Game Theory

ECON 159 - Lecture 9 - Mixed Strategies in Theory and Tennis

Chapter 1. Mixed Strategies: Definition [00:00:00]

Professor Ben Polak: So last time we saw this, we saw an example of a mixed strategy which was to play 1/3, 1/3, 1/3 in our rock, paper, scissors game. Today, we're going to be formal, we're going to define mixed strategies and we're going to talk about them, and it's going to take a while. So let's start with a formal definition: a mixed strategy (and I'll develop notation as I'm going along, so let me call it Pi, i being the person who's playing it) Pi is a randomization over i's pure strategies. So in particular, we're going to use the notation Pi (si) to be the probability that Player i plays si given that he's mixing using Pi. So Pi(si) is the probability that Pi assigns to the pure strategy si.

Let's immediately refer that back to our example. So for example, if I'm playing 1/3, 1/3, 1/3 in rock, paper, scissors then Pi is 1/3, 1/3, 1/3 and Pi of rock--so Pi(R)--is a 1/3. So without belaboring it, that's all I'm doing here, is developing some notation. Let's immediately encounter two things you might have questions about. So the first is, that in principle Pi(si) could be zero. Just because I'm playing a mixed strategy, it doesn't mean I have to involve all of my strategies. I could be playing a mixed strategy on two of my strategies and leave the other one with zero probability. So, for example, again in rock, paper, scissors, we could think of the strategy 1/2, 1/2, 0. In this strategy I assign--I play rock half the time, I play paper half the time, but I never play scissors.

So everyone understand that? And while we're here let's look at the other extreme. The probability assigned by my mixed strategy to a particular si could be one. It could be that I assign all of the probability to a particular strategy. What would we call a mixed strategy that assigns probability 1 to one of the pure strategies? What's a good name for that? That's a "pure strategy." So notice that we can think of pure strategies as the special case of a mixed strategy that assigns all the weight to a particular pure strategy. So, for example, if Pi(R) was 1, that's equivalent to saying that I'm playing the pure strategy rock, i.e. a pure strategy.

So there's nothing here. I'm just being a little bit nerdy about developing notation and making sure that everything is in place, and just to point out again, one consequence of this is we've now got our pure strategies embedded in our mixed strategies. When I've got a mixed strategy I really am including in those all of the pure strategies. So let's proceed. I'm going to push that up a little high, sorry. So now I want to think about what are the payoffs that I get from mixed strategies, and again, I'm going to go a little slowly because it's a little tricky at first and we'll get used to this, don't panic, we'll get used to this as we go on and as you see them in homework assignments and in class.

So let's talk about the payoffs from a mixed strategy. In particular, what we're going to worry about are expected payoffs. So the expected payoff of the mixed strategy P, let's be consistent and call it Pi, the mixed strategy Pi is what? It's the weighted average--it's a weighted average or a weighted mixture if you like--of the expected payoffs of each of the pure strategies in the mix. So this is a long way of saying something again which I think is a little bit obvious, but let me just say it again. The way in which we figure out the expected payoff of a mixed strategy is, we take the appropriately weighted average of the expected payoffs I would get from the pure strategies over which I'm mixing.

Chapter 2. Mixed Strategies: Examples [00:06:02]

So to make that less abstract let's immediately look at an example. So here's an example we'll come back to several times, but just once today, and this a game you've seen before. Here is the game Battle of the Sexes, in which Player A can choose--Player I can choose A and B, and Player II can choose a and b, and what I want to do is I want to figure out the payoff from particular strategies. So suppose that P is being played by Player I and P is let's say (1/5,4/5). So what do I mean by that? I mean that Player I is assigning 1/5 to playing A and 4/5 to playing B. And suppose that Q--so I am going to use P and Q because it's convenient to do so rather than calling them P1 and P2. So suppose that Q is the mixture that Player II is choosing and she's choosing a (½,½), so she's putting a probability 1/2 on a and a probability 1/2 on b. Just to notice I switched notation on you a little bit, for this example to keep life easy, I'm going to use P to be row's mixtures and Q to be column's mixtures.

And the question I want to answer is what is the expected payoff in this case of P? What is P's expected payoff? The way I'm going to do that is, I'm first of all going to ask what is the expected payoff of each of the pure strategies that P involves, the pure strategies involved in P. So to start off--so the first step is ask what is the expected payoff for Player I of playing A against Q and what is the expected payoff for Player I of playing B against Q? That will be our first question and we'll come back and construct the payoff for P. So these are things we can do I think.

So the expected payoff of A against Q is what? Well, half the time if you play A you're going to find your opponent is playing a, in which case you'll get 2, and half the time when you play A you'll find your opponent is playing b in which case you'll get 0. So let's just write that up. So I'm going to get 2 with probability 1/2 plus 0 with probability 1/2. Everyone happy with that? That gives me 1. Please correct my math in this. It's very easy at the board to make mistakes, but I think that one is right.

