 Welcome everyone. So, let us start with another example of a Nash equation. Let us look at a non-cooperative game of the following kind. So, there is a block here on a horizontal surface and there are two players and the players are exerting a unit force on that block. What the players have to choose is the direction in which they will pull the block. So, player 2 here, this is say player 2, we are going to measure this direction by an angle, let us call this angle U2, this is player 2 and likewise player 1 is measuring is pulling say in this direction, let us call this angle U1 for player 1. Now, the position of the block will now change, now that they are exerting this force on this. So, the equations of motion that we have for the block is that if you look, let us take this as the x1 axis, let us take this as the x2 axis. So, x1 double dot will now be the which is the acceleration along the x1 direction is going to be since these are unit forces, you can just take this as this is actually just cos of U1 plus cos of U2 and likewise x2 double dot is going to be sin of U1 plus sin of U2 and I will give you a initial conditions also, the block starts at rest. So, x1 of 0 is equal to x2 of 0 and let us take that point at as the origin and x1 dot of 0 is also equal to x2 dot of 0 and that is also equal to 0. So, it starts at rest at the origin and then these players exert these forces. Now, there is an assumption here which I need to make and I need to emphasize which is that these players are going to decide the angles at which they will pull, but then the angles once decided will remain fixed. The angles will not change with time although the block moves with time over time, the angles once decided are going to remain fixed. So, U1, U2 once decided are held fixed, this is the assumption. Now, what are the objectives of the players? These players want to have the following objectives. Player 1, what he wants to do is wants to minimize this value. This is x1 which is the x1 position of the block at time 1, x1 of 1. So, the position along the x1 axis at time 1 is what he wants to minimize. So, he wants to choose as U1 to minimize that. Player 2 wants to minimize x2 of 1, x2 of 1 is then the position along the x2, the vertical axis here. So, you can see basically the player 1 essentially wants to, player 1 wants to pull the block in this direction effectively that is where his, that is his attempt. Player 2 wants to pull the block in this direction. Now, obviously if player 1 was the only player in this picture, he would put all his force along this particular axis and that would, then that is what would and the block would end up at a unit distance in time 1. Actually at distance I think half at time 1. And so, you would get x1 equal to minus half here. Likewise, if player 2 was the only player in the picture, then he would just pull completely along this direction and you would get x2 of 1 equal to minus half again. Now, what we want to know now is, given that this is, now if these two players are engaged in this kind of a situation in a non, and they have to choose their U1 and U2 in a non cooperative fashion, what is the way to solve this problem and what should be the solution we should be, by which we should be analyzing this problem. So, this is again we are going to assume this to be a non cooperative setting. So, players are, will not communicate with each other and they will each want to choose their action with, to maximize their own payoff or minimize in this case their own, the coordinate that they are interested in. So, considering this, then we should be solving for the Nash equilibrium of this game. So, what we need to look for a Nash equilibrium U1, U2 or let us call this U1 star U2 star. So, now can someone tell me what would be the Nash equilibrium? So, P1 pulls against minus x1. So, U1 you know, yeah. So, P1 pulls towards the direction minus x1 that means, you can check this that U1 equal to basically U1 equal to pi and likewise U2 equal to minus pi by 2, U1 star equal to pi and U2 star equal to minus pi by 2 is a Nash equilibrium. Now, can you tell me where does the block end up in this situation? Yeah. So, where does the block end up? The block ends up. So, this is where U1 is this pi U2 is minus pi by 2 or in short this is U2 minus U1 star U2 star and you can check that what the block will end up at minus half, minus half. Now, what is interesting about this? The interesting thing is so, the block has basically moved in now as a result in this particular direction. Now, what is interesting about this is that if you see if these players work to in fact, instead of doing this, were they instead to choose the directions in which each pulled along the 45 degree line in this along in the third quadrant that means if they had instead chosen say U1 bar equal to U2 bar equal to minus pi pi by 4, then what would happen? Then the block would go along this direction, but even further down because both the forces would be aligned. So, they would actually each be better off. But because this is a non-cooperative situation, each player is basically optimizing his own payoff assuming the other payoff as fixed. So, consequently player 1 does not cannot guarantee that the other player is also going to play along this particular direction. What happens is essentially that as a result, each player basically pulls along their own respective action. So, now you can so, why you can convince yourself why about why this is not an ice equilibrium, why my pulling along minus 5 pi by 4 is not an ice equilibrium. So, if say suppose player 1 pulls along minus 5 pi by 4, thinking