0:00

Hi folks. This is Matt Jackson again and so now let's take a look at Mixed

Â Strategy Nash Equilibrium in practice and try and understand a little bit about

Â what it should tell us about what we should expect to see.

Â So let's start with situation of soccer penalty kick.

Â Kicks, and this is a quite natural application of mixed strategy equilibria

Â because they're ubiquitous in sports and competitive games, so situations where it

Â actually pays to be unpredictable. So, by not knowing what the other, the

Â opposition is going to do. It makes it a little more difficult to

Â for you to pick it up the most strategy in these games where one player wins and

Â the other player looses. And, in particular in soccer penalty

Â kicks, were, looking at a situation where a kicker has to try and kick the ball

Â into a goal, the goalie can try to move to deflect the ball, and this happens

Â very quickly, so it's essentially a simultaneous move game.

Â The kicker is choosing to go either in, in their simplified version say to to the

Â right or the left. The goalie is then going to dive to the,

Â to one side or the other side and try to deflect the ball and if the goalie,

Â guesses correctly and ends up in the same.

Â Direction as the, the kicker, then they have a high chance of, higher chance of

Â saving it. If they go in the opposite direction,

Â they have a lower chance of saving it. Okay, so, how, how are equilibria going

Â to adjust to the skills of the players? So, let's suppose, for instance, as a

Â kicker That I might be biased. I might be able to kick the ball more

Â accurately in 1 direction than the other. So if you ask me to kick it towards the

Â left side of the goal, it might be that I hit there very accurately.

Â If you tell me I have to kick it towards the right side of the goal, it could be

Â that I'm less accurate. And I have a higher chance of just

Â missing the goal altogether. So is the equilibrium going to change

Â when we change one of the players, in terms of their skills.

Â So let's have a, a, a peek at this. Should a kicker who kicks penalty kicks,

Â worse to the right than the left, kick more often to the left than the right?

Â So, if I, if I'm worse in kicking towards the right, does that mean that I should

Â kick in the opposite direction more often? Well that's have a peak.

Â So, let's start with just a simple version just to get, our ideas fixed, so

Â imagine that the, setting is one where the kicker and the goalie, if they, so

Â let's have the kicker on this side. so the kicker is the role player, the

Â goalie is the column player and if they end up kicking, if the kicker goes left

Â and the goalie also happens to go to the left, then the goalie saves.

Â And the goalie gets a payoff of 1. The kicker gets a payoff of zero, if

Â instead we're in a situation where say the kicker goes left and the goalie goes

Â right, then the kicker scores and gets a path of 1, and, and so forth.

Â Okay. So this is just a simple variation on

Â matching pennies, and in this situation, what's the equilibrium going to be.

Â The equilibrium's going to be quite simple.

Â It's just going to be that the kicker randomizes equally between left and

Â right. The goal randomizes between left and

Â right. Each person has a probability of half to

Â win from kicking to the left or right, Goalie to the left or right.

Â So it's a very simple game, and we're have a good idea of how to solve that

Â one. Okay, so now what're we going to do?

Â Let's change things and now we've got a kicker that sometimes misses when they go

Â to the right. So imparticular if the goalie happens to

Â go to the left. And the goals wide open to the right.

Â The kicker scores 75% of the time, but actually misses completely 25% of the

Â time. Okay, so this is the, a, a kicker who's

Â still . does well if, if they go left and the

Â goalie goes to the opposite direction. But now they have a lower probability of

Â winning when their kicking right and they have a wide open goal.

Â Okay, so how should this adjust, whats, what Should the new equilibrium look

Â like? So let's suppose let's first of all try and keep the kicker indifferent.

Â So let's think of the goalie going left with probability P, right with

Â probability 1 - P. For the kicker to be indifferent what has

Â to be true? Well what's their payoff if they go left? Their payoff to going left.

Â Left is just 1 x 1 - P. There kick-off, there payoff to going

Â right is .75 x P. These 2 things have to be equal, in order

Â to have this, thee kicker being different.

Â So what do we end up with? We end up with .75P is equal to 1 - P.

Â so we end up with, 1 = 1.75p. or p is = to 1 over 1.75.

