 Okay, so we want to predict how human individuals behave and we can think of different situations where these apply. So you have a different situation in which people may interact with another person or more persons and each of these persons is confronted with a strategic decision because the outcome does not only depend on what he does but also on how the other people behave. Okay, and so in some sense what you have to do is to predict how other people are going to behave and behave in an optimal manner. Okay, so this is what game theory is about. Okay, so how do you formalize a game? One way of thinking about a game is you look at the prototype system of a game which is chess. Okay, so in chess you have the players and there are some rules and these rules have to do with what actions, what possible actions can each player do at each point in time and with information what does each player knows at each point in time. Okay, and then what agents choose is actually strategies and a strategy is a contingent plan of action. So for example if you think about chess, so players do not choose just actions independently at each time but they follow a particular strategy or for example for the openings, there are different codified strategies and a strategy is a contingent plan of action which means it's a sequence of actions that depend on the action of the other players, the previous action of the other players and depend on also on the information that the agent has. Okay, so a strategy is a function in a very high dimensional space and then you have also to define the outcomes, so when is it that the game ends and what are the payoffs that each player receives when the game ends. So one important, very important thing that you assume is what is called common knowledge. Common knowledge is what everybody knows and what everybody knows that everybody knows and what everybody knows that everybody knows that everybody knows. So it's something that is common knowledge. Okay, and what we assume is that the rules of the game and the strategies and the outcomes and the payoffs, this is all common knowledge. So everybody, every player knows what the rules are. Okay, then also we assume that each player assumes that the other players is rational and they are rational themselves and they know that the other player knows that they are rational and they know that the other players knows that they know that the other player is rational etc etc. Okay, so rationality this common knowledge. So this is very important so and under this so a game theory is essentially a theory that predicts how players will behave, how players will play this game under these assumptions. Okay so let's see how it works. So this is a very simple game which is called matching pennies. There are two players, player one and player two. So player one chooses the side of a coin whether it's head or tail and player two also chooses the side of the coin whether it's head or tail. And player one wins if the two coins are different, if the two sides of the coins are different and player two wins if they are the same. Okay so one way of representing this game is what is called an extensive form game. An extensive form game can be represented as a graph where you have the action nodes, so the nodes where each of these nodes, one of the two players, have to take an action. Then links correspond to the possible action that an agent can take. So here is player one, he has to choose whether his coin will be head or tail. And then depending on two, it's time for player two to decide whether his coin will be head or tail. And then depending on this, each of them will receive payoff. And here this is the payoff of player one, which is minus one because the two coins are the same. And this is the payoff of player two. Okay so as you can see, so this is the way which you can represent any game. Of course, you can also represent chess in this way, but you will have a very, very complex three, now a very, very complex graph. Okay so the strategies are, as I told you, are contingent plans of action. So in this case, a strategy for player two will be to play head if the other guy, if player one played head, then to play tail if player one played tail. So you see that a strategy is a plan, is a way to respond that depends on the action of the other agent. Okay now in this case, we assume that player two knows, so information is also important, here we assume that player two knows what player one chooses before he makes his choice. But you could also think about actually matching pennies game is typically played simultaneously. So where player two does not know what is the choice of player one. Okay so in this case, in game theory, in extensive form game theory, one represents this fact that agent two does not know whether he's in this position or in this position by drawing these information sets. So these information sets, you should think of them as a cloud of fog that makes it impossible for player two to understand whether he's at this point or at this point. Okay so these are extensive form game, but then given an extensive form game, you can figure out what are the possible strategies. And then you can represent the same game in terms of what is called a normal form game. And the normal form game is this form, so you have a set of strategy, SI, that as I told you are plans of actions. And you list all of them. And then you list all of them also for your opponents. Here this s minus i means the strategies of all the other opponents. So you have a set of possible strategy for agent i. And then you can build the set of all possible strategies for all the other players. And then the strategy of the opponent will be an element of this set. And so you can fully define your game