 Okay, so let's start with a short recap of game theory. So, as we have said, so the game is a model of a strategic interaction between individuals. And it is formalized by a set of strategies for each of the players, my way of functions. The main result is a national team in 1950. It says that every finite normal form game, at least one national equilibrium in mixed strategies. Okay, so. So, so, so for example, you can consider the battle of the sexes and find out what are the national equilibrium as we said, one has to look at what is the best response of each player. The strategies of the opponent and by finding a fixed point of the best response correspondence, one finds a national equilibrium. Okay, so there is another way in which we can think about finding find. National equilibrium, which is what is called the backward induction. So let me explain that for induction for this very simple case. So this is a game where player one first have to choose L or R. And player two, if player one choose L, the game finishes and this is the payoff. If player one plays R, then player two is player two turn, you can choose X or Y. And if you choose Y, then it's again the turn of player one. And player one can choose A or B. These are the terminal notes. Okay, so in backward induction you proceed from the end of the game and you move upward. Okay, so you, you start from the end of the node. And then you ask yourself, what is the best action of player one at this node. Here it would get three, here it would get zero, so the best action here would be A. And then you can go up and now you can consider what player two has to choose. So if he chooses X, he will get one. If he chooses Y, he can anticipate that one will choose A and so it will get zero. So the best action for player two is to choose X. And then you can go one step up and now player one knows that player two will choose X. So he has to choose between payoff and this payoff and since two is larger than one, he will choose this payoff. So this L would be his best action. Okay, so you see that indeed this is the Nash equilibrium of this game. You can also find this out by writing this game in normal form. In normal form, you should write a table where you have the payoff of the two players, depending on the possible strategies of the two players. So in this case player one has two choices, this one and this one. So he has four possible strategies. And when you write this game in this form and you see what are the best response of each player to the strategy of the opponent. And then you'll find that the fixed point of the best response is either this one or this one. So essentially it's coincides with this Nash. Okay, so now, if you think a little bit more about this construction. However, there is some subtlety. So for example, should player two, you mentioned that player two is at this point. And then at this point he has to choose between X and Y. Should he still believe that player one is rational because if he were rational, he would not get you, he would not find himself at this point. Okay, because the player one should have chosen L. So then, if he does not think that player one is rational, then he may think that when he will choose at this point, he may choose B instead of A. And so you see this doubt on the rationality of player one. And this is really an interesting feature and actually it could be exploited also by player one because you could say that if player one actually chooses R instead of L. There is still this doubt on his rationality. And with the idea that maybe player two will choose Y instead of X. So he could get three instead of two. Okay, so again, there are a lot of this type of complications. I'm not going into too much detail but just want to give you an idea of the type of complication that may arise. Now, there is another situation that another comment I want to make which is about what is called sub game perfection. So let's analyze this game. So this is a game where a player one first moves. Then, if he chooses R then he has to move again. And then player two has to move. But player two does not know what player one has chosen between A and B. So, and then you can solve this problem, this game by backward induction and find out that there is a unique Nash equilibrium, which is three and four is this one. Because this is the best response of player two, both in this case and in this case. And then the best response of player one in this case is B, etc, etc. And now you can represent this as a normal form game. And if you do that, then you end up with this table here. Okay. Now, if you look at this table again, you see that three and four is a Nash equilibrium because this is the best response. Why is the best response of two to strategy RV and RV is the best response of player one strategy why. But if you look at what are the other best response, then you'll find out that there is another Nash equilibrium, which is essentially two six and two six. So it corresponds to this point here. She's not a Nash equilibrium that according to backward induction. But when you write this game in terms of a normal form game, then what you realize is that there is actually this Nash equilibrium. So it's a different nature than this one. And it is a different nation because the same theory jargon is not a sub game perfect because they a game, a national game is a game perfect. If it is consistent with every sub game. Okay, so if you consider here only the sub game that stems from this site here. You see that essentially the only Nash equilibrium that is the same in all sub games is essentially before. Now, however, you see that this national equilibrium is more convenient for a player to help because it would get six instead of four. So the question is, maybe, player two should say to player one. Look, I'm going to play X, no matter what. Okay, no matter what you do, I'm going to play X. And