Â Which is = to, 4 over 7. Okay? So that tells us that the.

Â Goalie should be going left with probability 4/7, and right with

Â probability 3/7. Okay? So we know what the goalie's

Â supposed to be doing, so, so now the fact that we changed, the goalie's payoffs

Â haven't changed, but the fact that we changed the kicker's payoffs, meant that

Â the goalie actually had to adjust. Right? So even though the goalie's

Â payoffs haven't changed at all in this game, the new equilibrium has a different

Â set of, of probabilities for the goalie in order to keep the kicker at different

Â now. Okay? So now let's, let's see what the

Â kicker's going to do. So how are we see what's so, so let's

Â suppose that the kicker goes left with probability q, right with probability 1 -

Â q, and let's solve for q. Well, for the goalie to be indifferent,

Â what is their payoff if they go left? If they're going left, they're getting and

Â they're getting a q probability that they match.

Â So they get q + 0.25(1 - q) if they go left.

Â If they go right instead what are they getting? They're just getting 1 minus q.

Â Right? So these 2 things have to be equal so we end up with, q = .75 * (1 - q).

Â So q / (1 - q) = .75. What does this tell us about q? It tells

Â us that q = 3/7. Okay, so what's going to happen, when we

Â work out this? We get 3/7 for the probability that the kicker's going to go

Â left. And 4/7 for the probability that the

Â kicker is going to go right. So overall what do we have now? We have

Â the strategies looking like this as we made this adjustment.

Â And we notice two sort of interesting things about this.

Â one is that the goalie's pay-offs didn't change, but they still had to adjust

Â their strategies. And the second is that the kicker is

Â actually kicking more often to the weaker side, right? So the, the, the right foot

Â got worse than it was before and they are actually going in that direction more

Â often. and why is that? It's because the kicker,

Â the, the goalie has also made an adjustment in this game.

Â And so, the, the comparative statics in mixed strategy Nash equilibria are

Â actually quite subtle, and somewhat counter intuitive in terms of what you

Â might expect you're, you're, you get a bias so that this becomes a weaker

Â direction, and the equilibrium adjusts. So that the player goes in that direction

Â more often. So, let's have a, a look just through the

Â intuition here. Again the goalie strategy must have the

Â goalie indifferent and so when we went through those payoffs, the kicker, the

Â goalie goes left more often than right, and the kicker, actually, so sorry

Â there's a type here. The kicker, actually goes right More

Â frequently, right, goes right with probability now 4/7, so they've increased

Â their probability on that. And when we end up, what we see is the

Â goalie's strategy is adjusting, but we also see that the kicker adjusts to

Â kicking more toward their weak side. so the, the goalie now actually has a

Â slight advantage. So if you go through and calculate the

Â probability that the goalie's going to win, they're going to win 4/7, so the

Â time in this, in this match. And, and we can think what would happen

Â if the goalie Actually just stayed with our old strategy of still going 50 50.

Â Then the kicker could always go left, and win 1/2 the time instead of 3/7.

Â So the, the fact that the goalie has to make an adjustment is because they have

Â to de, defend more to the left side to defer because now the, the, the kickers

Â has a, a better Chance of winning on that side.

Â So the goalie goes more in that direction.

Â That pushes the kicker towards their weaker side.

Â In order to make sure that the goalie is willing to go to the left side with

Â higher frequency. So, by adjusting to strategy to keep the

Â kicker indifferent the goalie takes advantage of the kicker's weak right kick

Â and wins more often. Often.

Â Okay, so just in terms of summary and, and mixed strategy in soccer penalty

Â kicks, in general. Players must be indifferent between the

Â things that they're randomizing over. that produces very interesting and subtle

Â Comparative statics. and you know there, there's a question

Â that might come up in your mind, do people really do this? I mean, this is

Â fairly complicated, right? So the, you know, 50-50 we can figure out one we get

Â these, to these games where a player has an advantage The advantage 1 way or

Â another, then the actual mixture becomes fairly complicated.

Â And it's not so obvious that players will actually do that.

Â so we'll, we'll take a look at that and see if, if this actually bears out in, in

Â practice.

Â