 So, this is a definition of the of a game for us to begin with, so a game comprises of the following. You have a finite set here which is called denoted by n which is called the set of players. Now, for each player or each i in n, you have a set let us call this Si is a set of strategies. Now, for the moment I am going to call these strategies and in a few lectures we will produce a more general term for this, but for the moment let us just call these strategies ok. So, for each player i you have a set of strategies, so if I take say X i in Si this is a particular strategy here particular strategy of player i. Now if I put together this collection X 1 till X n, this here you can think of this as some collection or you can think of it as a composite vector with these components whatever you however you want to think of it X 1 to X n here is what is called a strategy profile. So, 1, 2, now 3, so for each i for each player i you have a function let us call this function ui function ui that maps this product right to R, the ui maps the product of Si is to R. What is this is essentially saying that for each profile of each strategy profile that means each composite vector X 1 till X n the ui produces for me a real number and that ui is what is called the player's utility or payoff various names for this sometimes called utility sometimes called payoff. And now ui here is when I when I use the word utility or payoff it effective it connotes that it is something that you that the player gains from. So, this is something that the player is looking to maximize alternatively you can always flip the sign and of this u and say turn it into a cost that he is trying or a loss that he is trying to minimize. So, so utility or payoff or let us say cost say if it is utility or payoff the assumption is that he is looking to maximize this that he is looking for the largest possible value of this and if it is a cost then then he is looking to minimize this. You put together these three things then what you get is a game. So, a game basically comprises of a set of players and a set of strategies for each player and a set of you and a collection of utility functions for one for each player. This here is a game and as I said we are since we are focusing on non-cooperative games are underlying here I need to also alongside all this also specify what is what are the what are the assumptions regarding communication and so on. But so because we are focusing on non-cooperative games the it is in I am going to assume implicitly that there is basically no communication of being allowed between the players. So, this is a non-cooperative game with these specifications with this set of players, this set of strategies and these payoff functions is this clear. Now as since some amount of convex optimization and so on will be involved it is a it is usually convenient to whatever is the function it is convenient to minimize it rather than maximize. So, coming from you know usually in a control or machine learning and so on language we have we have a cost or a loss that we are trying to minimize whereas in the social sciences they tend to look at you something payoff that player is looking to maximize these are just conventions we will stick to the minimize convention alright. So, now this is what this is the formal definition of a game. So, now question is how are we going to solve this? So, what are we and to begin with what are we solving for by now again let us be clear I am not asking you how to compute. So, when I when I say solve I do not mean compute something ok compute comes after you tell me what is it that is to be computed alright. So, when I am asking for solving I am asking what is what should be the solution yeah. So, what is should be now if I just give you this and I tell told you the assumption that this is a non-cooperative game I want to I want something that I should say should be able to say is a solution of this right. So, if there was only one player involved here then the problem would become that well then the problem would be that there is one player who has a set of strategies and a payoff function which is trying to maximize then it is merely an optimization problem right and there is a natural notion of a solution. The natural notion is the optimal value of that of that of that of that particular problem you look for the global optimizer or if you are otherwise look for a local optimizer etcetera these are the natural solution concepts from which we derive our solutions it is a different matter how we go about computing them what algorithms we have and so on. But the logic is that well it is sort of self-evident that is what you are looking for you are looking you are looking for a solution of the optimization problem. Now you do not have one optimization problem but each player has an optimization problem right essentially there are n optimization problem and they are all interlinked the question is how do you solve or solve this and again how do you solve means what should be considered as a solution of this of this of this thing if this particular thing that we just built it is a function of the strategies of each player ok but what properties should that function satisfy without some without help from someone else ok interesting any other yeah so here one second just let me answer that since this point has been brought up a game as formulated does not necessarily have a winner or a loser ok this is all there is to a game which has we ok there could of course you can take certain special cases in which there would you can define such a thing as a winner so for example if you know if the payoff reaches a certain value then you win ok something like that that could be but here this is a far more general setting there is no no you know predefined notion of a winner or anything like that yeah sorry someone else was saying something ok with what