At schools that do forced-curve grading or grade normalisation. If what matters is your relative rank in the class, rather than your absolute score, then "everyone takes the 2" and "everyone takes the 6" produce the same relative ranks for everyone - i.e. the same result for you - while "you take the 2 while some other people take the 6" reduces your relative rank.
If there's a nonlinearity introduced by the mapping from points to a letter grade to a GPA, and you have an extremely detailed knowledge of your current grades, and you know 2 points won't bump you up a letter grade whereas 6 points will (i.e. 2 points and 0 points have the same effect on your GPA, only 6 points will improve your GPA).
Among people who believe punishing defectors to produce fair outcomes is more important than producing the best average outcome. If you believe there are inevitably people who will take the 6 points, and those people are assholes, and you'd rather punch an asshole in the face (make them lose 6 points) even if it hurts your hand (you lose 2 points). The same logic that makes people brake for tailgaters.
In parts of society that prize 'rational self interest' and believe everyone should look out for their own best interests and let market forces sort it out; or that it's inevitable that other people will do this and if you're not doing it, you're basically a sucker. The same logic that makes people take seven-figure salaries in finance instead of becoming school teachers - and makes salesmen push the product with the highest commission, not the one that's best for the customer.
To clarify, the theory behind the free market is that the seven figure salary in finance represents a greater contribution to the net social good than a teacher. You personally might not agree with this, but you are completely misrepresenting economic theory when you say that proponents of the free market consider it to be a prisoners dilemma situation. On the contrary they argue for the free market precisely because they believe that in the free market, individual and social goals are aligned
The "correct" answer is to generate a random number in [0, 1]. If less than 0.1, then select 6, else select 2. For safety use 0.05 (say) instead of 0.1. This means that everyone is guaranteed 2, but some (as many as possible) get 6. It works without coordination.
Everyone can work that out for themselves! In theory. Or else remember it from the course textbook. In context, "coordination" usually means communication after the game is set up, whereas the idea of the random number generator is a standard approach even in advance.
Students who are highly likely to get an A in the class should choose 6 points, to try to ensure that lesser students do not get a free opportunity to catch up.
Students who are likely to get less than an A should all choose 2 points, in the hopes that they'll get some free points.
Why should well performing students try to prevent poor performing students—'lesser' seems harsh—from 'catching up'? (This may be plausible in the classroom setting, but I think becomes more questionable if one thinks instead of, say, welfare grants.)
Also, setting aside any ethical considerations, I'm not sure that the mathematics checks out: if there are very few students getting an A, then they don't have significant power to block 'lesser' students from catching up; whereas, if there are very many students getting an A, then this behaviour is guaranteed to reward none of them, whereas they could instead grant themselves all an extra 2 points.
I don't think that this usage of 'guaranteed' is strictly correct (although I understand the informal sense in which it is meant). I don't know a formal definition of "random number" that would allow me to back up what I'm about to say, but it seems to me impossible to guarantee that exactly 10% of all generated random numbers are less than 0.1. For example, it can't possibly be true if (some, but) fewer than 10 numbers are generated. (For example, consider jmmcd (https://news.ycombinator.com/item?id=9874773 )'s hypothetical classroom with 2 students.)
The best guarantee that I can imagine is some sort of "law of large numbers", which won't help in a class of, say, 30 people, 4 of whom happen to generate a small number!
EDIT: Also, I think that the use of a random-number generator should probably count as co-ordination. Consider, for example, a situation where some outside observer picks floor(10%*$classsize) of the class, and the random-number generator is replaced by this person. (Let's say he has an auxiliary random-number generator, for which he generates a number up to 0.1 for the chosen students, and a number past 0.1 for the unchosen.) This is clearly co-ordination, but it is also indistinguishable to anyone who can't peer behind the curtain; so it seems to me that your situation should count as co-ordination, too.
EDIT 2: Also also, let's ignore the fact that there are only finitely many distinguishable floating-point numbers in [0, 1] (so that, in particular, it makes a difference what we instruct the student to do who generates 0.1 exactly), and suppose that there's some clever output scheme that gives infinitely many possibilities.
All of this is poorly defined (edit: I meant mathematically not well defined) as a game theory problem, from the premises to the suggestion that it's "the prisoner's dilemma with extra credit".
First, the prisoner's dilemma is defined for two people, which this is not. Furthermore, the prisoner's dilemma assumes perfectly rational actors; we cannot assume that of students.
