 Welcome everyone. So, this is a course called games and information. In IIT Bombay it runs under the course called SC 631. So, what is this course about? This course is about the theory of games, but with a particular emphasis on issues related to information, information in games. So, in that regard there are some other courses also on games, but this course is very different in the particular emphasis that it has on the role of information. So, what I will do is of course I will not assume any background in games. So, what I will be doing during this course is I will start off from the basics of game theory, introduce you to at least the elementary aspects of static games, where information is really trivial and does not pay as important a role. And then we will ramp up very quickly to games where information is an integral part of the game definition and the game solution. So, to begin with let us start with the what is a prototypical example, one of the sort of famous games that is often referred to in in books on game theory and even in popular literature which is the game called Prisoner's Dilemma. What is the Prisoner's Dilemma? Prisoner's Dilemma has a following situation. So, there are two prisoners that are held in solitary confinement, they have both been convicted or they have both been suspected let us say of a crime. Now, they have been held in solitary confinement and the judge offers them a certain deal you can say. The judge offers them a deal which is to say that well you can if you want to you can testify against the other prisoner. So, there are let us say two prisoners prisoners prisoner A and prisoner B and the judge offers them the following deal that if you testify against the other prisoner then so you have the option to testify against the other prisoner. Now, based on what happens in the sense that based on what both prisoners do you will get a certain number of years in jail that is going to that will determine your punishment. So, that so if I have to write out how many years in jail you are going to get it can be written out in a sort of like a matrix like this. So, the prisoners have the option to testify so they can either testify or stay silent. So, if prisoner A stays silent so he has two options to stay silent or to testify and likewise prisoner B also has the option to stay silent or to testify against the other prisoner and the deal is like this. So, now if both of you remain silent that neither testifies against the other in both of you would get one year in jail. So, the way I have written this is 1 comma 1 which means the first term here is the this the first one here is the number of years in jail for A and who is on the rows here and the second term here is the number of years in jail for B who is on the columns. So, if they both stay silent then they each get one year in jail if one of them stays silent and the other testifies. So, in this case for example, I am now writing if A testifies and B stays silent now if A testifies and B stays silent then the one who testifies goes free. So, he gets 0 years in jail and the one who stays silent gets 3 years in jail. Now we can of course debate on whether this is fair or not fair and so on but that is not our mandate here we want to be able to reason about this situation that is the that is what we want to do. So, symmetrically here so naturally now if A stays silent and B testifies against A then A will get 3 years in jail and B will go free and then the judge says that if both testify then both of you would get 2 years in jail. So, if each testifies against the other then each of you would get 2 years in jail. Now this is an example of a game. So, what is special about this? You can see the first thing you notice is that no player can a player obviously has a clear objective here which is that he wants to get as few years in jail as possible. But there is no the objective is not well defined for him because the it depends the number of years he will get in jail depends also on what the other one does. So, the number of years in jail that a player gets depends not just on what he chooses to do but also what on what the other player chooses and this is the case for each player. So, consequently there is a linking of decisions that is happening here where neither player can decide what he can do without taking into account what the other player is also going to do. But then remember the assumption that we started with we started with the assumption that these players are held in solitary confinement which means that they do not have the option to discuss between themselves on what their strategy is going to be. So, this is a classic example of a game now you might think that this is something something very contrived or artificial but this is exactly how decision making happens when say competing firms try to decide what their strategies are going to be or to generals from opposing armies would try to decide how they want to what their strategies are going to be and this is exactly how the payoff gets played out. The reward that one gets depends not only on what you do but also on what the other person does. Now what we want here is to answer certain answer this in a in some sort of a logical with some kind of a logical framework and that is what the theory of games attempts to do and it attempts to come up with a framework for reasoning about these situations. So, let me ask you here how would you reason about this problem? Okay, very good. So, here is one way of reasoning about this and this is what this is a special structure in this particular problem and it is also one of the reasons to start off with this problem. You can see there is the structure here is that if you look at the if you look at the strategy of any one player yes each player has two strategies. I will make this for more common but these are each player has basically two strategies either to stay silent or to testify. Now you compare the corresponding rewards that the player would get if he plays one strategy or the other strategy and what you observe is that testifying is always better than staying silent regardless of what the other player does. So, for example here, so if the other player is so if B is staying silent then testifying is better than staying silent for A. If B is going to testify then again testifying is better than staying silent for A and this is the situation is symmetric so both so naturally therefore similar reasoning can be applied for B also and so consequently it makes sense that both players testify. So, this is one line of reasoning this line of reasoning is essentially saying that what I am looking for is I am looking for a strategy that is uniformly better than all other strategies regardless of what the other player does. So, in other words I am making the interlinking of decisions irrelevant here. The fact that there is another player involved and what the other player is going to do affects my payoff all of that is becoming irrelevant what is happening what I am trying to do is find a strategy which outperforms the other strategies irrespective of what the other guy does. Now in this that is applicable in this game and that is applicable because of the way that these numbers are. You can imagine another type of situation in which firstly where these numbers are not so aligned in such that you know testify will always turn out to be better than the better than staying silent regardless of what the other person does. That