How Decision Making is Actually Science: Game Theory Explained

0:10 When you’re hanging out with your friends,

0:12 you probably don’t think too hard about

0:14 the math behind the decisions you’re making.

0:16 But there’s a whole field of math—

0:18 and science— that applies to social interactions.

0:21 It’s called Game Theory.

0:23 Game theory was pioneered in the 1950s by mathematician John Nash,

0:26 the guy from that Russell Crowe played in A Beautiful Mind.

0:29 But game theory isn’t about games the way we normally think about them.

0:33 Instead, a game is any interaction between multiple people in which

0:37 each person’s payoff is affected by the decisions made by others.

0:41 So, sure, that could apply to a game of poker.

0:43 But it could also apply to practically any situation where

0:46 people get together and get up in each other’s business.

0:49 Like, did you interact with anyone today?

0:51 Well, you can probably analyze the decisions you made using game theory.

0:56 Game theory is incredibly wide-ranging,

0:57 and it’s used all the time by economists,

1:00 political scientists, biologists, military tacticians,

1:02 and psychologists, to name just a few.

1:06 Game theory has two main branches:

1:08 cooperative, and noncooperative, or competitive, game theory.

1:11 Noncooperative game theory covers competitive social interactions,

1:14 where there will be some winners… and some losers.

1:18 Probably the most famous thought experiment

1:19 in competitive game theory is the Prisoner’s Dilemma.

1:22 The prisoner’s dilemma describes a game—

1:24 a social interaction— that involves two prisoners.

1:26 We’ll call them Wanda and Fred.

1:28 Wanda and Fred were arrested fleeing from the scene of a crime,

1:31 and based on the evidence the police have already collected,

1:34 they’re going to have to spend two years in jail.

1:37 But, the DA wants more.

1:39 So he offers them both a deal: if you confess to the crime,

1:42 and your partner does not, you’ll be granted immunity for cooperating.

1:46 You’ll be free to go.

1:47 Your partner, though, will serve ten years in jail.

1:50 If you both confess, and dish up loads of dirt about each other,

1:53 then you will both end up spending five years in jail.

1:56 But if neither of you confess, you’ll both spend only two years in jail.

2:00 Those are their options.

2:01 Then, Wanda and Fred are split up.

2:03 They don’t know what their partner is going to do.

2:05 They have to make their decisions independently.

2:07 Now, Wanda and Fred they- they’ve had

2:09 some wild times stealing diamonds or whatever,

2:11 but they don’t have any special loyalty to each other.

2:14 They’re not brother and sister; they’re hardened criminals.

2:16 Fred has no reason to think Wanda won’t stab him in the back, and vice versa.

2:20 Competitive game theory arranges their choices

2:22 and their potential consequences into a grid

2:25 that looks like this: If both Wanda and Fred choose not to confess,

2:28 they’ll both serve two years.

2:30 In theory, this is the best overall outcome.

2:32 Combined, they would spend as little time in prison as possible.

2:35 But… that immunity sounds pretty good.

2:37 If one of them chooses to confess,

2:39 and the other one doesn’t, the snitch gets to walk.

2:41 Then the math looks like this: That’s the problem:

2:44 Wanda and Fred have no reason to trust each other.

2:47 Wanda might consider not confessing, because if Fred doesn’t confess either,

2:50 they both only serve two years.

2:52 If they could really trust each other, that would be their best bet.

2:55 But Wanda can’t be sure that Fred won’t snitch.

2:58 He has a LOT to gain by confessing.

3:01 If he does decide to confess, and she keeps silent,

3:04 she’s risking ten years in jail while he goes free.

3:06 Compared to that, the five years they’d get

3:08 for both turning on each other doesn’t sound so bad.

3:11 And that is game theory’s solution:

3:13 they should both confess and rat each other out.

3:16 So, right now you’re thinking, “Wow,

3:17 game theory is a jerk.” But it actually makes sense.

3:20 That square in the grid where they both confess is

3:23 the only outcome that’s reached what’s known as Nash Equilibrium.

3:26 This is a key concept in competitive game theory.

3:29 A player in a game has found Nash Equilibrium when they make the choice

3:32 that leaves them better off no matter what their opponents decide to do.

3:37 If Wanda confesses, and Fred does not confess… she’s better off.

3:40 She gets to walk!

3:41 By confessing, she went from serving two years in prison to serving none.

3:45 If Fred does confess...she’s still better off.

3:49 If she’d kept her mouth shut, she’d be spending ten years in prison.

3:52 Now, she only has to serve five.

3:54 Sure, if she decides not to confess,

3:56 and Fred keeps his pinky promise too, they both get out in two years.

4:00 But that’s an unstable state.

4:02 Because Wanda can’t trust Fred- she doesn’t know what he’s going to do.

4:05 This is not a cooperative game:

4:07 all of the players stand to gain from stabbing each other in the back.

4:11 The Prisoner’s Dilemma is just one example of a competitive game,

4:13 but the basic idea behind its solution applies to all kinds of situations.

4:18 Generally, when you’re competing with others,

4:20 it makes sense to choose the course of action that benefits

4:23 you the most no matter what everyone else decides to do.

4:27 Then there are cooperative games,

4:29 where every player has agreed to work together toward a common goal.

