In the third week of working this problem (1/25/22), O asked “Can we just look up the solution?” and N commented again about how horrible these pirates are, but then the students solved it quickly with backwards induction. They determined not just the Nash equilibrium solution, but also which way each pirate would vote. Sadly, I forgot to take a screenshot of the student work, but we made a quick summary in week four (2/1) to recap for K, who was absent when we finished.

We moved on to the most famous of game theory problems:


You and your partner-in-crime get caught. The police separate you and offer you each the same deal:

  • If you both stay silent, you each get 1 year in jail.
  • If one of you confesses but the other stays silent, the one who confesses goes free and the other (silent) one gets 4 years.
  • If you both confess, you both serve 2 years.

What should you do?

The students struggled with this using a decision matrix for a while, until Z said “I’m having trouble understanding who you’re referring to.” (I was saying “you” and other pronouns and it was confusing!) The students then suggested naming the criminals; N proposed calling them Gallavan and Rubbert, which made things much easier to keep track of.

SPOILER: After a lot of work, the students determined that the dominant strategy for both Gallavan and Rubbert is to confess, that confessing is always better, that the outcome that’s best for the individuals is not the outcome that’s best for the group (which would be both staying silent.)

Can we do something to insure the outcome that’s best for the group? I asked.

“I don’t think so,” said O. Everyone discussed and then agreed with O, for now.

Sadly I (again!) forgot to take a screenshot before the 40-minute-Zoom break so I lost the students’ work. We quickly reconstructed a summary after the break, but this image understates the hard work that went into solving the problem.

At the start of the session 4, we watched a quick video of the problem to refresh memories and to introduce the problem to K, who was absent last week. K commented that “these rules make absolutely no sense in the real world!” (Another chance for me to make the point that math problems don’t have to be consistent with the real world; they only need to be consistent with themselves.)

Would it change things if the prisoners knew the payoff schedule in advance? I asked. Would both actors both still be best off by confessing? At first the students all said yes, but after more discussion, they thought that the answer comes down to whether you can trust your partner.

“Then don’t have a partner in the first place! I wouldn’t!” said N. I mentioned that changing the premises of the problem, as A also suggested doing in the Pirate Problem, can sometimes be a very useful math strategy.


There are two countries in the world: North and South. The world needs to prevent the world temperature from increasing by more then 2 degrees Celsius or life as we know it will be drastically harmed. Is climate mitigation good for everyone? (“Yes”) Mitigation will cost 6 in every region. If only one region does it, it will generate a profit of 4 in both regions. If both regions do it, there’s a profit of 8 in both regions. What do you think the question is?

“Should you do it?” said several students.

“Of course you should do it!” responded the other students immediately.

“Do the countries trust each other? What are the decision factors?” asked Z.

Let’s first assume countries don’t trust each other, but then talk about how it would play out if they do. Assume that profit is the only decision factor.

SPOILER: The group used a decision matrix to determine that the dominant strategy for each country is to pollute not mitigate. In other words, just as in the Prisoner’s Dilemma, looking out for one’s own best interests only is better mathematically than cooperation, even though cooperation gives a better outcome for the group as a whole. The students decided that you get a different answer depending on whether we are using the conventional game theory assumption of rational actors. “It comes down to trust,” they said.

Can countries trust each other, and if so, under what circumstances? I asked.

The students, some of whom are more knowledgeable about current events and history than I am, pondered how this problem might play out differently if the countries “North” and “South” represented the US, China, Russia, Korea, Bosnia, and others. I found it interesting that the students related Climate Talks to The Pirate Problem more than to The Prisoner’s Dilemma, the traditional model in the field of International Relations (IR).

“This is all politics!” said both N and K.

This is International Relations, where people in this field really do use math! I explained.


Now I was finally ready to ask the question that I had been excited for weeks to ask (four weeks of work to build up to this):

Is The Prisoner’s Dilemma a good model for climate talks?

I explained to students that some people who discuss the field of IR think it is and that things are pretty hopeless. Some people are unsure. Some people think it’s a bad model. What do you think?

The students had a huge discussion about this, with their many perspectives summarized below.

  • It’s a good model because
    • things do feel pretty hopeless
  • It’s hard to tell because
    • what it if leads to war?
    • What if countries can punish or threaten each other?
    • What if the threat of war is the only deterrent?
    • What if sanctions could work?
    • What if binding agreements could work?
    • What if countries do look at the long term, instead of the short-term reelection of their leaders? (Students pointed this out as an issue in The Pirate Problem too.)
  • It’s a bad model because
    • in reality, not all countries are equal (in their power/influence and in their contributions to global warming).
    • In reality, “not all leaders are scumbags” – this is what K really said when I wrote the bullet point “ethical leaders” on the slide. My in-class paraphrases in the interest of speed and brevity lost something in translation.
    • In reality, there are other decision factors; climate change is more complex than The Prisoner’s Dilemma.
    • In reality, people (governments and companies) realize that mitigation is a public good
    • In reality, it might not be so expensive to mitigate.
    • In reality, members of the public can/do put pressure on their leaders.
    • In reality, decision makers “are not robots” (i.e. they are not rational actors like game theory assumes).

In the end, K, Z, and N came down on the side of The Prisoner’s Dilemma being a bad model for Climate Talks.  O came down on the side of hopelessness. I didn’t want to leave students depressed from this topic. So I stressed that my belief is that there is hope.

Z concluded that “It’s cool that people try to apply math to real-world problems.”

Then I moved on to a more whimsical problem for some emotional relief, one of my old favorites.