by defining directly what is the payoff for each player i if you play strategy SI against strategy s minus i. And this is a real function that corresponds to these numbers here. So this is essentially how you define a game in game theory. So this is a celebrated example, which is called a prisoner s dilemma. And you can think this is a formalization of the situation of Bonnie and Clyde, which was in the first slide. And the story goes that, say, if player one and player two, imagine that player one and player two are, say, caught and they are charged of having committed a crime. So player one has the possibility of to confess to the police and to say that the other guy, player two, did commit the crime. And or to stay silent, not to say anything, not to confess. In this case, the strategy C is not to confess as the strategy D, which is called defect to defect and C stands for cooperate. I mean, cooperate with player two. So the GD is the one where player one confesses. And if they both don't confess, they stay in prison for a while, but then they are released. If player one confesses and player two does not, then he will get free. He will be free. But player two will go in jail for a long period, say nine years. Okay. But if they both confess, they both will go in prison for, say, six years. So you can represent, in this simple case, this play, this game in normal form with this table, where in each cell, you put the first number is the payoff of the first player, given the strategy that the first player chooses and the second player chooses. And in the second number, you put the payoff of the second guy. Okay. Okay. So now you can think clearly that if you look at this situation, well, it is clear what they should do. Because if they both do not confess, if they both cooperate, they would just spend one year in prison, say a small amount in prison, and then they will get free. But actually, what happens in this case is that whatever player one does, it is always best for player two to defect or to, in this case, to confess. Because imagine that player one does not confess, so he cooperates. Then if he cooperates, player two would get minus one. But if he defects, it will get zero. So it is better for him to play defect. Okay. Then what if player one defects, then if he cooperates, it would get minus nine. And if he defects, it would get minus six. Okay. So whatever player one does, player two will be better off by defecting. And the same is true for player one. So the outcome is that in spite of the fact that this would be the best outcome for both of the player, because they are rational, they will end up playing this, they will end up in this situation. Okay. And this is because this strategy C is dominated by strategy D, because whatever the opponent does, strategy D always gives a higher payout. So one idea of finding out how players play games is to look at dominated strategies and to eliminate these dominated strategies. And then the prediction would be that what remains is what players will play. And you can do this process alternatively and look for some strategy which is dominated by some other strategy and eliminate them alternatively. And in this way, you can think of solving the game. So maybe it's a good time to ask if there are questions. Yes, Professor, can you hear me? Yes. Okay. So I saw the lectures earlier and I'm loving them. I think they're really mind blowing, at least in my estimation. So regarding the prisoner's dilemma, for instance, but also other things that we discussed about, it struck me as a bit odd that we've now, this is maybe a naive question, maybe I'm getting a bit ahead of myself, but we never once mentioned the concept of a trustworthiness, trust. And it seems to me that by introducing this, a lot of the so-called dilemmas- So, Carlos, so now we are talking about Bonnie and Clyde. They love each other. Okay. And they say, and they can promise to them, to each other, look, I will, I will never betray you. I will never confess. Okay. Yes. But when they get into, when they are caught and they talk to the police, should they really trust the other guy or not? I mean, if they are really rational, and this is the assumption that we are making, then no matter what they promise to each other, they will behave in this way. I mean, the issue of trust is something that we will come back, we will come back, because it's a very important one. Yes. And also, it struck me also that, well, the argument for Bonnie and Clyde that seems to be true, but I think it can only hold up in isolated single instantiation, like in single games, because if the games are iterated, then this sort of argument cannot work, because if I work with someone and I snitch on them, exactly. Yes, exactly. You are precisely right. So we are talking about a single stage game, a situation where player one and player two, imagine they didn't meet before, and they will never meet again. Okay. Okay. And then in that case, everything that player one says to player two in this kind of game should not be trusted. Okay. There is no reason why player two should trust player one or vice versa. So it's not credible. Okay. Yes. Okay. Okay. So everyone is a bit psychopathic in these scenarios. Okay. Thank you. So concept of trust, there is a question from Ayush Gautam. Are all concepts concept like trust considered irrational? I assume rational decision take only the possible possible outcomes into account. So let me turn this question the other way around. So in game theory, you'll try to find whether there are incentives that support trust. So is trust supported by incentives? So imagine that say I may tell you, look, either you give me 100 euros or I will jump out of the window. Okay. And of course, you should not trust me