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 — andscience — that applies to social interactions.
It’s called Game Theory.
Game theory was pioneered in the 1950s bymathematician John Nash, the guy from that Russell Crowe played in A Beautiful Mind.
But game theory isn’t about games the waywe normally think about them.
Instead, a game is any interaction betweenmultiple 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 anysituation 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 decisionsyou made using game theory.
Game theory is incredibly wide-ranging, andit’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 competitivesocial interactions, where there will be some winners … and some losers.
Probably the most famous thought experimentin 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 fromthe scene of a crime, and based on the evidence the police have already collected, they’regoing to have to spend two years in jail.
But, the DA wants more.
So he offers them both a deal: if you confessto 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 yearsin jail.
If you both confess, and dish up loads ofdirt about each other, then you will both end up spending five years in jail.
But if neither of you confess, you’ll bothspend only two years in jail.
Those are their options.
Then, Wanda and Fred are split up.
They don’t know what their partner is goingto do.
They have to make their decisions independently.
Now, Wanda and Fred they- they’ve had somewild times stealing diamonds or whatever, but they don’t have any special loyaltyto each other.
They’re not brother and sister; they’rehardened criminals.
Fred has no reason to think Wanda won’tstab him in the back, and vice versa.
Competitive game theory arranges their choicesand 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 timein prison as possible.
But … that immunity sounds pretty good.
If one of them chooses to confess, and theother one doesn’t, the snitch gets to walk.
Then the math looks like this: That’s the problem: Wanda and Fred haveno reason to trust each other.
Wanda might consider not confessing, becauseif Fred doesn’t confess either, they both only serve two years.
If they could really trust each other, thatwould be their best bet.
But Wanda can’t be sure that Fred won’tsnitch.
He has a LOT to gain by confessing.
If he does decide to confess, and she keepssilent, she’s risking ten years in jail while he goes free.
Compared to that, the five years they’dget for both turning on each other doesn’t sound so bad.
And that is game theory’s solution: theyshould both confess and rat each other out.
So, right now you’re thinking, “Wow, gametheory is a jerk.
” But it actually makes sense.
That square in the grid where they both confessis the only outcome that’s reached what’s known as Nash Equilibrium.
This is a key concept in competitive gametheory.
A player in a game has found Nash Equilibriumwhen they make the choice that leaves them better off no matter what their opponentsdecide 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 yearsin prison to serving none.
If Fred does confess.
she’s still betteroff.
If she’d kept her mouth shut, she’d bespending ten years in prison.
Now, she only has to serve five.
Sure, if she decides not to confess, and Fredkeeps 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’tknow what he’s going to do.
This is not a cooperative game: all of theplayers stand to gain from stabbing each other in the back.
The Prisoner’s Dilemma is just one exampleof a competitive game, but the basic idea behind its solution applies to all kinds ofsituations.
Generally, when you’re competing with others, it makes sense to choose the course of action that benefits you the most no matter whateveryone else decides to do.
Then there are cooperative games, where everyplayer has agreed to work together toward a common goal.
This could be anything from a group of friendsdeciding how to split up the cost to pay the bill at a restaurant, to a coalition of nationsdeciding how to divvy up the burden of stopping climate change.
In game theory, a coalition is what you calla group of players in a cooperative game.
When it comes to cooperative games, game theory’smain question is how much each player should contribute to the coalition, and how muchthey should benefit from it.
In other words, it tries to determine what’sfair.
Where competitive game theory has the NashEquilibrium, cooperative game theory has what’s called the Shapley Value.
The Shapley Value is a method of dividingup 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 playeris 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, youand your friends are baking cookies.
When you get sick for a day, probably fromeating too many cookies, the group produces fifty fewer cookies than they did on the daysthat you were there.
So your marginal contribution to the coalition, every day, is fifty cookies.
Number two: Interchangeable players have equalvalue.
If two parties bring the same things to thecoalition, they should have to contribute the same amount, and should be rewarded fortheir contributions equally.
Like if two people order the same thing atthe restaurant, they should pay the same amount of the bill.
If two workers have the same skills, theyshould receive the same wages.
Number three: Dummy players have zero value.
In other words, if a member of a coalitioncontributes nothing, then they should receive nothing.
This one’s controversial.
It could mean that if you go to dinner withyour friends, but you don’t order anything, you shouldn’t have to chip in when the billcomes.
Which seems fair, in that case.
But it could also mean that if somebody can’tcontribute to the work force, they shouldn’t receive any compensation.
The thing is, there are good reasons why somebodymight 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 mightwant to pay something out to them in spite of them not being able to contribute.
The fourth axiom says that if a game has multipleparts, cost or payment should be decomposed across those parts.
This just means that, for example, if youdid a lot of work for the group on Monday, but you slacked off on Tuesday, your rewardson each day should be different.
Or if you ordered a salad one night, but asteak dinner the next, you probably should pay more on the second night.
In other words, it’s not always fair touse 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 ordivvying up payment to all of the players that satisfies all of those axioms, that’sthe Shapley value.
The Shapley value can be expressed mathematicallylike this: Which, yeah, is kind of complicated.
But we can break down the concepts into somethingless … mathy.
Let’s go back to looking at cookies.
You’re baking cookies, and your friend isbaking cookies.
In an hour, you can bake ten cookies whenyou’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 yourprocess.
One person can mix up all the batter at onceor 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’donly have made 30 cookies in the same hour.
Then you sell each of those cookies for adollar.
Now you’ve got forty dollars.
How do you divide up the loot? The Shapley value equation tells you to thinkabout it like this: If you take the fact that you can make tencookies an hour, and subtract them from the total, that gives your friend credit for theother thirty cookies.
That’s what happens when you remove yourfriend from the system: their marginal contribution to you is thirty cookies.
But if you take the fact that your friendcan make twenty cookies an hour, and subtract that from the total, that gives YOU creditfor twenty cookies.
Because if you’re removed from your friend’scookie-making system, your marginal contribution to them is twenty cookies.
In the first case, your value to the coalitionwas only ten cookies.
But in the second case, your value to thecoalition is twenty cookies.
According to the Shapley value equation, youshould average those two numbers together.
Ten plus twenty is thirty, divided by twois fifteen.
So, the Shapley value equation says that youshould get fifteen dollars, and your friend should get twenty-five.
This method can be scaled up to coalitionswith hundreds of players, by finding their marginal contributions to every other playerand then calculating the average of all of those numbers.
Interactions can get much more complicatedthan the Prisoner’s Dilemma or baking cookies, so there’s a lot more to game theory.
But it comes down to this: in a competitivesituation, game theory can tell you how to be smart.
And in a cooperative situation, game theorycan tell you how to be fair.
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