ECON 159: Game Theory
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ECON 159 - Lecture 18 - Imperfect Information: Information Sets and Sub-Game Perfection
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. So if Player 1 went up, Player 2 was playing this sort of “strictly competitive game” with Player 1 and Player 2 could pick a choice that gave 2 4 and 1 0. Conversely, if Player 1 chose middle, Player 2 could crush Player 1 by choosing up, which gave, once again, Player 2 4 and Player 1 0. So there was a good reason here to avoid going into the part of the game following up or middle, and the reason was 2 has a huge second-mover advantage in those parts of the game. Is that clear to everybody?
So I now want to consider a similar, but importantly different game, so I’m going to draw the game again, but before I draw it, let me say what I’m going to do. So I want to introduce a new idea, and the new idea is going to be that Player 2 will not be able to distinguish between up or middle. So let’s just say it again. So if Player 1 chooses down, Player 2 will observe that, just as we’ve done before in our standard perfect-information games, but if Player 1 chooses either up or middle, I want to capture the idea that Player 2 doesn’t know which of those two choices was made. That’s clearly going to change the game a lot and the first question is, how do we represent that idea in a tree?
So let me try and show a good way to represent that in a tree. So the game has the same structure to it. Player 1 is again choosing between up, middle, or down. And Player 2 once again is choosing here, up or down, up or down, and here left or right. And the payoffs haven’t changed. They’re still (4, 0), (0, 4), (4, 0), (0, 4), (1, 2) and (0,0). So that’s exactly the same as you have in your notes already, but now I want to adapt this tree to show how we indicate that Player 2 cannot distinguish–cannot observe whether 1 chose up or middle, but can observe if Player 1 has chosen down. The way we do that very simply, we draw a dotted line between the two nodes of Player 2 between which 2 cannot distinguish.
So this idea here, what this dotted line indicates is that these two nodes are set in the same information set. So our new idea here is the idea of an information set. The payoffs are meant be the same on the left as on the right. So the idea here is that Player 2 cannot distinguish these two nodes. Player 2 knows that she’s in this information set: she knows that Player 1 must have chosen either up or middle, she knows that Player 1 did not choose down, but she doesn’t know whether she’s really here or here. Now what happens in this game? This game is a very different game. Why is it a different game?
Well let’s try and apply that loose intuition we talked about before. We said previously, in the old game, that if Player 1 chose up, 2 knew that Player 1 had chosen up, and observed that by choosing down Player 2 could crush 1. And if Player 1 chose middle, Player 2 could observe that Player 1 had chosen middle and this time by choosing up could crush 1. The problem is that now in this new game Player 2 doesn’t know whether she’s here, in which case she would want to choose down, or here, in which case she’d want to choose up. So Player 2’s choice is not so obvious anymore. That simple backward induction argument has disappeared.
Moreover, Player 1 knows that Player 2 will not be able to observe between up or middle, so it isn’t necessarily the case that Player 1 will want to choose down anymore. It’s still true that if Player 1 did choose down that Player 2 would be able to observe that and will choose left, so that part of the argument’s the same. What do we think is going to happen here? Well we don’t know, but let me give a suggestion what might happen here. Player 1 might say, hey I could randomize between up and middle. I could choose half the time up and half the time middle. If I choose half the time up and half the time middle, Player 2 isn’t going to know–in general–isn’t going to know what I’ve done. It isn’t quite clear what Player 2’s going to do and since I’m randomizing between up or middle whatever Player 2’s going to do, I’m going to get half the time 4 and half the time 0 for an expected value of 2.
So to say it again, so Player 1 might decide in this game to randomize fifty-fifty between up and middle, knowing that half the time therefore he will get 4 and half the time he’ll get 0 for an expected value of 2, which notice is better than he got by choosing down. So this change in this game, change in the information in this game, not only led to a different game but led to a very different outcome. So here 1 might, for example, might randomize between up and middle, and over here we know exactly what 1 does, 1 chooses down.
So we get very different outcomes because of this change in information in the game, and the theme of today is that information is going to matter. The way we’re going to model information is by thinking about these information sets. And as we go through today, I want to start giving you some formal definitions. So this is the idea, now let’s look at the formal definition. There’s going to be a lot of writing today, so I hope you brought a notepad with some room on it. So the first formal definition of the day comes off that last example. The formal definition is the idea that I want to capture: I want to capture the idea that players down the tree may not know exactly what was done up the tree. And the formal definition is going to go through the idea of an information set.
So an information set of Player i–in this case involves Player 2 but more generally of Player i–is a collection–or a set if you like–is a collection of Player i’s nodes between which–I guess it can be more than two–so let’s say among which i cannot distinguish. Now it’s going to turn out that, by clever use of information sets, we’re going to be able to use our technology, our technology of drawing trees, to capture all sorts of interesting and increasingly complicated information settings.
In this particular game, it’s the case that Player 1 knew that Player 2 was not going to be able to distinguish between up or middle in this tree, and Player 1 knew that Player 2 would be able to distinguish in the left hand tree. We can even use information sets in a more elaborate tree to capture the idea that Player 1 may not know what Player 2’s going to know. But I won’t do that now: I’ll leave that later, and you’ll see some examples of that on the homework. So we have our formal definition. This is going to be the first of our big tools of the day, but let me just put down a few things that we have to be careful about: a couple of rules.