“You are exploring a land populated by hydrophobic vicious animals.  You are safely wading in a one-foot deep stream when you come to a fork in it.  Each branch leads to a different pond.  Each pond has a helicopter on the other side of it that can transport you to safety.  A sign at the fork tells you that the pond on one side has an average depth of 5 feet, and the other is 7.  Oh, and did I mention that you can’t swim?”

I presented the problem and said nothing. The students initially agreed that the answer is 5. But then someone eventually questions the word “average.” Students debated and calculated and asked questions such as Z’s: “How tall are you?” They came to the realization that distribution/spread actually matters more than average/central tendency. O concluded that if you’re guessing, you might as well assume that this is a typical pond that is more shallow around the edges. The others agreed.


In week 3 (1/25), O presented a function machine where the students had to guess the rule y=x/3. This led to a quick discussion of square roots, and that “divided by 3” not the same as square root

In week 4 (2/1) N asked to limit the domain to less than 100 but the students wanted me to present a rule for them to guess. First I gave a rule where the students derived a conditional function (“the next even integer if it’s odd, plus two if its even.”) This conditional rule does work. I challenged them to think about after class whether they can simplify, in other words, think of a way to express this rule as a single, not conditional, function? (Note from future me: we never came back to this; the unconditional rule I was thinking is that the output is the next larger even integer.)

With 2 minutes left, I gave another function in which some students defined the function as y=x-x and others as y=0x. I mentioned that different equations can give the same results, that even if the output is the same every time it is still a function.


N brought a nice math quote to class today:  “Every triangle is a love triangle when you love triangles” (Pythagoras)



The students have been enjoying The Pirate Problem on such a deep level that I’m sticking with game theory to build up to problems about using game theory to model real-life situations. The particular students are very interested in philosophical themes – cooperation, rational actors (is there such thing and how do they act), trust, and the validity (or not) of mathematical models.


NOTE: This reports includes references to human trafficking, Chernobyl, hypothetical violent actions, guns, and student reactions to these topics. Feel free to skip it and jump down to the next topic.

During Session 1 of this course, N was appalled by one of the Decision Factors of the Pirate Problem: that If the majority vote no, the proposer is thrown overboard and the next most senior proposes a distribution.  I had given the students a trigger warning but the shock value was still there. I didn’t anticipate that. There has been ongoing discussion in the Math Circle community on how problems involving life and death are highly motivating but … are they actually appropriate for students?  

During Session 4, when I showed the video of The Prisoner’s Dilemma, which was essentially a slide show of drawings, the first image generated a reaction from N: “Guns!” (The image was a drawing of a pirate holding a gun.) I didn’t anticipate that.

I also didn’t anticipate O’s reaction to the seeming hopelessness of climate change; nor did I anticipate the class discussing the possible need for force and threat to keep countries in line

A few years ago, I taught a Math Circle course on “The Mathematics of Social Change.” In this course, we discussed how math can be used to reduce human trafficking. I fortunately did anticipate student reaction to this and was able to inform and educate families in advance and also invite parents to sit in on class. I also polled parents in advance on whether to explore this topic. All said yes. In a course on the history of Fermat’s Last Theorem, I decided to not include anecdotes that involved violence. Planning helped a lot. I have used my judgment about what age groups to mention things to.

Unfortunately, I haven’t always anticipated which topics may generate a reaction, and these students are not adults! (In the current group, they are 11-13. They were a bit older in the Social Change course.) Sometimes the students, not me, bring up potentially-upsetting topics (like Chernobyl, or anecdotes involving violent behavior, which I immediately quashed). Sometimes something flies under my radar, like the gun in the Pirate video – I knew better because in a past course there was some family preference that I not teach the problem “The Truel” which involves guns. But I just didn’t notice, and that will happen sometimes.

Moving forward, I think my plan will be to always try to remember that these are children! I’ll continue to give trigger warnings and ask parents up front via email. Ideally, I want to have a list of things that I should warn people in my Math Circle community about: crime, guns, climate change, human slavery, war, international relations, etc  …..  I invite members of this Math Circle community to put in the comments your suggestion of topics and age groups for this list.

On the other issue: how to handle things students bring things up. I plan to continue to generally shut it down right away, saying just the mention of that thing could make someone the class feel bad. Nuclear war was very relevant to our discussion today, however, so didn’t shut it down, but again wish I had anticipated and warned people up front.


Just a quick note about things I’ve been learning about this specific platform of Zoom: one student got locked out after the break because he came back a bit late and I was sharing my screen, so didn’t see the “admit” button so easily. Also, ugh, twice I lost the work when I had to exit and reenter because of the 40-minute rule. (In the past, I was using a different version of Zoom without this limit.) Here’s my new system: when we are about to break, have the students remind me to take a screen shot! (Future me reports that this has worked extremely well.)


Function Machines: On student asked “can you give us a hint?” I said to ask the others, not me. The others did not want a hint. They said that if they didn’t solve it after a few more minutes, then a hint would be okay. I was so close to giving the hint and thereby lessening their joy of discovery. So glad I didn’t!


Differing Perspectives on Applying the Prisoner’s Dilemma to Climate Change Talks

One little detail: with my 20/20 hindsight, I would rewrite this problem to say “revenue” instead of “profit” (my economic training showing here – but the students didn’t question it).

Game Theory

I’m using these elements in a game – the actors, the rules, the decision factors, equilibrium, and the problem.

Nash Equilibrium

The Pirate Problem – Famous Game Theory Puzzle

The Prisoner’s Dilemma – the Most Famous Problem in Game Theory

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