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Game Theory: Lecture 5 Transcript

September 19, 2007

Professor Ben Polak: Okay, so last time we came across a new idea, although it wasn't very new for a lot of you, and that was the idea of Nash Equilibrium.

What I want to do today is discuss Nash Equilibrium, see how we find that equilibrium into rather simple examples. And then in the second half of the day I want to look at an application where we actually have some fun and play a game. At least I hope it's fun.

But let's start by putting down a formal definition. We only used a rather informal one last week, so here's a formal one. A strategy profile — remember a profile is one strategy for each player, so it's going to be S1*, S2*, all the way up to SM* if there are M players playing the game — so this profile is a Nash Equilibrium (and I'm just going to write NE in this class for Nash Equilibrium from now on) if, for each i –so for each player i, her choice — so her choice here is Si*, i is part of that profile is a best response to the other players' choices. Of course, the other players' choices here are S — i* so everyone is playing a best response to everyone else.

Now, this is by far the most commonly used solution concept in Game Theory. So those of you who are interviewing for McKenzie or something, you're going to find that they're going to expect you to know what this is. So one reason for knowing what it is, is because it's in the textbooks, it's going to be used in lots of applications, it's going to be used in your McKenzie interview. That's not a very good reason and I certainly don't want you to jump to the conclusion that now we've got to Nash Equilibrium everything we've done up to know is in some sense irrelevant. That's not the case.

It's not always going to be the case that people always play a Nash Equilibrium. For example, when we played the numbers game, the game when you chose a number, we've already discussed last week or last time, that the equilibrium in that game is for everyone to choose one, but when we actually played the game, the average was much higher than that: the average was about 13. It is true that when we played it repeatedly, it seemed to converge towards 1, but the play of the game when we played it just one shot first time, wasn't a Nash Equilibrium. So we shouldn't form the mistake of thinking people always play Nash Equilibrium or people, "if they're rational," play Nash Equilibrium. Neither of those statements are true.

Nevertheless, there are some good reasons for thinking about Nash Equilibrium other than the fact it's used by other people, and let's talk about those a bit. So I want to put down some motivations here — the first motivation we already discussed last time. In fact, somebody in the audience mentioned it, and it's the idea of "no regrets." So what is this idea? It says, suppose we're looking at a Nash Equilibrium. If we hold the strategies of everyone else fixed, no individual i has an incentive to deviate, to move away. Alright, I'll say it again. Holding everyone else's actions fixed, no individual has any incentive to move away. Let me be a little more careful here; no individual has any strict incentive to move away. We'll see if that actually matters. So no individual can do strictly better by moving away. No individual can do strictly better by deviating, holding everyone else's actions.

So why I call that "no regret"?

September 19, 2007

Professor Ben Polak: Okay, so last time we came across a new idea, although it wasn't very new for a lot of you, and that was the idea of Nash Equilibrium.

What I want to do today is discuss Nash Equilibrium, see how we find that equilibrium into rather simple examples. And then in the second half of the day I want to look at an application where we actually have some fun and play a game. At least I hope it's fun.

But let's start by putting down a formal definition. We only used a rather informal one last week, so here's a formal one. A strategy profile — remember a profile is one strategy for each player, so it's going to be S1*, S2*, all the way up to SM* if there are M players playing the game — so this profile is a Nash Equilibrium (and I'm just going to write NE in this class for Nash Equilibrium from now on) if, for each i –so for each player i, her choice — so her choice here is Si*, i is part of that profile is a best response to the other players' choices. Of course, the other players' choices here are S — i* so everyone is playing a best response to everyone else.

Now, this is by far the most commonly used solution concept in Game Theory. So those of you who are interviewing for McKenzie or something, you're going to find that they're going to expect you to know what this is. So one reason for knowing what it is, is because it's in the textbooks, it's going to be used in lots of applications, it's going to be used in your McKenzie interview. That's not a very good reason and I certainly don't want you to jump to the conclusion that now we've got to Nash Equilibrium everything we've done up to know is in some sense irrelevant. That's not the case.

It's not always going to be the case that people always play a Nash Equilibrium. For example, when we played the numbers game, the game when you chose a number, we've already discussed last week or last time, that the equilibrium in that game is for everyone to choose one, but when we actually played the game, the average was much higher than that: the average was about 13. It is true that when we played it repeatedly, it seemed to converge towards 1, but the play of the game when we played it just one shot first time, wasn't a Nash Equilibrium. So we shouldn't form the mistake of thinking people always play Nash Equilibrium or people, "if they're rational," play Nash Equilibrium. Neither of those statements are true.

Nevertheless, there are some good reasons for thinking about Nash Equilibrium other than the fact it's used by other people, and let's talk about those a bit. So I want to put down some motivations here — the first motivation we already discussed last time. In fact, somebody in the audience mentioned it, and it's the idea of "no regrets." So what is this idea? It says, suppose we're looking at a Nash Equilibrium. If we hold the strategies of everyone else fixed, no individual i has an incentive to deviate, to move away. Alright, I'll say it again. Holding everyone else's actions fixed, no individual has any incentive to move away. Let me be a little more careful here; no individual has any strict incentive to move away. We'll see if that actually matters. So no individual can do strictly better by moving away. No individual can do strictly better by deviating, holding everyone else's actions.

So why I call that "no regret"?

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