# Game Theory: The Science of Decision-Making

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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Prison break anyone?

bro, in Brazil, ifyou confess, 2 weeks later you get killed

John Van Noy pioneered game theory John Nash was his student.

@0:28, No. Game Theory was pioneered by John Von Neumann. John Nash, read his work and just identified a point in it- Nash Equilibrium. It was a great work and won him the Nobel Prize. However, credit for the subfield should go to Neumann.

So, what’s the real benefit in cookies for the patrons in this equation…? 🤔

(Apparently doesn’t fit in here)

why would we not pay people bonuses based on co-operative game theory (if we could truly measure what they do in equivalent units) instead of competitively as we do today ?

how does nash equilibrium arrive at it's 5/5 with the table of 2 options?

option 1 = 2 or 10

option 2 = 10 or 5

option 1 = 12 (average that = 6

option 2 = 15 (average that = 7.5

say she confesses, because girl prisons are sweet as pie, and the dude doesn't confess because dude prisons are bad.

that would be realistic scenario…

i think either , 'nash equilibrium' isn't working with simply probabilistic odds and outcomes.

do you destroy the world by working in a sky scraper? or go grow your own vegetables and farm chickens and rabbits in the wild?

nash equilibrium would suggest the road of most destruction?

perhaps the value is wrong here, as prison time is negative

if they shared the loot, depending on how the police heat was on them.

it might make sense in reality to both confess and find their loot in 5 years.

perhaps it's a 'spread' thing, reflecting a perception of suffering and paranoia.

2 is further from 10 so the risk is greater.

but 5 is closer to 10, so the risk is lower.

has anyone commented anything else on this

Climate change. haha. None of those countries are doing much to reduce their gas output.

Wanda and Fred (depicted in this video as criminals) look awfully familiar to Hillary and Bill Clinton

Or you could pay people their job market price. Fairness is not always competitive and may end up losing you, your business. But if you pay people what their worth, then you stay competitive, and in business. If you have an especially great, cookie batter mixer person, who can make your cookies more cheaply or tastier, then by all means, give that person a rase. But equations don't always equal profits, and profits is what keeps you in the cookie business.

Sharing this will make things better.

~~Wait you're not MatPat~~And this is why rightist and leftist fight each other. right-wing think value is come from the capital, while left-wing think value is come from work. so according to Shapley value, right-wingers pov people should rewarded what they invested (including labor market source) while left-wingers pov people should rewarded what they work (excluding income from just owning things)

The consequence is greater though than just these rules pointed out. Such as the Mafia rule. Snitching on your partner in crime carries with it a death sentence. In this case sacrificing your 10 year jail sentence to potentially allow your partner to go free would carry more weight in street loyalty. Therefore if both do not confess, 2 years is better than getting an ice pick shoved into your skull for ratting.

Pay off for outcome is generally greater than what is proposed. IN terms of a game if a loser can find enjoyment in the game even though they lost, then losing is not all that bad. Its a premise that I've often hated to hear why my opponent who just lost shrugged their shoulders and said so what. If Game Theory is the science behind the interaction of people then these outcomes must also be addressed. A player who cares nothing for losing can also impact their aggressiveness in the game which influences all outcomes because their aggression could lead them to victory since they had no fear of failure.

This isn’t operations research

The prisoners dilemma is flawed, at least in this example…because it supposes both confess at the same time, and that the offers are concurrent…reality, I would guess, is that as soon as the first one confessors, the offer would not be valid for the 2nd

That was very good and informative…thanks

7:50 isn’t it 30, since it’s 10 plus 20

But, that's just a theory…

It's so theoretical I feel bad for the people who's job it is to analyze THE WORLD in terms of GT. I would pull my hair out.

Couldn't a coalition of nations coming together to form solutions to climate change be considered competitive?

Supporting resolutions for certain nations to enact environmental regulations that directly impact particular fossil fuel industries, for instance, could directly benefit those nations willing to step in to fill the subsequent void, especially if they're profiting from said resolutions while having a relatively low impact on the environment to begin with.

Seems fairly competitive to me. Maybe I'm missing something here.

INGEN BRYR SEG OM MATTE IKKI GIDD Å PRØV Å FÅ FØLK TE Å LIKE MATTE SUG ME BAK

I thought the video was great.. Is not about math. Is about interactions with people. depending if it is adversary or cooperative interaction.

Nash would have to disagree with what you said on the shapely value. If you look at it from a purely productive standpoint, someone who is on maternity leave does not contribute to the group and does not add to the pot to divvy it up, and thus to have the maximum possible gain should be cut out. Granted, This is only the case if one wanted the maximum possible output and would get it at any and all costs, putting aside morality and legality for more gain, and if we don't take into account any other factors.

Thank you!

So many HUGE mistakes in this video… I don't know if I can trust SciShow after watching this. Biggest mistake: VERY wrong definition of Nash-equilibrium…..

Did someone say climate change? 😂😂😂😂

oh man I thought that this was a chess tactics .

I could understand every word he said, but I couldn't understand every sentence he said,wtf

I want cookies

No I didnt interact with anoyne today.