distribution ok but the opponent would be doing the same so ok so here this is here is another important point about games and about how we study games ok again once I post this particular thing to you and I say this is the game and we want to be able to solve it when we go about coming up with a logic for solving a game like this we have to first fix a point of view ok point of view meaning that who might be solving this for and whose who is solving this game is it is it a particular player in this situation who is solving the game are we solving say for example for player 1 or for player 2 or for player n are we solving for all of them together so whose I mean this is being solved from whose point of view so this is an important thing right so especially particularly when it comes to applying game theory when you apply game theory say for example you want to apply game theory to a security situation where you want to advise say the security of an airport about how it should be deploying its troops to protect the various you know terminals and so on in the airport are you thinking are you doing this from the point of view of the security company or the security agency or are you doing this from the point of view of the of a potential attacker or are you doing this from a completely different other point of view so this the this is part this is again this is not something that that is there in this problem definition this is again part of the or the approach that we want to take as far as solving a situation like this is concerned so it is it is it is a human choice we are making that we will take the view of not any particular player in the game but but the but the point of view of an observer of the game okay if you remember I I said initially that we are trying to develop something like a law like a bunch of laws so I should be thinking of these players as more like particles that are evolving according to a certain law of motion the law of motion is that they each looks to may have maximize his payoff or minimize his cost whatever and then there is this interaction between them that their costs are interdependent as an observer of this particular system seeing them seeing this particular system play out in the lab I want I would be I would want to know okay how should I be solving this or what should be the final notion of a solution of this so the point of view we take in game theory is not of any one particular player but that of an observer of the game so we do not do things like you know put ourselves in the shoes of the players imagine what the other player would do and so on we we take the neutral point of view of observing what of an observer of the game in which players are equipped with a certain set of assumptions and a certain set of capabilities we ask okay what should have what should they do or what must they do okay so the viewpoint is that so viewpoint taken in game theory is that of an observer of a game so so you will soon see in a few in a few lectures from now there could very well be games in which a player is getting bluffed for instance okay by another player and that if you were in that players position you would play differently from whereas you would reason about the situation very differently if you are an observer if you are an observer you would see that the you could get bluffed but if you are in fact that player you would avoid getting bluffed okay so this point the point of view matters it leads to very different sort of ways of applying game theory and ways of shifting through the the logic of game theory okay so the observe so here here so we are going to stick to the observer viewpoint now if you are observers of this game so okay now again we are we are back back to the same question we are observers of this game seem like we like the prisoners dilemma we are observing that there are these two prisoners with equipped with these options and these these these payoffs what must they do and more generally I mean now now that I have written out a general game you should tell me what should be the way to reason about this so the main the so one of the main landmarks in the theory of games is this is the methodology or the logic for reasoning about situations like this situations of with involving non-cooperative games and that and in fact even prior to that the recognition that that how you reason depends on whether the game is cooperative or non-cooperative or whether it is non-cooperative or non-cooperative okay so all of that that observation came from the form from Nash's thesis and so Nash basically said that if you have a non-cooperative game then what you should be doing is basically solving it using this particular concept and that is what I am going to define now. So in the Nash equilibrium is a strategy profile let us call it x1 star to xn star is Nash equilibrium is the following holds if ui of x1 star to xn star so I as I said I have assumed that ui is a player is trying to minimize so this is okay so ui of x1 star to xn star is less than equal to ui of so I will introduce a notation here less than equal to this for I will explain what this means for all xi in psi and and for all i now what does this notation here mean this notation here stands for xi so xi comma x minus i star this notation essentially stands for the following vector so x minus i star is the the the part of the strategy profile which in which I have taken all the other players except for player i so it okay so I have x1 star x2 star dot dot dot x i minus 1 star and now in place of player i i in place of the starred one I am going to insert this xi here so this is xi comma xi plus 1 star dot dot xn star so xi comma x minus i star is essentially this composite vector it fair I have taken the starred values from the first for all players except for player i and in place of player i is strategy I have put in this xi that is chosen for me