# Prisoner's dilemma formulation for children

I am preparing an introductory course on Game Theory for children (between 10 and 17 years old). In the course description, I want to include a prisoner's dilemma in order to catch children's attention and persuade them to attend the course. Therefore, I would like to find a short, direct and attractive-to-children formulation, a "slogan" for my course. The problem is that all of the formulations I have found falls into one of the following issues:

• It includes a reference to "+18" content (death, crimes, etc.).
• It is formulated capitalistically, making money the main protagonist.
• It is a complex and enmeshed formulation, or it is too unreal for a kid's mind.

Can anyone suggest a formulation of the prisoner's dilemma that does not fall into any of these issues?

• "Suppose two competing countries A and B can choose between arming or not arming. There are four possible outcomes, then. Both countries may choose not to arm. A may arm while B does not. B may arm while A does not. Or both countries may arm." — The Harmony of Interests Revisited Oct 9 '19 at 23:12
• I wouldn't consider an example about making money an "issue". They will have problems later in life if they don't learn about money between 10 and 17 years old. Oct 10 '19 at 12:50
• Death and crime are not taboo subjects for kids aged 10-17.
– user507
Oct 10 '19 at 15:36
• Why do you think prison is a taboo subject for kids? I'm sure most of them have read or seen Harry Potter and the Prisoner of Azkabaan, for example. Oct 10 '19 at 18:28
• Also, game theory requires a quantitative measure of how good outcomes are. And money is what we use to measure value. I don't understand the objection to have a quantitative measure of value. While money and utility aren't exactly the same thing, trying to explain the difference to 10-year-olds is going to be difficult. Oct 11 '19 at 17:51

## 8 Answers

Here's a silly example:

Give students collections of the same type of thing, where each collection contains "good" objects and "bad" objects -- for example, a stack of Pokemon cards with both rare and common cards. We might assume common cards are worth $$1$$ and rare cards are worth $$5$$. Have ready another stack of cards that are all common -- call this the "Free Stack".

Students pair up and play a trading game, as follows:

• Students want to build a nice deck of cards (in the sense of overall point value), and they can get new cards by trading.
• A trade is made when both students select a card from their own deck, placing it face down in front of them. When the cards are revealed simultaneously, the following takes place:
1. If both cards are rare, then the cards are exchanged and each student takes a card from the Free Stack, adding it to their own deck. This ends the trade.
2. If both cards are common, then the cards are exchanged and the trade ends.
3. If the players offer different types of cards (one common and one rare), then the player who offered the common card gets both cards. This ends the trade.

It is obviously advantageous for players to "cooperate" and both offer rare cards. They trade those cards, but they both get +1 from the free card from the Free Stack.

It is not helpful or harmful if they both offer common cards, so both players get +0 for that option.

If they offer different types, then the winner gets +5 because they received the rare card, and the loser gets -5 because they lost the rare card.

• If Pokemon isn't quite the thing it used to be (I don't know), I'm sure you could substitute many other manipulatives for a trading game like this. Oct 10 '19 at 5:37
• This is interesting. But Prisoners Dilemma with repeated play is different from the standard one time formulation. In this game the kids might well learn to cooperate, perhaps discover tit-for-tat. Oct 10 '19 at 16:36
• If you did this long enough the whole class might learn to cooperate. There's lots of research on repeated Prisoner's Dilemma - easy to find with a google search. Oct 10 '19 at 17:18
• Just to add input - while Pokemon itself is still kinda a thing, Pokemon cards almost certainly aren't. While this could be re-skinned easily (or implemented with regular playing cards), I feel like it regardless might be too abstract/arbitrary to keep most kids' attention. Oct 10 '19 at 20:21
• Note that, in card games (probably Pokemon, but don't know the game myself), a "good deck" is a complex thing. Cards have multiple features, and there's strategy to consider. Kids might get distracted thinking about the real Pokemon card game. Oct 11 '19 at 12:22

I found out about the Prisoners' Dilemma as a kid from a book about the Harry Potter phenomenon, which had a chapter about the problem, but presented as a story about Harry and Draco being accused of breaking school rules. Each was offered the same deal as in the original problem, formulated with House Points being taken away instead of a prison sentence. Maybe you could write a story using these or other characters that kids would be familiar with?

Alternately, the existence of crime and prison sentences are well-known to 10-17-year-olds, so you might not have to modify it at all. Maybe you could play up the second-person formulation ("You are accused of a crime...") to make it more engaging.

I like Nick C's idea more than modifying the typical formulation. The notion of snitching on a friend, regardless of the severity of the "crime", has real-world ramifications beyond the punishment put out by the authorities. Depending on your student population, that is possibly going to spur a conversation that will overshadow the objectives of your lesson.

Here's another non-crime formulation that students will be familiar with:

You and a fellow student are assigned to work on a team project that will give a significant grade in your class. You break the work in half and both agree to do your part over the weekend. Even though it is a major grade, the amount of work for each half of the project is reasonable considering the consequences of a good grade.