So these information sets have to obey certain rules and certain things are not allowed. So in particular the following is not allowed. Here’s a tree in which Player 1 moves first and Player 2 does not observe Player 1’s move. So these two nodes are Player 2’s nodes. They’re in the same information set, which means Player 2 is not meant to be able to distinguish between these two nodes. And suppose however the tree looked like this. Okay, so I claim that this is crazy. We couldn’t allow this. It wouldn’t make any sense to allow this. Can anyone see why? Why is this not really a sensible tree? Everyone see that? Why is that not a sensible tree?
Student: If Player 2 knows that he has three choices then he’ll know he’s at the top node.
Professor Ben Polak: Exactly, in this tree you haven’t got the payoffs in, but if Player 2 observes that she has three choices, she knows she must be at the top node. If she observes she has two choices she must be at the bottom node. So in this tree, it was supposed to be the case that 2 didn’t know whether she was here or here, but merely by observing how many choices she has, she could infer whether she was at the top node or the bottom node. So that can’t make any sense. So this is not allowed, so we’ll put a red cross through that one. Now the second thing that’s not allowed is a little bit more subtle, and actually is an interesting thing. This is just kind of bookkeeping, but the second thing is more interesting.
So let’s have a look at it. Here’s a more interesting tree. Player 1 moves first. Player 2 observes that move, and Player 2 moves second. And then at the bottom of this Player 1 may have another chance to move again. So again I haven’t put payoffs in here. Player 1 moves first; Player 2 moves second; and, if Player 2 chooses down here or up there, then Player 1 gets to move again. Now I claim again that this is not a sensible tree. It’s not a sensible arrangement of information sets. Can anyone see why this isn’t sensible? Why is this not sensible? Steven?
Student: Player 1 knows what node he’s at based on the first choice that he made.
Professor Ben Polak: Exactly, so to get to the upper node here for Player 1, Player 1 must have chosen up before, and to get to the lower node here, Player 1 must have played down before. So provided that Player 1 remembers his or her own move, she knows where she is. Is that right? So provided that Player 1 can recall what she herself did earlier on in the tree she should be able to distinguish these things. So we’re going to rule this out, but I just want to make a remark here. There’s an assumption in ruling it out and the assumption is we’re assuming perfect recall or perfect memory. And people don’t always–in the real world, players don’t always have perfect recall.
There are two reasons–and we’re always going to assume this, but let me just make a remark. There are two reasons why people might not have perfect recall. One reason is, like me, they’re getting old. They simply can’t remember what they did yesterday. So while I’m driving home I know roughly how many traffic lights I have to go through before I turn right, but I sometimes forget which traffic light I’m at and I turn right too early or too late. That doesn’t happen to you guys, but it happens to me as I’m getting a bit senile, so old age would rule out perfect recall.
A more important example, perhaps, is if players of games are themselves institutions. It’s sometimes useful, and we’ve often talked about it in this class, to imagine a player of a game being a firm or a country, or some kind of institution in which the actual decisions may be being taken by different actual people within the firm, institution, or country. This assumption of perfect recall is saying that the players within the institution knew what the other players within that same institution were doing. If we’re modeling General Motors as one player, this assumption is assuming that the chief financial officer and the chief executive officer at GM are observing each other’s actions: are on the same page. The left hand knows what the right hand’s doing.
We are typically going to assume that, but just to make the point: it is an assumption, and it’s quite interesting to see what happens if you relax it. So with that in mind, we can move to our next definition. and this is something I’ve referred to early on in the class, but I want to be formal now. Now we can be formal. We’ve talked earlier on in this class about the idea of perfect information. So, for example, when we talked about Zermelo’s theorem, we talked about games of perfect information.
We said informally what this was–a game of perfect information is a game where each player in the game can observe all previous moves. That was our informal definition, but we can now give a formal definition very simply. Perfect information is a setting where all information sets in the tree–games of perfect information are games where all information sets in the tree–contain just one node. I want to be clear here. What we’re saying here is, if we have a tree in which every information set is a singleton, we basically don’t have to bother with any dotted lines: that’s a game of perfect information. And that shouldn’t be a surprise to anybody here because that’s exactly how we drew trees since the mid-term. Is that right?
Of course, the novelty is we’re now going to be allowed to look at games of imperfect information. The reason we’re doing this is because it will be interesting–as in the example we’ve just seen–to think about games where information is not perfect. So what is the definition of imperfect information? Imperfect information’s formal definition is “not perfect information.” We’ve defined what perfect information is. Imperfect information is the rest. In the real world there’s a lot of games that turn out to have imperfect information. There’s lots of strategic situations where I’m going to be able to observe some things that you’ve done, but other things, I won’t know quite what you’ve done.
Chapter 2. Games of Imperfect Information: Translating a Game from Matrix Form to Tree Form and Vice Versa [00:18:56]
Okay, let’s go straight to an example. So I don’t think we really need to keep that definition very focal, so let’s get rid of that board. Let’s do an example. We’ll be doing many examples today. So this example is going to be a tree in which Player 1 moves first. Player 2 cannot observe this move; and sometimes rather than labeling both of these nodes with a 2, I’ll just put a 2 on the information set, just to indicate that both of these nodes belong to Player 2. So Player 2 moves second. And we’ll call Player 1’s move up or down; and we’ll call Player 2’s move left or right, kind of suggestively. So what’s the information set here? The information set is indicating the fact that Player 2 cannot observe whether Player 1 moved up or down. Player 2 cannot observe whether Player 1 chose up or down.