Conversely, what if I played B? What's the expected payoff for the row player of playing B against Q, where Q is 1/2, 1/2? So half the time when I play B, I'll meet a Player II playing a and I'll get 0 and half the time I'll find Player II is playing b and I'll get 1. So let's write that up. So I'll get 0 half the time and I'll get 1 half the time for an average of 1/2. That's the first thing I ask. And now to finish the job, I now want to figure out what is the expected payoff for Player I of using P against Q? That was the question I really wanted to start off with. What's the way to think about this? Well P is 1/5 of the time--according to P, 1/5 of the time Player I is playing A and 4/5 of the time Player I is playing B, is that right? So to work out the expected payoff what we're going to do is we're going to take 1/5 of the time, and at which case he's playing A and he'll get the expected payoff he would have got from playing A against Q, and 4/5 of the time he's going to be playing B in which case he'll get the expected payoff from playing B against Q.

Now just plugging in some numbers to that from above, so we've got 1/5 of the time he's doing the expected payoff from A against Q and that's this number we worked out already. So this number here can come down here, 1. And 4/5 of the time he's playing B against Q, in which case his expected payoff was 1/2, so this 1/2 comes in here. Everyone okay so far, how I constructed it so far? Is this podium in the way of you guys, are you okay? Let me push it slightly. So the total here is what? It's going to be 1/5 of 1 plus 4/5 of ½. 4/5 of 1/2 is 2/5, so I've got a total of 3/5. So the total here is 3/5. Everyone understand how I did that? Now while it's here let's notice something. When I played P, some of the time I played A and some of the time I played B. And when I ended up playing A, I got A's expected payoff. And when I played B, I got B's expected payoff. So the number I ended up with 3/5 must lie between the payoff I would have got from A which is 1, and the payoff I would have got from B which is 1/2.

Is that right? So 3/5 lies between 1/2 and 1. Everyone okay with that? Now that's a simple but very general and very useful idea it turns out. The idea here is that the payoff I'm going to get must lie between the expected payoffs I would have got from the pure strategies. Let me say it again. In general, when I play a mixed strategy the expected payoff I get, is a weighted average of the expected payoffs of each of the pure strategies in the mix, and weighted averages always lie inside the payoffs that are involved in the mix. So let me try and push that simple idea a little harder. Suppose I was going to take the average height in the class--average height in this class. So let me just, rather than use the class, let me just use some T.A.'s here.

So let me get these three T.A.'s to stand up a second. Suppose I want to figure out the average height of these three T.A.'s. So stand up close together so I can at least see what's going on here. So I think, from where I'm standing, I've got that Ale is the tallest and Myrto is the smallest, is that right? So I don't know instantaneously what this average would be, but I claim that any weighted average of their three heights, is going to give me a number that's somewhere between the smallest height of the three, which is Myrto's height, and the tallest height of the three, which is Ale's height, is that right? Is that correct? So that's a pretty general idea. Thanks guys I'll come back to you in a second.

Let's think about this somewhere else, let's think about the batting average of a team. The team batting average in baseball, let's use the Yankees, for example. We know that the team batting average, the average batting average of the Yankee's--I don't know what it is, I didn't look it up this morning--but I know it lies somewhere between the player who has the highest batting average which I'm guessing is Jeter, I'm guessing, and the lowest, the person on the team who has the lowest batting average, who is probably one of the pitchers who played, who batted a few times in one of those inter-league games. (It would have been better if I'd used the Mets but I feel I should take pity on Mets fans this week and not mention them.)

So this is a very simple idea, it's deceptively simple. It says averages, weighted averages, lie between the highest thing over which you're averaging and the lowest thing over which you're averaging. Everyone okay with that idea? Now this very simple idea is going to have an enormous consequence, and here's the enormous consequence. Simple idea, big consequence. So there's going to be a lesson that follows from this incredibly simple idea and this is the lesson. If a mixed strategy is a best response, so if a mixed strategy is the best thing you can be doing, then each of the pure strategies in the mix--I'm being a little bit loose here but I mean assigned positive probability in the mix, for those people who are nerdy enough to worry about it--each of the pure strategies in the mix must themselves be best responses.

So, in particular, each must yield the same expected payoff. So here's a big conclusion that follows from that incredibly simple idea about averages lying between the highest one and the lowest one. Let's draw ourselves from that lesson to this big conclusion. What is the conclusion? The conclusion is if a mixed strategy is a best response, if the best thing I can do is to play a mixed strategy, then each of the pure strategies which I'm playing in that mix, which I'm assigning positive probability to in that mix, must themselves be best responses. In particular, each of them therefore must yield the same expected payoff.

So let's go back to our example. Can I steal my three T.A.'s again? Suppose the game, suppose the thing I'm involved in--I should have made this easier before, let me come down a little bit. I'll stand above here, this is good. So suppose the game I'm involved in, the payoff in the game is, a game in which I have to choose the tallest group of my T.A.'s. So my payoff is going to be the average height of whichever subgroup of my T.A.'s I pick and these are my three choices. So if I pick more than one of them I'm going to get a weighted average, that's a mixed strategy. My aim here is to maximize the height of whatever subgroup I pick.

