 Hi folks, this is Matt Jackson again, and so now let's take a look at mixed strategy Nash equilibrium in practice and Try and understand a little bit about what it can tell us about what we should expect to see so Let's start with situation of soccer penalty kicks and this is a quite natural application of mixed strategy equilibria because they're ubiquitous in sports and competitive games so situations where it actually pays to be Unpredictable, so by not knowing what the other the opposition is going to do It makes it a little more difficult for you to pick an optimal strategy in these games where one player wins and the other player loses and In particular in soccer penalty kicks We're looking at a situation where a kicker has to try and kick the ball into the goal The goalie can try and move to deflect the ball And this happens very quickly, so it's essentially a simultaneous move game The kicker is choosing to go either in their simplified version say to the right or the left The goalie is then going to dive to the to one side or the other side and try to deflect the ball And if the goalie guesses correctly it ends up in the same Direction as the the kicker then a high chance of a higher chance of saving it if they go in an opposite direction They have a lower chance of saving it. Okay, so How how are equilibria going to adjust to the skills of the players? So let's suppose for instance as a kicker that I might be biased I might be able to kick the ball more accurately in one direction than the other So if you ask me to kick it towards the left side of the goal It might be that I hit there very accurately if you tell me I have to kick it towards the right side of the goal It could be that I'm less accurate and I have a higher chance of just missing the goal altogether So how is the equilibrium going to adjust when we change one of the players? adjustments in terms of their skills So let's have a peek at this should a kicker who kicks penalty kicks Worse to the right than the left kick more often to the left than the right So if I if I'm worse on and kicking towards the right does that mean I should kick in the opposite direction more often? Well, let's have a peek. So let's start with just the simple version just to get our ideas Fixed so imagine that the setting is one where the kicker and the goalie If they so let's have the kicker on this side So the kicker is the row player the goalie is the column player And if they end up kicking if the kicker goes left and the goalie also happens to go to the left Then the goalie saves and the goalie gets a payoff of one the kicker gets a payoff of zero if Instead we're in a situation where say the kicker goes left and the goalie goes right Then the kicker scores and gets a payoff of one and and so forth Okay, so this is just a simple variation on matching pennies and in this situation What's the equilibrium going to be the equilibrium is going to be quite simple. It's just going to be that the kicker randomizes equally between left and right the goalie Randomizes between left and right each person has a probability of half to win from kicking to the left or right goalie can left or right so it's a very simple game and and We're Have a good idea of how to solve that one Okay, so now what are we going to do? Let's change things and now we've got a kicker who Sometimes misses when they go to the right So in particular if the goalie happens to go to the left and the goals wide open to the right the kicker Scores 75% of the time, but actually misses completely 25% of the time. Okay, so this is the kicker who still does well if they go left and the Goalie goes to the opposite direction, but now they have a lower probability of winning when they're kicking right and They have a wide open goal. Okay, so how should this adjust? What's what should the new equilibrium look like? So let's suppose Let's first of all try and keep the kicker indifferent So let's think of the goalie going left with probability P right with probability one minus P For the kicker to be indifferent. What has to be true? Well, what's their payoff if they go left their payoff to going left is just one times one minus P Their kickoff their payoff to going right is point seven five Times P these two things have to be equal In order to have this the kicker be indifferent So what do we end up with we end up with point seven five P is equal to one minus P So we end up with One equals one point seven five P Or P is equal to one over one point seven five which is equal to Four over seven. Okay, so that tells us that the goalie should be going left with probability four sevens and Right with probability three sevens. Okay? So we know what the goalie is supposed to be doing So so now the fact that we changed the goalies payoffs haven't changed But the fact that we changed the kicker's payoffs meant that the goalie actually had to adjust right? So even though the goalies payoffs haven't changed at all in this game The new equilibrium has a different set of Probabilities for the goalie in order to keep the kicker indifferent now Okay, so now let's let's see what the kicker is going to do. So how are we gonna see what so let's suppose that the Kicker goes left with probability Q right with probability one minus Q and let's solve for Q Well for the goalie to be indifferent What is their payoff if they go left if they're going left? they're getting and They're getting a Q probability that they match so they get Q Plus point two five times one minus Q if they go left if they go right instead What are they getting? They're just getting one minus Q right, so these two things have to be equal so we end up with Q is equal to point seven five times one minus Q So Q over one minus Q is equal to point seven five What does it tell us about Q that tells us that Q is equal to? three sevens Okay, so what's going to happen when we work out this we get three sevens for the probability that the kicker is going to go left and Four sevens for the probability that the kicker is going to go right. So overall, what do we have now? We have the strategies looking like this as we made this adjustment And we notice two sort of interesting things about this One is that the goalies payoffs didn't change but they still had to adjust their strategies and the second is That the kicker is actually kicking more often To the weaker side, right? So the they their right foot got worse than it was before and they're actually Going in that direction more often And why is that it's because the kicker the goalie has also made an adjustment in this game And so the the comparative statics and mixed strategy Nash equilibria are actually quite subtle and Somewhat counter-intuitive in terms of what you might expect you're you get a bias so that this becomes a weaker direction And the the equilibrium adjusts so that the player goes in that direction more often. So let's have a Look just through the intuition here again and the goalie strategy must have a kicker indifferent and so When we went through that those payoffs the the kicker the goalie goes left more often than right And the kicker actually so sorry there's a typo here the kicker Actually goes right More frequently right goes right with probability now 47 so they've increased their probability on that and When we end up what we see is the the goalie strategy is adjusting, but we also see that the kicker Adjusts to kicking more to the towards their weak side So the goalie now actually has a slight advantage So if you go through and calculate the probability that the goalie is going to win They're going to win four sevenths of the time in this in this match and and we can think you know What would happen if the goalie actually just stayed with their old strategy of still going 5050? Then the kicker could always go left and win half the time instead of three sevenths so the the fact that the goalie has to make an adjustment is because they have to Defend more to the left side to defer because now the the kickers has a better Chance of witting on that side. So the goalie goes more in that direction that pushes the the kicker towards their weaker side In order to make sure that the goalie is willing to go and to the left side With higher frequency So by adjusting the strategy to keep the kicker in different the goalie actually gets advantage of the kickers weak white right kick And wins more often Okay So just in terms of summary and mixed strategy and soccer penalty kicks in general players must be indifferent between the things that they're randomizing over That produces very interesting and subtle comparative statics And you know there's a question that might come up in your mind. You know do people really do this? I mean this is fairly complicated, right? So the you know 50-50 we can figure out once we get these these games where a player has a Advantage one way or another then the actual mixture becomes fairly complicated and it's not so obvious that players will actually do that So we'll take a look at that and see if this actually bears out in practice