“Nash Equilibrium and the Prisoner’s Dilemma…Coordination Game and Self-Fulfilling Prophecy…Market Competition…Why Do People Come to Play Nash Equilibrium? Part I…Why Do People Come to Play Nash Equilibrium? Part II…Why Do People Come to Play Nash Equilibrium? Part III…Stylized Facts and Nash Equilibrium…Make Yourself Unpredictable: Mixed Strategy Equilibrium…Sports Games and Game Theory…Nash Equilibrium Exists in All Games…”
2-2 Coordination Game and Self-Fulfilling Prophecy
2-3 Market Competition
2-4 Why Do People Come to Play Nash Equilibrium? Part I
2-5 Why Do People Come to Play Nash Equilibrium? Part II
2-6 Why Do People Come to Play Nash Equilibrium? Part III
2-7 Stylized Facts and Nash Equilibrium
2-8 Make Yourself Unpredictable: Mixed Strategy Equilibrium
2-9 Sports Games and Game Theory
2-10 Nash Equilibrium Exists in All Games
2-1 Nash Equilibrium and the Prisoner’s Dilemma
Nash equilibrium is a situation where all players are doing their best against others.
Why do people play a Nash equilibrium? Okay so let’s go down to the first lecture.
Okay so what’s the definition of Nash equilibrium? Let me explain the case of a, in the case of two players.
So in Nash equilibrium player one is taking his equilibrium strategy a1 star.
Player two is taking his equilibrium strategy a2 star.
Take player one, and if everybody sticks to Nash equilibrium, player one’s payoff, payoff is represented by g one.
So what happens if player one deviates from equilibrium action a one start to something else.
Player was taking a one star, but now suppose he switches to something else.
Okay, but a two, player two, is still choosing equilibrium action.
If he deviates to something else, to the red strategy here, his payoff changes from here to here, okay? Nash equilibrium conditional says that player cannot increase his payoff by deviating, so this cynical deviate must be true, okay? And this should be true for all strategy, a1.
A lone star equilibrium strategy for player one, and equilibrium strategy for player two, a mutual best reply, a one star is a best reply, to player two’s equilibrium strategy, and player two is also taking a best reply to player one’s.
So Nash Equilibrium has a property of mutual best reply and the Nash Equilibrium has the following property, no single player can increase his payoff by deviating by himself.
So therefore players take their actions simultaneously, and then that’s the end of the game.
I confine my attention to this simple class of games, and the possibly in, in the last week we are going to see dynamic game with, with some timing, okay? So game theory says that in any simultaneous move game, player’s behavior can be predicted by.
Okay? So let’s apply the concept of Nash equilibrium to the famous game of Prisoner’s Dilemma.
Let’s suppose two individuals, say player one and two committed some crime.
Likewise player two can either cooperate or defect.
So what happens if both players remain silent, if both players chose cooperation? And then, they are put in prison for one year, Okay, they stole some moneys, so they are going to be put in prison, for one year.
So let’s say they are paying off this minus one, and minus one The number here, the first number here represents Player One’s payoff, the second number here represents Player Two’s payoff.
Okay, so if the game was simple, if there are only two players and if the number of actions is small, like, corporation defection, only two actions, a game can be summarized by a simple table what is called payoff table.
So what happens if both players defect to confess and then the police finds out that their original intention was to set fire to the house? Okay.
What happens if player one defects and where player 2 cooperates? Well, the defected player who confessed is set free, he’s rewarded for telling the truth.
On the other hand, player two, who remained silent, was harshly punished.
Well, it may depend on what player two is going to do.
So let’s suppose player two is going to cooperate.
Okay? And what happens if you, player 1, switches from cooperation to defection? Okay, so if you switch from cooperation to defection your payoff changes from minus one to zero.
Is it clear? Okay, now suppose player 2 is going to defect.
In a Nash equilibrium, players are taking mutual best replies, but the best reply is always to defect.
OKAY, on the other hand, mutual cooperation is better for the group of two players as a whole.
2-2 Coordination Game and Self-Fulfilling Prophecy
Okay, in the second lecture I’m going to tell you about what is called the coordination game and I’m going to talk also about intriguing phenomenon called self-fulfilling prophecy, okay, so let me start with a very simple example.
Okay, so the choice of keyboard one choice is what is called the QWERTY, this is a design which everybody uses nowadays, so this is the design widely used, and there’s another design called Dvorak, and it’s designed for faster typing.