okay so that is what that is what that is the notation for xi comma that is what this notation means xi comma x minus i star it is a notion notation specialized for game theory because this kind of a thing comes up all the time okay so now let us see what Nash is basically saying so what Nash is saying is that we should be looking at a point x1 star to x10 star is called a Nash equilibrium if the following holds so what what do we have on the left hand side on the left hand side you have the the cost of player i when this strategy profile is played when the players play x1 star to xn star n players all of them played a starred value what I have on the right is the strategy profile is the is the cost of player i when the strategy profile changes to xi comma x minus i star that means all the other players stick to the starred value the starred part of the the continue to play the strategy the starred strategy but player i switches to xi okay so what this is seeing is that the player player i is here this in what this inequality is saying is that a player i is better off playing the starred value than he is by switching to a some other xi provided the other stick to their starred values okay and this holds for every xi and moreover this holds not just for this particular player i that you have chosen this is symmetric it is holds for all player so so every player i refers to stick to his starred value starred strategy provided the others do not switch from do not deviate from their own star value so no player has an incentive to deviate from his from his strategy from the profile provided no one else deviates from their own strategy in the profile okay so this is this is called the Nash equilibrium now why why why is this such a such a big deal why why is this you know I said this is a landmark in the theory of games and why why is that the case the reason is basically this essentially what Nash said gave gave us is a is a essentially the a a watertight way of now reasoning about games what is this situation what is this inequality really saying what this inequality is saying is that if I want to call if I want a point x1 to x if I want a profile x1 star to xn star to be considered a solution then this must hold what should must what must hold well no player should have an incentive to deviate unilaterally from his from that profile so assuming the others stick to the their strategies in the profile I would not want to deviate from my own strategy from the profile now why is this correct or reasonable why is this logical for for a non-cooperative game the reason is logical is because see once a game is non-cooperative okay once a game is non-cooperative only unilateral deviations are feasible there is no way for a pair of players or a group of players to to collective to you know form a unit and then deviate together you know like it happened with the she was saying now for example you know you cannot have that kind of a possibility where a bunch of players deviate come together and deviate together what what is happening here so in a non-cooperative setting when there is no communication is allowed between players only unilateral deviations are feasible if only unilateral deviations are feasible well then what it means is if I if I if any point has to be considered a final solution then it has to be stable against such deviations because if not then the adjustments process is still going on it whatever you are at is not a final is not a solution anymore is not an outcome anymore so this is essentially a necessary condition for any any profile to be considered an outcome if it has to be an outcome it has to it has to be a Nash equilibrium right because otherwise that if it is not if it is not a Nash equilibrium then it means that there is at least one player who under the communication constraints can deviate from it and then if that is the case then whatever you were considering earlier from that point is not an outcome anymore right something else could potentially become an outcome the adjustment process continues you know the whatever the reasoning process continues and and and maybe there is some other outcome so what this basically does is essentially it gives us clarity on how we should be thinking about about non-cooperative situations non-cooperative situations how you are to think of them as something where players if they could unilaterally deviate they they would so if you want a final solution or if you want anything to be considered as an outcome then it has to be of the kind where unilateral deviations are not profitable for the player anyway all right so this this is a this is the this is the Nash equilibrium of a game okay and so this is the solution this is basically the most popularly applied widely applied solution concept in any non- cooperative situation you will you will you know as you go forward you will see how you know the kind of amazing properties it has and so on and how much it is able to explain about about strategic interactions and so on but for the moment let us just go back to the prisoners dilemma and let's see what is the Nash equilibrium for the prisoners dilemma yeah because in a non- cooperative game only unilateral deviations are feasible if you if you allow if you take the same situation in which you have a cooperative game in which a bunch of players could communicate and say well as a group we want to move out move from our from our present form a certain from one sub profile to another sub profile right that would be a different game and that would be a different type of situation and it would really lead to a different type of concept in fact the coalition formation games which is a kind of cooperative game has is there you solve the game using exactly these lines of reason okay the about what kind of coalitions are stable well stable the coalitions that are stable are the ones where in which no