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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%.
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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

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3 Answers 3

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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.

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  • $\begingroup$ @EnergyNumbers Please comment. $\endgroup$
    – Giskard
    Jan 4, 2019 at 8:38
  • $\begingroup$ @KitsuneCavalry Please comment. (Also how does one do carbon copies in comments?) $\endgroup$
    – Giskard
    Jan 4, 2019 at 8:38
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    $\begingroup$ 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. $\endgroup$
    – Kitsune Cavalry ♦
    Jan 4, 2019 at 18:30
  • $\begingroup$ 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. $\endgroup$
    – Kitsune Cavalry ♦
    Jan 4, 2019 at 18:31
  • $\begingroup$ @KitsuneCavalry I thought A,B,C (perhaps even D) grades all pass? $\endgroup$
    – Giskard
    Jan 4, 2019 at 18:39
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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.

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  • $\begingroup$ The idea was to use the Italian grading system, in which grades are in a 30-points scale, where pass is equal to 18. $\endgroup$
    – plastico
    Dec 4, 2018 at 16:00
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    $\begingroup$ So, again, a pass isn't quantitative (it's pass or fail). Is anyone claiming constant marginal utility here? $\endgroup$
    – 410 gone
    Dec 4, 2018 at 16:02
  • $\begingroup$ 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. $\endgroup$
    – plastico
    Dec 4, 2018 at 16:24
  • $\begingroup$ 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 $\endgroup$
    – 410 gone
    Dec 4, 2018 at 16:27
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    $\begingroup$ @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? $\endgroup$
    – Herr K.
    Dec 5, 2018 at 1:44
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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.

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