So in this game, here's my three pure strategies: my three pure strategies are to pick Myrto; Ale; or Jake. Those are my three pure strategies. And my mixture, I could mix these two, I could mix these two, I could mix all three. But remember my payoff here is to get the group, the average as high as I can. So how am I going to get the average as high as I can? I get the average as high I as I can, I'm going to kick out Myrto for a start because Myrto's just bringing down the average, is that right? Average height I should say, there's nothing--and actually I think I'm going to kick out Jake as well I think, I'm probably going to kick out Jake as well because that way I just have Ale.

So if it was the case that I was picking both of them, it would have to be they were equally tall but since they're not equally tall, I should just pick the best one. Let's go back to my Yankee's example, if I want to pick a sub-team of the Yankee's, I'm allowed to pick any number of people, to have the highest average, batting average, in that sub-team. The way to do it is to find the Yankee who has the highest batting average and just pick him. Let's do one more example. Let me use the front row of students here, so here's my, can I get this front of students to stand up a second? This is a part of the row.

And suppose my aim in life is to construct the highest average GPA. I'm not going to embarrass these guys and ask them what their GPA's are. So my aim in life here is to pick some sub-group of these one, two, three, four, five, six, seven, eight students, such that the average GPA of that sub-group is as high as I can make it. So what will I do here? So this being Yale I'll just find the people who have the 4.0 GPA's and just pick them. Is that right? You might think well why not include somebody who has a 3.9 GPA? That's pretty good. So why not? Because if there's anybody in this group who has a 4.0 GPA, I'd do better just to pick that person. The 3.9 person would just be pulling down the average.

Now suppose there's nobody with a 4.0 GPA and suppose it's the case that three of these people, let's say these three people have a 3.9 GPA. So these three have 3.9 GPA, imagine that, and these other people they've got horrible grades like B+ somewhere. These are our future law school students and these are the people--who knows what they're going to end up doing--being President probably. So to construct the group with the highest average GPA, what am I going to do? Well first I'll throw out all these guys with low GPA's, so they can all sit down and I'll look at these last three and these last three, if they're all in the group they better all have the same GPA. Why on earth?

If I'm trying to maximize the average of my group, if any of them had a lower GPA I should kick them out, and if one of them has a higher GPA than the other two, I should kick out both the other two. So if I'm including all three of them, in my constructing of the average all of them must have the same GPA, which I'm going to assume is 3.9, to assume you can still make into law school. Everyone understand that? Yeah? Okay, thanks guys.

So that's the way I want to think about this. So the idea here is if I'm using a mixed strategy as a best response, it must be the case that everything on which I'm mixing is itself best. And the reason is, if it wasn't, kick out the thing that isn't best and my average will go up.

Chapter 3. Mixed Strategies: Direct and Indirect Effects on the Nash Equilibrium [00:22:20]

So that leads us to the next idea, but before I do just for formality, let me add a definition. The definition is this, a mixed strategy profile--what I'm going to do now is I'm going to define Nash Equilibrium again, just so we have it in our notes somewhere. So a mixed strategy profile--there should be a hyphen there--(P1*, P2*, …all the way up to PN*), is a mixed strategy Nash Equilibrium if for each Player i--so for each Player i--that player's mixed strategy Pi* is a best response for Player i to the strategies everyone else is picking P -i*. So I'm exploiting, by now, a well developed notation for player strategies. So this definition of Nash Equilibrium, it's exactly the same as the definition of Nash Equilibrium we've been using now for several weeks, except everywhere where before we saw a pure strategy, which was an S, I have replaced it with a P. So the same definition except I'm using mixed strategies instead of pure strategies.

But an implication of our lesson is what? It's that if Pi* is part of a Nash Equilibrium--so if Pi* is a best response to what everyone else is doing, P-i* --, then each of the pure strategies involved in Pi* must itself be a best response. So an implication of the lesson is, the lesson implies the following. If Pi* of a particular strategy is positive, so in other words, I'm using this strategy in my mix, then that strategy is also a best response to what everyone else is doing. Okay, so from a math point of view this is the big idea of the day, this board. If you're having trouble reading this at the back, trust me I've written that up on the handout that will appear magically on the computer, at the end of class.

At the moment you're staring at this, it's all a bit new, and as well as being new, you're saying, okay but so what, why do I care about this seemingly mundane fact? The reason we're going to turn out to care about this seemingly mundane fact, is that this fact is going to make it remarkably easy to find Nash Equilibria. This fact, this lesson, this idea that if I'm playing a pure strategy as part of the mix, it must itself be a best response, that's going to be the trick we're going to use in finding mixed strategy Nash Equilibria. The only way I can illustrate that to you is to do it, so I'm going to spend the rest of today just doing that.

I'm going to look at a game and we're going to go through this game. We'll discuss it a little bit because it's a fun game, and we're going to find the mixed-strategy equilibria of this game. Everyone know where we're going? I want to make sure before I go on, are people looking very sort of deer in the headlamps? That was a lot of formality to get through in a short period of time. Does anyone want to ask a question at this point? Are you okay? Okay to go on? So just remember that the conclusion here comes from this very simple idea. The simple idea is, the payoff to a weighted average must lie between the best and worst thing involved in the average, and therefore if I'm including things in there as part of a best response, they must all be good. That's the simple idea, this is the dramatic conclusion.