then, if player one, first, what player two is saying, then what he should do is essentially to play LA or to play L. Okay, and then they will get into this. Okay, so the question is now. Do you think that player one should trust player two or not. Okay, so I hope the question is clear. Now, essentially what we can do is to go to this. I see yesterday, and you can add your your, your answer. Okay. Okay. So we have one answer up now. Two answers. Okay. Very good. 15 answers. Okay, so essentially, now we have almost 30 answers. So let's wait a little bit more. Okay, so we have 31 answers. And your what comes out of these answers is that essentially half of you think that player one should trust player two. Half of you think that it should not trust. However, say when player one actually trust players to does not trust player to threat, then essentially, he would play left. And so sorry, he will play right. And when when they are at this point, when this point, they are here. And this game has only one national equilibrium. So it is not credible that in this game, player two will play X. Okay, so this is not a credible threat by player. Okay, so thank you very much for talking to me. Okay, so what I want to So this idea of credible threats is is an important issue when you think about games that are repeated in time. Okay. Now, say when you have a game that is repeated in time imagine that you have players that play several times the same game then the situation becomes a little bit more complicated. Okay, because now really strategy become plans of action. So a strategy is a specification of what a player how the player would behave at any time depending on how he played before and how his opponent played before. Okay, so the space of strategy becomes really huge. Okay, it becomes very, very large. And also, while the payoffs now at each stage of the game, you have some payoffs which are given to agents. So the total payoff or the payoff that agent could see to maximize is essentially the sum of these payoffs. Now, usually one also introduces what is called a discount factor, this delta here. Delta is less or equal to one. With the idea that present payoffs are worth more than future payoffs. Okay. And why is this so? Well, one idea is that yes, if you have $1 today, then you can put it in the bank and it will become say more than $1 tomorrow. Okay, because of the interest rate. So that $1 tomorrow is worth less dollars today. Okay, the other way to interpret the discount factor is like a probability. Okay, so you can think that delta is the probability that the game will be played next time. Okay. And if the game is so that every time the game is repeated with a certain probability delta. And so that there is a probability that the game will end and then all the payoff will be zero from that point on. Okay. Okay, so let's see what happens if we consider a repeated prisoner's dilemma. Okay. So here is the table of the prisoner's dilemma. And as you can see here the strategy L dominates strategy R because essentially whatever player 2 plays strategy L always gives a higher payoff to player 1. Okay. Now you mentioned that this game this is the prisoner's dilemma is played one time, this is a second time, then actually the trigger is played three times. Okay. So there is a small window block in the upper right corner of the screen. So this is the mess. Is it okay now? No, we can still see the so so because otherwise I don't see your questions. Okay, so now is it okay? Is it okay now? No, we still see the you still see the what is this? What is this? Ah, this one. Is it okay now? Is it okay now? No, professor, there is a still a window a window in the right down right. Oh, this is really strange. The window title is build order. It is really no build effect. Okay, so so let me see okay, so let me try to show again and I hope it will not work fine. Is it okay now? Is it okay now? It's gone. Now it's gone. Yes, now it's okay. Very good. Okay. So let's go back to our repeated personas dilemma. Now this situation you can analyze it with backward induction. So imagine that you are at the last stage of this game. Then you know that the game will end and then the only possibility is to play the Nash equilibrium which is essentially to play L and L. But then if this is true for the last game then you go to the previous game and the previous game you know that in the next game you will play LL so you should play LL also in the previous game and you can go on like this and figure out that essentially the Nash equilibrium in this finally repeated game must be always the fact and the fact. Okay. So there is nothing really new. Okay. Now the things changes may change if you change a little bit the game. So now this is a game where I have added a new strategy you okay. And this strategy you such that now there are two Nash equilibria. So this is LL and you okay. And let's imagine that this game is placed twice. It's played at time T equal one T equal two. Okay. So if we have to play this game then well we can then of course two Nash equilibria we can essentially agree to play this Nash equilibria U and then we can get essentially a pay off of six. Imagine in this case delta is equal to one. A pay off of six. Okay. Very good. However we can also do better. Okay. Imagine the following strategy that are called strategy C. And the strategy C is that player playing the strategy will play R at the first stage. And then if the other person the other player also plays R then it will play U at the second stage. Okay. But if the other player does not play R then it will play L. Okay. So this is a strategy contains like a effect. Okay. So I'm telling you, look I'm going to play R. And if you play R I will play U at the second step. But if you don't I will play L. Okay. And it is a threat which happens to punish. So it suggests what the other player should do and if the other player does not