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ECON-159: GAME THEORY

Lecture 20 - Subgame Perfect Equilibrium: Wars of Attrition [November 14, 2007]

Chapter 1. Wars of Attrition: The Rivalry Game [00:00:00]

Professor Ben Polak: Last time we looked at how to apply our new idea of sub-game perfect equilibrium to a whole bunch of games, and our general idea for how to solve the sub-game perfect equilibrium is as follows. We looked at each sub-game. We solved for the Nash equilibrium in the sub-game, that is something we learned to do long ago. And then we rolled back the payoffs: we rolled them back up the tree. And towards the end we learned something interesting. I'm not going to go back to it today — I just want to emphasize it. We learned that strategic effects matter.

So in that investment game we looked at last time, when you're considering whether to rent a new piece of machinery, it made a very big difference whether you considered how this action would affect the actions of the other side; in this case, how it affects your competition. This is a very general idea, a very general point. So just to give you a couple of more examples, when you're designing tax systems — I mentioned this last time — when you're designing a tax system, to make some changes in the U.S. tax system, it's not good enough to look at how people are behaving in the old tax system and just calculate in an accounting manner how much more money you're going to raise, or how much money it's going to cost you. You have to take into account how that's going to lead to changes in behavior.

Once again, that's a strategic effect, and in the homework that you're handing in today, all of you will have had a nice example of that in the toll booth problem. So in the toll booth problem, when you're putting tolls on roads — or more generally, when you're building new roads, new bridges, new flyovers, new bypasses, you need to take into account how those new tolls, how those new roads will affect all of traffic flow. Traffic flow down the tree will form a new equilibrium and you need to consider that in designing your tolls and designing your road system. So that's another example of SPE.

So today I want to do something quite different, a little bit like what we did with duel, I want to play a game today, and probably spend the whole of today analyzing this one game. So it's quite a complicated game, but it's quite a fun game. So what's the game we're going to look at? The game is going to involve two players, and each player in each period, they choose — or each chooses I should say — each chooses, whether to fight or to quit. So F means fight and Q means quit, and they make this choice simultaneously. The game ends as soon as someone quits.

So there's good news and bad news for this game. Let's do the good news first. The good news is that if the other player quits first you win a prize. Generally we'll call this prize V, but we'll play for some cash in a minute. The bad news is, each period in which both fight — so each period in which both players choose to fight — each player pays a cost, so they pay -C. Just to keep things interesting let's fill in the other thing here which is if both quit at once — so if both quit at once then they get 0 that period. So this is a game we've seen a little bit before. We saw a little bit under the auspices of Hawk Dove. Those people in the MBA class saw a game a lot like this. But we're going to analyze this in much more detail then we did before.

As I said, we're going to spend the whole of today talking about it.

Lecture 20 - Subgame Perfect Equilibrium: Wars of Attrition [November 14, 2007]

Chapter 1. Wars of Attrition: The Rivalry Game [00:00:00]

Professor Ben Polak: Last time we looked at how to apply our new idea of sub-game perfect equilibrium to a whole bunch of games, and our general idea for how to solve the sub-game perfect equilibrium is as follows. We looked at each sub-game. We solved for the Nash equilibrium in the sub-game, that is something we learned to do long ago. And then we rolled back the payoffs: we rolled them back up the tree. And towards the end we learned something interesting. I'm not going to go back to it today — I just want to emphasize it. We learned that strategic effects matter.

So in that investment game we looked at last time, when you're considering whether to rent a new piece of machinery, it made a very big difference whether you considered how this action would affect the actions of the other side; in this case, how it affects your competition. This is a very general idea, a very general point. So just to give you a couple of more examples, when you're designing tax systems — I mentioned this last time — when you're designing a tax system, to make some changes in the U.S. tax system, it's not good enough to look at how people are behaving in the old tax system and just calculate in an accounting manner how much more money you're going to raise, or how much money it's going to cost you. You have to take into account how that's going to lead to changes in behavior.

Once again, that's a strategic effect, and in the homework that you're handing in today, all of you will have had a nice example of that in the toll booth problem. So in the toll booth problem, when you're putting tolls on roads — or more generally, when you're building new roads, new bridges, new flyovers, new bypasses, you need to take into account how those new tolls, how those new roads will affect all of traffic flow. Traffic flow down the tree will form a new equilibrium and you need to consider that in designing your tolls and designing your road system. So that's another example of SPE.

So today I want to do something quite different, a little bit like what we did with duel, I want to play a game today, and probably spend the whole of today analyzing this one game. So it's quite a complicated game, but it's quite a fun game. So what's the game we're going to look at? The game is going to involve two players, and each player in each period, they choose — or each chooses I should say — each chooses, whether to fight or to quit. So F means fight and Q means quit, and they make this choice simultaneously. The game ends as soon as someone quits.

So there's good news and bad news for this game. Let's do the good news first. The good news is that if the other player quits first you win a prize. Generally we'll call this prize V, but we'll play for some cash in a minute. The bad news is, each period in which both fight — so each period in which both players choose to fight — each player pays a cost, so they pay -C. Just to keep things interesting let's fill in the other thing here which is if both quit at once — so if both quit at once then they get 0 that period. So this is a game we've seen a little bit before. We saw a little bit under the auspices of Hawk Dove. Those people in the MBA class saw a game a lot like this. But we're going to analyze this in much more detail then we did before.

As I said, we're going to spend the whole of today talking about it.

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