Basics of game theory.
Most game theory courses start with the prisoner's dilemma. However, in my opinion that's too complex a topic to start with. Let's start with fair cake-cutting problem instead.
[[[name:chores section:44
Dear Grandma,
I hope you are well. Also, best wishes for you birthday last month.
I am writing you because I have a problem you may be able to solve.
You've been visiting us recently and so you know how we always figth with Judy about splitting of the chores. You may have also noticed that Judy is much louder than I am and also a big whiner. In the end that means that I am always assigned more work than Judy. And that's not fair.
I know that Judy -- although she ignores whatever she is said -- actually respects you and often does what you say. Can you please write her a letter and ask her to play fair?
With love, Toddy
Dear Grandma,
I've shown your letter to Judy. She read it and said that she recognizes the system. She said it was used by prisoners in concentration camps. When they had to split bread, one would cut it in two, the other one would choose the half to take. You could have told me. I am not a little boy anymore, you know.
Anyway, we've tried it with chores. I've spent two hours devising a perfect split. It went into smallest details assigning even tasks like vacuuming between radiator fins which we do, like, twice a year. If we had adopted that plan everything would have been well. However, Judy spoiled everything and choose the part I was planning to do. How is that supposed to be fair?
With love, Toddy
Dear Grandma,
We've tried re-splitting the chores maybe twenty or thirty times in a row. First I proposed a split and Judy chose the easier part. Then I proposed a different split and Judy again took the easier part. When that went on for a while I said it was not fair and that I want to choose as well. So Judy proposed a split and I chose the easy half (no vacuuming!) Then she proposed a different split with vacuuming of ground floor in one part and vacuuming of top floor in the other part. There was no way to avoid vacuuming altogether. Anyway, after like two hours we've got a split that we were both happy with.
So, thanks for the advice. It may work in the end. And the best thing is that Judy can't whine now. If she does I would just propose to exchange our chores. And she knows that so she doesn't even try.
With love, Toddy
Dear Grandma,
It turns out to be more complicated than I thought. I felt I've been doing more work than Judy so I proposed to add vacuuming of the guest room to her chores. I was right because she accepted it. The I proposed adding vacuuming of the small toilet by the guest room to her chores and she accepted it again. Then I proposed that she vacuums my room as well and she chose to exchange the chores. Now I was left with almost all of the vacuuming!
Now here comes the unfair part: I've backtracked and proposed that she doesn't vacuum my room, just the guest room and the toilet. And she REFUSED to exchange to chores back. That's unfair! She agreed with that before! Can you please write her and say that people should always keep their word?
Thanks! Toddy
Dear Grandma,
Yes, I understand that I've challenged the status quo and that I should bear the consequences. The things have moved on here though. As angry as I was I stopped caring about keeping the living room (to be cleaned up by Judy) clean. I mean, if she doesn't bother to keep her word why should I make life easier for here?
Judy yelled at me and made a mess in the kitchen (to be cleaned up by me).
This is getting ridiculous. Now we both have more work to do than we used to while we were figthing. Can you please write her a letter and ask her to stop?
Thanks! Todd
]]]
The example above is supposed to show how incentives of different people can be played one against another to achieve particular outcome. Also, by the end it's getting close to showing the difference between zero-sum and non-zero-sum game.
This is a pretty entertaining way to introduce prisoner's-dilemma-like problems: https://www.youtube.com/watch?v=S0qjK3TWZE8
[[[name:nick-and-abraham section:96
Host: "Abraham and Nick now have 13,600 pounds in front of them. We know they've got it, but the golden question is: Can they keep it?"
"Abraham and Nick, you now face a very straightforward choice, but it's a choice that could make one or both of you extra wealthy. But it could also lose both of you everything that you have fought for today."
To the audience: "They have to decide to SPLIT or STEAL!"
Muzak plays.
Host: "Abraham, Nick, you have two final golden balls in front of you. And they are the most important golden balls in the game. You each have a golden ball with the owrd SPLIT written inside. You both have a ball with the word STEAL written inside. You will know which is SPLIT and which is STEAL because you can have a look. If you both pick the split ball you split the 13,600 and you go home with 6,800 each. If one of you choses the STEAL ball and the other choses the SPLIT ball whoever choses the STEAL ball goes home with the whole lot, 13,600. But if you both chose the STEAL ball you'll leave today's game with what you came with. Nothing."