So that’s why it’s called the QWERTY keyboard, and on the other hand, the Dvorak was designed for faster typing, and one important characteristic of Dvorak is all vowels are located here, okay.
Okay, and so, QWERTY was not exclusively designed for optimal typing, fastest possible typing.
Okay, therefore the most efficient keyboard design is highly likely to be different from what we use today, okay? The most efficient design is likely to be different from QWERTY, okay? So, what about the Dvorak? Well, Dvorak was designed for a faster typing, but it may not be the optimal design.
Okay, so let’s consider choice between QWERTY and the optimal keyboard.
Okay the situation can be, again, summarized by a simple table, player’s table.
So let’s suppose their payoff’s are low, say zero, if they choose different designs, and if they both choose QWERTY they’re happier and their payoffs are one, and if they choose the optimal keyboard, they are even happier and their payoffs are two, okay.
Okay, so important lesson you should learn from this very simple example, if that game may have many Nash equilibria.
Okay, lesson number two, and moreover, one equilibrium say optimal equilibrium, can be better than the other, for all the players, ‘kay, so the same society may have good equilibrium for everybody, and the bad equilibrium for everybody.
Okay, so once the society is trapped in a bad equilibrium it’s very difficult to get out, but the very definition of Nash equilibrium, if you are the only person switching from QWERTY to the optimal design, it doesn’t pay.
Okay, so to move from bad equilibrium to good equilibrium, everybody should move simultaneously.
Okay, when new technology is invented, it may come in several different formats, okay.
One is High Definition DVD, HD DVD, the other is Blue-ray, okay? Those two formats are incompatible.
So if you have D, HD DVD player, you can’t really play Blue-ray discs and the same is true or vice versa, if you have Blue-ray recorder, player, you cannot play HD DVD. Okay, so the society has at least two equilibria.
In one equilibrium, everybody uses HD DVD. So here, HD DVD became de facto standard of the new generation DVD. The other equilibrium everybody uses is Blue-ray, okay? So there was a struggle for de facto standard, okay.
Pick the Blue-ray, buy Blue-Ray, and let’s see what actually happened, okay, eventually, people believed to this guy and Blue-ray dominated the market, okay.
Everybody uses HD DVD, everybody uses Blue-ray, and eventually Blue-ray prevailed, okay and which one eventually prevails depends on people’s expectations.
2-3 Market Competition
Okay, so far, we have seen many games, like a location game, prisoner dilemma, and coordination game.
Okay, so let’s look at the market for pencils, ‘kay, and the days that demand craft for pencil is the price is high demand is low, if price is low demand is high, and let’s suppose that unit cost of production is ten cents.
If there are lots of firms, and if price is above ten cents, okay, it’s profitable to produce a pencil, so lots of firms would like to produce pencils.
And economics textbook says that price and quantity in perfectly competitive market where there are lots of firms, okay, perfect competitive equilibrium is given by the intersection between demand and supply.
So if cost is c by a simple calculation, the total quantity of pencil is given by a minus c over b. Remember this value, it’s going to play an important role.
Okay, so game theory can show what’s going to happen if you have N firms, if na is equal to 3, game theory predicts what’s going to happen if you have three firms.
Okay, so it’s price minus quantity and it’s going to be given by this formula.
Q, small q here, is your output, okay, and large Q here is other firms’ total output.
So let’s draw a diagram of your profit, okay, so your profit is a function of your output, q, small q. So this is equal to constant times your profit, times your output, okay, so your pro, out, your profit is equal to constant times q minus constant times q times q. Okay, this is what is called quadratic equation, and the graph of quadratic function looks like this.
Your profit is 0 if your output is zero, okay, that’s clear, and let’s look at this part.
If your output is equal to this amount here, then this part here becomes 0 and your poor, profit is also 0, okay? So if your output is equal to this yellow number then your profit is 0, is it clear? Okay, and the your best reply is in between 0 and it is quantity, and since the graphic is symmetric, we have the same reply, it’s just the half of this yellow quantity.
Okay, so given this observation, let’s calculate the Nash equilibrium.
Since all firms are identical they have the same unit cost and Nash equilibrium has the property that every firm produces the same amount, okay? So let’s say q star is the Nash equilibrium output of each firm and the definition of Nash equilibrium says that this should be mutual best reply.
