How Decision Making is Actually Science: Game Theory Explained

Summary

This video script delves into the fascinating realm of Game Theory, pioneered by John Nash, which applies mathematical concepts to social interactions. It explains the two main branches: noncooperative, exemplified by the Prisoner's Dilemma, and cooperative, where the Shapley Value determines fair distribution of gains. The script illustrates how game theory guides decision-making in competitive and cooperative scenarios, emphasizing the Nash Equilibrium and the importance of trust and contribution in various contexts.

Takeaways

• 🧩 Game Theory is a mathematical and scientific field that studies social interactions and the decisions made by individuals within them.
• 🕵️‍♂️ Pioneered by John Nash in the 1950s, Game Theory is not about traditional games but about interactions where each person's outcome is influenced by others' decisions.
• 🤝 Game Theory has two main branches: cooperative and noncooperative (competitive), each with its own principles and applications.
• 🔑 The Prisoner's Dilemma is a famous noncooperative game that illustrates the conflict between individual and collective interests, leading to a rational choice that may not be the most beneficial for all.
• 🔄 Nash Equilibrium is a concept in competitive game theory where no player has anything to gain by changing their strategy unilaterally, given the strategies of the other players.
• 🤔 Trust plays a crucial role in game theory, as individuals must decide whether to cooperate or compete based on their expectations of others' actions.
• 🤝 Cooperative game theory focuses on how to divide gains or costs fairly among players, using concepts like the Shapley Value.
• 📏 The Shapley Value is calculated based on axioms that determine the fair distribution of benefits or costs, considering individual contributions and the value of players' marginal contributions.
• 🍪 An example of cooperative game theory is the cookie-baking scenario, where the Shapley Value is used to determine a fair division of profits based on individual contributions.
• ⚖️ Game theory is used in various fields, including economics, political science, biology, military strategy, and psychology, to analyze strategic interactions.
• 🌐 The script also mentions a Patreon campaign for SciShow, highlighting the role of community support in creating free educational content.

Q & A

What is Game Theory?

Game Theory is a field of mathematics and science that applies to social interactions, where the outcome for each participant is influenced by the decisions of others. It was pioneered by mathematician John Nash in the 1950s.

How is Game Theory different from traditional games?

Game Theory is not about traditional games like poker. Instead, it refers to any interaction between multiple people where each person's payoff is affected by the decisions of others.

What are the two main branches of Game Theory?

The two main branches of Game Theory are cooperative and noncooperative or competitive game theory.

What is the Prisoner's Dilemma?

The Prisoner's Dilemma is a thought experiment in competitive game theory that involves two prisoners, Wanda and Fred, who must decide whether to confess to a crime independently of each other, leading to different outcomes based on their choices.

What is Nash Equilibrium in the context of the Prisoner's Dilemma?

Nash Equilibrium is a key concept in competitive game theory where a player makes the choice that leaves them better off no matter what their opponents decide to do. In the Prisoner's Dilemma, it is the outcome where both prisoners confess, as it is the best individual strategy regardless of the other's choice.

What is the Shapley Value in cooperative game theory?

The Shapley Value is a method in cooperative game theory for dividing up gains or costs among players according to the value of their individual contributions. It is calculated by considering each player's marginal contribution to the coalition.

What are the axioms of the Shapley Value?

The Shapley Value is based on several axioms: 1) Each player's contribution is determined by their marginal contribution. 2) Interchangeable players have equal value. 3) Dummy players have zero value. 4) If a game has multiple parts, cost or payment should be decomposed across those parts.

How does the Shapley Value equation work in a simple example of baking cookies?

In the example, if you can bake ten cookies an hour and your friend can bake twenty, and together you make forty cookies, the Shapley Value would calculate your contributions to each other and average them out to determine how to fairly divide the earnings from selling the cookies.

What is the purpose of the Shapley Value in cooperative games?

The Shapley Value aims to determine what is fair in cooperative games by calculating each player's contribution to the coalition and ensuring that the division of gains or costs satisfies the axioms of fairness.

How can Game Theory be applied in various fields?

Game Theory is widely applied in fields such as economics, political science, biology, military strategy, and psychology to analyze and predict outcomes of interactions among multiple parties.

What is the significance of the Nash Equilibrium in competitive situations?

In competitive situations, the Nash Equilibrium provides a strategic guide for players to choose the best course of action that benefits them the most, regardless of what others decide to do.

Outlines

00:00

🎲 Introduction to Game Theory

This paragraph introduces the concept of Game Theory, a field of mathematics and science that applies to social interactions. It was pioneered by John Nash, known from 'A Beautiful Mind'. Unlike traditional games, Game Theory involves interactions where each person's outcome is influenced by others' decisions. It's used widely across various disciplines such as economics, political science, biology, and psychology. The paragraph explains two main branches: cooperative and noncooperative game theory, with the latter focusing on competitive interactions like the famous Prisoner's Dilemma. This thought experiment illustrates the lack of trust between two prisoners, Wanda and Fred, who must independently decide whether to confess to a crime, potentially leading to a scenario where both betray each other for the best individual outcome, known as the Nash Equilibrium.