4:34 This could be anything from a group of friends deciding how

4:36 to split up the cost to pay the bill at a restaurant,

4:39 to a coalition of nations deciding how

4:41 to divvy up the burden of stopping climate change.

4:43 In game theory, a coalition is what you

4:46 call a group of players in a cooperative game.

4:48 When it comes to cooperative games, game theory’s main question is how much

4:52 each player should contribute to the coalition,

4:54 and how much they should benefit from it.

4:57 In other words, it tries to determine what’s fair.

4:59 Where competitive game theory has the Nash Equilibrium,

5:02 cooperative game theory has what’s called the Shapley Value.

5:05 The Shapley Value is a method of dividing up gains

5:08 or costs among players according to the value of their individual contributions.

5:12 It works by applying several axioms.

5:14 Number one: the contribution of each player is determined by what

5:18 is gained or lost by removing them from the game.

5:21 This is called their marginal contribution.

5:23 Let’s say that every day this week, you and your friends are baking cookies.

5:26 When you get sick for a day, probably from eating too many cookies,

5:29 the group produces fifty fewer cookies than they

5:31 did on the days that you were there.

5:33 So your marginal contribution to the coalition, every day, is fifty cookies.

5:37 Number two: Interchangeable players have equal value.

5:40 If two parties bring the same things to the coalition,

5:42 they should have to contribute the same amount,

5:45 and should be rewarded for their contributions equally.

5:47 Like if two people order the same thing at the restaurant,

5:49 they should pay the same amount of the bill.

5:51 If two workers have the same skills, they should receive the same wages.

5:55 Number three: Dummy players have zero value.

5:58 In other words, if a member of a coalition contributes nothing,

6:01 then they should receive nothing.

6:03 This one’s controversial.

6:04 It could mean that if you go to dinner with your friends,

6:07 but you don’t order anything, you shouldn’t have to chip in when the bill comes.

6:10 Which seems fair, in that case.

6:11 But it could also mean that if somebody can’t contribute to the work force,

6:14 they shouldn’t receive any compensation.

6:16 The thing is, there are good reasons

6:18 why somebody might not be able to contribute: maybe they’re on maternity leave.

6:22 Or they got in an accident.

6:23 Or they have some kind of a disability.

6:24 In situations like that, the coalition might want to pay something

6:27 out to them in spite of them not being able to contribute.

6:30 The fourth axiom says that if a game has multiple parts,

6:34 cost or payment should be decomposed across those parts.

6:37 This just means that, for example,

6:38 if you did a lot of work for the group on Monday,

6:40 but you slacked off on Tuesday, your rewards on each day should be different.

6:44 Or if you ordered a salad one night, but a steak dinner the next,

6:47 you probably should pay more on the second night.

6:49 In other words, it’s not always fair to use the same solution every time.

6:53 The numbers should be reviewed regularly,

6:55 so that the coalition can make adjustments.

6:57 If you find a way of dividing up costs or divvying up

7:00 payment to all of the players that satisfies all of those axioms,

7:04 that’s the Shapley value.

7:05 The Shapley value can be expressed mathematically like this: Which,

7:08 yeah, is kind of complicated.

7:10 But we can break down the concepts into something less… mathy.

7:13 Let’s go back to looking at cookies.

7:15 You’re baking cookies, and your friend is baking cookies.

7:17 In an hour, you can bake ten cookies when you’re working alone.

7:20 Your friend though, is like, a cookie wizard,

7:22 and in the same hour, working alone, he can bake twenty cookies.

7:25 When you decide to team up.

7:26 When you work together, you streamline your process.

7:29 One person can mix up all the batter at once or whatever,

7:32 which saves you a lot of time.

7:33 So after an hour, you have forty cookies.

7:35 But if you’d each been working alone,

7:37 you’d only have made 30 cookies in the same hour.

7:39 Then you sell each of those cookies for a dollar.

7:42 Now you’ve got forty dollars.

7:43 How do you divide up the loot?

7:44 The Shapley value equation tells you to think about it like this: If

7:48 you take the fact that you can make ten cookies an hour,

7:50 and subtract them from the total,

7:52 that gives your friend credit for the other thirty cookies.

7:55 That’s what happens when you remove your friend from the system:

7:58 their marginal contribution to you is thirty cookies.

8:00 But if you take the fact that your friend can make twenty cookies an hour,

8:04 and subtract that from the total, that gives YOU credit for twenty cookies.

8:08 Because if you’re removed from your friend’s cookie-making system,

8:11 your marginal contribution to them is twenty cookies.

8:13 In the first case, your value to the coalition was only ten cookies.

8:16 But in the second case, your value to the coalition is twenty cookies.

8:20 According to the Shapley value equation,

8:22 you should average those two numbers together.

8:24 Ten plus twenty is thirty, divided by two is fifteen.

8:27 So, the Shapley value equation says that you should get fifteen dollars,

8:31 and your friend should get twenty-five.

8:33 This method can be scaled up to coalitions with hundreds of players,

8:36 by finding their marginal contributions to every other player

8:39 and then calculating the average of all of those numbers.

8:42 Interactions can get much more complicated

8:44 than the Prisoner’s Dilemma or baking cookies,

8:46 so there’s a lot more to game theory.

8:48 But it comes down to this: in a competitive situation,

8:50 game theory can tell you how to be smart.

8:53 And in a cooperative situation, game theory can tell you how to be fair.

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