It’s half of this number here, a minus c over b minus total output of the remaining firms, okay, so since there are n firms, the number of other firms is n-1, and each of those n-1 firms are producing q star of Nash equilibrium, so therefore this is what we denoted by large Q, the total quantity of other firms, and this is your best reply, and at the Nash equilibrium everybody’s best responding each other, okay? So this is a simple equation.
So 2 times q star is equal to a minus c divided by b minus n minus 1 and q star, okay, so you move this term from right to left, and what you have here is N plus 1 times q star, equals to this number here, a minus c divided by b, okay.
So therefore q star is equal to 1 over n plus 1 times this number here, a minus c, divided by b, okay.
The total output at the Nash equilibrium was N firms is N times this number, so N over N plus 1 times this number here, a minus c divided by b, okay? So let me just rewrite this part, N over N plus 1, so N is equal to N plus 1 minus 1.
If there are few firms in the market, price is very high and quantity is small, but as the number of firms increase, price goes down and eventually it converges to competitive, perfect competitive market equilibrium, okay? So, large number of firms actually implies perfect competition.
2-4 Why Do People Come to Play Nash Equilibrium? Part I
Now it’s time to ask the following crucial question, why do people play a Nash equilibrium? In the following few lectures, I’m going to give you some answers.
So if there are two players, only two Player 1 is taking best re, reply against, you know, two, Player 2’s action.
Okay? Mutual best reply is a property of Nash equilibrium.
The question is how do players find out such a point? This is a question I’m going to ask.
So there are several reasons why players might play a Nash equilibrium and I’m going to give you three main reasons.
Maybe players are very rational and they, if rational player calculate what player should do.
Player number two says that, well player communicate before playing the game, about how to play the game.
So players get together and talk, how they should play the game.
The third reasoning is that players might not be so rational and they may not have an opportunity to talk to each other.
So they just play the game and outcome may not be Nash equilibrium.
Then they play the same game again or similar game and as they accumulate the experience, eventually they convert to a Nash equilibrium.
So player plays the game only once and before playing the game, they really think hard what they should do.
Rational thinking sometimes leads to a Nash equilibrium.
If suppose you are playing player one and suppose you think very hard what you should do in this game before playing Prisoner’s Dilemma.
If your, your opponent player two is going to play C, Corporation and if you switch from C to D. Your payoff increases from minus 1 to 0.
So given that your appointed play is Corporation C, your best reply is to play D. Okay? And same is true for the case where player two is going to defect.
So rational reasoning leads to a Nash equilibrium in a Prisoner’s Dilemma game.
So this game has two players, a man and a woman and they are going for a date.
Everybody go to football is a Nash equilibrium.
So there’s another equilibrium, shopping equilibrium.
Okay? And intuitively, it should be clear that rational reasoning alone does not tell which equilibrium to play.
So rational reasoning alone usually does not, you know, lead to a Nash equilibrium.
Okay? Rationality and correct to beliefs about other player’s actions lead to a Nash equilibrium.
Player, player’s reasoning alone is not sufficient to form a correct to beliefs.
I’m going to give you two more reasons why people play Nash equilibrium.
2-5 Why Do People Come to Play Nash Equilibrium? Part II
Let me con, let me continue my lecture on why people might play a Nash equilibrium.
In the last lecture two elements are crucial for players to play a Nash equilibrium.
Okay, so the first explanation of why people might play Nash equilibrium.
Well, if players don’t know how to play a game, a reasonable thing for them to do is to get together and talk about what they should do, okay? So before playing a game, player might get together and talk about what they should do.
Okay, so this reasoning pre-play communication leads to Nash equilibrium says that pre-play communication gives you the correct beliefs about what others are going to do.
In the two equilibrium, everybody goes to, you know, man and woman, go together for a football game.
The other equilibrium they go together for shopping.
When they don’t talk, it’s not so clear which equilibrium to play.
Okay, so discussion before playing this game, tomorrow.
Discussion may, you know the discussion may leads to a particular equilibrium.
Promise of going to football by man and woman will be kept without any reward or punishment because it’s a Nash equilibrium.
This is called self-enforcing agreement, okay? So one important interpretation of Nash equilibrium is that Nash equilibrium represents self-enforcing agreement, as in the game of dating, Battle of the Sexes.