If you both do your best work, you will both get 100% but both of you have to do some work over the weekend. If neither of you commit to the project, you will both get 70% but you can both have a free weekend. If one of you commits but the other doesn't, then you will both get an 80% on the project but one of you will work over the weekend while the other doesn't. Your partner won't know whether you decided to do your share of the project until Monday morning, and you won't know what your partner decided. Will you do your share of the project?

This isn't a pure formulation of the Prisoner's Dilemma, because the outcome is being measured in two different currencies (grades and free time) which different students may value differently. But I think any sort of "tragedy of the commons" scenario applied to two players would play out like the Prisoner's Dilemma.

• I kind of like the idea of this example, but I think that the payoffs (grades + freetime) confuse the issue. The core of the dilemma is that the best possible for player A should occur when player A defects and player B cooperates. In you example, it is not clear what the best possible outcome is. Oct 10 '19 at 12:48
• @XanderHenderson The challenge of the Prisoner's Dilemma, especially when repeated, is that you need to behave well enough that your partner will believe that you will cooperate. You'd love it your partner cooperated while you defect, but your partner will also defect if they suspected you were thinking like that. That said, I agree that maybe not every student will see the grade/effort trade-off as something worth striving for. Oct 10 '19 at 14:31
• Perhaps the students have conspired to cheat on an exam. If they cooperate, the instructor will allow them both to retake the exam; but if one rats out the other, then the rat gets whatever score they earned, while the other cheater is expelled from the class. If they both rat the other out, they each get a zero on the exam. The problem that I see with this example is that it uses unethical behaviour (running awfully close to the original objection of criminality), but at least the incentives are clear. Oct 10 '19 at 15:51
• @XanderHenderson "To test limits of Johns Hopkins professor's scaled grading policy, all of his students boycott the final -- and all get As as a result." — Dangerous Curves Oct 11 '19 at 17:52
• @RustyCore That is a great example! Oct 11 '19 at 17:52

It sounds like you wish to protect your students from the violence and greed of the adult world, while still making the dilemma real enough to keep them engaged.

To that effect, I offer two solutions.

One, replace prison with detention. Make the crime something like using cell phones in class or throwing spitballs.

Two, have them arbitrarily grade each others' work (obviously don't make it their actual grades. It's up to you whether you want to let them think they're grading each other). If they both grade each other well, they both get high grades. If one grades the other poorly, but the other grades the first well, then the first gets a higher score, and the second gets 50%. And whatever numbers you want to formulate in there.

as a side note... I don't think money and prison are taboo, or too much for kids to handle. But I agree that they're kinda clichéd, and somewhat boring because we're desensitized and removed, except for the poor kids for whom the issue is far too close to home

# Edit

Following the original prisoner's dilemma, no matter what the opponent does, ratting them out gives you a lighter sentence. Not the best ethics lesson, but this is mathematics. So...

• If they're ratting you out, not ratting them out would give you a E, ratting them out would give you a C.
• If they're not ratting you out, ratting them out would give you an A, and not ratting them out would give you a B.
• Maybe utilizing some sort of curve, and an absentminded teacher - If you both give each other high grades, the teacher will curve you both to some average, like a B. If you give the other student an A and they point out your mistakes, then they'll get the A and you'll fail. If you both point out each other's mistakes, the teacher will notice that neither of you know the material, and so you both fail. Oct 10 '19 at 21:28
• I think that the obstacle course solution I gave works well if you're trying to avoid any themes of unethical behavior, but that this answer does the best+simplest job of actually following the prisoner's dilemma. The conflict between personal gain and justice may be an important part of the problem. Oct 10 '19 at 21:36

Use of performance-enhancing drugs in sports is a good example.

However, why the objection to money and capitalism? Real-life actors value money very much and it affects their behavior greatly.

A very realistic and applicable example involving money is what happens when a group of friends goes dining, depending on whether each friend pays for themselves or the bill gets split equally. Suppose a pair of friends are offered a cheap snack that costs 5 units of money or an expensive dish for 25. The friends consider the dish overpriced, valuing it at 20. The snack is fairly priced in their opinion.

If the friends agree to split the bill evenly, the payoff matrix is as follows: $$\begin{array}{|l|c|c|} \hline & \text{Snack} & \text{Dish} \\ \hline \text{Snack} & 0,0 & -10,5 \\ \hline \text{Dish} & 5,-10 & -5,-5 \\ \hline \end{array}$$ which is typical of the prisoner’s dilemma.