Second (and related to the number of players), the 10% metric has different meanings based on the number of players (students). With two people, one person opting for six points constitutes 50%, which might be the case for an advanced elective at a small school. A first year biology class at a major university might have hundreds of students, in which case we can treat the 10% condition as stated.
Third, the outcomes of the prisoner's dilemma are either absolute gain or absolute loss - either jail or freedom. In this case, no possible outcome constitutes loss relative to the starting state; a player can fail to gain, but cannot be worse from where he or she started before playing. A premise that would make this closer to the prisoner's dilemma would be that if more than some percentage of the students opt for the six points, then all students lose six points.
Overall, this seems like something this instructor is doing for fun rather than as an experiment in a formal extension of the prisoner's dilemma.
For my part, I'd opt for the 2, based on what I remember about undergraduate students. I have no formal reasoning to support this, however.
tl;dr As stated, this is neither an extension of the prisoner's dilemma, nor is it well-defined in any case. It seems to me that any suggested solution will require either additional data or additional assumptions. Either way, I'd choose 2 points, and I would be interested to see the outcome of this instructor's survey.
N-player prisoner's dilemma is well-known. Neither 2-player nor N-player assumes rational actors. EDIT: It assumes that payoffs are in a reasonable unit of utility (so we don't need to think about whether $1000 is worth 10 times as much as $100, or more, or less). That's a related point but not the same.
The second point doesn't make it undefined. If there are two people, and one opts for 2 points, then the 10% is exceeded and the condition is triggered. No confusion.
>The prisoner's dilemma is a canonical example of a game analyzed in game theory that shows why two purely "rational" individuals might not cooperate,
The generalized prisoner's dilemma is indeed well-known; the name "prisoner's dilemma", without further qualification, does, however, refer to the case of two perfectly rational players (hence the title of the wiki article you linked). In either case, the generalized prisoner's dilemma generalizes the payoffs and penalties, but still assumes two players.
Perhaps you were thinking of the iterated prisoner's dilemma since it does deal with more than two players; however, the iterated prisoner's dilemma deals with playing the game more than once in succession, while here we have a single round with an unknown number of players.
As it stands, the given problem is not a version of the prisoner's dilemma, but an entirely different sort of problem altogether. We could perhaps impose additional constraints in order to reduce it to some generalization of the prisoner's dilemma; in any case, as stated, we cannot say that this is "the prisoner's dilemma with extra credit" in any meaningful sense.
(significantly edited for clarity and organization)
Further edit: I am not aware of a canonical statement and solution of an N-player prisoner's dilemma. I would be interested in a reference.
The players don't have to be rational. It is interesting to study PD with non-rational players, e.g. when it arose as a model of the nuclear arms race. It's true that the PD also "shows why two purely 'rational' individuals might not cooperate" but that is not part of the defn of PD.
I was not confusing iterated and n-player versions.
All students get 2 points automatically, and the outcomes are +4, 0, and -2. That satisfies your demand for a negative outcome (and shows that your demand is unnecessary)
18 comments
[ 2.5 ms ] story [ 51.8 ms ] threadAny guesses about where (in society) one can find those who answer "I take the six"?
If there's a nonlinearity introduced by the mapping from points to a letter grade to a GPA, and you have an extremely detailed knowledge of your current grades, and you know 2 points won't bump you up a letter grade whereas 6 points will (i.e. 2 points and 0 points have the same effect on your GPA, only 6 points will improve your GPA).
Among people who believe punishing defectors to produce fair outcomes is more important than producing the best average outcome. If you believe there are inevitably people who will take the 6 points, and those people are assholes, and you'd rather punch an asshole in the face (make them lose 6 points) even if it hurts your hand (you lose 2 points). The same logic that makes people brake for tailgaters.
In parts of society that prize 'rational self interest' and believe everyone should look out for their own best interests and let market forces sort it out; or that it's inevitable that other people will do this and if you're not doing it, you're basically a sucker. The same logic that makes people take seven-figure salaries in finance instead of becoming school teachers - and makes salesmen push the product with the highest commission, not the one that's best for the customer.
The "correct" answer is to generate a random number in [0, 1]. If less than 0.1, then select 6, else select 2. For safety use 0.05 (say) instead of 0.1. This means that everyone is guaranteed 2, but some (as many as possible) get 6. It works without coordination.
Students who are highly likely to get an A in the class should choose 6 points, to try to ensure that lesser students do not get a free opportunity to catch up.
Students who are likely to get less than an A should all choose 2 points, in the hopes that they'll get some free points.
