EnergyNumbers: I don't recall any requirement in the Allais paradox involving constant marginal utility. Could you provide a source on how exactly constant MU is at play in the paradox?
Herr K.
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2018-12-04T15:56:26.667Z
2018-12-04T15:56:26.667Z
410 gone
<p>The Allais paradox is a challenge to the <strong>independence</strong> axiom of expected utility representation. With monetary outcomes in the lotteries, you might have utility representation that was risk averse, risk neutral, or risk loving. Regardless of your utility, your preferences should not violate the independence axiom. If you are risk averse or risk loving, clearly the marginal value of money is not constant, but this is not a requirement to show independence can be violated. Independence rather is a prerequisite for expected utility representation in the first place.</p>

<hr>

<p>The problem with the experiment above, as is, is that the lottery outcomes are rankings rather than on a scale. You can no longer geometrically represent lotteries on a simplex.  An easy amendment to this is to just have some numerical grade assigned instead of letter grades. Preferences among simple lotteries in this case however might now violate continuity. A "low" B+ is the same as a "high" B+, so a lot of number grades might have the same preference until a particular jump. The comments in EnergyNumbers's answer also suggest that the grading scale might be discrete rather than continuous.</p>

<p>So if anything, that is another problem altogether.</p>

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2019-01-04T01:59:04.300Z
2019-01-04T01:59:04.300Z
Kitsune Cavalry
<p>For a project in experimental economics, I thought of doing something related to expected utility theory/prospect theory, but using grades instead of money.</p>

<p>Is this reformulation of the Allais paradox conceptually right or not?</p>

<p><em>Problem 1.1:<br>
Consider the following scenario:<br>
A.1 – you can get an B+ with probability 100%<br>
B.1 – you can get A with probability 10% or get B+ with probability 89% or not passing the exam with probability 1%.<br>
+<br>
A.2 – you can get B+ with probability 11% or not passing the exam with probability 89%,<br>
B</em>.2 – you can get A with probability 10% or not pass the exam with probability 90%.*</p>

<p>Then I will modify the problem to see if for higher stakes student's preferences change.</p>

<p>Edit:
If we consider that not passing the exam gives utility = 0, (as in the Allais paradox we have the same case because it corresponds to receiving 0$) the graph of the utility function, assuming constant marginal utility of grades would look like this:
<a href="https://i.sstatic.net/G0sdT.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/G0sdT.jpg" alt="utility"></a></p>

<expected-utility><risk><risk-aversion><experimental-economics>
Can I recreate an experiment on Allais paradox using student grades as payoffs?
plastico
# Can I recreate an experiment on Allais paradox using student grades as payoffs?

(Asked by: username_0 on 2018-12-04)

For a project in experimental economics, I thought of doing something related to expected utility theory/prospect theory, but using grades instead of money.

Is this reformulation of the Allais paradox conceptually right or not?

*Problem 1.1:  
Consider the following scenario:  
A.1 – you can get an B+ with probability 100%  
B.1 – you can get A with probability 10% or get B+ with probability 89% or not passing the exam with probability 1%.  
+  
A.2 – you can get B+ with probability 11% or not passing the exam with probability 89%,  
B*.2 – you can get A with probability 10% or not pass the exam with probability 90%.\*

Then I will modify the problem to see if for higher stakes student's preferences change.

Edit:
If we consider that not passing the exam gives utility = 0, (as in the Allais paradox we have the same case because it corresponds to receiving 0$) the graph of the utility function, assuming constant marginal utility of grades would look like this:
[![utility](https://i.sstatic.net/G0sdT.jpg)](https://i.sstatic.net/G0sdT.jpg)

**Tags:** expected-utility, risk, risk-aversion, experimental-economics

---

## 3 answers

### Answer by username_1 on 2019-01-04. Score: 1

I am not sure I understand the problems outlined in the other answers. Seems to me that if we assume students maximize their expected utility and lotteries A1 and B2 (or B1 and A2) are chosen by someone, we would have

\begin{align\*}
100\% \ U(B) & > 10\% \ U(A) + 89\% \ U(B) + 1\% \ U(F) \\
\\
11\% \ U(B) + 89\% \ U(F) & < 10\% \ U(A) + 90\% \ U(F)
\end{align\*}
which are contradictory. There is no need to assume anything about the function $U$ for these to be contradictory.

