 Hello and welcome once again to this course on Game Theory and Mechanism Design. In this course we will be discussing these two topics, Game Theory and Mechanism Design as mentioned in the title, almost in equal proportions. This lecture is intended to give you a very broad overview of what this course is all about. Now, our focus in this course will be more on mechanism design and game theory will be a preparatory step to learn the mechanism design part of this course. Now what is mechanism design? It is often called the engineering approach to economic theory and this is defined by Eric Maskin who is one of the recipients of the 2007 Nobel Prize in Economics for his contribution in mechanism design theory. And indeed mechanism design is an engineering approach. So what do we typically do in usual engineering courses? So typical engineering courses always have two components to it. One is the analysis component and other is the synthesis component. So let us look at certain examples. So let us say the classical engineering course of circuit analysis and synthesis. This is a very popular course in electrical engineering. In the analysis part of this course, what is typically done is you are given a bunch of resistors, capacitors and they are connected over the circuit and you are asked to find out what is the voltage or current in different parts of this circuit or the network. On the other hand, when you go for a circuit synthesis course, it tries to give you the objective that we want certain desired voltage or current in some part of this network of the circuit and you will have to synthesize the circuit using those resistors, capacitors and other equipments, other components to get that desired voltage. Similarly, when we talk about algorithms courses, typically in design of algorithms you are given a specific algorithm and you are trying to analyze that algorithm and find out different parameters, different metrics of that algorithm. For instance, complexity is one such metric and in the design part of the algorithm, you are trying to design an algorithm which will give a certain desired complexity. Now using these two examples, you can clearly see what I mean by these two complementary approaches and indeed game theory and mechanism design are complementary approaches to each other. We will see that in the game theory part, we are going to give a specific strategic form of different players and we will try to find out by analyzing the game part is the probable outcome which is the analysis part and in the mechanism design part, we will try to design a game, a synthesizer game such that we get certain desirable outcomes. Now game theory is defined as the theory where there are multiple individuals whom we will call agents or players and who have potentially conflicting interests with each other and we are trying to give a kind of a predictive guarantee. So the setup in game theory is the following. We have these multiple individuals and the important part is that they have conflicting objectives. We will soon discuss an example and this will make clear what we mean. So these terms like agents, players will be using interchangeably in this course and this setup of multiple players or agents with their conflicting objectives or their payoffs is what we will be calling a game throughout this course. As I mentioned before, in game theory part, which is the first part of this course, we will be giving a specific game or essentially start with a given game, we will try to find what is the most probable outcome of that game or maybe outcomes of that game which will depend on the responses of these agents or players and this is the first part which is called the game theory part and quite naturally this is the analysis component and the guarantee that we are going to give here is essentially predictive, we are predicting what is going to happen in this game. Now if you take the inverse approach where we want a specific reasonable outcome and we want to find or build the game that yields those kind of desirable outcomes as the probable outcomes probable as we have defined in the first part that is the game theory part. This type of approach we will be calling that mechanism design and quite naturally this is the synthesis part as we have already discussed in the classical engineering courses and the approach is there will be prescriptive that is we are going to give a specific prescription how you should design this game such that you get all those reasonable outcomes of that game. Now let us start with an example to illustrate all these points that I have made so far. So this game is called the neighboring kingdoms dilemma if you search for it you can search with prisoners dilemma I have just renamed it slightly in order to make it more interesting in this context. So suppose we go back a little little back in history so this was the times when kingdoms used to rule so suppose there are two kingdoms A and B and they have very limited options or resources to invest they either can invest on agriculture and therefore save people from starvation or they can invest in warfare and for the time being let us assume that they cannot invest on both although there are possibilities and we will also discuss such situations but for this example let us imagine either they can spend entirely on agriculture or entirely on warfare so if they invest on warfare what happens is they are capable of defending themselves because they are raising a powerful army and also they can attack other kingdom to ransack their produce whatever and get their land and wealth for everything. Now one important point that I would like to make here is that when you look at this kind of scenario it is not only sufficient to conclude what will happen based on your own actions so maybe if you are one of these kingdoms and you are investing on agriculture you cannot be sure what will be your payoff because you do not know what the other kingdom is going to do if they invest on warfare then they might possibly attack your kingdom and you will have no defense