Chapter 4. Mixed Strategies and the Nash Equilibrium: Example [00:27:05]

So the only way to prove this to you and the only way to prove to you that this is useful is to go ahead and do it. So what I'm going to do is I'm going to clean these boards and I'm going to start showing an example. Again don't panic, I think a lot of people at this part of the class have a tendency to panic, because it's a new idea, it seems like a lot of math around. None of it's very hard math, it's all kind of arithmetic. It's just this idea of not panicking. So the example I want to look at is going to be from tennis, and I'm going to consider a game within a game, played by two tennis players, and let's call them Venus and Serena Williams.

So a couple of years ago we used to use Venus and Serena Williams for this example, and then for a while I worried, that you wouldn't even remember who Venus and Serena Williams were, and so we picked any two random Russians, but now we're back. Seems like we're back to picking Venus and Serena. So the game within the game is this, suppose that they're playing and Serena is at the net and the ball is on Venus' court, and Venus has reached the ball and Venus has to decide whether to try to hit a passing shot past Serena on Serena's left or on Serena's right. Notice I'm going to exclude the possibility of throwing up a lob for now, just to make this manageable. So basically the choice facing Venus is should she try to pass Serena to Serena's left, which is Serena's backhand side or to Serena's right, which is Serena's forehand side.

People are familiar enough with tennis to understand what I'm talking about? So we're going to assume this is Wimbledon, otherwise no one would be at the net to start with I guess. So this is at Wimbledon. Let's try and put up some payoffs here. So these are going to be the payoffs. I think that this example is originally due to Dixit, but it's not a big deal. I think this example is due to Dixit and Skeath. So here's some numbers and I'll explain the numbers in a minute. So this is 50, 50, 80, 20, 90, 10 and 20, 80. So what are these numbers? So first of all let me just explain what the strategies are, so I'm assuming the row player is Venus and the column player is Serena.

I'm assuming that if Venus chooses L that means she attempts to pass Serena to Serena's left, we'll orient things from Serena's point of view, and if she hits right that means she's attempting to pass Serena on Serena's right. If Serena chooses L that means she cheats slightly towards her left: not cheats in the sense of breaking the rules, but cheats in terms of where she's standing or leaning. And if she chooses right that means she cheats slightly towards her right. So this is cheating towards her backhand and this is cheating towards her forehand, assuming she's right handed, which she in fact is. Okay, what do these numbers mean? So let's start with the easy ones. So if Venus chooses left and Serena chooses right, then Serena has guessed wrong. Is that correct? In which case Venus wins the points 80% of the time and Serena wins it 20% of the time.

Conversely, if Venus chooses right and Serena chooses left, then again, Serena has guessed wrong and this time Venus wins the points 90% of the time and Serena wins the points 10% of the time. This should be a familiar idea by now, but why is it the case these nineties and eighties are not a 100%? Why is it the case that if Serena guesses wrong Venus doesn't win 100% of the time? Anybody? Perhaps we can get a show of hands, get some mikes up. Why isn't it 100% here? Somebody? Patrick? Wait for the mike.

Student: Sometimes she hits it out of bounds when she serves.

Professor Ben Polak: Right, this isn't even a serve, this is a passing shot but the same is true. So sometimes you're successfully going to hit it past Serena but the ball is going to sail out. So that happens 10% of the time here and 20% of the time here. Look at the other two boxes, if Venus hits to Serena's left and Serena guesses left, then we're going to assume that Serena's going to reach the ball and make a volley, but her volley only manages to go in--go over the net and go in--half the time, so the payoffs are (50, 50). Half the time Venus wins the point and half the time Serena wins the point. Conversely, if Venus hits the ball to Serena's right and Serena guesses correctly and chooses right, then we're in this box. Once again, Serena has guessed correctly and she's going to successfully reach the volley and this time she gets it in 80% of the time, so Venus wins the point 20% of the time and Serena wins it 80% of the time.

So just to finish up the description of the game here, notice that we're assuming that Serena is a little better at volleying to her right than she is volleying to her left. So this is her forehand volley and we're going to assume that that's stronger than her backhand volley. Conversely, we're assuming that Venus' passing shot is a little better when she shoots it to Serena's left than when she shoots it to Serena's right. This is her cross court passing shot and this is her down the line passing shot. So none of that fine detail matters a great deal, but just if you're interested that's where the numbers come from. I'm not claiming this is true data by the way, I made up these numbers. Actually I think Dixit made up these numbers, I forget where I got them from.

So okay, everyone understand the game? So now imagine, either imagine you are Venus or Serena, or imagine perhaps more realistically, that you've become Venus or Serena's coach. Do I have any members of the tennis team here? No. Well imagine you've become their coach, so you take this class and then you apply to replace their father as being their coach. That's a tough assignment I would think. So an obvious question is, you're coaching Venus before Wimbledon, you know this situation's going to arise and you might want to coach Venus on what should she do here? Should she try and pass Serena down the line or she should try and hit the cross court volley, cross court passing shot? Notice that this is a question of should you, Venus, play to your strength which is the cross court passing shot, or should you play to Serena's weakness, which would be to hit it to Serena's backhand. Playing to your strength is to choose right and playing to Serena's weakness is to choose left.

