I heard this from Marco who heard it from Tzachi. Not sure what to make of it, but that will not deter me from ruminating publicly
There is a sack of chocolate and you have two options: either take one piece from the sack to yourself, or take three pieces which will be given to Dylan. Dylan also has two options: one pieces for himself or three to you. After you both made your choices independently each goes home with the amount of chocolate he collected.
Write down the payoff matrix for this game and you’ll get the Prisoner’s Dilemma. But where is the dilemma here ? In the few occasions that I presented the Nash solution to the prisoner’s dilemma to a friend who didn’t know about game theory, the response was always `something is wrong here’. I am pretty sure that if I try to do the same with the Chocolate Dilemma the response will be `well, duh’. Obviously every sensible person will prefer one chocolate piece to himself than three pieces to somebody else.
One can say that this hypothetical response vindicates the Nash solution also in the Prisoner’s Dilemma. After all, these are the same games. If people find the solution intuitive in the Chocolate Dilemma they should also accept it in the Prisoner’s Dilemma. But this argument is not very convincing since these are the same games only because we game theorists identify a game with its payoff matrix. The rest of the world might disagree.
The Leisure of the Theory Class 
17 comments
March 5, 2010 at 1:11 pm
Jeff Darcy
Some social games like Farmville have a similar dynamic – you can often send better items as gifts to your friends than you can buy for yourself. Many people do, and even go out of their way to recruit new friends with whom to do such exchanges. In fact, the games depend on this to expand their user base. I think the difference between that case and this one is the obviousness of cooperation as an option. In the social games it’s very clear that you’re in it with others; in the chocolate example not so much. I’ll bet if I went up to a colleague and said “let’s go get three pieces of chocolate each” it’d be an easy sell and defection would hardly ever occur.
March 5, 2010 at 5:40 pm
Eran
I should have wrote that after the game is played both you and Dylan go home with your chocolate and never see each other again. This is also the assumption behind the prisone’s dilemma.
in long-term relashionship (as with your colleauges) the Nash solution is less counter-intuitive since it can supprot cooperaion in repeated prisoner’s dilemma
March 5, 2010 at 2:01 pm
Ori
The prisoner’s dilemma is usually described using slightly more complicated payoffs. In your description the game is a sum of two games and the analysis of each is trivial, from a layman’s point of view.
March 5, 2010 at 2:47 pm
Vladimir Nesov
It’s still way better for both to have 3 pieces by cooperating than for both to have only 1 piece by defecting. If the “right solution” leaves the players worse off, it’s probably not the right solution, even if it’s arrived to with propriety and is supported by a seemingly inevitable argument.
March 6, 2010 at 4:11 am
afinetheorem
Do people really say “something must be wrong here” when presented with the prisoner’s dilemma? Perhaps the difference is that since there is a direct mapping from chocolate amounts to the payoff matrix (in utility), the payoff matrix is easier to understand. The prisoner’s dilemma implicitly assumes that you get a better deal from the prosecutor only if you snitch and your partner doesn’t, so perhaps people think of a conditional plea bargain is a strange concept?
March 6, 2010 at 6:05 pm
Eran
Yes (to `do people really say…’) maybe not in these words, but from my experience, when encountered with the story for the first time, people try to explain why defecting is actually irrational even in a one shot game.
March 7, 2010 at 3:11 am
Eilon
I think that many people who do not know game theory do not understand the prisoner’s dilemma. The well known story about prisoners is misleading. There are two reasons for that:
1) Presenting this situation as a one-shot game is incorrect: if I defect, my comrades will kill me or make my life miserable.
2) The outcome includes only the utility from imprisonment, and not the utility from other factors, like how I feel from betraying my comrade. (in fact, (2) includes (1)).
I suspect that this is why people do not understand this game: we theoreticians invented a theoretical story that does not fit the image that people have in their mind when they hear our story.
In fact, in my view the prisoner’s dilemma is a good example for why cooperation arises in game: repetition leads to cooperation.
In the chocolate game the Nash equilibrium is indeed the natural solution. When I presented this game to my older kid a couple of years ago (he was back then 10 or 11) he immediately said that he will take one chocolate for himself. And when I asked him what he will do if I told him that I am going to give him 3 pieces, he still said that he will take one for himself. I must admit that the chocolate story fits much better the payoff matrix of the prisoner’s dilemma than the story about prisoners. Maybe in the fifties chocolates were not that popular.