05:04

🍪 The Shapley Value in Cooperative Game Theory

The second paragraph delves into cooperative game theory, focusing on the Shapley Value, a method for fairly dividing gains or costs among players based on their individual contributions. It's determined by several axioms, including the marginal contribution of each player, equal value for interchangeable players, zero value for those who contribute nothing (noting the potential controversy in real-world applications), and the need to adjust contributions based on varying contributions over time or across different scenarios. The Shapley Value is exemplified through a cookie-baking scenario, where two individuals with different baking capacities collaborate, and their earnings are divided according to the Shapley value equation. This approach can be scaled for larger groups and helps to determine what's fair in cooperative situations. The paragraph concludes by emphasizing the broader applications of game theory in both competitive and cooperative contexts, and its utility in guiding smart and fair decision-making.

Keywords

💡Game Theory

Game Theory is a mathematical and scientific discipline that studies strategic interactions among rational decision-makers. It is central to the video's theme, illustrating how decisions in social interactions can be analyzed mathematically. The script uses Game Theory to explain various scenarios, including the Prisoner's Dilemma, to demonstrate how individuals make choices based on the potential actions of others.

💡John Nash

John Nash, referenced in the script, is a pioneering figure in Game Theory, known for his work on non-cooperative games. His concept of the Nash Equilibrium is foundational and is explained in the script as the point in a game where no player has anything to gain by changing only their own strategy unilaterally.

💡Nash Equilibrium

Nash Equilibrium is a key concept in Game Theory, defined as a state in which each player of a game has chosen their strategy, considering the strategies of the other players, and no player has anything to gain by changing only their own strategy unilaterally. The script explains this concept through the example of the Prisoner's Dilemma, where both prisoners confess to achieve the best outcome for themselves individually.

💡Prisoner's Dilemma

The Prisoner's Dilemma is a classic example of a Game Theory scenario where two individuals, despite mutual cooperation being beneficial, act in their self-interest and end up with a less favorable outcome. The script uses this thought experiment to illustrate the concept of non-cooperative game theory and the decision-making process under conditions of mistrust.

💡Cooperative Game Theory

Cooperative Game Theory is a branch of Game Theory that deals with situations where players can form binding agreements, and the focus is on how the total payoff can be divided among the players. The script contrasts this with noncooperative game theory and introduces the Shapley Value as a method for determining a fair distribution of gains or costs in cooperative games.

💡Shapley Value

The Shapley Value is a solution concept in cooperative Game Theory that provides a fair way to distribute the total payoff among the players based on their individual contributions to the coalition. The script explains this concept by using the axioms that define the Shapley Value, such as marginal contribution and the equal value of interchangeable players.

💡Marginal Contribution

Marginal Contribution refers to the additional benefit that a player brings to a coalition when they join. In the script, this concept is used to explain how the Shapley Value calculates the value of each player's contribution to a cooperative game, such as the example of baking cookies together versus individually.

💡Interchangeable Players

Interchangeable Players in the context of the script refers to the axiom in the Shapley Value that if two or more players contribute equally to a coalition, they should be rewarded equally. This concept is used to illustrate fairness in the division of gains or costs in cooperative games.

💡Dummy Players

Dummy Players are members of a coalition who contribute nothing to the group's outcome. The script discusses this concept as part of the axioms of the Shapley Value, noting that such players should receive nothing, which can be controversial in real-world applications where non-contribution may be due to circumstances beyond an individual's control.

💡Patreon

Patreon is a crowdfunding platform mentioned in the script where patrons can support content creators like SciShow. The script uses Patreon as an example of a platform that allows patrons to contribute voluntarily, contrasting with the compulsory nature of the contributions discussed in the context of cooperative games.

💡SciShow

SciShow is the name of the educational YouTube channel that produced the video script provided. It is mentioned in the script as the entity that will use the funds raised on Patreon to create new content, specifically a new series under the SciShow brand.

Highlights

Game theory applies to social interactions and was pioneered by mathematician John Nash.

A game in game theory refers to any interaction where each person's payoff is affected by others' decisions.

Game theory is used by economists, political scientists, biologists, military tacticians, and psychologists.

Game theory has two main branches: cooperative and noncooperative or competitive game theory.

Noncooperative game theory covers competitive social interactions with winners and losers.

The Prisoner’s Dilemma is a famous thought experiment in competitive game theory involving two prisoners.

In the Prisoner’s Dilemma, prisoners face a choice between confessing or staying silent with varying outcomes.

The Nash Equilibrium is a key concept in competitive game theory where players make the best choice for themselves regardless of others.

Cooperative game theory focuses on how much each player should contribute and benefit in a group.

The Shapley Value is a method in cooperative game theory for dividing gains or costs based on individual contributions.

The Shapley Value applies several axioms including marginal contribution, equal value for interchangeable players, and zero value for dummy players.

The Shapley Value can be calculated by averaging the marginal contributions of each player to the coalition.

Game theory helps determine the best course of action in competitive situations and fairness in cooperative situations.

The Prisoner’s Dilemma illustrates the concept of mutual defection being the rational choice despite the collective best outcome.

Cooperative games involve players working together towards a common goal, unlike competitive games.

Game theory can be applied to various real-life scenarios, from splitting restaurant bills to international climate change agreements.

SciShow is funded by Patreon patrons, and their contributions are being used to create a new series on YouTube.

Transcripts

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.

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