Is the class still curved? Because if so, then choosing 6 points would always be the correct strategy.
Student’s Dilemma, a riff of the Prisoner version with extra credit
11–19 of 19 posts
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#12> What's your answer? I take the two. Any guesses about where (in society) one can find those who answer "I take the six"?
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…
Braking for tailgators reduces the probability and cost of a crash, at the expense of time to destination.
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#13All 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…
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)
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#14SPOILER? 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.
Better strategy:
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.
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#15SPOILER? 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.
> This means that everyone is guaranteed
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.
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#16SPOILER? 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.
Better strategy: 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.
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#17> What's your answer? I take the two. Any guesses about where (in society) one can find those who answer "I take the six"?
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…
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
Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#18> What's your answer? I take the two. Any guesses about where (in society) one can find those who answer "I take the six"?
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…
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Re: Student’s Dilemma, a riff of the Prisoner version with extra credit
#19Earlier quoted context omitted.
I'm afraid you need to re-read your notes :) 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,…
From your link, >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, t…
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.
For a ref, eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.... and it gives some further citations.