Pirates, Prisoners, and Psychopaths (Math Circle Teens 5 and 6)
December 3 and 10, 2013
This report details some of the decision-theory questions we covered in two weeks of math circle, and notable student response to them. The footnotes at the bottom link to explanations and solutions.
The Pirate Problem
“There are 5 rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.
The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.
The pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates, including the proposer, then vote on whether to accept this distribution. If the proposed allocation is approved by a majority or a tie vote, it happens. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.
Pirates base their decisions on three factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins he receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal. The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from the main proposal.”1
The students promptly renamed the pirates Anna, Blackbeard, Connor, David, and Edith. An hour-long mathematical struggle ensued. Actually, the struggle was much more than mathematical. Some of the students were troubled by the pirates’ rules of distribution. There was a desire for more ethical rules among some students, and a countervailing attitude among others of c’mon dudes, they’re pirates!
I asked the students what was different about this question from the others we’ve been exploring in this course, and the kids immediately knew what I was thinking: up until now, the decision actors only needed to think about themselves (Decision Theory). With the pirates, however, all of the actors decisions affect each other (Game Theory). BTW, Wikipedia defines Decision Theory as the “economics, psychology, philosophy, mathematics, and statistics concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision.”
Once our group was fully able to accept (but not like) the Pirate rules of distribution, the struggle moved into the arithmetic of the distribution. For some, this was compelling. After about 15 minutes, we had tried 5 different distributions, quickly dismissed a bunch of others, but had not found a workable distribution. One student said “I’m bored,” but I think the rest wanted to keep on going. I started to talk with the group about what aspect of a math question elicits in us that “I’m bored” reaction. I had hoped to get into a conversation about our (normal!) emotional reactions to math questions, and also the emotional baggage that we come to math problems carrying in the first place, but sadly, time was up.
My intention today had been to begin with the Pirate Question and related discussion the next time, but the following week I had a different mix of kids so I decided to move into a new question. Note, however, that we did not end up solving the Pirate Question. If you do play with this question at home, I think it would be fun to get into the idea of what it means to be “rational.”
The Prisoner’s Dilemna
“Two prisoners...have been arrested for vandalism and have been isolated from each other. There is sufficient evidence to convict them on the charge for which they have been arrested, but the prosecutor is after bigger game. He thinks that they robbed a bank together and that he can get them to confess to it. He summons each separately to an interrogation room and speaks to each as follows: "I am going to offer the same deal to your partner, and I will give you each an hour to think it over before I call you back. This is it: If one of you confesses to the bank robbery and the other does not, I will see to it that the confessor gets a one-year term and that the other guy gets a twenty-five year term. If you both confess, then it's ten years apiece. If neither of you confesses, then I can only get two years apiece on the vandalism charge..."2
The group enjoyed working on this problem, and all of the related discussion. Once again, the conversation turned to the challenge of putting yourself in the mind of another for exploring decision-theory questions. It’s tempting to reject the given assumptions when they don’t fit your own world view. For instance, in our group, one student was adamant that these foolish criminals should be taking into account (1) the additional negative cost of having a crime on your record, and (2) the moral costs of vandalism and of falsely confessing.
Be patient with me here as I go a bit off topic – or maybe this is not off topic. Conversations like the above always make me think of that maxim about mathematics – that we humans have an instinct to seek structure, and that mathematics is an approach we can use to understand our world – maybe even to make sense of things when they seem patternless/random/chaotic. I just read that we use stories in the same way. In Salman Rushdie’s memoir Joseph Anton4, he states “two unforgettable lessons: first, that stories were not true… but by being untrue they could make him feel and know truths that the truth could not tell him, and second, that they all belonged to him… Man was the storytelling animal, the only creature on earth that told itself stories to understand what kind of creature it was.” So, do we also do math to understand what kind of creature we are? In our study of decision theory, are we are using stories to understand math to understand ourselves and our world? Are storytelling and mathematics flip sides of the same coin? And on a side note, are we also the mathematical animal, the only creature on earth that seeks architectural structure to understand our world? Hmmm….
Enough of me thinking aloud; back to the math the kids did…
The students used the classic decision matrix of causal decision theory to conclude that the prisoners should confess. I introduced the evidential decision theory approach by asking whether the outcome could be different if the prisoners had been twins. In a prior session, we had seen how different approaches (CDT vs. EDT) could lead to different decisions in the Newcomb’s Box problem – remember, Omega the alien? We briefly discussed whether Newcomb’s problem is a paradox because different approaches yield different outcomes. This discussion could have been much longer and deeper, but not everyone was interested. I do encourage those of you who came to the class to think about it on your own if you’d like.
The Psychopath Button
“Paul is debating whether to press the “kill all psychopaths” button. It would, he thinks, be much better to live in a world with no psychopaths. Unfortunately, Paul is quite confident that only a psychopath would press such a button. Paul very strongly prefers living in a world with psychopaths to dying. Should Paul press the button?”3
This question elicited, IMHO, the most deeply mathematical thinking from the group. The discussion started out, of course, with comments about each student’s personal decision factors. Then we shifted into Paul’s mentality and ended up with my favorite three lists on the board: Questions, Assumptions, and Conjectures. Honestly, though, I cannot remember whether the group ended up coming to a definitive yes/no answer to the question.
The following week we were snowed out, then attendance at our snow make-up session was nil because of bad roads. If you’re still reading, you might be interested in scheduling yet another make-up for that final session. I’ll list here what I came to class prepared to discuss, and ask that you email me directly if you’d like to come together in a group to discuss them. We would need three students to make it a true conversation.
People/Topics of Interest
Nash’s Equilibrium This is the next logical step in our Decision/Game theory course. It was developed by Nobel Prize winning John Forbes Nash (the subject of the book/movie A Beautiful Mind). I’d like to talk to the kids about whether the Pirate Problem applies here. I have read that some mathematicians consider that problem to not be a true decision/game-theory question, yet others describe its solution as a Nash Equilibrium.
Julia Robinson My goal for every Math Circle course we do at Talking Stick is to present some historical information about female mathematicians too. Boy, was that tough in this course on Decision Theory. I had planned to talk about Julia Robinson, whose personal background is fascinating, and who took the concept of Nash’s Equilibrium a bit further.
The Stag Hunt You are going on a hunt with another hunter a very long time ago. You can both cooperate to kill a stag, or both individually kill a bunny (sorry, bunny lovers!), or one of you go for the bunny and the other the stag. Each requires you to bring different equipment, and you don’t know what the other hunter is bringing. Which hunting equipment should you bring?5 This question gets into Nash Equilibrium, and another important concept in decision theory – dominance. If you’re curious right now, look at William Spaniel’s site “Game Theory 101,” which explains these concepts well.
If you are interested in more Math Circle this winter, our next one starts January 7 (weather permitting) for ages 11 and up. Teens are welcome.
Thanks to all of you for a great 6 weeks!
2 Resnik (1987, pp. 147-148 ), as reported on the LessWrong wiki here: http://lesswrong.com/lw/gu1/decision_theory_faq/#newcomblike-problems-and-two-decision-algorithms Section 11.1.10
3 ibid, Section 11.1.6
4 Salmon Rushdie, Joseph Anton, Random House 2012. page 19