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 noncooperative, or competitive, game theory. Noncooperative 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 every
player 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. Thanks for watching this episode of SciShow,
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