Coalition sounds a lot like Communism

Popped up in my recommended. Misread as Game of Thrones Theory. Got excited. Re read, got sad. Watched anyways.

DA=Dumb Attorney?

Can you do the Byzantine Fault Tolerant problem?

Where is muscle Hank?

Not very competent. See comments below.

But that's just a theory! A GAME THEORY

Nicely explained

where is the jacket hank?

awesome vid, thank you. i'm definitely a follower of your channel now.

Matpat: This is illegal

I'm glad you guys went with SciShow Psych. I've been loving those vids!

I came up with a game the other day that I think is interesting in that when each person pursues their own enlightened self-interest, it is also in the best interest of the group. The game can be played by 2N players, where N is a natural number, but I will explain it for the case of only two players:

The game is played in rounds, and the object of the game is to accumulate as many coins as possible. Each player starts with some number of coins, e.g. 100.

In each round, each player rolls a dice. The combination of the dice set what I call the 'exchange rate." In each round a flip of the coin decides which roll each player will take, giver or getter. For an exchange to take place, both players must agree to the exchange.

Say in the first round, person A is designated as the 'giver' and rolls a 3. Person B is the 'getter' and rolls a 4. In this case, person A can elect to give up three of their coins and person B will get 4 coins.

Now, say in the second round, person B is the 'giver' and rolls a 6 and person B is the 'getter' and rolls a 1. In this case, person B could elect to give up 6 of their coins so person A gets 1.

OK, so that's the game. So, clearly if both players decide that they will never give up a single coin, the game is stagnant and each player just keeps their 100 coins. Interestingly, if both players are overly generous, and refuse to accept any coins and only offer to give them up, we arrive at the same point.

Now an obvious strategy is, if you are paired up with someone who is fair, is to only take and never give. This might seem to be the optimal strategy, but at the end of our game, the greedy person will end up with 200 coins and the non-greedy person will end up with nothing.

Compare this with the optimal strategy which is to agree to any exchange that introduces more coins into the game. This translates to agreeing to any exchange where the 'getter' rolls a number higher than the 'giver'. In this case, the number of coins in the game is unbounded and the number of coins that each player has approaches infinity as the number of rounds approaches infinity.

To me, this is a fairly realistic game as each of us are confronted with dozens of scenarios like this every day.

Can anyone tell me if this is a well-known game?

Non-cooperative is just avoiding worst case scenario, as we assume worst from others. For cooperative cookie example, we can split $40 as below:

First, slower cook gets $10 and faster cook gets $20, that's their per hour speed.

Now we have $10 left as 10 extra cookies were made. Now faster cook is twice as fast as slower cook so should get twice benefit. So faster cook gets $6.5 and slower gets $3.5. Thus, faster cook gets $26.5 and slower cook gets $13.5.

In your example, slower cook unnecessarily got benefited as he got $15. That method of money splitting is promoting slower work, eventually non quality work and laziness. Smarter and faster people should benefit depending on their skills, how fast and smart they really are as compared to others.

Also, game theory should support progressive method of decision making. When whole scenario is not fully known beforehand and final outcome is unsure, could be probabilistic or having range.

Example: Grain for cultivating it is to be distributed over entire country to farmers. Some regions are likely to get less rain and we don't know which ones. Half portion of grain is to be delivered before rain starts, and remaining over next 3 months. How to maximise total cultivation keeping transport load minimum? Rainfall value will get updated on 1st of every month. OR something similar example where there are unknown factors and dynamic decisions.

"But that's just a Theory. A Game Theory."

(The Game Theorist)

Tip: Don't bake with the Cookie Monster. It doesn't work out well for your profit margins.

John Nash doesn't have a 'Beautiful Face' 🙂

I missed 2 SECONDS and he totaly lost me….. This is a horrible video. It explains SOMETHING noeone asked an explanation FOR …. I STILL don't know what this video is about!??!

scishow psych is a hit!! now time for scishow life!!

Something wrong with his stomach ? Or is that just a bit of fat ?

我觉得这玩意儿就是中国先秦那会的：纵横捭阖啊，哈哈哈。好玩好玩。Where is human interaction, there is game theory.

Huh, every other description of the prisoners' dilemma I've ever watched mentions that you're always better off defecting, but then states the optimal known strategy to score the fewest years (or the most points) on average to be tit for tat, or some variant thereof.

Gotta say I'm a little disappointed in you.

Hey everyone, I made this video on Game Theory seen from a new perspective, about how it relates to cold war and arms races. I think you'll find it really interesting.

https://youtu.be/Q4ohFOB3ny4

I don't think I would apply the Shapely value equation in real life.

If I were the cookie wizard, I'd be pissed that my unskilled partner got a 50% raise and I only received a 25% raise. In actuality, cookie wizard is 2 times better and his or her revenue should also be 2 times higher. I would ratio it out.

game theory does not specify what should be done. in fact, based on the design, they should both not rat each other… what game theory says is that they WILL rat each other out

this is an awful video.

Where is scishow Bob?

I think it says a lot about our society that of the many times I have heard about the prisoner's dilemma, no one has ever mentioned that not only does it convict a pair of innocent people with the 'game' mentality, if one person is guilty and the other is innocent the innocent person goes up the river for 10 years.