So let’s suppose they reached an agreement that they are not going to confess in police interrogation, okay? This is a wonderful agreement, better for them, but it’s not enforced because it’s not a Nash equilibrium.
If an agreement is not a Nash equilibrium, it cannot be fulfilled.
It’s a very good promise but it’s not a Nash equilibrium.
I presented three reasoning why people might play a Nash equilibrium.
Dynamic adjustment by means of trial and error leads to a Nash equilibrium.
They won’t play Nash equilibrium, but if they encounter similar or same game again, they have better idea about how to play the game.
So accumulating experience in the same game or similar game eventually leads to a Nash equilibrium.
So by means of this trial and error adjustment, eventually you may go to a Nash equilibrium.
So we have seen a wonderful example where Nash equilibrium can be explained in this way.
So in the first week, we have seen the traffic allocation around this city, was closely you know, Nash equilibrium, closely explains the allocation of traffic around Hamamatsu city.
This Nash equilibrium didn’t come from rational reasoning of drivers.
Obviously, all those drivers didn’t have any opportunity to get together and talk about which equilibrium to play.
This Nash equilibrium emerged from trial-and-error adjustment of players.
Then people play Nash equilibrium of this traffic game.
2-6 Why Do People Come to Play Nash Equilibrium? Part III
And in the last lecture, I showed that dynamic adjustment by means of trial-and-error, may lead to a Nash equilibrium.
By means of trial-and-error, you can form correct belief about what others are going to do.
You can find out what is rational for you to play, okay.
By means of the local currency, 1,000 points is roughly equal to $10. Okay, in this auction game nine buyers simultaneously submit their bids, okay? And their bid is either 0, 5, 10, up to 1,000.
That’s the value of the object, okay? So each player selects one number from here and sum, submits this number bid simultaneously with others.
Then the buyer with the highest bid wins, and pays his bid.
So that’s the nature of this auction game.
Okay before I showing the example I have to say that in case of ties, okay, the winner is chosen randomly with an equal probability.
So let’s suppose player one submit that bid 500, player two submitted 990, and so on.
Okay in this example, who the winner is? Well the highest bid is 990, so player two is the winner and he obtains t-shirt and pays his bid.
Gain, is 10, okay? So that’s the nature of this auction game.
Is it clear? Okay, obviously this is not a Nash Equilibrium.
The winner, winner’s payoff, is five, okay? So, if you deviate from 995 to 1000, you are the winner.
If you bid, if you bid down from 995, you become loser and your pay-off goes down to 0.
Obviously there is another Nash equilibrium, where everybody bids 1,000, okay? So, in this case everybody is getting 0.
Nobody can possibly increase their payoff if everybody is bidding 1000.
By the same token, or more generally, if at least two buyers bid 1000, it’s a Nash equilibrium.
Let me show you the results, okay? So this is experimental result in the United States, okay.
Okay, so let’s see what happened to the experiment in the United States.
Okay, first they are bidding very low because bidding very low potentially they can get very large profit.
Then they are bidding up and up and up and eventually they converge to a Nash equilibrium around, in eighth round, okay? So, accumulating experience they learned how to play Nash equilibrium here.
This is an example where trial and error adjustment forces led to a Nash equilibrium, okay.
Well, in Tokyo, the convergence to Nash equilibrium was faster, okay.
2-7 Stylized Facts and Nash Equilibrium
So I have been giving you lots of reasons why people might play a Nash equilibrium, but now I have to address the issue that people might not play Nash equilibrium.
Okay, so probably I gave you three reasons why people might play at Nash equilibrium.
People might not be so rational, they may not play at Nash equilibrium but by accumulating experience in the same game, or similar games, they eventually might play a Nash equilibrium, okay? So, if adjustment process ever converges to a certain point, the outcome ought to be a Nash equilibrium.
Well, if the situation is not Nash equilibrium, there’s at least one player who can deviate to increase his payoff.
So if the adjustment ever settles on a certain point, that should be Nash Equilibrium, okay, that’s fine, but there is no guarantee that, you know, this adjustment process always converges, okay, so in that case, you know, the behavior might be chaotic, and even if adjustment process eventually does converge to a Nash equilibrium, it may take a very long time.
If there is no guarantee at Nash, that the Nash equilibrium always emerges, is the concept of Nash equilibrium really useful? Okay, and so we start to meet, that people might not play a Nash equilibrium so let me address this issue, okay.