Now, why does that matter? I haven’t put the payoffs in yet but I will in a second. It matters because had this game been a game of perfect information, had this information set–had there been two information sets here, this dotted line not been there –, then Player 2 could have chosen separately whether to choose left or right at this node, or left and left or right at this node. But since Player 2 doesn’t know whether she’s at the upper node or the lower node–she doesn’t know whether Player 1 chose up or down–she really only has one choice to make here. She’s either choosing left at both nodes or she’s choosing right at both nodes. And just to pull it back to our first example in the class, we saw the same feature there.
When we moved from a game of perfect information to a game of imperfect information we reduced the choices available for Player 2. Here Player 2 could choose separately up or down, at these two different nodes. But here Player 2 only makes one choice that has to apply to both nodes because Player 2 cannot distinguish those two nodes. So let’s have a look and see, once we put some payoffs on, what it does in this particular game. So here’s some payoffs: (2, 2), (-1, 3), (3, -1) and (0, 0). So once again Player 2 cannot separately choose at the upper node or the lower node, she’s either choosing left or she’s choosing right.
But it turns out that this game is a little easier than the game we started with. Why is it easier than the game we started with? It’s easier than the game that we started with because, from Player 2’s point of view, whether she thinks she’s up here or whether she thinks she’s down here, she has the same best choice in either case. If she thinks she’s at the upper node then by choosing left she’ll get 2 and right she’ll get 3, so right seems better. If she thinks she’s at the lower node then choosing left gets -1 and right gets 0, so once again right is better.
So in fact, in this particular game, regardless of whether Player 2 thinks that Player 1 chose up, and hence she’s at the upper node, or whether Player 2 thinks that Player 1 chose down, and hence she’s at the lower node, Player 2 is going to make the same choice in this game, namely right. So notice that this particular game actually solves out rather like backward induction. Even though Player 2’s choice is a little bit more complicated–she doesn’t know where she is–it’s actually clear what Player 2 will do at this information set.
Now if we push this forward a little bit harder, we can see why. Player 1 in this game has two strategies up or down. And Player 2 has two strategies, she either chooses left or right. Notice that she only has two strategies because she has to choose the same thing at these two nodes, she doesn’t know where she is. Okay, so let’s draw up the matrix for this game and see if it looks familiar. So Player 1 is choosing between up or down. And Player 2 is choosing between left or right. And the payoffs are as follows: (up, left) is (2,2); (up, right) is (-1, 3); (down, left) is (3,-1); and (down, right) is (0,0). So what game is this? It wasn’t meant to be a trick question. Somebody waved their arm in the air. What game is this?
Student: Prisoners’ Dilemma.
Professor Ben Polak: This is Prisoners’ Dilemma. This is an old friend of ours. This is Prisoner’s Dilemma, a game we saw the very first day. But notice what have we seen here? This is Prisoner’s Dilemma that we’ve seen many, many times, that’s almost unbearably familiar to most of you. Now here’s Prisoner’s Dilemma as represented the way in which we talked about games before the mid-term. But here is the same game. This is also Prisoner’s Dilemma, but now I’ve drawn it in a tree. Here I drew it in a matrix, and here I drew it in a tree. Now that we have information sets we can represent all the games that we’ve studied before the mid-term. All the games that were simultaneous move games we can study using trees by building information sets. What’s the key observation here?
It doesn’t really matter whether Player 1 moves first or Player 2 moves first. It doesn’t really matter what’s happening temporally in this game. What matters is information. When Player 1 makes her move she doesn’t know what Player 2’s going to do. She doesn’t know what 2 is doing. And when Player 2 makes her move she doesn’t know what 1 is doing: and that’s a simultaneous move game, even if time is passing. The key is information, not time. Now, on the way here I snuck something in and I should just tell you what I snuck in. I snuck in what a strategy is.
I went from a tree, or an extensive form game, to a normal form game, and we’ve already done that a couple of times before in the class. We did it with the entry game, for example, about a week ago. But there all we did was we defined what a strategy was in a game of perfect information. And just to remind you, a strategy in a game of perfect information is a complete plan of action. It tells the player in question what they should do at each of their nodes. But now we have to be a bit more careful. We can’t have a strategy–once we move to imperfect information–we can’t have a strategy tell you what to do at each of your nodes, because you yourself can’t distinguish between those nodes. So we need to adapt our definition of a strategy to make it appropriate for these more complicated games.
So let’s just adapt it in the obvious way. Definition, I’ll just define pure strategies for now. A pure strategy of Player i is a complete plan of action–so this is the same as before. But what does it mean to be a complete plan of action? It can’t tell me what to do at every single node. That can’t be the right definition because I can’t distinguish nodes. So all that it can be doing is telling me what to do at each information set. So it specifies what Player i should do–should perhaps is the wrong word, let’s just say will do at each of i’s information sets.
So if you go back about a week you’ll see almost exactly the same definition of a strategy, but the previous definition told i what to do at each node and this one just tells i what to do at each information set. What I’m doing is tidying up the previous definition so we can apply it to the more interesting games we’re going to look at from now on. So now we have the definition of a strategy, we can carry on the idea we’ve just seen here. So what’s the idea here? Any game you give me in the form of a tree, I can rewrite the game in the form of a matrix.
So let’s see some other examples of that idea. A lot of new ideas today, but some of them are just tidying up and kind of bookkeeping, and some of them are more interesting. So let’s start with a tree. Let’s make it a slightly more interesting tree than the one we’ve seen before. Actually that’s too interesting, let’s go a little bit slower. Let’s have Player 1 have two choices, and Player 2 have three choices. So here’s a simple tree, and let’s put some payoffs in, but let me just put some letters in for payoffs rather than put in numbers. So we’ll call these actions up and down. And we’ll call this action left, middle, and right; and left, middle, and right. And we’ll call the payoffs (A1,A2), (B1,B2), (C1,C2), (D1,D2), (E1,E2), and (F1,F2), so just to keep track of it.
