Game Theory for Dummies

Game theory is the fascinating study of how people interact. It is the foundation for economics, law, politics, psychology, and artificial intelligence. Under the simple assumption of rationality (which is more reasonable than you might think), it’s possible to build models for behavior.

The simplest models don’t quite look right, in the same way the simplest physics problems ignore friction. But as they grow more complex, patterns of behavior, or “strategies”, emerge, and they grow more and more reasonable as copmlexity increases.

Many game theory concepts are directly applicable to our lives. Concepts like repeated games, the endgame problem, matching games, and more have direct correlations to our world. Let’s examine one simple game.

You’re Bonnie, and Clyde is your partner in crime. You’ve been caught by the police after robbing a bank, and now you’ve been stuck into an interrogation room with no escape and no way to get a message out.

You’re caught. But you won’t rat out Clyde. The interrogator presents you with an option. If you snitch on Clyde, they’ll go easy on you. Clyde is in another room, presented with the same option. What would you do?

If you’ve studied game theory before, you’ll immediately recognize this as the famous prisoner’s dilemma, one of the most important games. Let’s examine how to play the prisoner’s dilemma.

If neither of you snitch, you both get three years in prison. If one of you snitches, and the other doesn’t, then the one who didn’t snitch gets nine years, and the one who snitched gets off scot free. If you both snitch, then you both get six years.

From now on, to match game theory parlance, we’ll call snitching “competing” and not snitching “cooperating”. And we’ll call the “joint payoff” the negative sum total time you both spend in prison. If both cooperate, then the joint payoff is -6. If both compete, then the joint payoff is -12. Which is worse. If only one snitches, then the joint payoff is -9. If you both compete, you get the worst outcome! You should clearly both cooperate, you’ll both get the least possible amount of jail time.

But you’ll compete (snitch) every single time.

Go back to the interrogation room. Now imagine you have those individual and joint payoffs in front of you (or, the game is of perfect information), and you know that Clyde is making his decision at the exact same time as you (or, the game is static). You’re in a static game of perfect information. Remember, there is no communication at any time after your arrest.

You’re sitting there, the decision in front of you, and with the knowledge that Clyde is making the same decision. You know you should both cooperate. He knows that too. Let’s suppose you think he’ll cooperate. You could get -3 by cooperating. But you could get that sweet, sweet 0 by competing! Okay, let’s suppose you think he’ll compete. If you cooperate, you’re going to get a -9, but if you compete you could turn that into a -6! No matter how bad -6 is, it is better than -9.

No matter how Clyde acts, it is always in your best interest to compete!

The kicker is, Clyde has figured this out too. You two are tragically fated to compete with one another and make yourselves both worse off.

What we discovered there was a Nash equilibrium. There is never any reason for one player in the game to deviate from the compete-compete outcome, so the game has reached equilibrium — you’ll always compete. That’s the evil of the prisoner’s dilemma. It was designed to get you both to rat each other out. Without some way to change the payoffs of the game, even communication before you are arrested cannot change this outcome. Even if the game is made dynamic, where Clyde (or Bonnie) chooses first, and Bonnie (Clyde) is told what decision their partner made before making their decision, you will still always compete-compete.

People simply cannot be trusted, it seems.

This game, made more complex, becomes the tragedy of the commons — public resources will always be overutilized (there is also a tragedy of the anti-commons, where rightholders will exclude one another from using a resource, to the point it provides value to nobody).

A short side note, most formal literature on game theory will talk about socially efficient outcomes, Pareto efficient, or Kaldor-Hicks efficient, but these all essentially mean the “best” outcome — the one with the highest joint payoff. This isn’t necessarily the best outcome for you, but it’s the best outcome for everyone. It’s the same way that greedy algorithms find local optima, but almost always fail to find the global optima.

Now you probably feel pretty bad about this. Nobody can really, truly be trusted when it comes down to the wire. And nobody can trust you.

But this game could be slightly different.