Okay in general, stylized facts are likely to be Nash equilibria, why? ‘Kay, so if people are not following a Nash equilibrium by the very definition there’s always someone who can gain by deviating.
So if you find a stable mode of behavior, well which is repeatedly observed in a society, a stylized fact that is highly likely to be a Nash equilibrium, and the very important goal of any social science is to explain stylized fact and the stylized facts are likely to be Nash equilibrium.
2-8 Make Yourself Unpredictable: Mixed Strategy Equilibrium
So if you play rock paper and the scissors, you can choose one of the three strategies.
Rock strategy and paper strategy and the scissors strategy, okay? So, paper defeats rock, because paper can wrap up, rock.
Obviously, scissors cuts, paper, so scissors defeats paper, and the rock is stronger than scissors.
There is no mutual best reply in stone paper, and rock paper and scissors game.
So in this game of rock, paper, and scissors it’s very important to be unpredictable.
So let me explain what it is by means of rock, paper, and scissors game.
It’s probability one-third that you choose rock, it’s probability one-third the paper, it’s probability one-third the scissors as taken.
Okay, so let’s you know, write down the payoff table of two player rock, paper, scissors game, okay? So, if two players choose the same action, it’s a tie situation.
Let’s say in, in a situation of tie, they get zero payoff, okay? So let’s suppose so this, let’s suppose you are choosing rock, paper and scissors and your opponent is choosing rock, paper and scissors and let’s see what happens if your strategy is rock, okay? And if your opponent chooses paper, you lose, so your payoff is minus 1 and your opponent is getting 1.
So let’s examine what happens to you if your choice is rock, and let’s all say examining what happens to you if you choose this paper.
With probability of one-third, your opponent chooses paper and you lose.
Okay, so if you choose rock and if your opponents are equally mixing rock, paper and scissors.
If you take paper strategy, situation is very similar and by similar calculation, your expected payoff is also 0.
Okay, so given that your opponent is equally mixing rock, paper, and scissors, no matter which strategy you choose, your accept, expected payoff is always equal to 0.
Okay, so if your opponent chooses rock, paper, scissors with an equal probability.
So any mixture of rock, paper, and scissors provide you with exactly the same payoff of 0.
So anything is optimal for you given that your opponent is equally mixing rock, paper, and scissors.
Choosing rock, paper and scissors with an equal probability is one of the many best replies.
Is it clear? So therefore, your opponent, equally mixing rock, paper and scissors and you are also equally mixing rock, paper and scissors.
Okay, to sum up this observed behavior, equal mixing of rock, paper, and scissors is a Nash equilibrium in random strategy, and it’s called a mixed strategy equilibrium.
2-9 Sports Games and Game Theory
Probably a good example is penalty kick in football or soccer game.
Penalty kick is a part of soccer game where one player, kicker, is playing this game of penalty kick against the other player, the goalkeeper, or goalie.
The strategies could be very complicated but usually either the kicker kicks to the right side, or the left side.
So we can safely assume that kicker, one of the players, have, has two strategies, kick to the left and kick to the right.
Ideally, the goalkeeper can see which direction the ball is coming and he can jump to the right side, okay? But in practice, the ball is coming very fast, so the goalkeeper has to guess in advance, okay? So simultaneously with the kicking, goalkeeper, goalie can easily jump to right or jump to the left.
So it’s a simultaneous move game with two players, goalkeeper, goalie and the kicker simultaneously choose one of the two strategies.
If kicker always kicks to the same side to, to the left for example, it’s predicted by goalie, and his goal is lost.
So let’s analyze the Nash equilibrium in penalty kick.
The kicker has two strategies, kicking to the left and kicking to the right.
Goalie has two strategies, jumping to the left or jumping to right, okay? And then you can analyze or, or you can examine the probability that kicker wins.
What is the probability that kicker wins when kicker kick, kicks the ball to the left side, and goalie jumps to the same side? You have certain number a here, and you have different numbers b, c, d for other combinational strategies, okay? So first task for you to analyze mixed strategy equilibrium to, is to get those numbers.
The kicker is trying to maximize the probability that he wins, okay? So, this is the payoff for kicker.
On the other hand, the goal keeper, goalie, is trying to minimize the probability that kicker wins.