Conversely, for Serena, should you lean towards your strength, which I guess is leaning to the right or should you lean towards Venus' weakness, which I guess is leaning left? When you look at coaching manuals on this stuff, or you listen to the terrible guys who commentate on tennis for ESPN--oh no I'm getting in trouble again--very nice guys who commentate on tennis for ESPN, they say just incredibly dumb things at this point. They say things like, you should always play to your strengths and don't worry about the other person's weakness. I think it won't take much time today to figure out that's not great advice. But can people at least see that this is a difficult problem, this is not an immediately obvious problem, is that correct?

One reason it's not immediately obvious is not only is no strategy dominated here, but there is no pure strategy Nash Equilibrium in this game, in this little sub game. There is no pure strategy Nash Equilibrium--and notice that I added the qualifier now. Previously I would just have said Nash Equilibrium, but now that we have mixed strategies in the picture, I'm going to talk about pure strategy Nash Equilibria to be those that are the only involving pure strategies. Okay, so why is there no pure strategy Nash Equilibrium? Well let's have a look. So if Venus--If Serena thought that Venus was going to choose left then her best response, not surprisingly, is to lean left and if Serena thought that Venus was going to choose right, then her best response is to cheat to the right, so 50 is bigger than 20, and 80 is bigger than 10. And conversely, if Venus thought that Serena was cheating a bit to the left then her best response is to hit it to Serena's right, and if Venus thought Serena was leaning to the right then Venus' best response is to hit it to Serena's left.

So I think that's not at all surprising when you think about it, not at all surprising, you're going to get this little cycle like this, but we can see immediately that these best responses never coincide, so there is no pure strategy equilibrium. So that leaves us a bit stuck except I guess you know what the next question's going to be, and I shouldn't leave it in too much suspense. The next question's going to be, okay there's not pure strategy Nash Equilibrium, but we've just introduced a new idea which was what? It was Nash Equilibrium in mixed strategies. Maybe there's going to be a mixed strategy Nash Equilibrium. In fact, there is, there is going to be one. So our exercise now is, let's find a mixed strategy Nash Equilibrium, and before we find it, let's just interpret what it's going to mean.

A mixed strategy Nash Equilibrium in this game, is going to be a mix for Venus between hitting the ball to Serena's left and Serena's right, and a mix for Serena between leaning left and leaning right, such that each person's mix, each person's randomization is a best response to the other person's randomization. Since these players are sisters and have played each other many, many times, not just in competition but probably in practice, it seems like a reasonable idea that they might have arrived in playing each other, at a mixed strategy Nash Equilibrium. That's what we're going to try and do, now how are we going to do that? So what we're going to do is we're going to exploit the trick that we have here, the lesson here.

The lesson we have here says if players are playing a mixed strategy as part of a Nash Equilibrium, each of the pure strategies involved in the mix, each of their pure strategies must itself be a best response. We're going to use that idea. So let's try and do that. So I'm hoping that by doing this, I'm going to illustrate to you immediately, that this idea is actually useful, at least useful if you end up coaching the Williams sisters. Alright, I want to keep this so you can still read it. Ill bring it down a bit. Can people still read it? Okay, so what I want to do is, I want to find a mixture for Serena and a mixture for Venus that are equilibrium. Having put it up there let me bring it down again. This was not so intelligent of me.

I actually want to bring in some notation, so as before, let's assume that Serena's mix is, let's use Q and (1-Q) to be Serena's mix and let's use P and (1-P) to be Venus' mix. Let's establish that notation. So here's the trick, So this is the slightly magic bit of the class, so pay attention, I'm about to pull a rabbit out of a hat. Trick, what should I do first, to find Serena's Nash Equilibrium mix, so that's (Q, (1-Q)), what I'm going to do is I'm going to look at Venus' payoffs. So to find Serena's Nash Equilibrium mix the trick is to look at Venus' payoffs, that's going to be my magic trick. Let's try and see why.

So let's look at Venus' payoffs, Venus' payoffs against Q. So if Serena is choosing (Q, 1-Q), what are Venus' payoffs? So if she chooses left then her payoff is 50 with probability Q--and I'm going to use the pointer here, and hope that the camera can see this too. She gets 50 with probability Q and she gets 80 with probability 1-Q. If she chooses right then she gets 90 with probability Q and she gets 20 with probability of 1-Q. I meant to point to that. So what? So what is this: we're looking for a mixed strategy Nash Equilibrium, so in particular, not only Serena is mixing but in this case what we're claiming is, Venus is mixing as well. So if Venus is mixing as well, that means that Venus is using the strategy left with some probability P and using the strategy right with some probability 1-P. Since Venus sometimes chooses left and sometimes chooses right as her best response to Q, her best response to Serena, what must be true of the payoff to left and the payoff to right?