What is climate change? That tells me nothing about the narrative of this story.

This leads to confessions even though maybe both are innocent.

SciShow is to science what TMZ is to filmmaking. i.e. irrelevant and unreliable.

You probably don't know there's a crack in the math system. Lol.

Life don't have to be this complicated, just bake da damn cookies and enjoy 😂💀

Can someone teach me about game theory for political case 😞🙏

Thanks before

And hey, it’s just a theory, A GAME THEORY

Balls

How would it be fair for the 2 bakers? While the one that makes 10 cookies has 50% gain from his usual and the more productive one only gains 25%. The less productive one definitely gain more than the other.

Personally i think it should be based on their the solo performance before they group up.

You have

gotto be related to Devin Townsend somehow!https://cdn.mos.cms.futurecdn.net/3TyARfjwtwfay7jjZvT7eS.jpg

Ha, if the only way a person got paid was per contribution, then everyone should contribute. But the system doesn't work that way. I don't see my moral obligation to play a game with different rules for different people.

Game theory is a jerk! Loved it.

Depends of the country. In some countreys if both don't confess each one will take 10 years cause justice will lie them about the other.

This is why US-china trade war is a lose-lose scenario

Game theory sounds like a pick up artists vlog name

Wow

Game theory proves simply how communism is neither possible nor achievable by/for humans.

So… Why people keeps trying what is clearly a complete failure?

Video also did not acknowledge the seminal game theory work of John von Neumann. See his book Theory of Games and Economic Behavior.

Game theory is where matpat gets triggered by Scott cawthon

What about Solitaire?

Nice video!!!!

Good job 👍

Game theory is a YouTube channel idiot

But that is Just a theory a game theory ANDDD CUT

Here is an issue.

Bob works in a factory. He CAN contribute, but his contribution has been nullified because it has become more cost effective to give his job to a robot.

So, Bob no longer contributes to society. But, Bob, theoretically could. Just not competitively in an automated capitalist market.

What do you do with Bob?

If you let him stick around then he becomes a threat. Homeless and impoverished he naturally socializes with countless others like him. Hungry, angry, and homeless they vote to take the money from the rich. At least in a democracy the mob robs the rich. But, the rich can see this coming and they take prevention measures and Bob and his friends are left with insurection.

You could pay Bob to just sit at home and read. But, in the long term strategy this only works out for Bob if there remains a majority of unemployed dependants like himself. So, Bob and friends have as many kids as possible. Which means the rich will have to support more and more Bobs.

Alternatively, the rich can wait until they have enough of production automated that they can do without the poor. Then they kill them all off and leave only a minority of workers.

Game theory appears to suggest that the best best option for Bob is to revolt as soon as possible before he losses his strategic edge, and the rich should kill off Bob as soon as they can survive without him.

Revolution seems to be the most rational course, perhaps inevitable here. And the one who waits for the other to attack during their strategic advantage, losses.

Yeah but is just a a Theory…

everybody seems to be correcting the video, so I will also join the chorus. FOr once that I can. The winning strategy in the prisoner's dilemma is to defect only if the game is played only once. If you play another version which is repetitive prisoner dilemma, the winning strategy end up TfT (tiit for tat) where each player starts to cooperate and then copies the behaviour of the other player in the previous round.

the color code is wrong. Rows are the 2nd numbers and columns are the first numbers. Otherwise when you compute the best responses its wrong.

Well rule 2 definitely doesnt apply to the UN & the Paris Agreement

BS , they both serve 2 years . Any criminals have to have enough comaradrie to engauge in the crime game to begin with etc etc . This is probably why Amerika lost Iraq … so smart .

Yugioh………yugioh

It doesn't make sense!!

I mean in the cooky example he said that both should get 5 cookies from the extra 10 cookies, but calculating that the second guy is faster ( he can make 20 cookies in his own), the first guy can produce only 50 % working alone, so I think a fair split would be the first guy to take 33. 33% of the ten extra cookies and the second to take 66.66 because he contributed more.. The split would be 10+3.33 = 13.33 for the first guy and 20+6.66 for the second guy, now if you divided 26.66 on 13.33 you will get 2 which is the exact number we started with, the se ond guy makes two times the first guy, this is why he deserves two times (6.66) what the first guy deserves (3.33)… Makes sense?

But what if they both know game theory and are smart to guess that the other person will choose to confess and think that they would expect the same?

Finite & infinite play..3rd possibility

8:50 Damn! That's true Man…

8:32 Thats wrong. If the 20c (cookie) guy is making twice as much as the 10c guy, that's a ratio of of 21.

The extra $10 should be split with 20c getting 2 thirds of that and and 10c getting only one third of the extra $10.

$13.33 & $26.67

At least i think.

Game theory is inherently empowering or not. How can Prisoners Dilemma be unbiased? As you said, it is affected by other decisions. What influenced that decision, and so forth? So, what matters is if you are empowered, or only interested in power.

I've just realised Nash is an idiot.

Thanks

The lecture is just a tour on Game and a good introduction. We can dig deeper