because I mean jumping out of the window is against my own interest. Okay. Then of course, I mean, I'm not considering all the psychological dimension, the fact that I may be crazy, et cetera, et cetera. But if you just think of a rational behavior, then you should not trust me. Okay. Sorry, there was another question. So now there is a, there is a rajat asking why are, and precisely why they are not using backward induction. Okay. So this is going ahead. I mean, backward induction is one of the strategy in which you can solve in which you can solve games like this one, for example. I mean, if you start from the top of the tree, you can find out what is the best option for player two and then figure out what, what is the payoff that player two will get in this point and in this point. And then you can figure out what player one will do. Okay. So this is called backward induction. Now, in prison as dilemma, things are very easy because there is a dominated strategy. So you don't need to use backward induction. And if you use backward induction, you get at the same situation, at the same outcome. So players cooperate and they are charged for what they were caught for. Okay. So you are answering to another question. So, other questions? Let me see. So what is minus six, minus six? Yes. So you have been asked, answered by people. So we think about these numbers here as being a payoff. So you can think of this as a monetary reward or as anything, anything else. Okay. So, very good. So the problem with iterated elimination of dominated strategies is that it doesn't work always. So, and so what John Nash did was a certain point to formalize what is a good notion of a solution for a game. And this solution is called Nash equilibrium. So you say that a strategy profile, profile means that this is a, this S is a list of strategies for each of the players. This is a Nash equilibrium. If for all players and for all strategies of all these players, the payoff of player Y, if he plays another strategy, which is different from the strategy he plays in the Nash equilibrium, cannot be larger. So this means that each player has no incentive to deviate from a Nash equilibrium strategy. Okay. That's the early presented game concept applied for games with low amount of payoffs when the game is based on a morality question, based on experience, which participation with this types of game, it's easy to select the morality correct decision if the payoff difference is not large, especially monetary. So with respect to this, we will come to that about discussing about say morality. At the moment we are not considering morality, we are just considering what should rational players do. Okay. Okay. So Nash equilibrium is a situation such that no players have incentive to deviate from this setting. Okay. And one way in which you can find this Nash equilibrium is by finding the best response. So you define the best response of agent I to the strategy of her opponents as the strategy that maximizes easy utility or heavy utility, given the strategy of the other. So this is the best response. Okay. And if you think a little bit about it, so the Nash equilibrium is equivalent to saying that my best response. So I should play the best response to the best response of my opponents. Okay. Or in other words, that the Nash equilibrium is a fixed point of this best response correspondence of this best response mapping. Okay. And so, for example, in the prisoner's dilemma, you see that, well, the best response for player two is always the factor is the fact and the factor where player one plays C or D. And the same is true for player one. So that you see this is a fixed point of the best response correspondence. Okay. Very good. So now I want you to play a game. Just look at this game and find what is the Nash equilibrium. Then you should go to this site. And you should insert this code and submit your answer. So let's see whether you've got the concept of Nash equilibrium. So I'll wait until we have collected enough responses. Okay. See, we are at the moment, we are 118. Let's see, I want to see at least 50 responses. I think probably you also need to add. The screen has disappeared. Hello. The screen has disappeared. Yes. Let me put it again. What is it? Okay. Here it is. So here is the game. So I still see five answers only. So come on. Only five answers. You can do better. Okay. Six answers. Come on, seven. Okay. You are thinking very hard on this problem. So 12, 13. Come on. Okay. So, okay. Now I think you can vote. Sorry. It's the first time I'm using this thing. Okay. We have just two answers for the moment now. Okay. So seven answers. Things are a little bit changing. So let me stop when there are 20 answers. We are 15. Okay. So maybe what I can do is to share with you, if I can share with you the results. Okay. So I think we are, so let me summarize for you what the results are. We have 34 answers. So the correct answer is this one. It's 33. Because this is the best response. You see, if player one plays B, then player two, the best option for player two is to play R. Okay. And if player two plays R, then the best response for player one is to play B. So this is the Nash equilibrium. So this got 44% of the votes. Okay. So at least the majority, I mean, almost the majority of you got it right. Then there was a large part of you, 36% said that the Nash equilibrium is this one. But you see that say if, for example, player one plays top, this strategy T, then player two would be better off by playing C. Okay. So this is not the best strategy of player two. And if player two plays L, player one would like to switch to five. Okay. So this is not a Nash equilibrium. Although the payoffs are larger than in the Nash equilibrium. Then Professor. Yeah. But