Also, setting aside any ethical considerations, I'm not sure that the mathematics checks out: if there are very few students getting an A, then they don't have significant power to block 'lesser' students from catching up; whereas, if there are very many students getting an A, then this behaviour is guaranteed to reward none of them, whereas they could instead grant themselves all an extra 2 points.
I don't think that this usage of 'guaranteed' is strictly correct (although I understand the informal sense in which it is meant). I don't know a formal definition of "random number" that would allow me to back up what I'm about to say, but it seems to me impossible to guarantee that exactly 10% of all generated random numbers are less than 0.1. For example, it can't possibly be true if (some, but) fewer than 10 numbers are generated. (For example, consider jmmcd (https://news.ycombinator.com/item?id=9874773 )'s hypothetical classroom with 2 students.)
The best guarantee that I can imagine is some sort of "law of large numbers", which won't help in a class of, say, 30 people, 4 of whom happen to generate a small number!
EDIT: Also, I think that the use of a random-number generator should probably count as co-ordination. Consider, for example, a situation where some outside observer picks floor(10%*$classsize) of the class, and the random-number generator is replaced by this person. (Let's say he has an auxiliary random-number generator, for which he generates a number up to 0.1 for the chosen students, and a number past 0.1 for the unchosen.) This is clearly co-ordination, but it is also indistinguishable to anyone who can't peer behind the curtain; so it seems to me that your situation should count as co-ordination, too.
EDIT 2: Also also, let's ignore the fact that there are only finitely many distinguishable floating-point numbers in [0, 1] (so that, in particular, it makes a difference what we instruct the student to do who generates 0.1 exactly), and suppose that there's some clever output scheme that gives infinitely many possibilities.
First, the prisoner's dilemma is defined for two people, which this is not. Furthermore, the prisoner's dilemma assumes perfectly rational actors; we cannot assume that of students.
Second (and related to the number of players), the 10% metric has different meanings based on the number of players (students). With two people, one person opting for six points constitutes 50%, which might be the case for an advanced elective at a small school. A first year biology class at a major university might have hundreds of students, in which case we can treat the 10% condition as stated.
Third, the outcomes of the prisoner's dilemma are either absolute gain or absolute loss - either jail or freedom. In this case, no possible outcome constitutes loss relative to the starting state; a player can fail to gain, but cannot be worse from where he or she started before playing. A premise that would make this closer to the prisoner's dilemma would be that if more than some percentage of the students opt for the six points, then all students lose six points.
Overall, this seems like something this instructor is doing for fun rather than as an experiment in a formal extension of the prisoner's dilemma.
For my part, I'd opt for the 2, based on what I remember about undergraduate students. I have no formal reasoning to support this, however.
tl;dr As stated, this is neither an extension of the prisoner's dilemma, nor is it well-defined in any case. It seems to me that any suggested solution will require either additional data or additional assumptions. Either way, I'd choose 2 points, and I would be interested to see the outcome of this instructor's survey.
N-player prisoner's dilemma is well-known. Neither 2-player nor N-player assumes rational actors. EDIT: It assumes that payoffs are in a reasonable unit of utility (so we don't need to think about whether $1000 is worth 10 times as much as $100, or more, or less). That's a related point but not the same.
The second point doesn't make it undefined. If there are two people, and one opts for 2 points, then the 10% is exceeded and the condition is triggered. No confusion.
The prisoner's dilemma arises just as well if the payoff matrix is all positive. You just need T > R > P > S [https://en.wikipedia.org/wiki/Prisoner%27s_dilemma].
>The prisoner's dilemma is a canonical example of a game analyzed in game theory that shows why two purely "rational" individuals might not cooperate,
The generalized prisoner's dilemma is indeed well-known; the name "prisoner's dilemma", without further qualification, does, however, refer to the case of two perfectly rational players (hence the title of the wiki article you linked). In either case, the generalized prisoner's dilemma generalizes the payoffs and penalties, but still assumes two players.
Perhaps you were thinking of the iterated prisoner's dilemma since it does deal with more than two players; however, the iterated prisoner's dilemma deals with playing the game more than once in succession, while here we have a single round with an unknown number of players.
As it stands, the given problem is not a version of the prisoner's dilemma, but an entirely different sort of problem altogether. We could perhaps impose additional constraints in order to reduce it to some generalization of the prisoner's dilemma; in any case, as stated, we cannot say that this is "the prisoner's dilemma with extra credit" in any meaningful sense.
(significantly edited for clarity and organization)
Further edit: I am not aware of a canonical statement and solution of an N-player prisoner's dilemma. I would be interested in a reference.
I was not confusing iterated and n-player versions.
For a ref, eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.... and it gives some further citations.