There is actually an empirical study conducted by a professor now at the LSE, Ignacio Palacios Huerta, and he collected a large number of penalty kick data.
It’s, the data set contains 1,417 penalty kicks in professional soccer games in Europe, Spain, Italy, UK, and other countries between September 1995 and June 2000.
Okay, so by looking at those large number of penalty kicks you can estimate those four numbers, probability of winning of the kicker.
So if kicker kicks the ball to the left of side of goalkeeper, and the goalie jump to the, the same side, it’s likely that goalkeeper catches the ball.
So the winning rate of the kicker is not so high, okay? So winning rate is 58%. 58.30%. One the other hand, if kicker kicks the ball to the left and goalie jumps to the wrong side, then kicker can win.
Okay, computing mixed strategy equilibrium in penalty kick.
So let’s as in rock, paper, scissors game, given that Goalie is choosing left and right with this probability, let’s examine the kicker’s expected payoff when he kicks to left.
Okay, so if left is kicker’s choice with probability p, he wins with this probability.
Since kicker is mixing left and right, that means, you know, left and right are equally good for him.
Okay, you can perform similar calculation for kicker’s behavior.
Okay given the pay-off table, Goalie has a higher chance of jumping to right, and Kicker also has a higher chance of kicking to right.
Just as in rock, paper, and scissors game, goalies and kickers in soccer game are trying to be unpredictable, and they are playing, mixed strategy equilibrium.
2-10 Nash Equilibrium Exists in All Games
So John Nash discovered that any game has a Nash equilibrium.
Okay? Nash equilibrium exists under certain conditions.
Okay? So precise statement about the existence of Nash equilibrium is as follow.
The first conditions says, there should be finitely many players.
The second requirement each player has finitely many strategies, so let me examine those two conditions one by one.
The first condition, there are finitely many players, this condition is always, almost always satisfied.
Second condition, each player has finitely many strategies.
Okay, so the location game had a two players, two ice cream vendors, A and B. And they are choosing a location on a street, okay? So, location is a continuous object.
Okay, so even if a game has many strategies such a game can be well approximated by a model with finitely many strategies, okay? So therefore this second condition is not so restricted, okay? So the two conditions are practically, always satisfied, so therefore Nash discovered, that any game has a Nash equilibrium point, where players are doing their best, against others.
Nash’s great insight is finding generally, applicable principle in any social situation, what people is going to do, okay? It’s given by the analogy between people optimal strategy, and a vortex point on the surface of coffee.
This, see, this point here is the original mode of behavior of players.
This point here, is, the situation where players take better replies, against the original behavior of other people.
Matching pennies has two players, let’s say player 1 and the 2, 1 and 2.
If two players chose the same side, heads on heads, then if you have same sides, player 1 wins.
If the 2 players chose, or showed different sides, say, heads and tails.
Different sides, player 2 wins, ‘kay? It’s a very simple, game which is very similar to rock, paper, and scissor.
Let’s consider how we can represent people’s behavior, or player’s behavior in matching pennies game.
So let’s the about the probability of choosing say, heads, and the probability of choosing tails, for each player.
Okay, so in this graph I’m going to measure by the horizontal axis probability, that player 1 chooses heads.
Okay? The probability is in between 0 and 1.
I’m going to, you know, represent player 2s probability to choose heads by the vertical axis.
Okay, so the square here is the set of all possible random behaviors of player 1 and 2.
So at this point both players are choosing heads, it’s probability one.
That this middle point most player, as mixing heads and the tails with the equal probability, and there are lots of other possibilities.
Okay, so as you can see in this set here, square here is very similar to coffee service.
Okay that’s the starting point of an Nash’s discovery.
Okay? If two players show the same side then player 1 wins.
So the vertical axis measures player 2s probability of choosing heads.
So if the probability is high, your opponent, player 2 is choosing heads without, with very larger probability.
Okay? So your best reply is going in this direction.
Okay? By the same token, if the probability that other player choosing heads is low, that means is a larger probability your opponent is going to choose tails.
So this is the nature of the better reply for player 1.
If player 1 is choosing heads, is a large probability, player 2 is like to choose tails, okay? So, it’s good for player 2 to decrease his probability of choosing heads.
So this is the nature of player 2s best, better reply.
The opposite is true, when the probability of a heads by player 1 is small.
A middle point here, where both players are equally mixing heads and tales, is a mixed strategy Nash equilibrium.
