# Yale - ECON-159 GAME THEORY Lecture 19 - Subgame Perfect Equilibrium Matchmaking and Strategic Investments

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ECON-159: GAME THEORY

Lecture 19 - Subgame Perfect Equilibrium: Matchmaking and Strategic Investments [November 12, 2007]

Chapter 1. Sub-game Perfect Equilibria: Example [00:00:00]

Professor Ben Polak: So last time we covered a whole bunch of new ideas, and it was really quite a lot of ideas for one class. Here's some of the ideas we covered. We talked about information sets, and these were ways to allow us to model imperfect information. So what's imperfect information? It's a way of being able to capture both simultaneity in moves and sequential moves in the same game. So it's a way that's going to allow us to meld the lessons from before the mid-term and after the mid-term.

Then we talked about what strategies meant in this context, and the basic idea is strategies are instructions — strategies for each player — give them an instruction at each of their information sets. Then we talked about what sub-games were, and, leaving aside technicalities, sub-games were just games within games. And finally we introduced the idea of sub-game perfection which is our new solution concept that refines the idea of Nash equilibrium. What sub-game perfection is going to do is it's going to instruct the players to play a Nash equilibrium in every sub-game. Another way of saying it is, a sub-game equilibrium is a Nash equilibrium in the whole game, but in each sub-game it induces Nash play as well.

Now, we're going to see today examples. If we have time I'll go through three different examples, and I'll tell you at the end of each example what it is I'm hoping to be able to take away from that example. So, last time was a lot of formal stuff. Today is going to be a lot of examples. Okay, that's our agenda. Here's a game. Here's our first example. And I call this example, I call this game, "don't screw up," for reasons we'll see in a minute.

So this is a game in which Player 1 has to choose between Up and Down. If Player 1 chooses Up then Player 2 gets to move and chooses between left and right. And if Player 2 chooses left then Player 1 gets to move again and Player 1 chooses between up or down. Everyone looking at that game? So why don't we play this game since we haven't played a game for a while. We'll play a couple of games today.

So what I'm going to do is let's divide the class in two. So if I just draw a line down the middle of the class, everybody to my left (to your right), everybody on this side of the class is a Player 1. Okay you're all Player 1's. And everyone on this side of class you're Player 2, including you guys hiding from the camera, you're Player 2's. Okay, so let's figure out what we're going to do. Everyone had the time to look at the game? So Player 1's you get to move first, those of you who are going to choose Down raise your hand now. Raise your hand. Wave it in the air. Keep it up so the camera can see you. And those of you who are going to choose Up raise your hands.

Lots more Ups. Those of you who chose Up why don't you all stand up. I don't want to do all the exercise here, so all those who chose Up, stand up. So you can see that choosing down ends the game, so this many people are still playing the game. Everyone who is still sitting down, everyone who sat down here has exited. All right, Player 2's you get to move now. So Player 2's, those of you who choose right, including the people on this aisle, those people who choose right raise your hand now — one right over there. Those of you who choose left raise your hands.

Lecture 19 - Subgame Perfect Equilibrium: Matchmaking and Strategic Investments [November 12, 2007]

Chapter 1. Sub-game Perfect Equilibria: Example [00:00:00]

Professor Ben Polak: So last time we covered a whole bunch of new ideas, and it was really quite a lot of ideas for one class. Here's some of the ideas we covered. We talked about information sets, and these were ways to allow us to model imperfect information. So what's imperfect information? It's a way of being able to capture both simultaneity in moves and sequential moves in the same game. So it's a way that's going to allow us to meld the lessons from before the mid-term and after the mid-term.

Then we talked about what strategies meant in this context, and the basic idea is strategies are instructions — strategies for each player — give them an instruction at each of their information sets. Then we talked about what sub-games were, and, leaving aside technicalities, sub-games were just games within games. And finally we introduced the idea of sub-game perfection which is our new solution concept that refines the idea of Nash equilibrium. What sub-game perfection is going to do is it's going to instruct the players to play a Nash equilibrium in every sub-game. Another way of saying it is, a sub-game equilibrium is a Nash equilibrium in the whole game, but in each sub-game it induces Nash play as well.

Now, we're going to see today examples. If we have time I'll go through three different examples, and I'll tell you at the end of each example what it is I'm hoping to be able to take away from that example. So, last time was a lot of formal stuff. Today is going to be a lot of examples. Okay, that's our agenda. Here's a game. Here's our first example. And I call this example, I call this game, "don't screw up," for reasons we'll see in a minute.

So this is a game in which Player 1 has to choose between Up and Down. If Player 1 chooses Up then Player 2 gets to move and chooses between left and right. And if Player 2 chooses left then Player 1 gets to move again and Player 1 chooses between up or down. Everyone looking at that game? So why don't we play this game since we haven't played a game for a while. We'll play a couple of games today.

So what I'm going to do is let's divide the class in two. So if I just draw a line down the middle of the class, everybody to my left (to your right), everybody on this side of the class is a Player 1. Okay you're all Player 1's. And everyone on this side of class you're Player 2, including you guys hiding from the camera, you're Player 2's. Okay, so let's figure out what we're going to do. Everyone had the time to look at the game? So Player 1's you get to move first, those of you who are going to choose Down raise your hand now. Raise your hand. Wave it in the air. Keep it up so the camera can see you. And those of you who are going to choose Up raise your hands.

Lots more Ups. Those of you who chose Up why don't you all stand up. I don't want to do all the exercise here, so all those who chose Up, stand up. So you can see that choosing down ends the game, so this many people are still playing the game. Everyone who is still sitting down, everyone who sat down here has exited. All right, Player 2's you get to move now. So Player 2's, those of you who choose right, including the people on this aisle, those people who choose right raise your hand now — one right over there. Those of you who choose left raise your hands.

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