I want to show you how we take this tree and turn it into a matrix. So how do we turn it into a matrix? Well we look and say how many strategies has Player 1 got and how many strategies has Player 2 got? So Player 1 here just has two strategies, up or down. And Player 2 has three strategies, either left, middle or right. Again they can’t choose separately at these two nodes, so they really just have three choices: left, middle, or right. Leave a space here in your notebook, leave a space to the right here, and let’s draw the matrix for this tree down here. So here’s my matrix. Player 2 is choosing left, middle, or right and Player 1 is choosing up or down. The payoffs go in the obvious way. So (A1,A2), (B1,B2), (C1,C2), (D1,D2), (E1,E2), and (F1,F2).
So everyone understand that was just a simple exercise to show we can go from an extensive form, a tree, to the normal form, a matrix? That was easy right. However, there’s an interesting thing here. It isn’t obvious that, if I just gave you the matrix, it isn’t obvious that this is the tree from which it came. Let me draw another tree that I claim corresponds to that same matrix. Here’s another tree. So this other tree instead of having Player 1 move first, it’s going to have Player 2 move first. Player 2 better have three choices and we better call them left, middle, and right. And it better be the case that Player 1 is in one big information set and Player 1 only has two choices, which we’ll call up and down because that’s what this matrix is telling us. It’s telling us Player 2 had three choices and Player 1 had two choices.
So that’s true in the matrix I’ve drawn. And let’s be a little bit careful where the payoffs are. So (left, up), that’s easy: that’s going to be (A1,A2). (Left, down) is going to be (D1,D2). (Middle, up) is going to be (B1,B2). (Middle, down) is going to be (E1,E2). (Right, up) is going to be (C1,C2) and (right,down) is going to be (F1,F2). So I have to be a little bit careful where I put in the payoffs, but I think that’s right what I just did. Notice that what I did here: I started from this tree. It was an easy operation to construct the matrix, so easy that it was kind of boring, and it’s not that hard to see that I can go the other way and construct this other tree from the matrix. This is also a tree in which Player 2 has three strategies and Player 1, this is all Player 1, has two strategies.
So what, what have we learned from this? Well let’s look at this more carefully. This tree is a tree in which Player 1 moved first and Player 2 didn’t observe Player 1’s choice. Is that right? This is a tree in which Player 2 moved first and Player 1 didn’t observe 2’s choice. What are we noticing here? They’re really the same game. There’s no difference between these two games. They’re really the same game. It doesn’t matter whether it’s Player 1 moving first and Player 2 who’s unable to observe 1’s choice, or whether it’s Player 2 who is moving first and Player 1 who is unable to observe 2’s choice. All that matters is that neither player could observe the other person’s choice before they got to move, they both correspond to exactly the same game.
So what’s the message here? The message is something we’ve talked about before in the class, but I’m trying to be a bit more formal about it. The message is that what matters is information, not time. Clearly, time isn’t an irrelevant thing. I couldn’t know something that hasn’t happened yet, so time is going to have effects in information, but ultimately what matters is information. What do I know and when did I know it? So the key idea that we’re trying to capture with these information sets, just to repeat, is what did the player know and when did they know it, that famous expression from the Watergate Trials.
Chapter 3. Games of Imperfect Information: Finding Nash Equilibria [00:35:11]
Okay, let’s look at a more interesting example, and see if we can actually talk about what’s going to happen in these games. So by the end of today I want to have enough machinery so we can actually start analyzing these games and predicting what’s going to happen. So as we go on we’ll get more complicated, so let’s get a little bit more complicated now. Once again, here’s a game in which Player 1 is going to have two choices and we’ll call those choices Up or Down. It’s getting to be a familiar theme. Once again, Player 2 is going to move next and now, this time, just to keep things simple, we’ll have Player 2 just have two choices left or right, or left or right.
But now I want to make things more interesting, let’s have Player 1 move again. So if Up, right happens then Player 1 gets to move again, in which case Player 1 is going to choose up or down. I’ll use a [little] u and a [little] d to distinguish it from the ones further to the left of the tree. So this is a very simple tree, Player 1 moves first. Player 2 moves second–I forgot to put a 2 in here. And then if (Up, right) has occurred then Player 1 gets to move again. Let’s put some payoffs in. So let’s have this be (4,2), (0,0), (1,4), (0,0) again and (2,4). Let’s just carry on analyzing this game using exactly the methods we’ve been talking about in the class today so far.
So the first thing I’m going to do is I want to turn this into a matrix. The first thing to do on that route is to try and figure out how many strategies does Player 1 have, and how many strategies does Player 2 have. Before we even do that let’s try and figure out how many information sets they have. So I claim that Player 2 just has the one information set, is that right? But Player 1 has two information sets. This information set at the beginning of the game and then potentially this second information further down the tree.
A strategy must tell you–a strategy must tell the player what to do at each of their information sets. So the strategies for Player 1 are what? Well one strategy is Up and then up again, another strategy is Up and then right, another strategy is Down and then up, and a fourth strategy is Down and then right. Notice something which we’ve seen already in this class before: there’s a little bit of a redundancy here. These two Down strategies force the game into a part of the tree where this node will not arise. To put it less grandly, if Player 1 chooses Down, she knows that she won’t have to make a choice of up or down later on.
Yes. Sorry–thank you. Let me start again since that was the wrong notation. So Player 1’s choices are Up and then up; Up and then down; Down and then up; and Down and then down. Thanks Jake, sorry.