Let’s imagine a new game. Let’s say you’re a construction company, and you have a client. Your client forwards you ten thousand dollars to build a shed.

You never signed a contract (more on this later!), so if you just walked away with his money, you could net yourself a neat ten thousand! Let’s call this action “breach” (contract parlance here, but for now let’s just ignore any legal ramifications). Or you could “perform”, and build his shed. This will net you about one thousand dollars in the end. Let’s say that the client values this shed at eleven thousand dollars, netting him one thousand dollars as well.

This is starting to look bad. There’s nothing to stop you from walking away with the money! And that’s exactly what would happen — if the game were played once.

But what if the game repeats? Do you still steal his money (and conversely, he may never give you the money to begin with!), or do you build his shed?

The answer depends on how the game repeats.

Let’s say this game is played ten times, and you both know this. Let’s say the first nine times, you play nice and build his shed, to make sure he’s giving you the money. But the tenth time, what’s to stop you from stealing his money?

This is the endgame problem, and it’s a real life problem for the people who are associated with someone near death. The dying person (barring any belief in judgment from a higher power), suddenly has no incentive to be nice to you, and instead will just take what they want. They have no long term, no reputation, to worry about.

But in our case with ten games, we can take this further. Since you and the client both know what’s going to happen the last game, it isn’t really a game at all — it’s a predetermined outcome. So there are only really nine games. But the ninth game is now the last game! That one is now predetermined as well! Inductively, this peels away all the games until we’re back where we started.

The world just sucks. Every game so far has ended in deceit. Deterministically repeated games of perfect information are just the same.

But there’s a hope on the horizon — Stochastically repeated games of perfect information!

Let’s say that instead of 10 games, each game has a 10% chance of being the last game. You only find out it is the last game after it is played (or before, it’s not all that important). On average, that’s ten games. But it could be seven, eleven, or just one.

All of a sudden, the game changes! Your client will always give you money, and you’ll always build the shed! Remember that one word, reputation?

You’ve gotta maintain that now. Because nobody knows when the game will end, it’s suddenly not in their benefit to cheat each other. To see this, imagine that the game extends infinitely. If you cheat the client the first time, they’ll never give you opportunities to build sheds ever again. You’ll make ten thousand the first time around, but after eleven games, you’d have been better off performing each time. You’ll never get a contract to build something from anyone ever again, and suddenly that ten thousand looks sort of small. You could have a potentially infinite payoff by performing every time, because you don’t know when the last game is!

This logic extends to our randomly ending game. You don’t know how many more games there will be, so you continue to perform because it could yield better opportunities in the future. In other words, you’ve begun to think long term.

Fascinatingly, this mirrors real-life. Any day, you have a (hopefully) miniscule chance of dying. It almost never informs your decisions (except during the endgame problem), so every interaction is unlikely to be the last. That’s why trust forms — you don’t know when you’ll need it again.

The magic of stochastic games also solves the prisoner’s dilemma — suspend reality for a moment, and try the stochastic repetition method on the Bonnie and Clyde scenario. Suddenly, instead of compete-compete, you’ll actually find a new equilibrium at cooperate-cooperate!

We saw what games are, and studied the bare basics of equilibria and strategies. But we haven’t even scratched the surface.

Each game has been about perfect information. But what about information asymmetry, like in buying a used car? Or in the job market? What about changing the payoffs by leveraging the judicial framework with contracts, remedies, and damages? What about criminal activity and deterrence?

Game theory is a fascinating way to examine your own decisions and the decisions of others. I’ll leave you with one other way to solve the prisoner’s dilemma.

What if the payoffs for ratting are lower? By not snitching, you incur the wrath of the police, but what if by snitching you incur the wrath of someone else, lowering your payoffs for snitching?

You know what they say. Snitches get stitches.

People have come up with their own solution, organically, to the prisoner’s dilemma. They’ve imbued a negative payoff into the very idea of ratting someone out. They call it tattling, snitching, and compare the action to being a rat. And if you do it? You’re going to need medical attention.