if player one plays top, and player two plays center, then player one wins. So why would player two prefer C? No, because, because then T is not a best response. Sorry. C is not a best response to T. So the best response of player two, if player one plays T is L. Oh, yes. Okay. But if player two plays L, player one would like to play M. Okay. And if player one plays M, player two would like to play C. And if player two plays C, player one would like to play T. Okay. Okay. So, and you are in this, in this loop. Okay. So there is no, there is no Nash equilibrium in this sense. Okay. Is it clear? Yes. Yes. Yes. Thank you. Okay. So thank you very much for this exercise. I think you, since you liked it, then I think we are going to go to the issue of, okay. So this was the right response. And these are the best responses. You can see that there is no fixed point here. Now, let's go to another simple exercise, which is the ultimatum game. And this brings me to the issue of morality. Okay. So now the ultimatum game is as follows. So there are two people. So Alice and Bob. Alice has the possibility to get 100 euros. And the choice she has to make is to give an amount of this to Bob. Now Bob can decide either to accept or to refuse. So if Bob refuses, then they will both get nothing. But if Bob accepts, then he will get what Alice gives you. And Alice will keep the rest. Okay. So you should think again that Alice and Bob are rational and that they never met each other and they will never meet again. They are not brothers and sisters. They are not wife and husband or anything. And this is the payoff of, I mean, it is the payoff matrix. Okay. This is the game. So the strategy of Alice goes from zero, one, two, up to 100. This is how much she decides to give to Bob. And the strategy of Bob is either to accept or to refuse. So again, think about what is the Nash equilibrium, and then go to Slido and submit your answer. Okay. So very good. So we have five answers, 18 answers. Okay. This is very interesting. Okay. So we have almost 33 answers. But I think the result is very clear. Okay. So the vast majority of you thinks that the Nash equilibrium should be 50-50. So Alice should give to Bob 50 euros and keep 50 for herself. 66% think that this should be the rational outcome. Now, actually, the Nash equilibrium is when Alice gives zero to Bob. And Bob accept it, because essentially you see in this situation where Alice, what is the best response of Alice? What is the best response of Bob if Alice gives zero? Well, the best response is both accept and refuse. Okay. So that this, as a matter of fact, this is a Nash equilibrium. Okay. So the best response of Bob is always to accept whatever Alice offers, whatever Alice offers, because whatever is better than nothing. Okay. Is this clear? Now you see the tension between what is rational and what is fair. You may say, well, this Nash equilibrium is not very fair. Okay, because Alice is very stingy. So it is not behaving properly. Okay. So indeed, if you do experiments like what we have done on the ultimatum game, you find out that most of the people will not play the Nash equilibrium. And there is a lot of people who have been thinking about why is it that people do not play the Nash equilibrium? Aren't they rational? Aren't they play rationally? And there are a lot of answers to this. One reason is that actually when we are in a strategic context, we do not really think hard about how to behave, but we apply social norms. And as a social norm, being generous is a good social norm. Because essentially, and we have learned this because of reciprocity. So because if I'm kind with you, you will be kind with me. And if I'm kind with a person in the street, a person in the streets will be kind with me. This is a norm of the society. Okay. So but this has nothing to do with rationality or needs different explanation. If you want to explain this behavior, that is not explained by rationality. Excuse me. Yeah, please. In this game, does Bob know about the deal he knows about the amount of money? Yes, that is common knowledge. So he knows. He knows. Yes. So you would think that Bob refuses because he gets offended or because he wants to punish Alice. So these are not rational behaviors. But if Bob knows about the amount, it is very probable that refusing the deal when Alice gave him nothing. Yes. So indeed, the issue is exactly this. So that in this type of situation, Alice should not assume that Bob is rational. And so she should not behave in a rational manner. Okay. So that probably she should think, well, maybe if I give him 10, he will not refuse. Ah, I see. And so if I give him, say, then of course, maybe Alice can frame this offer in a particular way and Bob will accept. I don't know. But the point is that if Alice knows that Bob is rational and Bob knows that Alice is rational, then Bob should not get offended by Alice's behavior because she's behaving rationally. And he should accept even if he gets zero. I see. Okay, thank you. The issue is really game theory deals with describing the behavior of rational individuals. Okay. But there are situations where it is not reasonable to believe that people are behaving rationally. Okay. I have a question. Yeah, please. Can Bob use this model to actually bargain about the money Alice gives her? For example, if she gives me under 15, I always say no. Yes. So this is the same as in the as in the prisoner's dilemma. No. So Bob could threaten Alice and say, if you give me less than 50, I will refuse. Okay. Then Alice may offer him 49. Should Bob refuse or not? I mean. He should not accept. Sorry. He should not accept. If she gives her 49.9, he should not accept. Yes, but this is not rational. This is not rational. And it is not. I mean, if