Let's go through it again, so we're going to assume that Venus is mixing. So sometimes she chooses left and sometimes she chooses right and she's going to be, she's in a Nash Equilibrium, so she's choosing a best response. So whatever that mix P, 1-P is, it's a best response. Since she's playing a best response of P and that sometimes involves choosing left and sometimes involves choosing right, it must be the case that what? It must be the case that both left itself and right itself are both themselves best response. If she's mixing between them, it must be that both choosing left or choosing right are themselves best responses. If they weren't she should just drop them out of the mix, that would raise her average payoff. Right, just like we dropped out the short T.A.'s to get a high height and we dropped out the failing Yale students to get a high GPA.

So if Venus is mixing in this Nash Equilibrium then the payoff to left and to right must be equal, they must both be best responses, both left and right must be a best response, so in particular, the expected payoffs must be the same. Is that right, is that correct? So what does that allow me to do? It allows me to put an equals sign in here. Since left is a best response and right is a best response, since they're both best responses, they must yield the same expected payoff. Here's their expected payoffs, they must be the same. Now, I've got one equation and one unknown, and now I'm down to algebra. So let me do the algebra. I claim this expression is equal to that expression, so simplifying a bit I'm going to get--you should just watch to make sure I don't get this wrong--I'm going to get 40Q, so this implies 40Q is equal to 60(1-Q). So I took this 50 onto this side and this 20 onto that side, so I have 40Q is equal to 60(1- Q) and that implies that Q is equal to .6. So those last two steps were just algebra.

So what was the trick here? The trick was I found Q, which is how Serena is mixing by looking at Venus' payoffs, knowing that Venus is mixing and hence I can set Venus' payoffs equal to one another. Say that again, I found the way in which Serena is mixing by knowing that if Venus is mixing, her expected payoffs must be equal and I solved out for Serena's mix, this is Serena's mix. Let's do it again. Here I'm wishing I had another board. I don't want to lose those numbers entirely, so I'm going to try and squeeze in a bit. I know what I can do. Let's get rid of this one entirely. There we go, that works. Let's get rid of this one entirely. I can still see my numbers.

Let's do the converse. Let's do the trick again, this time what I'm going to do is I'm going to figure out how Venus is mixing. I know how Serena is mixing now, so now I'm going to work out how Venus is mixing. Now, to figure out how Serena was mixing, I used Venus' payoffs. So to find out how Venus is mixing what am I going to do? I'm going to use Serena's payoffs. So to find Venus' mix, which is P, 1-P, --let's be careful it's her Nash Equilibrium mix--use Serena's payoffs. Here we go, so if Serena chooses, this is S's payoffs, if Serena chooses L then her payoffs will be what? So again, just watch to make sure I don't get this wrong and I'll point to the things to try and help myself a bit. So with probability P she'll get 50. So 50 with probability P, and with probability 1-P she'll get 10. And if she chooses to lean to the right, to lean towards her forehand, then with probability P she'll get 20 and with probability 1-P she'll get 80.

We know that Serena is mixing, so since Serena is mixing what must be true of these two payoffs? What must be true of the two payoffs? The payoff to l and the payoff to r, what must be true about them since Serena is using a mixture of these two strategies in Nash Equilibrium? It must be the case that both l is a best response and r is a best response, in which case the payoff must be, someone shout it out, equal, thank you. They must be equal, these must be equal. They must be equal since Serena is indifferent between choosing left or right and hence is mixing over them. So again, using the fact that they're equal reduces this to algebra, and again, I'll probably get this wrong but let me try. So I claim, let's take 20 away from here, I've got 30P equals 70(1-P). I hope that's right, that looks right. Again, this is just algebra at this point. So I took 20 away from here and 10 away from there, and this implies that P equals .7.

So I claim I have now found the mixed strategy Nash Equilibrium. Here it is. The Nash Equilibrium is as follows. Let's be careful, this is Venus' mix. So if Venus is mixing .7, .3, .7 on left and .3 on right, and Serena is mixing .6, .4, so this is Venus' mix and this Serena's mix. Venus is shooting to the left of Serena with probability of .7 and Serena is leaning that way with probability of .6. So we were able to find this Nash Equilibrium by using the trick before. Now let's just reinforce this a little bit by talking about it. So suppose it were the case that Serena, instead of leaning to the left .6 of the time leant to the left more than .6 of the time. So suppose you're Venus' coach, and suppose you know that Serena leans to the left more than .6 of the time, what would you advise Venus to do?

Let me try it again. So suppose your Venus' coach and suppose you've observed the fact that Serena leans to the left more than .6 of the time, what would you advise Venus to do? Pass to the right, shout out.

Student: Pass to the right.

Professor Ben Polak: Pass to the right, exactly. So if Serena cheats to the left more than .6 of the time, then Venus' best response is always to shoot to the right. That maximizes her chance of winning the point. Conversely, if Serena leans to the left less than .6 of the time, then Venus should do what? Shoot to the left all the time. So if Serena doesn't choose exactly this mix, then Venus' best response is actually a pure strategy. Say it again, if Serena leans to the left too often, more than .6, then Venus should just go right and if Serena leans to the left too little, then Venus should always go left. We can do exactly the same the other way around. If Venus shoots to the right, so that's her cross hand passing shot more than .7 of the time, and you're Serena's coach, what should you tell Serena to do? Go that way all the time.