play R there is a punishment because instead of U I will play L. Okay. So now if you think that if you are playing against this strategy you already know what will happen and so you can essentially summarize this two stage game into a single stage game because you know that if you play R then at the second stage we will play U. Okay. Which is the best thing. And so we will get four plus three which is equal to seven. Okay. If instead you play anything else at the first stage if you play say L or U then the opponent will play L at the second stage and it will be better for you to play L. Okay. So essentially the payoff if you don't play R at the first stage is what you would get by adding one to your payoff at the equal one. So you end up with this game and now if you look at this game you find that there is a new Nash equilibrium which is essentially seven seven. So essentially playing strategy C against strategy C is a Nash equilibrium that allows you to get a higher payoff than what you would get otherwise. Okay. By just playing the single stage Nash equilibrium. Okay. So this idea of credible threats is essentially what also allows you to find a similar Nash equilibrium for infinitely repeated games but let me ask first if there are questions on this or is there a few questions so is everything clear? Okay. It looks like that everything is clear. I don't see any questions. Okay. Very good. So let's go ahead. So now let us consider again this resource dilemma but now we are going to play it a number of times and essentially as I told you so the payoff is going to be the sum of all the payoffs and the strategy is going to be a sequence of a plan of actions. Okay. Now consider this trigger strategy here. Okay. So this trigger strategy is like a contract and you should think about it as a contract and say I'm going to play R at the first the first time at equal one and I'm going to play R at time T if you have been playing R at all previous times but if you deviate from this if you don't play R at all previous times I I will play L forever. Okay. So essentially I'm going to play I'm going to be nice to play R as long as you play R but as soon as you deviate I'm going to play L forever. Okay. So the idea is again the one of like a say a threat which is triggered by the deviation of the other player and this deviation of the other player triggers a punishment which is playing L forever. Okay. Now you can think about computing what is the utility function of an agent of player one if you place this trigger strategy against a trigger strategy. Okay. So if both players play a trigger strategy it means that they will always play R. Okay. Because no one deviates so they will continue playing R. Excuse me professor. Yeah. Is trigger strategy common knowledge? The strategy. Okay. Thank you. Yes. So the idea is that you can think that these two players will talk to each other and say player one will tell the player to look. I'm going to play this trigger strategy. And then the issue for player two is to is to understand whether what player one says is credible or not. Okay. So if the strategy if the plan that is a strategy is compatible with incentives of player one. Okay. Does not go against his own interests. And here you can see that it does not because essentially the punishment correspond to going back to a Nash equilibrium. And a Nash equilibrium is an equilibrium. Okay. So but you can think that so these strategies are and so the players discuss before playing this game about how to play it. And one of the two players says I'm going to play this trigger strategy. And the other player decides whether it should how we should best respond. Okay. Now, so if you if you compute the payoffs of player one if he plays the trigger strategy against the trigger strategy then you find that he gets a payoff of four divided by one minus delta not because especially he would get a payoff of four at any time. Okay. Now let's think about what is the payoff of player one if he plays R for T time steps but then he plays L at time T plus one and then well if he plays against the trigger strategy then he knows that player two is going to play L forever so then his best response would also be to play L forever. Okay. Then in this case his payoff is four for the first three periods then at time T you will get a payoff of five because he plays L and player two plays R and then after that we get the payoff of one because they're going to play this much. Now if you do the calculation then what you find is that the payoff of this strategy is equal to the payoff of player trigger strategy again trigger strategy plus delta to the T times one minus four delta divided by one minus delta. So this means that if delta is larger than one fourth then the payoff of this deviating from a trigger strategy at any time is going to give you the payoff which is smaller than the one of the trigger strategy. So essentially playing a trigger strategy against the trigger strategy is an ash equilibrium or a trigger strategy is the best response to our trigger strategy. Okay so you see the remarkable thing here is that because of this threat of this trigger strategy you are going to achieve when you repeat this game many times you are going to be able to achieve a payoff which is four four which is not an ash equilibrium of the single state game. Okay and this is one of one of the important inside of say repeated games that essentially when you repeat the same game the outcome can be very very different from the outcomes from the outcome of the single state game. Okay and also the other thing which is important that so you see that this trigger strategy is the best responsive for this delta is large enough. Okay now if you interpret this delta as the probability that the game will continue so this tells you that you expect this type of cooperation this type of agreements between agents to be possible when say in