One runs into the usual problem of trying to elicit true preferences in a poll, but so does the original paradox.

> **Comments:**
> **username_1**: @EnergyNumbers Please comment.  
> **username_1**: @KitsuneCavalry Please comment. (Also how does one do carbon copies in comments?)  
> **username_2**: Idk if there is a feature for cc pinging. I did not get this ping b/c I have to comment here first in order to be replied to.  
> **username_2**: Given OP's comments, it seems that getting a score of 0-17 is all failing, but then there is a jump in benefit for when you pass. The range of B grades we'd expect indifference until a jump to B+ or something like that. So it appears preferences are discontinuous, and you need continuous preferences to have Von-Neumann Morgenstern utility representation. I may not be thinking about this rigorously enough though.  
> **username_1**: @KitsuneCavalry I thought A,B,C (perhaps even D) grades all pass?  
> **username_2**: That is my impression too. I am assuming though that one prefers a higher "pass" than a lower one, at a discontinuity. But I think maybe my error is that I am confusing discontinuous utility for discontinuous preferences.  
> **username_1**: @KitsuneCavalry I am not well versed in topology but I think that everything here is continuous, only the set of alternatives is not finite, and hence in some sense not [dense](https://en.wikipedia.org/wiki/Dense_set).  
> **username_2**: I'm just tryna do the little epsilon balls in my heads to think of preference continuity but my brain is too tired atm lmao  

---

### Answer by username_3 on 2018-12-04. Score: 0

No, it is incorrect to reformulate it like that. You've formulated something, but it's not the Allais paradox.

The point of the Allais paradox is that the rewards are (in the theory being tested) quantitative and of constant marginal utility. Grades are not quantitative, and are not claimed to have constant marginal utility.

> **Comments:**
> **username_0**: The idea was to use the Italian grading system, in which grades are in a 30-points scale, where pass is equal to 18.  
> **username_3**: So, again, a pass isn't quantitative (it's pass or fail). Is anyone claiming constant marginal utility here?  
> **username_0**: But any score above the minimum score which is required to pass the exam has constant marginal utility. What if we consider that any score below 18 still has constant marginal utility because not passing the exam, having a score 5 or 17 is not the same? I means, if you take 17 it means that you will have to study less than someone who has taken 5 in order to pass the exam the next time you take it.  
> **username_3**: I suggest you plot what you think is the marginal utility, and edit it into your question. Also, add in all this extra information which might be relevant, about study effort and opportunities to retake the exam  
> **username_0**: I think you are right  
> **username_4**: @EnergyNumbers: I don't recall any requirement in the Allais paradox involving constant marginal utility. Could you provide a source on how exactly constant MU is at play in the paradox?  

---

### Answer by username_2 on 2019-01-04. Score: 0

The Allais paradox is a challenge to the **independence** axiom of expected utility representation. With monetary outcomes in the lotteries, you might have utility representation that was risk averse, risk neutral, or risk loving. Regardless of your utility, your preferences should not violate the independence axiom. If you are risk averse or risk loving, clearly the marginal value of money is not constant, but this is not a requirement to show independence can be violated. Independence rather is a prerequisite for expected utility representation in the first place.

---

The problem with the experiment above, as is, is that the lottery outcomes are rankings rather than on a scale. You can no longer geometrically represent lotteries on a simplex. An easy amendment to this is to just have some numerical grade assigned instead of letter grades. Preferences among simple lotteries in this case however might now violate continuity. A "low" B+ is the same as a "high" B+, so a lot of number grades might have the same preference until a particular jump. The comments in EnergyNumbers's answer also suggest that the grading scale might be discrete rather than continuous.

So if anything, that is another problem altogether.


---