could very well happen I can come up with numbers like that. So, your line of reasoning therefore cannot change case by case or game by game you need a universal logic because then it is a logic otherwise it is a hack. You need a logic which will be applicable across all games regardless of what the kind of coincidences that are present in the numbers of that particular game. So, now tell me what how would you reason about this now. So, that is that is more or less the same reasoning that you had earlier. So, there is also another subtlety here. So, I am not asking you how you would play this game. So, there are there are two different types of questions we can ask in this in a game. You can ask the following first question is you can ask the question what will each player do. The emphasis being on the word will. The other question is you can ask is what must each player do? What is the difference between these two? The difference between these two is what will each player do is asking for a prediction. I am asking you to predict what each player would do. It is like asking you to tell what the future is going to look like. It is what will each player do. What must each player do is not asking for a prediction. What must each player do is asking you for what the law is. What should be the right thing for the player to do. So, in game theory we are not interested in the first question. We do not intend to predict although game theory can be used to predict, but we do not intend to predict what is going to be the outcome of a game. What we can we do not really know what each player would do. I mean if I gave you two human prisoners like this or two armies or two companies or any of that, we really do not have in a position or not in a position to say what will each of them do. We can only guess what each of them will do. No, not necessarily, not necessarily, but the point here is this is in fact the subtlety here. What will each player do is really not our business. They may they are they have free will they may do whatever they want. What each player ought to do assuming it assuming a certain set of axioms is what we want to try and answer. So, it is in in in many ways game theory what it is trying to do is coming come up with a sort of a set of laws a set of laws and a reset of reasoning rules by which we can reason about situations like this. So, given those given the bunch of reasoning rules if you follow through those rules you should be able to say what each player must must be doing. And hopefully the answer is sharp enough that it actually tells us that here is exactly this is the precise thing that each player must be doing and therefore that is what we think is the solution of the game. So, now what one of the most of course popular use cases for game theory is to use it as a prediction. But whether the prediction works or not depends on whether your laws align with the with the situation that is actually prevalent in the game or not essentially. So, the so what the the the the central quest in the theory of games is not to come up is not to is not to predict it is rather to reason. So, what what this means is we cannot say what is what will happen. So, we confine ourselves to saying what is logical. So, that is and what should be the train of logic is the thing that we are asking ourselves that is the that that is the that is essentially the question at hand. So, now that we know that this is what we want to answer we would like to ask what what when we if you want to answer what must each player do we basically would like to see what are the what should be our reasoning rules for this. So, this this requires us to basically set set up a set of axioms. One axiom is actually automatic here implicit in this problem formulation and that axiom is that each player is looking for the least number of years in jail. So, that is that is effectively one axiom and in fact this is the this is a underlying basic axiom across all of game theory that there is there is there is a function or payoff function or a cost function or something like that which each player is is looking to optimize and the what what a player is basically interested in is the least value of that particular function. Now, of course as I said this function is not well defined until I also tell you what the other player is other player is doing. So, it is not really a function but it is a family of functions that each player. The other thing that is implicit in the problem formulation is that each player these prisoners were held in solitary confinement which means that under the underlying assumption here the axiom here is that players cannot communicate with each other. So, there is no way for players to communicate and or discuss what they what each of them would do and the absence of a communication medium actually change is an underlying assumption in this. So, suppose if I told you so just to see why this is the case suppose if I told you the other case suppose I told you that the players could discuss amongst themselves and come up with a strategy on what to do what would they what would they play. Now, let us say let us say they want to the problem is symmetric. So, if you want to take say let us say the sum total of the years in jail or something like that. So, the question is if both players were allowed to communicate with each other would they necessarily play silent silent. Now, in this case that answer is correct but I will tell you there is a certainty here it could very well happen that both players talk to each other each player convinces the other that I am going to stay silent. But when it comes to actually you know pressing the button he goes and goes back and presses testify having convinced the other person to stay silent no no no. So, the so pre-play communication has this important role in this because I by talking you into it it is I could potentially influence you to stay silent and then having done that it then becomes optimal for me to testify. So, again all what this so this is so there is there are two there is a subtlety here which is that just communication is not enough I can say I am going to stay silent but what is the guarantee that I will stay silent. The issue is not just communication communication is important but in alongside that what is also important is a mechanism to enforce an agreement what is called a binding agreement some mechanism by which when by which players when they say that this is what they would do that is what they would actually do. So, if you have such a mechanism what such a mechanism means how do you make that formal etc all of that is part of the theory of games and that needs development. This is not I am not telling you a sort of sociological story this is such a thing can actually be made formal what exactly is such a mechanism what exactly do we mean by replay communication how much is that communication etc all of this can be made formal but once that is made formal it does turn out if you give players both the option to communicate with each other and to enter into binding agreements it then yes it is indeed the case that players that they should both remain silent. That is the point on which that is the agreement that they will end up signing which is that I will stay silent and you will also stay silent. Now, the mechanism to enter into binding agreements requires sometimes a mediator so I need to introduce a third player C here who will be the one that creates these