Now, why are there four strategies? It’s a bit of a surprise perhaps because if Player 1 chooses Down then she knows she will never have to make a choice at her second information set. Nevertheless, when we write down a strategy we have to write down an instruction for every single information set. So we include both of those strategies. Strategies for Player 2 here are a little bit easier. Strategies for Player 2 are just left or right. With that in mind, let’s draw up the matrix. So Player 1 here has four strategies and they are (Up, up), (Up, down), (Down, up), and (Down, down). Player 2 has two strategies, and they are left or right. Everyone okay so far?
We’re just basically transferring things across, and then we have to transfer the payoffs across so (Up, up) followed by left is going to be (4,2). (Up, up) followed by right is going to be (0,0). So Up, right, up is (0,0). ((Up, down), left) is the same as Up, left, down so it’s going to be (4, 2). ((Up, down), right) is going to be Up, right, down so it’s going to be (1, 4). ((Down, up), left) is the same as saying Down, left so it’s going to be (0, 0). ((Down, up), right) is going to be (2, 4). ((Down, down), left) is once again going to be (0, 0) and ((Down, down), right) is once again going to be (2, 4). Does everyone see how I got the payoffs? I just used those strategies to tell me which way I’m going through the tree. If I combine them, it gives me an entire path and gets me to an end node.
You can see this redundancy we talked about. We pointed out that these things are kind of the same thing, and you can see in the matrix that the bottom four squares of the matrix have repetition. This row is the same as that row. Everyone happy with that? Okay, we have a matrix. Let’s analyze it by finding the Nash equilibria in this game. So to find the Nash equilibria in this game, we’re going to find best responses. So let’s just start by asking what is the best response to left? So if Player 2 chooses left, Player 1’s best response is either (Up, up) or (Up, down). If Player 2 chooses right then Player 1’s best response is either (Down, up) or (Down, down). Everyone okay so far?
If Player 1 chooses (Up, up) then Player 2 is going to choose left. If Player 1 chooses (Up, down) then Player 2’s best response is to choose right. If Player 1 chooses (Down, up) then Player 2’s best response is to choose right. And if Player 1 chooses (Down, down) then Player 2’s best response is to choose right. So this is kind of slow and I just want to be careful. I’m going slow for a reason. We’re going to gradually get harder, and I want to be a little bit careful. I can see people are looking a little sleepy around the room. I know it’s kind of lunchtime. If you see your neighbor getting sleepy give them a good sharp elbow because I think this isn’t a good time to fall asleep. Sometimes I’m worried you’re going to miss something, and it’s only going to get harder and you’re going to miss things.
Alright, so what are the Nash equilibria in this game? We know how to do that. The Nash equilibria must be (Up, up) followed by left. Make sure I get them all. (Down, up) followed by right; and (Down, down) followed by right. I want these three Nash equilibria. Okay, so that wasn’t such a big deal. I’ve got three equilibria in this game. And if I’d simply given you this game in the first half of the semester, I hadn’t shown you the tree–you’ve never seen this tree–I just gave you this game and said find the Nash equilibria in this game, and that had been a question on the mid-term, we’d have stopped here. We’d have said: okay, I found these Nash equilibria. Maybe you’d have gone on and found mixed ones I don’t know but essentially we’d be done at this point.
Let’s say again. If we’d started as we would have done before the mid-term with me giving you a payoff matrix and asking you to find the Nash equilibria, then at this point we’d be done. We’d have found the three Nash equilibria, at least the three pure-strategy Nash equilibria. The problem is if we go back to the tree, to the dynamic game–the game that has some action going on in it–and actually look at this game, it’s not clear that all of these Nash equilibria are really equally plausible. Can anyone see what might be a bit implausible about some of these Nash equilibria? What’s implausible about them? Any taker on this?
Well, let’s look at this game again. This game is a little bit complicated. It’s not clear what one should do here perhaps, and perhaps it’s not clear what Player 2 should do here, because after all, Player 2 doesn’t know where he is and he doesn’t know whether Player 1, if Player 1 gets to move again is going to choose up or down but: what’s the but? Can we get a mike on Patrick?
Student: So if you look at it backwards, you can cross out Player 1’s second choice. He’s always going to choose down so that’s (1, 4) at that node. So then you know Player 2 is always going to choose right because his payoff is always 4. So then Player 1’s not going to have–I mean Player 1 knows which to choose then. He’s going to choose down.
Professor Ben Polak: Good. So let’s just walk through what Patrick just said. That’s very good. So if we just analyze this game the way we’ve been taught to analyze trees, essentially using backward induction, we first of all observe that if Player 1 gets to move again here she’ll know where she is, and she’ll know she’s choosing between 1 and 0. She’s going to choose down. Is that right? She’s going to choose down. But knowing this, Player 2, even though Player 2 doesn’t know where he is, Player 2 actually has a pretty easy choice to make. He knows that if he chooses left, he either gets 2 or 0, but if he chooses right he gets 4. 4 is bigger than 2. 4 is bigger than 0. So Player 2 is actually going to choose right. And given that, given that Player 2 is going to choose right, Player 1 is essentially choosing between 1, if she chooses up, which would be followed by right and down; and 2, which is what happened if she chooses down followed by right.
So this game we can essentially analyze through backward induction. It’s not quite backward induction because we had to add in this little piece about 2 not knowing where she was, but it turned out, no matter where she was, she had a dominant strategy, she had a better strategy once she figures out that Player 1 is going to choose down. Is that right? If we go back and look at these Nash equilibria, the prediction that we just got, which is what? Down for Player 1, right for Player 2 and then down again for Player 1. That strategy is this one. So one of these Nash equilibria corresponds to our sensible analysis of this tree, but the other two do not. These two Nash equilibria are inconsistent with backward induction.