that that is that is what would you know is the that is what is the cooperative solution. So, if he pulls along that along that direction, player 1 can so, player if player 1 pulls along this direction player 2 can basically deviate to any direction here. So, he can basically play any direction here and get a better result than what he gets by playing minus 5 pi by 4. So, minus 5 pi by 4 comma minus 5 pi by 4 is not sustainable in a non-cooperative setting because in the absence of communication each player has an incentive to deviate from this. So, player if player 1 sticks to this player 2 would would lean more towards the vertical axis downwards. Likewise if player 2 sticks to this player 1 would lean more towards the horizontal axis and then eventually as a result it is better for player 1 to go in this direction and player 2 to go in this direction. So, the reason I brought this example up is because this helps you sort of visualize what is going on in a strategic situation. Essentially incentives are such that and the pay off functions or utilities are such that they are pulling players in various directions. You know there is a certain direction probably wherein they are all better off but that is not sustainable or not enable under the communication constraints that we have. The communication constraints require that the players without communicating with each other they have to make their decisions. So, as a result only the incentive the unilateral incentives at play matter and in that case what the players would play is actually along these two axes. So, this is one more example of a game and again how cooperation versus competition as a trade off plays out in game theory. Let me end with a few other remarks about the Nash equilibrium as a concept. So, remember the Nash equilibrium is just a concept. It was something that Nash proposed as a way of solving games. Now the reason it has such an important status in game theory is because of the context of the times and the direction that the field took up after he introduced the concept. Basically he brought clarity to the issue of communication in games which was not clear at that time. He also clarified a whole bunch of other ideas that were sort of seen as axioms back then in the theory of any kind of multi-agent decision making including economics and so on. Now because this is only a concept it is not possible to actually derive the Nash equilibrium in general at least. I will show you some cases where it can be but it is not possible actually to derive the Nash equilibrium as a concept. You can of course compute a Nash equilibrium and derive that a particular point is a Nash equilibrium but you cannot derive the definition of a Nash equilibrium from any first principle. So, consequently we can only try and justify the Nash equilibrium as something that we find reasonable. Sorry, derive means that you know starting from a certain set of assumptions can I derive that the only point that must satisfy all of these is what Nash defined as the Nash equilibrium. So, certain set of assumptions. So, what should be those set of assumptions and so on is itself a question and that needs deliberation that needs some in some cases it can be done as I will show you but in general this cannot be done. What you can do is you can try and justify it. So, of having posed the concept you can expose trying to justify what you know your post hoc justification for why it makes sense. So, you can justify it for instance one of the main justifications is as I said and it is if you want to call anything an outcome it has to be a Nash equilibrium under because otherwise it is under the communication constraints there is an incentive for players to deviate and if the players deviate then it is not an outcome anymore. So, that is one the other is this idea that you can think of the Nash equilibrium as something where if players are dropped at that point then from there there can be there is no further need for movement. So, it is so if you were if somehow you are able to suggest them to play this they would agree to that suggestion because assuming the every assuming the other player also agrees. So, under the communication constraints if players are dropped at this particular point where they are they are both been suggested to play this strategy and the other also sticks to that suggestion then you would also not want to do this. So, this is also another way to justify the Nash equilibrium. The Nash equilibrium is also somehow it is it is remarkable that it is also seen in nature. There are good number of examples in biology and you know particularly adaptation and so on where a version of the Nash equilibrium basically is can be seen as playing out. So, that is that could be one more reason why we find it you know attractive to think about the Nash equilibrium. Any questions? Yeah, okay. So, both the games or all three games actually that I have talked about the Prisoner's dilemma the the dear Rabbit game and this one all three of them have this property that there was a Nash equilibrium. The Hunter Rabbit game had two Nash equilibria and we have not yet seen a game in which there is no Nash equilibrium. But you can construct games where there are there is no Nash equilibrium. So, the now it is all well and good to give a concept but what if there is no point that satisfies that concept then how do you even apply that concept becomes a question. So, this is this is actually a valid question but you will soon see that the reason there is no Nash there are games where there are there is no Nash equilibrium is because we have not