alternatives gets them to agree etc which is all of this is too early to get into this force at this moment but remember there is this subtlety involved here and what this is also telling you effectively is the again the second thing the thing that I mentioned about the course which is the role of information when there is no communication channel between them the problem is has a very different character from the when the there is a communication channel between and moreover when the communication channel passes through a mediator or does not pass through a mediator. All of these are these are this is art of the axioms of a game of the game. Now, the way I presented this particular problem to you I said that players are obviously self-interested they would each like their the least number of years in jail and that they cannot communicate with each other at all. So, actually if you see the theory of games bifurcates itself into two different categories so games can be categorized two into two broad categories the first is non-cooperative games non-cooperative games means a game is said to be non-cooperative players have no players cannot communicate so this is when players cannot communicate with each other at all that is so the prisoner's dilemma game that I just showed you is a game is a non-cooperative game the other extreme this is not the negation but the another extreme is when there are cooperative games cooperative games are when players can communicate with each other and enter into binding agreements these are remember two different extremes they are they are not negation of each other just because you cannot communicate does not mean that you can communicate to any extent and also get into binding agreements. So, these are two different extremes in between these two extremes lie a whole number of sort of there is a huge gray region here in between okay the gray region is where you know some amount of communication is allowed some kind but binding agreements are not allowed or agreements are allowed but mediators are not allowed etc etc various types of you know exotic combinations are possible in between these two and all of those are very interesting very very fascinating but also quite a bit of that remains open okay so it is not you know fully resolved as well but these two being extreme cases in in science it is often the case that when you cannot solve a general problem but you can solve corner cases the corner cases of are become the way you to anchor your further theory so these are extreme cases that have been fairly well resolved now okay so most for most part of the course we will actually focus on only non cooperative games okay so the I will we will we will consider games of of various kind that are but all of them will have a non cooperative flavor yeah no no so are cooperative games mere optimization problems not necessarily okay so because of course they can be posed as optimization problems but so one and one of the criteria for the solution solution of a game in a cooperative setting is is that they there is some kind of a let us say a combined optimal being asked for okay but that is not the only that is not the only criterion okay so see for the particular emphasis here is the issue of binding agreements right so again this is a slight digression but I will just mention this this is an important issue because we need to also factor in into our way of reasoning about the game what when we reason about this we need to factor in what happens if there is no agreement reached so what is the disagreement fallback option that both players have once you factor that in it becomes a much more sophisticated thing than just you know trying to do a collective some you know optimization of the sum of the pay off so it is not a straight forward optimization no no not necessarily so this is part so so if they can enter into binding agreements they need not right so what happens if they do not agree is is is something that they are always mindful of while while trying to agree right so this is part of the part of the part of the way we go about reasoning about this again so I should also mention since you are on this topic so I should also mention another another thing here see cooperative versus non-cooperate does not necessarily mean conflict or non-conflict okay so corporate cooperation non-cooperation when non a non-cooperative game just means that players cannot communicate but it does not mean that there is necessarily a conflict in the in the game all right so they cannot communicate with each other but that does not mean they are out to you know kill each other okay does not mean the that there is a there is a conflict in the game so for example an a game where where there is no conflict but there is still you can still think of it as a non-cooperative situation is is a game where industry standards are being decided okay now industry associations are usually have you have industry associations for each segment you know there will be an association that that creates say the hardware another association that makes the software but unless they there is they agree on a standard they will not they will not be able to each get the best best outcome for both of them so for example like so you for you guys probably this is not you you are probably not as exposed to this but there was a time when when when cd drives and dvd drives were being introduced what should be the the radius of such a drive the physical radius someone has to define this standard right that it is 3.3 and half centimeters or something like that is the radius that has to that is so cd the cd manufacturer may prefer one kind of radius the the the one who makes the the reader of that drive would prefer another type of radius they would each want to arrive at some some standard because unless they work together nobody is going to be able to use either of their devices so they have to coordinate on a certain standard so coordination is is important but then they the situation could very well be non-cooperative in the sense that they do not want to be able want to be discussing with each other because they will be end up ending up revealing what what you know say for example their weak points or something like that okay so a situation where where industry wide standards are being determined is a situation which is often non-cooperative in nature but there is no underlying conflict in world okay likewise cooperative does not mean that everything is you know there is Bonhami either cooperative could also there could be conflict alongside cooperation for example cooperative games so cooperative game with conflict is a game say for example all these geopolitical conflicts in which people are trying to trying to come up with an agreement on say what should be the border or and so on and so these are actually two situations of conflict all right but you but you are you are trying to but it is because you are discussing and because there is there is eventually a pact to be signed and so on this is actually a cooperative game okay so so neither of these is true so cooperative does not mean so the correct way to put it is cooperative game does not mean cooperation necessarily okay does not is not equal to no conflict okay so now with with this now I think we can start with a formal definition of what a what a game is so this is of course a simple simple definition because I am not I have not yet introduced any exotic aspects of games