situations where which are stable when you expect to interact with other people with the same people many times. Okay and that being a very volatile environment where essentially you are not sure you will survive more or which essentially corresponds to a delta which is very small then essentially it is not possible to achieve this cooperation and you cannot okay and you are back to the outcome of the single state game. So other questions on this do players play the study at a certain time simultaneously so essentially so the idea here is that players decide their strategies before starting the game and then when they play the game they just unfold their strategies and so in this case for example this discussion here between the two players of course before the start playing okay and then also player one can do this calculation before starting the game and figure out that actually playing a trigger strategy again is the best is the best that we can do and then after that essentially they they play this game just as mandated by the study how can we think of trigger study no no no no in what sense primary level I don't understand your question no because here the strategy is already given so if we have a new game how can we think of as trigger strategy that is also an equity okay so this is in a situation where is a very idealized situation where the game is not going to change and it's common knowledge that these two players will play this game over and over again okay now if the game changes at a certain time then this is not discussed by this by this in this setting okay but maybe I've seen these what is important here is essentially this recognition that strategic interaction if it is limited to a short term then it can have very different outcomes then if it is if it extends over a longer longer and there is a video of a TED talk we'll see what are the implications of this or geopolitics and I think it's interesting so Colin says that it is very confusing he doesn't understand what is delta okay so delta is it's called the discount factor okay so delta is is the way in which future payoffs are weighted with respect to present payoffs okay so if you look at this formula you'll see that tomorrow's payoffs one dollar tomorrow is worth delta dollars today okay and you can interpret this as an interest rate okay because if you have one dollar tomorrow that is equivalent to less than one dollar today or you can interpret it in terms of probability that eventually you have tomorrow I mean there is a probability delta that player one dies and then one dollar tomorrow is worth in expected is worth delta dollars today okay because delta is the probability that you will still be aligned with this okay very good so but what I'm doing here is a very short say summary of very broad field so it's mostly meant to be like an enticement for you to interpret this subject so what if player one of these are and player two reach the standard place and then again both play and at the end of the game player two will have better results so shouldn't this be really a study before player two so so so you are saying okay so if they play this trigger strategy say player two plays this trigger strategy and player one at a certain point defects okay and you are right that this payoff will be larger than the payoff that player two will get against this strategy here but this is not the important thing I mean it's not whether player one gets more than player two doesn't really matter what it matters is what player one gets if he plays one strategy against what he gets if he plays another strategy okay and here so the comparison is between player one is the trigger strategy and player one when he plays a different strategy okay okay so now the other interesting observation is that you can think of other trigger strategy think for example to these other trigger studies player one tends to play a two look I'm going to play R at all times and I'm going to play L at even times and these as long as you play R and if you stop playing R I will play L forever okay from that time onward okay so it's exactly the same calculation and you can find out that if this delta the discount factor is sufficiently close to one then this is a national trigger so this is a national trigger in the sense that player two playing so if player two plays a trigger strategy then player one should play R or should always play R because otherwise if he deviates he will get a lower payoff okay so this is a kind of exploitation because if you go back to this table player one is saying I'm going to play R you have to play R all the time and I'm going to play R half of the time and L half of the time so half of the time you are going to get zero and I'm going to get four and but that's the deal okay and what you can find is that if you do the same calculation as here you can find that this is actually a national equilibrium so if player one plays this trigger strategy then player two should always play R okay and so you can find that actually there are many other trigger strategies on top you can think well I'll play R once every five rounds and otherwise I will play L and you always have to play R okay so this again is a trigger strategy that if delta is large enough it's close to one then okay and this is the content of this theorem which is called the folks theorem and which says essentially the following thing that if you take this payoff table for the present dilemma in this case and you put essentially here the payoff of player one and the payoff of player two and then these four points then you have four points in this graph so the point one one which is the national equilibrium the point zero five which is this one point five zero and the point four four okay and then you can go over this polygon and you can take this from this national equilibrium you can go over this line and essentially identify this shaded region