adequately exploited the strategic alternatives available to players. Here right now the way we have defined the strategies for the players is that players are required to play either this action or that action and so on. Right. If you it turns out that it is possible to generalize this in which you allow players to not just play a particular action but play an action randomly and to randomize over their choices of actions once you allow this randomization it turns out a Nash equilibrium always exists. And this was one of one of the main main points in his paper that not only is this concept meaningful there is always a point that satisfies such a this concept. So, we will come to that later in the course. Right now what I want to do is I want to show you an instance of when how a Nash equilibrium can actually be derived. And this sort of takes us back to what one of the things that came up in the prisoner's dilemma. So, if you recall in the prisoner's dilemma the situation was that you had we had these this sort of a matrix right for the prisoner's dilemma we had so there were two choices for the players silent testify this was prisoner's dilemma. Now one of this when I asked you about how you would solve this game one of the observations that one of you made was that if you look at the strategy testify for any player then it has the property that it is better for that player regardless of what the other player played. So, for example let us take for player A. So, for player A testifying is always better than staying silent irrespective of what the other player played. So, because 0 is less than 1 and 2 is less than 3. Now and likewise same for player 2 and hence as a result you could say that therefore irrespective of the strategic consideration that play it is logical for me for a player to play testify and therefore both players would play testify. Now this is one way by which you can solve for you can solve for the game without even invoking the Nash equilibrium. You do not need the Nash equilibrium concept to reason about this because the numbers are such that that is what the whole this works out this particular logic works out perfectly. So, what we will talk about today is a little bit more general version of this that goes into the concept of what is called dominance. Now there is two goals of about in this one is that I will of course teach you about what dominance is and how you actually use dominance in solving for games. The other is that you will also see why being careful about the assumptions of a game is really matters and you will see that there are pitfalls if you do not make the right assumptions. So, let us take this I will write out a game now. So, let us take this game with two players. So, this is player 1. Player 1 has three choices which is up middle and down and player 2 has three choices left middle and right. I will write out the payoffs here for the players. So, this is 4, 3, 5, 1, 6, 2, 2, 1, 8, 4, 3, 6, 3, 0, 9, 6, 2, 8 and both players are we will make the assumption here that this both players are maximizing. If both players are maximizing the number. So, each player is basically interested in the maximum number in its own along its own along its own axis. Now, we want to analyze this particular game. So, we need to start off with a few assumptions. So, what did I tell you about the assumptions of the prisoner's dilemma? What have we what have we assumed about the prisoner's dilemma? Yeah. So, we assume that players cannot communicate with each other. So, that was one assumption. What did we assume about the payoffs this matrix? Yeah. Of course, players were looking for the smallest number. Yes, but what what did they know about this matrix? It is a static but apart from that, essentially it was known to them. Both players knew that this is the game that they are that they are playing. So, if you remember the narration that I had said that these players are held in solitary confinement and then there is a judge that now gives the tells them these options. So, essentially he is telling them that this is the matrix that that they are looking at. We will make the same assumption about this particular matrix. We that both players are are aware of this. We also implicitly are assuming that players are are interested in the maximum payoff for themselves or or in the here since this was yours in jail, they are implicitly we have assumed that players are interested in the least number of years in jail. In this in this case, each player is interested in the largest number for themselves. Okay. Now, this is this assumption is what is called the assumption of rationality. Now, rationality basically means that rationality refers to is essentially saying that given given a set of choices. So, rationality means basically given a set of choices a player chooses the one with the highest payoff. It seems very reasonable and logical that essentially what we are assuming here is that players are rational. That means given whatever be the choices that they have, they would pick they would not pick pick something that is that is that gives them a lower payoff than something else. So, they would pick the one with that with the highest payoff. If there are multiple of the ones with the highest payoff, they could pick any one of those, but they will not pick anything that has giving them a payoff that is strictly lower than the highest. This is the assumption. So, now from this assumption, let us see if we can derive derive that the players will actually play the Nash equilibrium. Okay. So, let us we will come to the Nash equilibrium, what the Nash equilibrium