They’re perfectly good Nash equilibria, if we’d given you this matrix at the mid-term you’d have thought that it’s just fine. But it turns out both of these Nash equilibria involve Player 1 choosing a strategy up, that we know that Player 1 is not going to do if reached. And one of these Nash equilibria involves Player 2 choosing a strategy left, that, in fact, she’s only choosing because she thinks Player 1 is going to choose up, which, in fact, we just argued Player 1 is not going to do. The people at the back, there’s a little bit too much volume bouncing off the wall there, so just keep it down in the balcony. Thank you.
So these two Nash equilibria, they’re perfectly good Nash equilibria to the game, but they don’t make any sense. They’re completely inconsistent with the way we’ve learned to talk about games. Now we’ve seen this before. We saw it in the entry game. This is a much more complicated, much more interesting example, but we saw in the entry game when there was one entrant entering into a market, that in that game there were actually two Nash equilibria and one of them we argued was incredible. Here it’s a bit more complicated, but nevertheless, these two equilibria seem like bogus equilibria or phony equilibria, or equilibria we wouldn’t really believe in. The reason we don’t believe in them is that they don’t correspond to backward induction and our common sense intuition is about backward induction.
So we need some new notion, the aim of the class has been what? We want to be able to model games that have both sequential moves and simultaneous moves, and we want to be able to look at the games and use our techniques from both halves of the class. We want to be able to use the idea of Nash equilibrium from the first half of the class, and we want to be able to use the idea of backward induction from the second half of the class. But what we’re learning here is that Nash equilibrium, if we just take the notion of Nash equilibrium and plunk it down on these sequential move games, it will produce equilibria that don’t make any sense. So we need a more refined notion of equilibrium, a better notion of equilibrium than just Nash equilibrium to deal with these settings where we have both simultaneity and sequential moves. We have both some perfect information and some imperfect information.
Chapter 4. Games of Imperfect Information: Sub-games [00:49:59]
That was one example. Let me give you a second example if that example wasn’t yet convincing. Let me leave that example up. So far we’ve seen that Nash equilibrium gets us into trouble in these games and we’ve seen it got us into trouble because it basically conflicted with our backward induction intuitions. Now I’m going to show you a different game and we’re going to see again that Nash equilibria is going to get us into trouble. This is a going to be a three-player game. We will get more complicated as we go along. So another example, this time with three players. So as the examples get harder I need you to be more alert to see if you can follow them.
So this is a more complicated tree. Here’s a tree in which Player 1 moves first and chooses between A or B and if Player 1 chooses A the game is over, she gets 1, and the other two players get nothing. If she chooses B then Players 2 and 3 get to play a little game down here in which 2 moves first in this little sub-game and 3 moves second, and the payoffs in this sub-game are as follows. Again, using Player 1’s payoff first. So there’s (0, 1, 1), (0, 0, 2), (0, 0, -1) and (2, 1, 0). So this is quite a complicated game, it’s got three players for a start, so it’s going to be a little bit hard to draw it up in a matrix, but nevertheless, let me try and do that.
So I claim that we can model this game as follows. It’s a game in which Player 1 is choosing which matrix, let’s call this Matrix A and Matrix B. Player 1 is choosing the matrix, Player 2 is choosing, let’s call them up and down: Player 2 is choosing up or down. And Player 3 is choosing left or right. Notice in this game Players 2 and 3 actually can observe the choice of A or B to start with. So let’s try and put in the payoffs in the correct places. It’s not always easy to do, but let’s try. So A is easy, if Player 1 chooses A, then the payoffs in this matrix are somewhat trivial. Because if Player 1 chooses A, whatever anyone else does the payoff is 1, 0, 0. Somewhat uninteresting matrix over there.
But if Player 1 chooses B then life gets more interesting. Then if Player 2 chooses up and Player 3 chooses left, we end up here so that’s (0,1,1). If Player 2 chooses–this is 2 and this is 3–if Player 2 chooses up and Player 3 chooses right then we’re at (0,0,2), so this is (0,0,2) going in here. If Player 2 chooses down and Player 3 chooses left then we’re at (0,0,-1). Everyone okay with that? If Player 3 chooses down then we’re down here which is (2, 1, 0). Okay so here’s a little game. Player 1 is choosing the matrix, Player 2 is choosing the row in the matrix, albeit trivially on the left hand side and Player 3 is choosing the column in the matrix, again, albeit trivially on the left hand side. We don’t really care about this picture very much.
Okay, so now what? Well, once again we could look for Nash equilibria in this game. It turns out there are lots of Nash equilibria in this game. Let me just show you one Nash equilibrium and then we’ll talk about it. So I claim that there are lots of Nash equilibria, and one of them is the Nash equilibrium (A, up, left). So let’s just see where that is in the tree first of all. So Player 1 chose A, Player 2 chose up and left, but it followed A so we end up here, we end up at (1, 0, 0). So (A, up, left) is this box in the tree. Now, let’s just check that that actually is a Nash equilibrium.
