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Hank: When you’re hanging out with your friends, you probably don’t think too hard about the math behind the decisions you’re making. But there’s a whole field of math — and science — that applies to social interactions. It’s called Game Theory. Game theory was pioneered in the 1950s by mathematician John Nash, the guy from that Russell Crowe played in A Beautiful Mind.

But game theory isn’t about games the way we normally think about them. Instead, a game is any interaction between multiple people in which each person’s payoff is affected by the decisions made by others. So, sure, that could apply to a game of poker. But it could also apply to practically any situation where people get together and get up in each other’s business.

Like, did you interact with anyone today? Well, you can probably analyze the decisions you made using game theory. Game theory is incredibly wide-ranging, and it’s used all the time by economists, political scientists, biologists, military tacticians, and psychologists, to name just a few.

Game theory has two main branches: cooperative, and non-cooperative, or competitive, game theory. Non-cooperative game theory covers competitive social interactions, where there will be some winners... and some losers. Probably the most famous thought experiment in competitive game theory is the Prisoner’s Dilemma.

The prisoner’s dilemma describes a game — a social interaction — that involves two prisoners. We’ll call them Wanda and Fred. Wanda and Fred were arrested fleeing from the scene of a crime, and based on the evidence the police have already collected, they’re going to have to spend two years in jail.

But, the DA wants more. So he offers them both a deal: if you confess to the crime, and your partner does not, you’ll be granted immunity for cooperating. You’ll be free to go. Your partner, though, will serve ten years in jail. If you both confess, and dish up loads of dirt about each other, then you will both end up spending five years in jail. But if neither of you confess, you’ll both spend only two years in jail.

Those are their options. Then, Wanda and Fred are split up. They don’t know what their partner is going to do. They have to make their decisions independently. Now, Wanda and Fred they- they’ve had some wild times stealing diamonds or whatever, but they don’t have any special loyalty to each other. They’re not brother and sister; they’re hardened criminals. Fred has no reason to think Wanda won’t stab him in the back, and vice versa.

Competitive game theory arranges their choices and their potential consequences into a grid that looks like this: If both Wanda and Fred choose not to confess, they’ll both serve two years. In theory, this is the best overall outcome. Combined, they would spend as little time in prison as possible. But, that immunity sounds pretty good. If one of them chooses to confess, and the other one doesn’t, the snitch gets to walk.

Then the math looks like this! That’s the problem: Wanda and Fred have no reason to trust each other. Wanda might consider not confessing, because if Fred doesn’t confess either, they both only serve two years. If they could really trust each other, that would be their best bet. But Wanda can’t be sure that Fred won’t snitch. He has a LOT to gain by confessing. If he does decide to confess, and she keeps silent, she’s risking ten years in jail while he goes free. Compared to that, the five years they’d get for both turning on each other doesn’t sound so bad. And that is game theory’s solution: they should both confess and rat each other out.

So, right now you’re thinking, “Wow, game theory is a jerk.” But it actually makes sense. That square in the grid where they both confess is the only outcome that’s reached what’s known as Nash Equilibrium. This is a key concept in competitive game theory.

A player in a game has found Nash Equilibrium when they make the choice that leaves them better off no matter what their opponents decide to do. If Wanda confesses, and Fred does not confess... she’s better off. She gets to walk! By confessing, she went from serving two years in prison to serving none. If Fred does confess... she’s still better off. If she’d kept her mouth shut, she’d be spending ten years in prison. Now, she only has to serve five.

Sure, if she decides not to confess, and Fred keeps his pinky promise too, they both get out in two years. But that’s an unstable state. Because Wanda can’t trust Fred- she doesn’t know what he’s going to do. This is not a cooperative game: all of the players stand to gain from stabbing each other in the back.

The Prisoner’s Dilemma is just one example of a competitive game, but the basic idea behind its solution applies to all kinds of situations. Generally, when you’re competing with others, it makes sense to choose the course of action that benefits you the most no matter what everyone else decides to do.

Then there are cooperative games, where everyone has agreed to work together toward a common goal. This could be anything from a group of friends deciding how to split up the cost to pay the bill at a restaurant, to a coalition of nations deciding how to divvy up the burden of stopping climate change.

