# Yale - ECON-159 GAME THEORY Lecture 18 - Imperfect Information Information Sets and Sub-Game Perfection

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ECON-159: GAME THEORY

Lecture 18 - Imperfect Information: Information Sets and Sub-Game Perfection [November 7, 2007]

Chapter 1. Games of Imperfect Information: Information Sets [00:00:00]

Professor Ben Polak: So today we have a lot of stuff to get through, but it's all going to be fairly formal. We're not going to have time to play a game today. So it's going to be a day where we have to learn some new ideas. So the reason we need to go through some new formal ideas today is we've kind of exhausted what we can do with the ideas we've gathered so far. So, just to bring us up to date with where we are: in the first half of the semester — so before the mid-term — we looked at simultaneous move games. And one way to think about those simultaneous move games were games where, when I make my choice, I don't know what you've done, and, when you make your choice, you don't know what I've done.

Since the mid-term we've been looking at simple examples of sequential move games — sequential move games under perfect information — in which I typically do know what you did when I get to make my choice. And you know I'm going to know what you did when I get to make my choice. What I want to be able to do moving forward is I want to be able to look at strategic situations that combine those two settings. I want to be able to analyze games which involve both sequential moves and simultaneous move games. In particular, I want to see how we can extend the technique we've been focusing on for the last few weeks which is backward induction. I want us to see how we can extend the notion of backward induction to cope with games where some parts are sequential and some parts are simultaneous.

So we're going to look at a lot of examples and we're going to introduce some new ideas, and I'm going to try and walk you through that today. So that's our goal. Let's start with an example. So here's a very simple game in which Player 1 moves first, and has three choices. Let's call them up, middle, and down. And then Player 2 moves, and Player 2 has two choices from each of these nodes, and we'll call the choices suggestively, up and down — up and down. And here we'll just call them left and right. The payoffs are as follows, (4, 0), (0, 4), (0, 4), (4, 0), (1, 2), (0, 0). So this is just a standard game of perfect information, much like all the games we've seen since the mid-term. In fact, it's a relatively easy one.

So we know how to solve this game. We solve this game using what? Using backward induction, and that isn't so hard here. We know that if Player 2 finds herself up here, she will choose 4 rather than 0; if she finds herself here, she'll choose 4 rather than 0; and if she finds herself here, she'll choose 2 rather than 1. So Player 1 won't want to go up here because he'll get 0, and he won't want to go into the middle because he'll get 0, and he won't want to — but if he goes down Player 1 will choose left and Player 1 will get 1. So Player 1 will choose down. So backward induction predicts that Player 1 chooses down and Player 2 responds by choosing left.

Just staring at this a second, notice that the reason in this game — taking a step back from backward induction a second — the reason Player 1 did not want to choose either up or middle was because that move was going to be observed by Player 2, and in either case Player 2 was going to crush Player 1.

Lecture 18 - Imperfect Information: Information Sets and Sub-Game Perfection [November 7, 2007]

Chapter 1. Games of Imperfect Information: Information Sets [00:00:00]

Professor Ben Polak: So today we have a lot of stuff to get through, but it's all going to be fairly formal. We're not going to have time to play a game today. So it's going to be a day where we have to learn some new ideas. So the reason we need to go through some new formal ideas today is we've kind of exhausted what we can do with the ideas we've gathered so far. So, just to bring us up to date with where we are: in the first half of the semester — so before the mid-term — we looked at simultaneous move games. And one way to think about those simultaneous move games were games where, when I make my choice, I don't know what you've done, and, when you make your choice, you don't know what I've done.

Since the mid-term we've been looking at simple examples of sequential move games — sequential move games under perfect information — in which I typically do know what you did when I get to make my choice. And you know I'm going to know what you did when I get to make my choice. What I want to be able to do moving forward is I want to be able to look at strategic situations that combine those two settings. I want to be able to analyze games which involve both sequential moves and simultaneous move games. In particular, I want to see how we can extend the technique we've been focusing on for the last few weeks which is backward induction. I want us to see how we can extend the notion of backward induction to cope with games where some parts are sequential and some parts are simultaneous.

So we're going to look at a lot of examples and we're going to introduce some new ideas, and I'm going to try and walk you through that today. So that's our goal. Let's start with an example. So here's a very simple game in which Player 1 moves first, and has three choices. Let's call them up, middle, and down. And then Player 2 moves, and Player 2 has two choices from each of these nodes, and we'll call the choices suggestively, up and down — up and down. And here we'll just call them left and right. The payoffs are as follows, (4, 0), (0, 4), (0, 4), (4, 0), (1, 2), (0, 0). So this is just a standard game of perfect information, much like all the games we've seen since the mid-term. In fact, it's a relatively easy one.

So we know how to solve this game. We solve this game using what? Using backward induction, and that isn't so hard here. We know that if Player 2 finds herself up here, she will choose 4 rather than 0; if she finds herself here, she'll choose 4 rather than 0; and if she finds herself here, she'll choose 2 rather than 1. So Player 1 won't want to go up here because he'll get 0, and he won't want to go into the middle because he'll get 0, and he won't want to — but if he goes down Player 1 will choose left and Player 1 will get 1. So Player 1 will choose down. So backward induction predicts that Player 1 chooses down and Player 2 responds by choosing left.

Just staring at this a second, notice that the reason in this game — taking a step back from backward induction a second — the reason Player 1 did not want to choose either up or middle was because that move was going to be observed by Player 2, and in either case Player 2 was going to crush Player 1.

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