So we all know how to do this from the first half of the class. So to check that that’s a Nash equilibrium we better check that no individual player can do better by deviating. So let’s start with Player 1. If Player 1 deviates holding Player 2 and 3 fixed, then Player 1 will be switching the matrix from Matrix A to Matrix B. Is that correct? So we’ll move from this box in the left hand matrix to the equivalent box in the right hand matrix. Player 1’s payoff will go from 1 to 0, so Player 1 doesn’t want to deviate. Everyone happy with that? Player 1 doesn’t want to deviate here. How about Player 2? If Player 2 deviates, holding Players 1 and 3 fixed, then Player 1 is going to switch rows in this matrix, so we’ll move from this box to this box. Player 2 was making 0 before. She’s still making 0, so she has no incentive to deviate. And the same argument applies for Player 3 because she will be choosing the column holding the row in the matrix fixed, so once again she gets 0 in either case. So everyone happy with that?
So that actually is a Nash equilibria, and again, if this had been on the mid-term I could have set this up, I could have given you these matrices or the story behind them, and you’d have found–I could have asked you whether this was a Nash equilibrium and the answer would have been yes. But I claim that once again this is just not a believable Nash equilibrium. It is a Nash equilibrium, formally it’s a Nash equilibrium, but it’s not a plausible prediction for how this game is going to be played. Why is it not a plausible prediction for how this game’s going to played? Anyone see? Stare at the tree a bit.
So in the information here, the pre-mid-term information, it’s fine. But knowing about the actual structure of this game I claim this makes no sense at all. Why does it make no sense? Well, notice that if Player 1 were to switch her action from the prescribed action A to action B, then we’d be here. Notice that the tree from here on in looks like a little game, is that right? The tree from here on looks like a little game. So this thing here, let’s put it in green, is a little game within the game. It’s a sub-game. This sub-game really only involves two players. The two players that it involves are Player 2 and Player 3. Player 1 is done, Player 1 has put us into this game, but now this little sub-game, it’s a little sub-game involving just Player 2 and Player 3.
So we can analyze this little sub-game. If we analyze this little sub-game, what will it give us? What will we find? So let’s look at this sub-game. So look at the green sub-game, the game that would have happened had Player 1 chosen B. This is a sub-game involving just Players 2 and 3. So why don’t I just forget Player 1. We know–I mean Player 1 is part of the game, he’s getting payoffs, but Player 1 has made their move, they’re not really involved any more. So let’s just look at this game as a game involving Players 2 and 3, and let’s look at the matrix for Players 2 and 3. So it actually corresponds to the matrix above, it’s a matrix in which Player 2 is choosing up and down. There it is up and down and simultaneously Player 3 is choosing left or right. There it is left or right at this information set and the payoffs are (1, 1), (0, 2), (0, -1) and (1, 0).
So this is, I claim, a representation of this little green game. Perhaps we should put this in green as well. This thing corresponds to that thing. Everyone okay with that? So if Player 1 had chosen B rather than A, then we’d be involved in a little game, a game within a game, or a sub-game involving just Players 2 or 3. And we can analyze that game. That’s a straightforward game. Here it is. And what would we do with that game? We’d look for the Nash equilibria in that game. So let’s look for the Nash equilibrium of this game. So what do we notice about this game? So if Player 3 chooses left then Player 2 would rather choose up. If Player 3 chooses right then Player 2 should choose down. If Player 2 chooses up then Player 3 would rather choose right, because 2 is bigger than 1. And if Player 2 were to choose down than Player 3 would choose right again, because 0 is bigger than -1.
So in fact, in this little sub-game, actually, Player 3 has a dominant strategy. If it turned out that we got involved in this little sub-game, Player 3 has a dominant strategy which is to play right, and moreover, this sub-game has just one Nash equilibrium. If I’d given you this sub-game on its own, it’s clear that the Nash equilibrium of this sub-game–this game within a game–is (down, right). So what’s that telling us?
It’s telling us if Player 2 and 3 ever get called upon to play in this game–and that only happens when Player 1 chooses B–if Player 2 and 3 ever get called upon to play in this game, we know from when we were young or at least from before the mid-term, we know that they’re going to play Nash equilibrium in that sub-game. And the Nash equilibrium in the sub-game is going to have Player 3 choosing right and Player 2 choosing down. But the equilibrium we talked about, this equilibrium we argued before about, (A, U, L)–the equilibrium we talked about before, (A, U, L), doesn’t involve Player 2 choosing down. In fact she chose up. And it doesn’t involve Player 3 choosing right. In fact she chose left.
So let’s sum up. We found an equilibrium of this game, this equilibrium of this game was (A, U, L) but I claim this is not a plausible equilibrium. It’s not a plausible equilibrium because it predicts that if we actually were to play the game within the game, we wouldn’t play equilibrium. Let me say it again, in the whole game (A, U, L) is an equilibrium. But I claim it’s a silly equilibrium because it involves the prediction that, if in fact we ever got into the game within the game, we would no longer play equilibrium, and that doesn’t seem right. If we’re going to believe in equilibrium, we should be consistent and believe in equilibrium throughout.
So this brings us to a new idea and the new idea is going to have two parts to it. The first part is kind of on the board already. It’s something we talked about informally. It’s the notion of a sub-game. What’s a sub-game? It’s a game within a game. I’ve been using that informally but we need to start thinking about, more formally, what it means. So I talked about it informally. I said that green object is the game that would be played were Player 1 to choose B. We talked about other sub-games in this class. We talked about the sub-game that would happen in the entry game if one of those rival pizza companies moved in, in the Miami market or something, the game within a game. When we talked about the Tour de France we talked about there being a game within a game that is about when you break away.
