# Yale - ECON-159 GAME THEORY Lecture 13 - Sequential Games Moral Hazard, Incentives, and Hungry Lions

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ECON-159: GAME THEORY

Lecture 13 - Sequential Games: Moral Hazard, Incentives, and Hungry Lions [October 22, 2007]

Chapter 1. Sequential Games: Backward Induction [00:00:00]

Professor Ben Polak: Okay, so I want to set up a new topic today, and to get us started I thought we'd play a game. So the game is going to be called cash in a hat, and we'll see what this game is like in a minute. So the idea is this, there's going to be two players. I'm going to pick on two of you. I hope I can find people who brought some money to class. So Player I will have a choice, Player I can put nothing, $1 or $3 into a hat. The hat will then be given to Player II and Player II can look at what's in the hat and either she can match what's in the hat, i.e., add the same amount in so 1 if it's 1, or 3 if it's 3, or she can simply take the cash.

The payoffs of this game will be as follows. Player I, if they put nothing in, they're not going to get anything. If they put 1 in and it's matched then they double their money, so they get 2 back. If they put 3 in and it's matched then they double their money, so they get 6 back so they'll net 3. However, if they put 1 in and Player II takes it they've just lost their dollar, and if they put $3 in and Player II takes it they've lost $3. Everyone understand that? Pretty simple. From Player II's point of view, if Player II matches then they get their investment back plus $1.50 if they matched 1, so the get 2.50 back in all; and they get their investment back plus $2 if they match 3, so they get 5 back in all. Of course, if they simply take the money out of the hat and put it in their pocket then that's what they get in the game.

So once again, if Player II matches with $1, there's $1 in the hat and she matches, then she gets back 2.50 for a profit of 1.50. If there's $3 and she matches 3, she gets back 5 for a profit of 2; and if she simply takes the money out of the hat and puts it in her pocket then she gets how ever much money was in the hat. So I here will be the provider of the hat, we'll use the same hat we used before, and if necessary I'll provide sheets of paper, we can write IOU's on, but I'm hoping we can play for real cash. Later on — it would be nice to get a chair here — but for now, let me just go down here and I'm going to grab this mike. Who here wants to be Player I? Who brought some money to class today? I'm quite willing to pick on somebody. All right good, so Player I is, what's your name again?

Student: Justin.

Professor Ben Polak: Justin. So hold onto that hat for a second and who wants to be Player II? All right, so this gentleman here whose name is?

Student: Nate.

Professor Ben Polak: Nate. All right so we have Justin and Nate. So Justin you have to decide how much money you're going to put in the hat. Did you bring some money too Nate? Better make sure you have, otherwise borrow some from your neighbors.

Student: I'm going to put in a buck.

Professor Ben Polak: All right, so the money's being put in the hat and delivered to Nate.

Student: I'm going to put in a buck too so that I get 2.50.

Professor Ben Polak: All right, so if we empty out the hat, here's our beautiful pink hat. I should really have the hat passed across the room rather than have me do it, but we have $2 in the hat so I'm going to pay each - I'm going to pay — what am I going to pay? I'm going to pay Player I $2, I'm going to give them back their original $1 and give them $2 in a second, and I'm going to pay Player II $1.50.

Lecture 13 - Sequential Games: Moral Hazard, Incentives, and Hungry Lions [October 22, 2007]

Chapter 1. Sequential Games: Backward Induction [00:00:00]

Professor Ben Polak: Okay, so I want to set up a new topic today, and to get us started I thought we'd play a game. So the game is going to be called cash in a hat, and we'll see what this game is like in a minute. So the idea is this, there's going to be two players. I'm going to pick on two of you. I hope I can find people who brought some money to class. So Player I will have a choice, Player I can put nothing, $1 or $3 into a hat. The hat will then be given to Player II and Player II can look at what's in the hat and either she can match what's in the hat, i.e., add the same amount in so 1 if it's 1, or 3 if it's 3, or she can simply take the cash.

The payoffs of this game will be as follows. Player I, if they put nothing in, they're not going to get anything. If they put 1 in and it's matched then they double their money, so they get 2 back. If they put 3 in and it's matched then they double their money, so they get 6 back so they'll net 3. However, if they put 1 in and Player II takes it they've just lost their dollar, and if they put $3 in and Player II takes it they've lost $3. Everyone understand that? Pretty simple. From Player II's point of view, if Player II matches then they get their investment back plus $1.50 if they matched 1, so the get 2.50 back in all; and they get their investment back plus $2 if they match 3, so they get 5 back in all. Of course, if they simply take the money out of the hat and put it in their pocket then that's what they get in the game.

So once again, if Player II matches with $1, there's $1 in the hat and she matches, then she gets back 2.50 for a profit of 1.50. If there's $3 and she matches 3, she gets back 5 for a profit of 2; and if she simply takes the money out of the hat and puts it in her pocket then she gets how ever much money was in the hat. So I here will be the provider of the hat, we'll use the same hat we used before, and if necessary I'll provide sheets of paper, we can write IOU's on, but I'm hoping we can play for real cash. Later on — it would be nice to get a chair here — but for now, let me just go down here and I'm going to grab this mike. Who here wants to be Player I? Who brought some money to class today? I'm quite willing to pick on somebody. All right good, so Player I is, what's your name again?

Student: Justin.

Professor Ben Polak: Justin. So hold onto that hat for a second and who wants to be Player II? All right, so this gentleman here whose name is?

Student: Nate.

Professor Ben Polak: Nate. All right so we have Justin and Nate. So Justin you have to decide how much money you're going to put in the hat. Did you bring some money too Nate? Better make sure you have, otherwise borrow some from your neighbors.

Student: I'm going to put in a buck.

Professor Ben Polak: All right, so the money's being put in the hat and delivered to Nate.

Student: I'm going to put in a buck too so that I get 2.50.

Professor Ben Polak: All right, so if we empty out the hat, here's our beautiful pink hat. I should really have the hat passed across the room rather than have me do it, but we have $2 in the hat so I'm going to pay each - I'm going to pay — what am I going to pay? I'm going to pay Player I $2, I'm going to give them back their original $1 and give them $2 in a second, and I'm going to pay Player II $1.50.

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