subset of players can actually deviate from but here because there is this is a non-cooperative setting only individuals can individual player can deviate yeah so what is the Nash equilibrium of the of the prisoners dilemma to to right testify testify is the Nash equilibrium of the prisoners dilemma and the reason for that is is that if you see what what how are we arguing here we need to argue that assuming the other player plays this his component from this profile I would not want to deviate from my component right so assuming the other player plays testify I would not want to be assuming the place testify a would not want to switch to silent and likewise if assuming if it plays testify be would not want to switch to side now this game is symmetric and that is why this computation is this this is very easy okay and you know but of course in general the number of strategies that players would have could be different for each player and so on alright so let us take another example he has another example this is also symmetric this is okay an example where there are two hunters okay hunter 1 and let us say hunter 2 and they are both poised to hunt down a deer okay they they are both poised to hunt down a deer I mean they let us assume that they are sort of two different two different they there is a deer in some central location and they are both viewing that deer from from two sides they cannot communicate with each other I mean of course there could be visual cues and this and that but let us let us assume that is not possible they cannot communicate with each other they are both seeing this poised to hunter deer ready to hunter deer but then suddenly there is this option that a rabbit runs by okay and then there is the possibility to also hunt a rabbit so and what is the dilemma the dilemma is the following that a deer is much bigger and much takes much more effort to hunt okay so if both go for the deer they will be able to hunt down the deer and they would each get let's say a payoff of 2 but if only one of them goes for the deer and the other goes for the goes for the rabbit the fellow who is going for the deer will not get will not be able to hunt the deer alone okay so the one who is going for the rabbit gets gets the rabbit but the one who is going for the deer doesn't get the deer okay so and if both go for a rabbit the rabbit is much smaller than the deer and so what they will each get is you know whatever half okay so now again what's the reasoning here so one thing you will notice right away is that this is not a situation like the prisoner's dilemma where there is one strategy that is uniformly better than the other one it's not true that regardless of what the other guy does it's better for me to go for the deer right in fact this is more akin to the coordination problems that I was just talking about we we are both better off if we coordinate but coordinating on a better thing is better for both what is the what's the logic here again we don't want to get into the trap of trying to predict what players will do because we again don't know what they will each do because that gets into a whole bunch of other issues like like how hungry each hunter is how selfish they are and so on so forth the amount of trust they have in the other individual and so on that is not our mandate that is not our competence what we are going to do is come up with a bunch of reasoning rules by which we can say what what is the logical outcome not necessarily and the reason is because the whatever the other player does arises you know your distribution of what the other player does for example is is part of the strategic poster of the other player so it is not the point is you may see this is a this is a temptation in games all the time right that instead of trying to say well I can think from the point of view of a player and I say okay maybe the other hunter 30% of the time goes for the deer and 70% of the time goes for the rabbit and then based on that I will try to I will try to decide what I I should do but remember that 30 and 70 is not if you know he is not just some fact of nature it is part of his decision how often he should be going for the rabbit versus deer is part of his strategy all you have done by just bringing in these probabilities is just lifted the same problem to the space of probabilities now each player has to decide not whether to go for rabbit or to go for deer but rather how often he goes for rabbit one or with what probability he goes for rabbit versus what he goes whether as opposed to deer so this this you know trying to model the other player using some probabilistic model basically just gets you to one the same reasoning sort of conundrums that you had even in the strategy space okay you just need but so you have the same issues just in a different space look I mean look at the classic case of the Normandy landings in Normandy right so you could you the during the world war there was this this very epoch making event where where the Allied forces landed in France to liberate France and from Germany from which was occupied by Germany at that time there were they picked two possible locations for landing on the coast on the northern coast of France one was called the name is Normandy the other is Calais the Germans also knew that these were the two potential landing locations alright what they did not know was where they would where they would actually land okay you could say well I do not know where they would land but I think there is I have a certain model based on which I I think that that is where they were likely to come but where they are likely to come depends and how well you will succeed depends also on what you are planning to do just like you are making a model about just like the Germans were making a model about where the Allies would land allies would be making a model about how how the Germans would where the