gamma so the folks theorem tells you essentially that for any point in this shaded region you can find a trigger study such that you can achieve this expected payoff an expected payoff that are exactly defined with these points here okay and this trigger study will be national equilibrium if delta is sufficiently close to one okay so now you see that well this was the national equilibrium that we derived simple trigger study and so if you see all the points to this side for response to points where essentially player one exploits player two exposition is possible because there is always the threat to play the national equilibrium which would give a lower payoff to both the players okay this is called folks theorem and tells you that essentially repeated games are really really complicated because the space of strategy is really huge it's very very loud and so in this space of strategy is the number of national equilibrium becomes say a dense so you have an infinite number of possible national equilibrium and it's called folks theorem because essentially well it has been proven by Friedman in 71 but it was essentially known to be true my game theorist well before 71 so many people understood that this was the case okay so excuse me professor please yes exactly for this for this graph how do you find the light green lines that you traced are they parallel to the axes are they parallel to the axes yes okay thank you perfect so these are these are the this gamma is a set of points in this polygon where the payoffs of both players are at least as large as those of the national equilibrium of this national equilibrium okay okay okay okay make sense thank you very good other questions okay so if not so let me finish by mentioning what happens so this is the case of repeated games where agents have complete information about the game they know who they are playing with they know what are the rules they imagine that instead you have a situation where you don't know exactly the payoffs or you don't know exactly who are you playing with okay this is the subject of repeated games of incomplete information and this is a fascinating subject and so let me give an example so imagine that there is a game between two people and every day they have a cake and this cake has a cherry in the middle and player one has to split this cake into two and player two has to choose which side to pick so player one does not know how much player two likes the cherry so but he may imagine that player two likes the cherry so what he may want to do is to cut the cake in such a way that the part which has the cherry is a little bit smaller okay and essentially with the idea that essentially when say the part of the cake which contains the cherry is small enough that player two is indifferent between having a larger part of the cake or having the cake the part with the cherry then then so he can make the part with the cherry this small okay now however they are playing this game every time and the issue is whether the player two picks the smaller part of the cake with the cherry so he is revealing information to player one of how much he likes the cherry and what is the consequence of this the consequence of this is that tomorrow the part of the the part of the cake with the cherry will be smaller will you get the point and so the more player two reveals information on his true preferences the more player one can exploit this to get a better payoff okay and so this is essentially the type of situations that repeat the game complete information it's a theory that has been developed by game theorists in the Cold War because essentially there was exactly this type of situations so the US and the Soviet Union were essentially discussing about treaties on nuclear armament each of them didn't know exactly what was the arsenal of the other side and of course when you write a treaty you also reveal information that you don't like to repeat okay so it was a group of game theorists that were engaged by the US government to study what is the optimal way to play this game okay and there is a very very nice one of these guys was Robert Oman and if you go on the website there is a there's a lecture of Robert Oman that explains one say a little part of this of this of this field of game theory of the game solving and it is really fascinating, I really recommend it to all of you to have a look so with this I think I am especially done if you have questions I am so I would have a question regarding the folks theorem you showed us the shaded region and okay but so I thought is there maybe some sort of let's say a line maybe connecting the Nash equilibrium with the optimum point the 4-4 the sort of property of this region for which we could kind of agree that there is exploitation but it's a sort of a friendly exploitation I don't know if that makes sense I mean yes you can so like you're not at the Nash equilibrium which is the safest possible point outcome but you're not really at the 4-4 a coordinate where everyone is maximally kind of earning but maybe there's some line or some point there some trajectory where we can kind of say okay well we're mutually exploiting each other and kind of making a compromise I'm just taking like stabs in the dark and yeah so my my discussion of exploitation was say very simple and naive was essentially saying that if you have a threat if someone tells you I mean this trigger strategy where you say I'm going to create once every week with you and otherwise I'm going to exploit you and you have to cooperate so this looks like essentially exploitation right but of course psychologists and game theorists and they have discussed all these issues in much more detail so I'm going to give you a sense of say the interesting things that happen when you model strategic interaction in a repeated way okay so other questions so if not then we take a break and we'll see you in the next lecture