December 12, 2015

I have to admit that, despite all this science-y stuff I do, I still have my vices.

You know when you’re just sitting there waiting for the train to take you home, and you wish you had a game to play on your iPhone? Then you go home and download that game, fully intending to only play it when you’re bored or waiting? Then you end up sacrificing the time you should have spent doing important things playing that game instead?

That’s **The Simpsons: Tapped Out** for me (along with new contender Marvel Contest of Champions, but that’s another story).

Every holiday, and even sometimes when it’s not a holiday, The Simpsons has a special holiday update. This year’s Christmas special lets you play a classic game originally known as the Prisoner’s Dilemma.

It is the canonical example of a “game” in **Game Theory**. **John Nash**, the subject of the movie **A Beautiful Mind**, made fundamental contributions to the theory. It has applications in military strategy, politics, and even sports.

## So how does it work?

In Tapped Out, every player has their own town with their own characters and houses, etc. You can visit other towns to get points.

In the Christmas update, when you visit a friend’s town, you drop a present off, and you are asked if you want to make the present “nice” (Lisa), or “naughty” (Bart). Naturally.

When you receive a present, you also have to choose nice/Lisa, or naughty/Bart.

So there are 4 possibilities:

- You choose Bart, other player chooses Lisa
- You choose Bart, other player chooses Bart
- You choose Lisa, other player chooses Lisa
- You choose Lisa, other player chooses Bart

But your choice affects how many points you receive.

This is a picture of the game that explains how it is played.

So as you can see, the point system is as follows:

- You choose Bart, other player chooses Lisa – You get 25, other player gets 1
- You choose Bart, other player chooses Bart – You get 5, other player gets 5
- You choose Lisa, other player chooses Lisa – You get 15, other player gets 15
- You choose Lisa, other player chooses Bart – You get 1, other player gets 25

In the original prisoner’s dilemma, the problem is posed as follows:

2 prisoners are given a choice of whether to cooperate with or betray the other. If they both cooperate, they both serve 1 year. If they both betray each other, they both serve 2 years. But if one betrays and one cooperates, then the one who betrays goes free, while the one who cooperated gets 3 years.

The problem can be summarized in a table like this:

Where P, R, S, and T are called “payoffs”. To be a prisoner’s dilemma game, we must have T > R > P > S, which is true for The Simpsons: Tapped Out version also.

Once you understand the problem, the question becomes – how do I make a choice that will maximize my winnings?

## How do I maximize my winnings in prisoner’s dilemma?

The Prisoner’s Dilemma is a dilemma because it seems obvious that to maximize both our winnings (altruism), we should cooperate.

But if I know you’re rational and will cooperate with me, and I am selfish, then I will defect, so I win more and you get nothing.

The problem is, if we both think that way, then we will both defect and we will both end up with less.

I would also be hesitant to cooperate because if you are selfish, then you will defect and I will end up with nothing.

The only “rational” solution is to be selfish and betray. Why? Because it is the only choice for which choosing the other thing will result in a worse outcome.

Regardless of what you do – if I choose to betray, it is better than if I had chosen to cooperate.

If you choose cooperate – I choose betray, I get 25 points. I choose cooperate, I only get 15.

If you choose betray – I choose betray, I get 5 points, but if I choose cooperate, I only get 1.

So no matter what you choose, my choice of betray is better than my choice of cooperate.

Of course, since I can drop off multiple presents to my neighbors’ towns, there isn’t only one “game” to play. In fact, multiple or iterated games make the prisoner’s dilemma even more interesting.

## Repeated Games

There is a bigger social element to repeated games.

If you betray me, then now I know you are a betrayer, so I will not cooperate with you.

If you cooperate with me, then perhaps I can trust you, and we can cooperate with each other until the end of the holiday special. That will result in more points for both of us.

But not as many for me compared to what I’d get if I continually betrayed you.

Then again, if I kept betraying you, you would stop cooperating with me.

You see how this gets complex?

## What is the best strategy?

Research done by Robert Axelrod in his 1984 book **The Evolution of Cooperation** helps to shed light on this issue.

He studied competing strategies in a tournament and discovered that purely selfish strategies performed very poorly in the long run, and that more altruistic strategies performed better, even when measured from a selfish perspective (how many points did I win?).

In terms of natural selection, this may explain why we have evolved over time to be both altruistic and self-interested.

The winning strategy was called “tit for tat”, which in short, means you just play the move your opponent previously played. The tit for tat strategy can be improved upon slightly by adding a small random chance that you will forgive your opponent if they betray you, and cooperate.

Interestingly, Axelrod’s analysis of the top strategies have a very “human” component. The conditions for a successful strategy can be summarized as:

- Be nice. Cooperate unless your opponent betrays you.
- Have a spine. If your opponent betrays you, don’t keep cooperating with them.
- Forgive. Don’t fall into a cycle of repeated betrayals.
- Don’t be jealous. Don’t strive to get more points than the other player and don’t be outcome dependent.

## Limited number of games

In The Simpsons: Tapped Out, you don’t have an infinite number of games to learn from or an infinite number of gifts to give. You are limited to 5 per day. Thus your ultimate strategy will probably be adaptive as well – choose only the 5 players you want to mutually benefit from your gift giving and choose the ones who have a high probability of cooperating.

## Conclusion

Let’s be real though. I don’t think the people who are playing this game are really considering this stuff. For the first few hours I just choose Bart, because that was the naughty one.

So yeah.

#a beautiful mind #game theory #john nash #prisoner's dilemma #tapped out #the simpsons #tstoGo to comments