In game theory, a coalition is what you call a group of players in a cooperative game. When it comes to cooperative games, game theory’s main question is how much each player should contribute to the coalition, and how much they should benefit from it. In other words, it tries to determine what’s fair.

Where competitive game theory has the Nash Equilibrium, cooperative game theory has what’s called the Shapley Value. The Shapley Value is a method of dividing up gains or costs among players according to the value of their individual contributions. It works by applying several axioms.

Number one: the contribution of each player is determined by what is gained or lost by removing them from the game. This is called their marginal contribution. Let’s say that every day this week, you and your friends are baking cookies. When you get sick for a day, probably from eating too many cookies, the group produces fifty fewer cookies than they did on the days that you were there. So your marginal contribution to the coalition, every day, is fifty cookies.

Number two: Interchangeable players have equal value. If two parties bring the same things to the coalition, they should have to contribute the same amount, and should be rewarded for their contributions equally. Like if two people order the same thing at the restaurant, they should pay the same amount of the bill. If two workers have the same skills, they should receive the same wages.

Number three: Dummy players have zero value. In other words, if a member of a coalition contributes nothing, then they should receive nothing. This one’s controversial. It could mean that if you go to dinner with your friends, but you don’t order anything, you shouldn’t have to chip in when the bill comes. Which seems fair, in that case. But it could also mean that if somebody can’t contribute to the work force, they shouldn’t receive any compensation.

The thing is, there are good reasons why somebody might not be able to contribute: maybe they’re on maternity leave. Or they got in an accident. Or they have some kind of a disability. In situations like that, the coalition might want to pay something out to them in spite of them not being able to contribute.

The fourth axiom says that if a game has multiple parts, cost or payment should be decomposed across those parts. This just means that, for example, if you did a lot of work for the group on Monday, but you slacked off on Tuesday, your rewards on each day should be different. Or if you ordered a salad one night, but a steak dinner the next, you probably should pay more on the second night.

In other words, it’s not always fair to use the same solution every time. The numbers should be reviewed regularly, so that the coalition can make adjustments. If you find a way of dividing up costs or divvying up payment to all of the players that satisfies all of those axioms, that’s the Shapley value.

The Shapley value can be expressed mathematically like this... Which, yeah, is kind of complicated. But we can break down the concepts into something less... mathy.

Let’s go back to looking at cookies. You’re baking cookies, and your friend is baking cookies. In an hour, you can bake ten cookies when you’re working alone. Your friend though, is like, a cookie wizard, and in the same hour, working alone, he can bake twenty cookies.

When you decide to team up. When you work together, you streamline your process. One person can mix up all the batter at once or whatever, which saves you a lot of time. So after an hour, you have forty cookies. But if you’d each been working alone, you’d only have made 30 cookies in the same hour. Then you sell each of those cookies for a dollar. Now you’ve got forty dollars. How do you divide up the loot?

The Shapley value equation tells you to think about it like this: If you take the fact that you can make ten cookies an hour, and subtract them from the total, that gives your friend credit for the other thirty cookies. That’s what happens when you remove your friend from the system: their marginal contribution to you is thirty cookies.

But if you take the fact that your friend can make twenty cookies an hour, and subtract that from the total, that gives YOU credit for twenty cookies. Because if you’re removed from your friend’s cookie-making system, your marginal contribution to them is twenty cookies. In the first case, your value to the coalition was only ten cookies.

But in the second case, your value to the coalition is twenty cookies. According to the Shapley value equation, you should average those two numbers together. Ten plus twenty is thirty, divided by two is fifteen.

So, the Shapley value equation says that you should get fifteen dollars, and your friend should get twenty-five. This method can be scaled up to coalitions with hundreds of players, by finding their marginal contributions to every other player and then calculating the average of all of those numbers.

Interactions can get much more complicated than the Prisoner’s Dilemma or baking cookies, so there’s a lot more to game theory. But it comes down to this: in a competitive situation, game theory can tell you how to be smart. And in a cooperative situation, game theory can tell you how to be fair.

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