Alice knows that Bob is rational and Bob tells him, look, I'm not going to accept anything below 50. Then Alice should not find this statement credible. Okay. It is not credible because it is in the it is against the interest of Bob himself to behave in this manner. Thank you. Excuse me, Professor. So I only I can understand totally the argument that well, something is better than nothing. But so if I would if I would have guessed, I would have maybe put a one and 99 because I don't see. I don't know. I don't know. I don't see why zero and 100. So the box with zero and 100 is rational. But the box with one and 99 is irrational. Well, I don't know if the best is not the best response. Sorry. It's not the best response of Alice to the strategy A of Bob. If Alice thinks that Bob is playing A, the best response of Alice is to play zero. Okay. Could you repeat that, please? I was. If Alice, if Bob is playing A, the best response of Alice is to play zero. Oh, yes. Yes. Yes. Yes. Okay. Yes. And so the fixed point of the best response is zero 100. Okay, I get it. When when you have to compute a Nash equilibrium in a game like this, you have to look at the fixed point of the best responses. Then you can think whether this makes sense, whether this you can make all the speculation, but the Nash equilibrium is defined in this way. And this is the way in which rational agents, rational individual will behave. Okay. I got it. Thank you. Awesome. Okay. So very good. So in the lecture notes, in the lecture that you find on the website, there is also this discussion about the tragedy of the commons, which is essentially the same situation as the prisoner's dilemma, but in a different context when there are many agents. And again, it's a situation where many players interact. And the social optimum is very different from the Nash equilibrium. So rational individuals end up overexploiting a system. And at the end of it, getting an outcome, which is much worse than what they would get at the social optimum. And this is essentially what happens whenever you have problems which involve common goods. Okay. Common goods are goods that essentially everybody can use. And everybody can exploit. Okay. But if everybody does so, then the quality of this good deteriorates. And nobody, there is no provision of this good by individuals. Okay. So this is a very interesting example and very relevant for economic behavior. Say for example, climate change is one of these problems, where essentially carbon emissions so the atmosphere is one and essentially everybody has incentives to emit because of industry production. But if everybody does so, then the atmosphere will get very bad and we will get global warming and okay, and we will have a lot of problems. Okay. So that is a situation where essentially the Nash equilibrium, the individual behavior, rational individual behavior leads to very bad outcomes. Okay. Okay. So now you can think about the Nash equilibrium that we define is always unique and the answer is clearly no. And one example of this is what is called the battle of the sexism. So this is the game between two persons and she wants to go to the opera and he wants to go to boxing max. And but they love each other. So they better go hang out together then just one go to the opera and the other goes to boxing. Okay. So the payoff matrix can be something like this. And so if this is she, she prefers to go to the opera to the boxing and he prefers to go to boxing and to the opera. And you see that there are two Nash equilibria. One is where both go to the opera and the other is where both go to boxing. Okay. And so there is another situation where you see that Nash equilibrium as we have defined it does not always exist. And this is the situation of matching pennies. So matching pennies is well the problem we have defined at the beginning. And you see that if I play head, then the best response of player two is to play tail, but the best response of player one is to play tail, then head tail, etc. So there is no fixed point of best responses. Then you ask why is this situation. So there is no Nash equilibrium here. Okay. But what you can realize is that when this is think you should expect this because this is the same situation as when you have a penalty kick in football. Okay. So the player has to choose whether to kick on the left or on the right. And the goalkeeper has to decide whether to go on the left or on the right. Okay. And the goalkeeper wins if he goes in the same direction as the player. And the player wins in the other situation. Okay. So what is the best strategy for the player in this case? Well, it should behave in a way that is as random as possible. So he should not, he should play this, he should kick this penalty in a way that the goalkeeper has no clue of whether he's going to go on the left or on the right. Okay. And he did, this is the idea also in this matching pennies. So then you should introduce the random randomized strategies. These are called mixed strategies. And a mixed strategy is nothing but a probability distribution on the possible strategies. Okay. So these strategies now are called pure strategies and these are probability distribution called mixed strategies. So you can define a mixed strategy of an agent, the mixed strategy of the opponent. And you can generalize the notion of the payoff matrix to the expected payoff. Okay. So this is the expected payoff under mixed strategy sigma i. Again, mixed strategy sigma j. Okay. And likewise, you can generalize the notion of the best response. It's just the best response to a mixed strategy sigma minus i is what maximizes the expected payoff. Okay. And then you can define and now what a Nash equilibrium is. And it's exactly the same as before. It's essentially