1-8 Nash Equilibrium of Example Games

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Stanford University

博弈论

1964 個評分

Popularized by movies such as "A Beautiful Mind," game theory is the mathematical modeling of strategic interaction among rational (and irrational) agents. Beyond what we call `games' in common language, such as chess, poker, soccer, etc., it includes the modeling of conflict among nations, political campaigns, competition among firms, and trading behavior in markets such as the NYSE. How could you begin to model keyword auctions, and peer to peer file-sharing networks, without accounting for the incentives of the people using them? The course will provide the basics: representing games and strategies, the extensive form (which computer scientists call game trees), Bayesian games (modeling things like auctions), repeated and stochastic games, and more. We'll include a variety of examples including classic games and a few applications. You can find a full syllabus and description of the course here: http://web.stanford.edu/~jacksonm/GTOC-Syllabus.html There is also an advanced follow-up course to this one, for people already familiar with game theory: https://www.coursera.org/learn/gametheory2/ You can find an introductory video here: http://web.stanford.edu/~jacksonm/Intro_Networks.mp4

從本節課中

Week 1: Introduction and Overview

Introduction, overview, uses of game theory, some applications and examples, and formal definitions of: the normal form, payoffs, strategies, pure strategy Nash equilibrium, dominant strategies

Introductory Video8:20

1-1 Game Theory Intro - TCP Backoff 11:23

1-2 Self-Interested Agents and Utility Theory 3:53

1-3 Defining Games 10:36

1-4 Examples of Games 5:59

1-5 Nash Equilibrium Intro 4:06

1-6 Strategic Reasoning 10:00

1-7 Best Response and Nash Equilibrium 2:43

1-8 Nash Equilibrium of Example Games 6:14

1-9 Dominant Strategies 7:32

1-10 Pareto Optimality 8:56

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Matthew O. Jackson
Professor
Economics
Kevin Leyton-Brown
Professor
Computer Science
Yoav Shoham
Professor
Computer Science

Let us now look at some examples and, of games and Nash equilibria in those games.

So here's the first game, a familiar game, this is of course the prisoner's

dilemma. The if both prisoners cooperate and then

they get a light punishment, and if they do not cooperate, they get a most severe

punishment. If the one cooperates and the others does

not, then the cooperator gets a terrible punishment and the one that does not

cooperate gets off scot, gets off scotfree.

and of course this game has a dominant strategy to defect no matter what the

other agent does, you are better off not cooperating.

And so of course, the only dominant strategy outcome is this one of both

defecting, and indeed, that is the only Nash

equilibrium in this game. So, it's a Nash equilibrium, it's the

best response. If the other person defects, then it's

the best response to defect but in fact, it's much stronger than that, it's best

to defect no matter what the other the other agent does.

So this is an example of one unique Nash equilibrium that happened to be a very

strong one, a dominant strategy, Nash Equilibrium.

So so here's another game. This is a game of pure coordination.

I think of it as walking towards each other on the sidewalk and you both can

decide whether to go to your respective lefts or respective rights.

In both cases, you will do fine and you will not collide,

and of course, if you miscoordinate, if you one goes to the left and the other

to the right, you will collide. So this is a natural game.

And, in fact, you see that you have two Nash equilibria,

the one that I wrote down here. If one one of the players go to the left,

it's the best response to go to the left. And conversely, if the the other player

goes to the right, you're best off going to the right as well.

And the others are not Nash equilibria. So here's an example of a game where

there are two Nash equilibria or two specifically pure strategy Nash

equilibria. Again, we'll discuss why we call these

pure strategy later on. Here's a very different game.

this is often called the game of the battle of the sexes.

Imagine a a couple and they want to go to every two movie and they are considering

two movies. One of them, a a very violent movie

Battle of the Titans, and the other, a very relaxed movie about flower growing,

call this B and F. the wife of course, would prefer to go to

Battle of the Titans, and the the husband would prefer to watch flower growing.

But, more than anything else, they would want to go together and so here are the

paths. If they both go to Battle Of The Titans,

then they're both probably happy the wife more than the husband.

If they go, both go to the flower growing movie, then the husband is slightly

happier than the wife, but if they go to different movies

neither of them was happy. That's that's, that's the that's, that's

the the game. how many how many equilibria we have

here? Well again we have two pure strategy Nash equilibria.

why is that? Well, if either of them goes to the Battle of the Titans, then the

other one wouldn't want to go there to, because if they go to a different one,

they would get zero rather than whatever they get here, one or two depending on

whether the husband or the wife. And then conversely, on the, on the flower

watching movie, flower growing movie and so, in both cases, they, the best

response is to go to the movie selected by the other party.

So, on the face of it, it looks very similar to the game of pure coordination

that we have here, but we do see a slight difference here, and we'll revisit that

later on when we speak about not pure strategies, but mixed strategies.

Here is here is another example, the last one we'll look at this is a game called

matching pennies. Imagine each of us, two players, needing

to decide, needing to decide on some side of a of a coin, heads or tail.

If we decide on the same sides, heads or tail, but we decide on the same one then,

then I win. If we decide on different sides, you

heads and me tails or vice versa, then you win.

and so we see this here, if we both decide on heads or we both decide on

tails, I win, and otherwise, you win. By winning I mean I get one and you get

minus one, so this is a zero-sum game. The sum of our payoffs is zero.

what is a pure strategy Nash Equilibirum here?

Well, let's think about it. Suppose I pick head, what is your best

response? Well, your best response then is to pick tails, because you get one

rather than minus one. But if you pick tails, then my best

response is now to play tail, because I want to coordinate with you, because

then, I will get one rather than the minus one that I would be getting here.

But, now if I play tails, you'd rather play heads, because you'd get one rather

than the minus one you're getting here. But again, if you're playing a tails, I

want to if you're playing heads, I want to play heads to match.

So we have this cycle where the best responses are leading us in the cycle.

And so there is no pure strategy Nash equilibrium in this game of matching

pennies.

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