I will end by mentioning that a variation of this story (as a TV show where each of two players writes on a note that is put in an envelope whether the host should give $1000 to him or $3000 to the other player) appears as an exercise in the Maschler-Solan-Zamir game theory textbook, soon to appear also in English.
March 7, 2010 at 12:47 pm
JGWeissman
Ignoring the fact that I haven’t eaten chocolate since giving up refined sugar, what if I care about Dylan, who is, after all, a fellow human being.
Try constructing a True Prisoners’ Dilemma: http://lesswrong.com/lw/tn/the_true_prisoners_dilemma/
Also, Vladimir Nesov is right. If you can pull it off, Cooperate – Cooperate for the win.
March 7, 2010 at 5:07 pm
Douglas Knight
If this is in a textbook, the authors must know whether it gets a different reaction. I would like to know!
I think Ori’s reason (factoring as two games) is more likely than Eilon’s (isolation). You could try to test this by first presenting the agglomerated game (about chocolate, not jail) and then the split version. There are still psychological differences in physically dealing with a bag vs, say, writing down an abstract choice, but I think it would be a useful experiment.
March 13, 2010 at 12:37 pm
Eran
Update: Marco actually tried it in his class. Students didn’t know whom they are playing with and still most of them chose to give away the chocolate. i view it as another proof that education kills rationality
March 13, 2010 at 1:02 pm
Vladimir Nesov
If most of them gave away the chocolate, on average they got more than if they all kept the chocolate for themselves. If it is correct to behave in a way that results in a smaller reward, then the goal of the game is something other than winning.
April 1, 2010 at 12:21 am
Ori
Do you think they would’ve done the same if you replaced each chocolate piece with $1000?
March 16, 2010 at 1:55 pm
Two Instructive Ways to Frame the Prisoners’ Dilemma « Cheap Talk
[…] 16, 2010 in Uncategorized | Tags: education, game theory | by jeff The first comes from Eran Shmaya: I heard this from Marco who heard it from Tzachi. Not sure what to make of it, but that will not […]
March 25, 2010 at 10:46 am
Jeremy
Hello,
Fascinating discussion by the way.
I just wanted to say that I’m a maths teacher and I was recently looking for a way to present the prisoners dilemma to some 16 year old pupils and I came up with this exact same matrix and using real chocolate as the payoff! So I was very surprised and pleased to see the same matrix and pay off here, I must be doing something right.
Interestingly when I did it in the classroom with 2 pupils (restricted from communicating with each other) whilst 1 pupil kept defecting the other kept co-operating (I didn’t set this up). It amazed me how many chances the co-operater was willing to give the repeat defector! It couldn’t have gone better because this really got the class going. I think initially they really wanted to shout out “just co-operate, it’ll be better for both of you” as the game progressed and they witnessed the defector amassing a wealth of cholcolate, alot of raised eyebrows and puzzled faces started to appear as they started to come to terms with the fact that the situation wasn’t quite so simple.
April 2, 2010 at 11:31 pm
Jonathan Weinstein
Arriving late to the discussion…
Definitely agree the chocolate is a less convoluted way to present PD to students. I think the only reason the prisoner story has stuck around is that we are permanently stuck with the phrase “Prisoner’s Dilemma,” and then we feel obliged to explain why it’s called that. I do think “Tragedy of the Commons” is more poetic.
I am a dedicated utility maximizer…but for me utility chocolate (or money.) I like chocolate, but I like even more for people not to think I’m a jerk. I also like for them not to decide that game theorists are jerks when I explain my actions. And more importantly, even presuming complete anonymity, I enjoy thinking of myself as not a jerk. So…with such low stakes, I will certainly give away 3 pieces, without thinking very hard. Yes, even if I will never see him again. Make it $3000 and it is harder. I am perfectly “self-centered” in the obvious sense that I only have direct knowledge of my own feelings, but am endowed with a tendency to be happy when I feel I am doing the right thing. In particular, I feel good when I think that I have increased the amount of goodwill and trust in the world. That feeling is worth some amount of money, just a question of how much.
To be fair, I also prefer to think I am not a chump, so that contributes some disutility to the C,D outcome.
I like JGWeissman’s point in his link: to truly be in the world of the payoff matrix, we must not care at all about the other person’s number, generally impossible unless he/she/it is a paper-clip maximizer from another universe.
April 2, 2010 at 11:33 pm
Jonathan Weinstein
In line 7 there was a symbol that the site deleted…should say utility \neq chocolate
April 9, 2010 at 11:05 am
micropile
Logic falls way short when it comes to satisfying your hunger for chocolate :-)