Game Theory: The Science of Decision-Making

Game Theory: The Science of Decision-Making

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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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100 thoughts on “Game Theory: The Science of Decision-Making”

  1. Damn shame that Anatol Rapoport died before he could collect the Peace Prize: he provided the US and the Soviets with the language that got us all through the dangerous atom-bomb years.

  2. i think that the cookie distribution is a bit faulty.
    given that the second person makes twice as many cookies as the first person
    no.of cookies by the first person = 13.33
    no. of cookies by the second person = 26.66

  3. @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.

  4. So, what’s the real benefit in cookies for the patrons in this equation…? 🤔
    (Apparently doesn’t fit in here)

  5. 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 ?

  6. 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

  7. 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.

  8. 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)

  9. 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.

  10. 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.

  11. 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

  12. 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.

  13. 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.

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

  15. 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.

  16. 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…..

  17. 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?

  18. 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.

  19. 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!??!

  20. 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.

  21. 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.

  22. 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.

  23. 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.

  24. 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

  25. 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.

  26. 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.

  27. You have got to be related to Devin Townsend somehow!

  28. 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.

  29. 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.

  30. 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?

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

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