a Nash equilibrium is a situation where whatever you play, you cannot increase your expected payoff. Okay. And then the final thing. So, okay, so this is an example of matching pennies, how you compute this for matching pennies. And maybe let me discuss this and then I'll just flash the result, the main result. Okay. So how do you compute the Nash equilibrium mixed strategy? So imagine that you have matching pennies and player one plays head with probability r and tail with probability one minus r. Player two plays head with probability q and tail with probability one minus q. Okay. So then what you have to do is to find out what is the expected payoff of player one if you place a mixed strategy r against a mixed strategy q. And you do this by computing what is the expected payoff of playing head against q or playing tail against q. And what you find is essentially that this is two r minus one times two q minus one. And this is the opposite of what player two gets. Okay. So given this function here, you can find out what is the best response for player one if he plays against q. Okay. And you find that this has this shape. So the best response of player one against q is this line here is essentially playing r equal to zero. So playing tail as long as player two plays head with probability less than one half because then it will get this one with higher probability. And then to play tail with probability to play head sorry with probability one if q is larger than one half. And the best response is exactly this line this yellow line here. Okay. You can do the same for player two and find what is the best response of player two against r. Then you plot the best response of player two on these axes and against r on the other axis. And here you get another line like this. And you find that the fixed point is exactly at the point one half one half, which means that the two players should randomize. And this is the only Nash equilibrium. Okay. So and to finish with this essentially so Nash proved in 1950 that any finite normal form games admits at least one Nash equilibrium in mixed strategies. So if you have a finite game there will always be a Nash equilibrium in mixed strategies. Okay. So this is what I wanted to tell you about today. So as an exercise you can compute Nash equilibria all the Nash equilibria in the battle of the sexes. And I think we have to stop here maybe if there are a couple of questions we can address them. Let's see what is in the chat. So there are still discussing about the ultimatum games. I think this discussion can be endless. So is there any burning question? I remind you that you can find these things discussed more in detail on the website. And but if there is a burning question then we can address them or otherwise take a little bit of break. Professor I have a question. Yes please. I was just wondering what happens when our concept of rationality instead of relies on optimizing for the individual becomes objective for the coming out. Okay so this is so what I'm discussing here is actually called non-cooperative game theory. Okay non-cooperative because essentially players do not cooperate. Okay there is another branch of game theory which is called the cooperative game theory which is the branch of game theory that tells you how you should for example design a treaty between different parties between different nations for examples. And that is a different subject that I'm not discussing here. So what we are discussing here is non-cooperative games. Thank you. Thank you for the answer for the question because this is very important. Professor. Yeah. I have a question. I remember studying Nash equilibrium in the context of networks. Is there a difference between Nash equilibrium in networks and in this context of game theory or is it basically the same thing or is there a plus in studying with networks or something like that? Okay so this is the definition of games that you find in books of game theory. Okay so this is how a game is defined. Then there has been a lot of research and a lot of work by different people by studying games on networks. Okay so putting games on network means that you have to define this way of matrices and these strategies for agents on the network. Okay and which is not very so I mean what they would like to say is that as you have seen is that already in a simple situation the mathematics can be very complicated. Okay so if you and you can extend what I've been saying here to any complicated situation but the question is whether you will be able to compute Nash equilibrium or not. Okay in many cases people have studied simple processes like for example you can study what is called the water model. So the water model is a model of opinion dynamics and you can study it on the on the network on a network. Okay so this is not a game. I mean it's not a it's not a model that you can really define as being a really a game because if you think about it well there is no utility function there is no rational really rational behavior behind it. Okay so yeah so I don't know whether I upped something or I made more confusion. No I think I think it's it's it's clear right now thank you. Okay so for mixed strategies the Nash equilibrium is now dependent on the probability Q of using a strategy yes the Nash equilibrium specifies what is the mixed strategy. So by the way the strategies S is a particular case of a mixed strategy when Q is equal to zero or Q is equal to one. Okay so mixed strategy when you go from strategy to mixed strategy you enlarge the space of possible strategies. Okay but the original strategies are still there. Okay so I think we need to take at least five minutes break before Guido's lecture. So thank you very much and we'll meet again tomorrow.