"It's the ultimate test of faith, trust, and let's face it, greed. Take a moment to look at the balls in front of you so that you know for definite which is SPLIT and which is STEAL, but obviosuly, keep them concealed from each other. Just have a look."
Abraham and Nick are checking the balls.
Host: "OK.It's the easiest choice but the most difficult one. What I want you to do is to spend half a minute talking to each other about what you both should do. Nick. Abraham."
Nick: "Abraham, I want you to trust me. One hundred percent I am going to pick STEAL ball."
Abraham: "Sorry, you gonna..."
Nick: "I am going to choose the STEAL ball. I want you to do split and I promise you that I will spilt the money with you."
Abraham: "After you picked STEAL?"
Nick: "Yeah."
Abraham: "You can [inaudible] STEAL. I am going to take SPLIT. So you take the money..."
Nick: "And I will split it with you."
Abraham: "After the show."
Nick: "Yeah."
Audience laughs and applauds.
Nick: "I promise you I'll do that. If you do STEAL we'll both walk away with nothing. I'm telling you I am one hundred percent going to do it."
Abraham: "I'll give you another alternative. What if we just both pick SPLIT?"
Nick: "I'm not gonna pick SPLIT. I'm gonna STEAL. Abraham, honestly, one hundred percent, I'm gonna STEAL.
Abraham: "It's [inaudible] STEAL?"
Nick: "No. I am honest and I am going to tell you..."
Abraham: "You are honest?"
Nick: "I am. That's why I'm telling you I'm gonna STEAL. If you do SPLIT then I will split the money."
Abraham: "I can't see what's your [inaudible]."
Nick: "OK, so I am gonna STEAL so we're gonna leave with nothing."
Abraham: "Where is your brains coming from?"
Nick points finger to his head. Audience laughs.
Abraham: "I can't work a..."
Nick: "I know. I am a decent guy and I will split the money with you."
Abraham: "No, we should just pick SPLIT then."
Nick: "No. I'm gonna do STEAL."
Host: "There is no reason for him to give you the money."
Abraham: "Of course."
Abraham: "If I gave you my word... Now let me tell you what my word means."
Nick: "OK."
Abraham: "My father once said to me: A man who doesn't keep his word is not a man. He's not worth nothing. Not worth a dollar."
Nick: "I agree."
Abraham: "So..."
Nick: "Abraham, I'm gonna STEAL so you've got the choice."
Audience laughs.
Nick: "You either STEAL and we'll both walk away with nothing cause, you know, I've told you my intention and I've told you that I will split the money with you afterwards."
Abraham: "If you took my word I was gonna split I would split... and you gonna take STEAL so..."
Host: "The only way you can guarantee to walk away with 6,800 is that you both put SPLIT bullet and I do now have to push you for a decision. It's a tough one."
Abraham: "It's lost. We've lost everything."
Nick: "OK."
Abraham: "We'll walk out without the money because you are an idiot."
Nick: "No, that's not..."
Abraham: "You are an idiot. That's what you are. You are an idiot."
Host: "This can go on all night and this people gonna get up for breakfast... Nick, choose SPLIT or STEAL! Abraham, choose SPLIT or STEAL! Now, please! Choose a ball!"
Both Abraham and Nick reach out for the balls.
Abraham to Nick: "Right, I'll tell you what. I'm gonna go with you."
Nick: "OK."
Abraham: "I'm gona go with you."
Nick: "I promise you. I will split it."
Applause from the audience.
Host: "You cannot change your balls now."
Host (in dramatic voice): "SPLIT OR STEAL?"
Abraham and Nick both open the balls and show them to the audience.
Abraham's ball says "SPLIT".
Nick's ball says "SPLIT".
Host: "Congratulations! You've both split and each receives 6,800 pounds!
Everybody's laughing.
Abraham: "Why did you put me through that?! Why did you do that to me?!
Audience cheers.
Abraham and Nick shake hands.
Abraham: "You are the worst person I ever met."
]]]
It also hints at something interesting: In the end, Nick chooses 'split'. He does so because it doesn't really matter. However, what if the game was to be repeated? Then he should choose 'steal'. This hints at the fact that repeated and one-off games have different optimal strategies.
There could also be a story about vampire bats. Kids are going to love that. Moreover it allows to investigate game-theoretic scenarios associated with repeated games (and reciprocal altruism in general): https://en.wikipedia.org/wiki/Reciprocal_altruism#Vampire_bats