• @NickC see edit. Oct 10 '19 at 22:50
• Sorry Roman, this doesn't seem realistic: why would I split a bill with someone who insists on a meal 5x costlier than mine, and which we agree is poor value? It stands out that there would have to be a big external pay-off for me, or that I'm being somehow forced to share, and that contaminates the pay-off matrix because there's stuff going on that's not reflected in the numbers. You could make the numbers closer, but then nobody cares and the point is again lost; perhaps more people to absorb the loss, but again, the payoffs become blurred. The scenario seems too soft to drive the point... Oct 11 '19 at 10:46
• @SusanW How is it unrealistic when studies show splitting the bill causes a statistically significant increase in spending? How is it too soft when the difference between X years behind bars and Y years does not mean anything to a child, while paying 50 shekels instead of the expected 40 could mean quite a lot? But maybe the doping example is indeed better. Very similar to the joke where two cowboys in turn bet each other they can eat the entire heap of buffalo dung, only to realize that in the end they have eaten a lot of it to no monetary gain. Oct 11 '19 at 13:56
• :-) Actually the cowboy thing is rather good - nice one! And sorry, when I said "too soft", I was unclear, I didn't mean "not nasty enough"; I just meant I didn't think it drew the edges of the scenario sharply enough, because I'd feel I'd sidestep the whole thing by saying "fine, we'll pay separately". In the original PD, a key aspect is that they can't escape and they can't negotiate with each other due to the 3rd party coercive captors; I'm trying to think how to introduce such separation in this scenario without breaking it. Interesting! Oct 12 '19 at 13:13

Perhaps:

You and another classmate are together in an obstacle course.

If you both make it to the end within a minute, you each get a free day off from school. If just one of you makes it to the end in a minute, they get a whole week off.

You know that if you had the other student's help, you could easily finish in a minute, but without it, you won't finish in time. Likewise, the other student would complete the course with your assistance, but not without it.

You don't know what the other student will do - even if you helped them, they might not return the favor.

Do you help them?

• What do you mean by obstacle's course? I'm thinking of sports, but in that case, it would be impossible not to know whether the other is helping you... Oct 12 '19 at 20:52

My suggestion would be to look at the existing literature.

Blake, Rand, Tingley, and Warneken (2015) "introduce a novel implementation of the repeated Prisoner's Dilemma (PD) designed for children to examine whether repeated interactions can successfully promote cooperation in 10 and 11 year olds."

Dealing with younger children (ages 6-11), Fan (2000) reports on a study of "children’s behavior in a prisoner’s dilemma game."

A classic treatment is Tedeschi, Hiester, and Gahagan (1969), which modified the Prisoner's Dilemma Game for a study of children ages 8 to 10.

Multiple papers have been written on how children with autism do or do not cooperate in the Prisoner's Dilemma, if that's a concern.

A journal search for "prisoner's dilemma" children "game theory" came back with about 1000 references. You could limit that to recent studies, or include additional search terms such as "teenager" if you wish. Pick a few studies and read the methodology section for ideas.

References:

Blake, P. R., Rand, D. G., Tingley, D., & Warneken, F. (2015). The shadow of the future promotes cooperation in a repeated prisoner's dilemma for children. Scientific Reports (Nature Publisher Group), 5, 14559. https://doi.org/10.1038/srep14559

Fan, C.-P. (2000). Teaching children cooperation — An application of experimental game theory. Journal of Economic Behavior & Organization, 41(3), 191–209. https://doi.org/10.1016/s0167-2681(99)00072-4

Tedeschi, J. T., Hiester, D., & Gahagan, J. P. (1969). Matrix values and the behavior of children in the Prisoners Dilemma Game. Child Development, 40(2), 517. https://doi.org/10.2307/1127419

Perhaps the problems of "make up" and "traffic" could also be used.

Make-up: Assuming that the chemicals etc., damage your skin in the long run, we may state that constant make-up is "bad". And contrarily no make-up is better in the long run since your skin remains "naturally beautiful".

Given any two individuals with the option of applying make-up or going natural almost everyone will succumb to the lure of make-up. Even though everyone is better off without it in the long run, everyone thinks they'll look better (read attractive) than everybody else - this "race to the bottom" is highly profitable for companies at the expense of the consumers, so to speak. Only a tight cooperation amongst consumers can break out of this cycle and that's very hard to do.

Traffic: Let's say you're in a country where people are semi-literate and if marginally violating a traffic law/discipline gets them going faster, they are going to violate it. Example, lane discipline or "right of way" (i.e., whose turn is it). Those flouting the rules will seem to "move faster" through the system and slowly everyone will do it leading to major traffic issues.

Why did I emphasize semi-literate? You can get the cooperation of the individuals (players in the game) by having them take exams or disseminate information that decreases the likelihood of such traffic violations leading to overall improvement in traffic conditions. Hence, you'll find traffic a lot better in Europe or Canada or USA vs. say the Indian sub-continent, certain African countries and the like.

You could try the same with candies - bring a box of candies to class along with some "health treats". Ask each student to take a single item from the box - either a candy or a healthy option. The candy should be more attractive. To do this though you'll have to have some good candy/chocolates and not some crappy candy i.e., Candy > healthy treat by taste/value. You could add a twist by saying that if you take one candy, the next person after you has the option to take an extra candy.

Another possibility is allowing them to "throw paper balls" in the middle of the class, without penalty or fear of punishment. The option is of a cleaner class vs. a messy one. Even if one student does it, the whole class will devolve into it.

The goal of the Prisoner's dilemma IMHO is simply this: selfishness is worse than cooperation in the long run, even though the former may come out ahead in the short run. Given this POV one could construct numerous examples to show case prisoner's dilemma in a classroom setting.