So if Venus is hitting it to Serena's left more than .7 of the time, Serena should just always go to her left, and if Venus is hitting to the left less than .7 of the time, so to the right more than .3 of the time, then Serena should always go to the right. So that's how this kind of comes back into the sort of the coaching manuals if you like. Okay, so how am I doing so far? Have I lost everyone yet or are people still with me? How many of you play tennis, ever? So all your tennis is going to dramatically improve after today, right? So now let's make life more interesting. Let's go back to the start.

We've figured out this is an equilibrium, this is how Venus and Serena play, Venus and Serena know each other perfectly well, they know that they mix this way, they're going to best respond to it, this is going to be where they end up. But in the meantime, Serena hires a new coach and Serena's new coach is just very, very good at teaching Serena how to play at the net, and in particular, how to hit the backhand volley. So Serena's new coach, let's say it's Tony Roche or somebody, it's just a brilliant coach and Tony Roche is able to improve Serena's backhand volley and that changes these payoffs. So you should rewrite the whole matrix but I'm going to cheat. So the new game is exactly the same as it was everywhere else, except for now when Serena gets to the backhand volley, she gets in it 70% of the time. So there used to 50, 50 in that box and now it's 30, 70.

So the game has changed because Serena has got better at hitting backhand volleys. We want to figure out how is this going to affect play at Wimbledon? Now it doesn't take much to check that there is still no pure strategy Nash Equilibrium. It's still the case, in fact even more so, that Serena's best response to Venus choosing left is to lean to the left. So it's still the case that the best responses do not coincide, there is still no pure strategy equilibrium. What we're going to do of course is we're going to find a mixed strategy equilibrium, but before we do so, let's think about this intuitively. Let's see if we can intuit an answer.

I'm guessing we can't, but let's see if we can intuit an answer. So Serena has improved her backhand volley, and hence when she reaches it she gets it in more often. So one effect, you might think, is what we might want to call a direct effect and I think there's two effects here. There are two effects, one of these I'm going to call the direct effect, and by effect, I mean in particular an effect on how Serena should play the game. So since Serena has improved her backhand volley, when she reaches that volley she gets it in more often, so one might say in that case--your Serena's coach--in that case you should lean to the left more often than you did before, because at least when you get that backhand volley you're going to get it in more often. So the direct effect says Serena should lean left more, in other words, Q should go up.

Is that right? So Serena's now better at playing this backhand volley, so she may as well favor it a bit more and hence Q will go up. So that's the direct effect, but of course there's a "but" coming. What's the but? Again, let's see my tennis players here, raise your hands if you play tennis. Suddenly nobody plays tennis, come on raise your hands okay. What's the but here? We think Serena's backhand has improved so she might be tempted to play towards her backhand a bit more often, what's the but? So I claim the but is this--you tell me if I'm wrong--the but is that Venus (she's her sister after all, right, so Venus knows that Serena's backhand has improved) so Venus is going to hit it to Serena's left less often than before. Is that right? So since Serena's backhand has improved, Venus is going to hit it to Serena's backhand less often than before, and that might make Serena less inclined to cheat towards her backhand because the ball is coming that way less often.

So this is a indirect or a strategic effect. The strategic effect is Venus hits L less often, so Serena should reduce the number of times that she leans to the left because the ball is coming that way fewer times. Now notice that these two effects go in opposite directions, is that right? One of them tends to argue that Q would go up, that's the direct effect and the other one is more subtle, it says we now think about not just how my play has improved, but also how the other person's going to respond to knowing that my play has improved, that's the more subtle effect and that's going to push Q down. That's going to make it less likely, that's an argument against leaning to the left.

So imagine you're going to be Serena's coach, which of these effects do you think is going to win, let's have a poll. Which of these effects do you think is going to win? The direct effect or the indirect effect? The direct effect or the strategic effect? Who thinks the direct effect? Who thinks Serena, who'd advise Serena to play to her strength a bit more and lean left a bit more, who thinks the direct effect? Raise your hands, let's have a poll. Who thinks the indirect effect, the effect of Serena hitting it that way less often is going to win? Who's abstaining and basically refusing to be a coach? Quite a number of you, all right. Well we're going to find out by re-solving for the Nash Equilibrium.

What we're going to do is redo the calculation we did before starting with Serena. So to find Serena's mix, to find Serena's new equilibrium mix, what do we have to do? The question is, in equilibrium, is Serena going to lean to the left more (so Q is going go up) or less (so Q's going to do down). So I need to find out what is Serena's new equilibrium mix. What's the new Q? How do I go about finding Serena's equilibrium Q, what's the trick here? Shout it out. Use Venus' payoffs. So to find the new Q for Serena, use Venus' payoffs. Now let's do that. So from Venus' point of view, if she chooses left then her payoffs are now, and again I should use the pointer, 30 with probability Q, this is the new Q and 80 with probability 1-Q, 30 with probability Q plus 80 with probability 1-Q.