But now I want to be formal about this notion of a game within a game and introduce some nomenclature. So the formal definition is this. A sub-game is a part of a game, informally that looks like a game within the tree, and it has three properties. It satisfies the following three properties. So one since it looks a game itself the sub-game must start from a particular point. So it starts–the sub-game must start–it starts from a single node. Let’s just look at the example. In the example we just looked at, the sub-game started from this node here. Second, it comprises all successors to that node. So in our example, here’s our sub-game. Here’s our green sub-game. Here’s the node it starts from. Here are all the nodes that are successors of that node: these are the children, and these are the grandchildren. If you have this grandparent node you have to have all of his children, and all of his grandchildren. So it comprises all the successors of that node. And finally–this is important–it does not break up any information sets. So a sub-game, informally, it’s just a game within the game. But slightly more formally, I can’t put one node that’s part of an information set into this sub-game unless I’m going to put all the nodes that are part of that information set into the sub-game.
Let’s have a look at some examples. We’ve got one example up there, the entry game looked something like this. So what are the sub-games here, no secrets here, this is a sub-game. There’s actually another sub-game, can anyone see what the other sub-game is? The whole game is a sub-game. The whole game is itself a sub-game, somewhat trivially. So this particular game, which is the schematic of the entry game, it has actually two sub-games but only one proper sub-game.
Here’s a more complicated example. This is actually going to be quite a complicated example just to make life interesting. So 1 is going to move, then 2 is going to move, and then 1 is going to move again. This is all one big information set for Player 1. And 1 is going to move like this. So again, without payoffs, this is a little tree and the key point here is this is an information set. Let’s look– let’s stare–at this tree a second and figure out what are and aren’t sub-games. So first of all, this was a sub-game and this was a sub-game. What about this thing here? Is that a sub-game? It’s not a sub-game. What rule does it break? It’s breaking up an information set, so that’s no good because of rule three. What about this thing? That doesn’t break up an information set. I’ve got the whole information set in there. Is that any good? No that’s no good because it doesn’t start from a singleton node, so that’s no good it violates 1. If we do this, we look at this piece. That piece there, that’s also no good. Why is that no good? Again it breaks up an information set, so this is no good again because of rule 3. So you can practice at home drawing trees and trying to identify what are and what are not sub-games.
Chapter 5. Games of Imperfect Information: Sub-game Perfect Equilibria [01:10:17]
So with the definition of a sub-game now formal–it’s basically just formalizing something we’ve talked about before which is the idea of a game within the game–I want to introduce our new–what’s going to be our new solution concept. And this is going to be the solution concept we’re going to use essentially almost until the final. Definition, so just remember what our task is. Our task is to come up with a solution concept that picks up the idea from the first half of the semester, namely Nash equilibrium, but does so in a way that respects what we’ve learned in the second half of the semester, namely that games have sequential elements and people move by backward induction.
So, in particular, what we want to rule out are those Nash equilibria that instruct players down the tree to play in sub-games according to strategies that are not Nash equilibria. Say it again, we want to rule out those Nash equilibria that instruct people way down the tree to play according to something which is not a Nash equilibrium. We want our new notion to say: wherever you find yourself in a tree, play Nash equilibrium and that’s exactly what the definition’s going to say. So a Nash equilibrium, S1*, S2* all the way up to SM* is a sub-game perfect equilibrium–so that’s a SPE–it’s a sub-game perfect equilibrium if it induces a Nash equilibrium in every sub-game of the game.
So to be a sub-game perfect equilibrium, it has to itself be a Nash equilibrium of course, but it also has to instruct players to play a Nash equilibrium in every sub-game. Let’s take that immediately back to our examples. In this example, we know that this is a sub-game. We know that, in this sub-game, there is only one Nash equilibrium; and that Nash equilibrium involves Player 2 choosing down and Player 3 choosing right. So we know that Player 2 is going to choose down according to that equilibrium, and Player 3 is going to choose right according to that equilibrium. So if we now have to look for an equilibrium of the whole game–let’s go back to Player 1’s choice–Player 1 if they choose A will get 1, if they chose B then they know that this Nash equilibrium will be played so they’ll get 2. They prefer 2 to 1, so the sub-game perfect equilibrium here is Player 1 chooses B, Player 2 chooses down, and Player 3 chooses right.
This is an equilibrium of the game and it induces an equilibrium in the sub-game. So in that example, the sub-game perfect equilibrium is found by first of all looking in the sub-game; find the equilibrium in the sub-game; and then go back and look at the equilibrium in the whole game. The equilibrium that we end up with, it is a Nash equilibrium in the whole game, but more importantly, it induces a Nash equilibrium in the sub-game.
Let’s just go back to our other example then I’ll stop. So our other example was here. Here was our other example. And we claimed, hang on everybody, we claimed that the good equilibrium here, the one we believed in was (Down, down, right). Where are the sub-games in this game? Where are the sub-games in this tree? Anybody?
So I claim there’s only one real sub-game here and that’s this piece. This is a sub-game. What’s the Nash equilibrium of this somewhat trivial sub-game? The Nash equilibrium of this somewhat trivial sub-game is that Player 1 must choose down. So for a Nash equilibrium to be a sub-game perfect equilibrium–here are three Nash equilibria: one, two, three. For this Nash equilibria to be a sub-game perfect equilibrium, it’s got to instruct Player 1 to choose down in the trivial sub-game and here it is. This is our sub-game perfect equilibrium in this game. Now I know today was a lot of formal stuff, a lot of new ideas, and when we come back on Monday we’ll first of all give you a game that refreshes these ideas and then we’ll go straight to applications. So trust me: there will be applications. It will be useful. See you on Monday. There’s a homework to come in, there’s another on the web going out.
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