is, but let us see where this assumption actually takes us. So, we have this, this is we have this situation that players are now exposed to this matrix. They are and we know that the players the player what we know as observers of the game that the player is that the players are rational. Okay. So, now let us see what can you say about this? Yeah. So, if you look at for if you look at player 2 and the strategy R for player 2, strategy R is always better than strategy M for player 2. Right. What is how do you conclude that? Well, if you look at the the the payoffs for player 2 are written in this are the second coordinate here. So, 2 is better than 1, 6 is better than 4 and 8 is better than 6. Right. So, R is better than M regardless of what player player 1 does. Okay. So, consequently from player 2's point of view, player 2 is rational. So, we know we can play from player 2's point of view player player 2 will never play M. Okay. And as observers of the game, we can also conclude that player 2 will never play M. Okay. So, let me mark that out then. I am going to remove this. So, player 2 is not going to play M. So, M is gone from the matrix that player 2 is looking at is this matrix formed from just these since player 2 is never is not going to play M what player 2 can see is basically effectively for him the problem is now about choosing between L and R. Okay. So, I have eliminated M for player 2. Alright. This is the matrix that player 2 is looking at. Is this the matrix that player 1 is looking at or is player 1 still looking at this matrix? So, we just said that players are rational. We did not say player is know that the other player is rational. Each player knows that he the player himself is rational. It is quite a different thing to say that I also that the player also knows that the other one is rational. Okay. So, if I assume okay if I assume now assuming that player 1 knows that player 2 is rational, then player 1 can also eliminate from player 2 strategies. So, this made the matrix here came about by eliminating M for player 2 but it was player 2 who could eliminate it because he knows he is rational he is rational he can eliminate it. Okay. So, player 2 eliminates M because player 2 is rational. Now, if you assume also that player 1 knows that player 2 is rational then player 1 is also looking at this at this particular matrix now. Okay. Now what? Yes. Right. So, now if you look at this if you look at this matrix now player 1 is now looking at this 3 by 2 matrix the one the 3 by 2 matrix that I have here. Right. If I assume that player 2 knows that player sorry player 1 is knows that player 2 is rational then player 1 is also looking at this particular matrix and in this matrix now U is better than both M and D for player 1. So, player 1 can therefore now eliminate M and D. Okay. So, we can just verify this if you want. So, see 4 is better than 2 and 3, 6 is better than 3 and 2. Okay. So, I can eliminate both M and D. Okay. So, now player 2 eliminated this and he is now looking at this just this matrix. So, player 1 is now has eliminated this and player 1 is now looking at this matrix. Okay. Now which matrix is player 2 looking at? P2 knows that P1 is rational. Okay. Is that enough? P1 knows that P2 is rational. Yes. That we have already assumed. Exactly. Exactly. So, just like we assumed here that P1 knows that P2 is rational we can symmetrically assume that P2 knows that P1 is rational but that is not enough. You also need that P2 knows that P1 knows that P2 is rational. Right. So, if you assume assuming again assuming that that P2 knows that P1 knows that P2 is rational then P2 can also eliminate both M and D M comma D from P1 strategy. Okay. So, in fact, I will just make this more general instead of saying that instead of saying P1 knows and so on. Let me just write it that each player knows that each player knows that each player is rational and then each player knows that each player knows that each player is rational. Okay. I can come to that as well. Each player knows that each player knows that each player is rational. Okay. We will come to non-rationality in a moment. So, now assuming this I have now been able to eliminate m and d not just from player once view, but also from player to view. So, hence if as an observer of the game, if I have if I know all of this about the players, I can now conclude safely conclude that both players are now looking at this reduced matrix here. Now, here again I can here of course, there is just one strategy for one strategy for player one. Now, you can from here it is rather trivial there is nothing left for player one to choose. So, he is obviously going to play you and then from and you know you can use whatever argument you want you can put one more level of assumption like this if you want for player two or you can just simply argue that well this is basically not a game anymore it is trivial because there is just one choice for player two and therefore player one and therefore player two will obviously play all. So, the point is that this is one way by which you can go about solving a game from first principles without you know creating a concept. This is and what do you mean by solving from first principles we need to put in assumptions about what players know and what players do and then from that like a almost like a puzzle go about reasoning what is exactly going to be what exactly is going to be the outcome of such a game. And that leads you to this outcome of U comma r. Now, this is this way of solving is what is called the elimination of dominated strategies. So, I will just make this more precise.