Again, this is the new Q, I should really give it, put Q prime or something but I won't. If she chooses right then her payoff is what? It's going to be 90 with probability Q and 20 with probability 1-Q. What do we know about these two payoffs if Venus is mixing in equilibrium? We know she's mixing in equilibrium because we saw there was no pure strategy equilibrium, so what we do know about these two payoffs since Venus is using both these strategies in equilibrium? They must be the same. Since she's using both these strategies, these strategies must be equally good. They must both be best responses so these two payoffs are equal.

Since they're equal all I have to do is solve out for Q, so let's do it. So I'm going to get 90 minus 30 is 60Q, is equal to 80 minus 20 which is 60(1-Q), so Q equals .5. If I did the algebra too quickly just trust me, I think I got it right. From here on in, it was just algebra. So what have I found out? Did Q go up or go down? Well it used to be, Q used to be what? .6 and now its .5, so let me ask what I think is an easy question, did it go up or down? It went down. Q went down, the equilibrium Q went down. So which effect turned out to be bigger? The direct effect of playing more to your strength or the indirect effect of taking into account that your opponent is going to play less often to your strength. Which effect turned out to be the bigger effect? The indirect effect, the strategic effect.

Of course I really did want the strategic effect to be bigger because this is a course about strategy, but the strategic effect actually won here. The strategic effect, the indirect effect is bigger. That's good news for me because it says the slightly dumb coach who didn't bother to take Game Theory would have stopped at this direct effect and they'd have told Serena to go the wrong way, but the smart coach who takes my class, and therefore somehow contributes to my salary, in an extraordinarily indirect way, gets it right.

Now we can also solve out for Venus' new mix and we'll do it in a second. But before I do it, let me just point out that we actually, we really can now intuit Venus' effect. It may not be exact numbers but we can intuit here. As I claim, I claim if we think this through carefully, we know whether Venus is shooting more to the left, than she was before, or less to the left, than she was before. Notice that in the new equilibrium Serena is going less often to her left even though she's better at hitting the backhand, she's better at hitting the ball when she gets there. So since Serena is leaning left less often what must be true about Venus in this new equilibrium? It must be the case that Venus is hitting the ball to the left less often. Does that make sense? We have enough information already on the board to tell us that, nevertheless, let's do the math. Let's go and retrieve a board to do the math. Just to complete this, let's figure out exactly what Venus does do.

So to figure out what Venus is going to do, what's our trick? I want to figure out how Venus is going to mix. I'm going to find out Venus' new P, how do I find out Venus' new equilibrium mix? I look at Serena's payoffs. So if Serena chooses left, her payoff is, and I'll read it off quickly this time, is 70P plus 10(1-P) and if Serena chooses right her payoff is 20P plus 80(1-P) and I'm praying that the T.A.'s are going to catch me if I make a mistake here, and I know these have to be equal because I know that in fact Venus is mixing--sorry, I know that Serena is mixing, so I know these must be equal. So since they're equal I can solve out and hope that I've got this right, so I've got 50P equals 70(1-P), so P is equal to 7/12. So again, that's just algebra, I rushed it a bit, it's just algebra. Same idea, just algebra. So 7/12 is indeed smaller than what it used to be, because it used to be 7/10, so that confirms our result.

So the strategic effect dominated. Venus shot to Serena's backhand less often, and as a consequence, so much so, that Serena actually found it worthwhile going more to the right than she used to before. Now let's just talk this through one more time. This was a comparative statics exercise. We looked at a game, we found an equilibrium, we changed something fundamental about the game, and we looked again to look at the new equilibrium, that's called comparative statics. Let's talk through the intuition. Before we made any changes Venus was indifferent. She was indifferent between shooting to the left and shooting to the right. Then we improved Serena's ability to hit the volley to her left, we improved her backhand volley. If we had not changed the way Serena played then what would Venus have done?

So suppose in fact Serena's Q had not changed. If Serena's Q had not changed, remembering that Venus was indifferent before, how would Venus have changed her play? Somebody? If we started from the old Q and then we improved Serena's ability to play the backhand volley, and if Q didn't change, what would Venus have done? She'd never, ever have shot to the left anymore, she'd only have shot to the right which can't possibly be an equilibrium. So something about Serena's play has to bring Venus back into equilibrium, it brings Venus back into being indifferent, and what was it? It was Serena moving to the left less often and moving to the right more often. To say it again, if we didn't change Q, Venus would only go to the right, so we need to reduce Q, have Serena go to the right, to bring Venus back into equilibrium.

Conversely, if Venus hadn't changed her behavior, if Venus had gone on shooting exactly the same as she was, P and 1-P as before, then Serena would have only gone to the left and that can't be an equilibrium. So it must be something about Venus' play that brings Serena back into equilibrium, and what is it? It's that Venus starts shooting to the right more often. So just two reminders, before you leave two reminders. Wait, wait, wait. First, in about five minutes time a handout will magically appear on the website that goes through these arguments again, all of them in two other games, so you can have a look at the handout. Second thing, a problem set has already appeared by magic on that website that gives you lots of examples like this to work on. Play tennis over the weekend for practice and we'll see you on Monday.

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