Germans would keep their defense you are effectively just elevated the problem to deciding how should we distribute our troops rather than trying to decide which where we should be keeping our troops so that is all there is the these these things only changes this particular flavor now to be to be fair there is there is another type of class of games called repeated games where things tend to get a little more intricate where where we are where we are playing this game not just once but multiple times okay but with the knowledge that this is being played multiple times and then you can try to sort of bake in some of the issues some of the things that you have in mind but again this is for the first lecture we are that is too complicated for to get into that okay yeah so what's the what's the logic then that's a very different game that's a very different game because there the what there that's part of the assumptions then that you know certain things about the other player yeah so this the assumption here the way I have stated this game you know in this year is that there is no such information all you know is this so in fact in fact the an important thing is okay you which some of you should have asked me before what do players actually know right so these players are choosing their actions without communicating with each other right but what do they know well the only thing that they know is this table okay so in the case of the prisoners dilemma they know this table that this is what has been offered they are being they are choosing their actions without the knowledge of what the other player is doing okay so I as good as simultaneously so they're so they do not know what the other player is doing in the in the Germany in the World War example essentially we do not know where the where the allies are going to land if that is known then that's a very different game or if that is known even with some noise that's a very different game or if the even if I or if I know even that there is a certain probability with which they are likely to land that's also very different game that's not the game that's been discussed no no no it's not see the that so so information changes the character of the of a game so if I know so if I know that you know say for example there is a point of no return after which the allies are going to land in a particular location right then that's a very different game my how I strategize will change very different so let's come back to this then players know this table they know that each player wants to get the maximum payoff they also know that this is how the the payoffs of dear and dear and rabbit and so on are are you know this is what the values are now how do we what's the logic for this so you cannot obviously there is no strategy here which is which dominates the other strategy which means that regardless of what the other guy is doing this is always better that kind of strategy is not doesn't exist we can try to reason about this through the Nash equilibrium so if you see essentially what Nash is saying is whatever is the final you know each guy each player would think think think okay decide okay should I go for dear should I go for rabbit etc etc whatever is the final if I if at all there is a resolution of this as if we were observing these two hunters what they would do is what they ought to do is that either both go for dear or both go for rabbit okay now how why is that the case so if you look at the if you look at let's say dear dear now dear dear is a Nash equilibrium because assuming the other guy is going for dear I would want to continue to play dear okay likewise rabbit rabbit is also a Nash equilibrium because if the other guy is going for the rabbit it doesn't make sense for me to try and get the dear because I won't be able to get the dear alone okay so it's better for me to stick to rabbit and dear rabbit is not a Nash equilibrium or rabbit dear is not a Nash equilibrium so there are two Nash equilibrium in this game okay dear dear is a Nash equilibrium rabbit rabbit is also a Nash equilibrium let's complete with today's lecture conclude today's lecture with one just this global picture so essentially if you want to position position a non-cooperative game theory and you could you could position it in the following way if you here you can think of the single player setting and this is the multiple player players setting single player and multiple player now the first setting and the kind of games that we have seen so far is what is what are called static games static games is when where both where all players where no player has any information about anything else apart from what is the game what he starts off so he does not know what the others have done does not know what else has happened and so on if anything else if any other evolution has happened and so on he does not have any such information those are what are called static games so player just plays once and with the same with a with a sort of a null information except with only the information about the payoffs of the players those are called static games now in the static setting if it's just one player or it just reduces to optimization and if it's if it's multiple players then it leads to static game theory on the other hand if you if there are if if the sitting if the setting is dynamic which means that there is there is a either a time evolution or there is a an aspect of information and so on then in that case the one player case reduces to is what we know as optimal control optimal control with under various names stochastic optimal control or whatever and the in the in the in the case of multiple players it leads to what is called dynamic game theory so my effort is going to be to quickly run through this so that you are you are familiarized with the basics and then then we will spend about 70% of the course on dynamic games.