because you have not invested in warfare so you will not be able to defend yourself so all your produce agricultural produce will be taken away and also your land will be lost so the outcome in this kind of a scenario depends not only the action picked by one of these players or agents it is dependent on the action profile which means that the actions picked by both these agents. So to represent this more concretely and in a more quantitative manner let us look at a matrix and this is how we will be representing this kind of games for a while now so what does this matrix say so there are you can see there are two rows here and these two rows represent the two available actions of Kingdom A the first player. Similarly these two columns represent the actions that are available to the second player which is player B so and the numbers in each of these boxes actually represent by some number I mean we are just using some representative number to denote how satisfied each of these individuals are so if suppose Agent A is picking agriculture and Agent B is also picking agriculture then we are assuming that both of them will save their kingdom from starvation the people of their kingdom from starvation at the same time because both of them are choosing agriculture nobody will attack each other and therefore their produce will remain with themselves and both of them get some amount of payoff which we are representing by this number five fair enough now if both of these kingdoms choose war then there won't be any agricultural produce because they have not invested anything on agriculture so okay so just to give you the context that this each of this so in each of these boxes we are putting two numbers the first number is actually representing the utility or payoff this is how we are going to call this utility of the first player in this case player A and the second number in this double is going to be the utility the second player which is player B and so and that is common in all these boxes now when we are considering the war situation for both these players then they will the only option they have is to attack the other kingdom and to loot their resources but the trouble is that now the other kingdom is also investing in war so they will be able to defend themselves so they will get some amount of payoff in the sense that they will not lose everything they will at least save their people and save their resources but they will get much less payoff if they were investing in agriculture so that is represented by one and one in this case but imagine the situation one of this kingdom let's say kingdom A is investing in war and the other kingdom is investing in agriculture in that situation what will happen is this kingdom which chooses war that will go and attack the other country other kingdom and not only they will get their agricultural produce they will also get their wealth their land and all those things so possibly they will get a little higher payoff than what they were getting if they are choosing agriculture alone on the other hand the second player will choose will get zero payoff because not only their agricultural producers were lost they were not able to defend themselves so they have actually lost their their produce as well as their land and perhaps human resources as well the similarly opposite thing happens when A chooses agriculture and B chooses war this is a very symmetric game now what is the predictive guarantee that we can give in this kind of a scenario so you can see that if the if a specific agent let's say agent A is thinking about whether to go for agriculture or to war it looks at what happens if the other agent is choosing agriculture versus or war look in this in this column the column of agriculture when player B is choosing agriculture you see that for player A it is getting a payoff of five when it chooses agriculture but it gets six if it chooses war so clearly war is a better option for for this kingdom kingdom A when the other player is choosing agriculture now let's look at the the other situation player B is choosing war then if it chooses agriculture then it gets nothing essentially it loses everything but if it goes for war at least it can defend itself and save its resources so again war is a better option so we can conclude that no matter what the other player is choosing going for a war is a better option for both for this player and you can use the same argument it's a symmetric game the same argument for for player two that when player A is choosing agriculture it is better for player B to choose war and similarly if the other player chooses war of course it is better for for it to choose war so this is the way you should be comparing so putting everything together in summary we can say that a war comma war is the most predictable outcome even though it's not the most optimal outcome this most optimal outcome would have been five comma five which is both of this kingdom's choosing agriculture but personal greed and no communication between these two agencies essentially making a worse outcome and that is what we are going to predict at least from this kind of a game so this is this is the reasonable outcome you might not like the outcome but this is what is what is predicted via game theory a game theories will tell that this is the most predictable outcome now we have already discussed this so let's make the the terminology a little more formal because we'll be using this over and over again in this course so a game is a formal representation of the strategic interaction between multiple agents called players the the choices that are available to these players are called actions in this example the agriculture or war was to possible actions now we'll sometimes call what is the strategy so the strategy I mean at this point it is a little more abstract because we have used both strategy in actions as same but in in principle and we'll see this in a in some later examples that it is just a mapping from the state of the game to the set of actions so here we have the set of actions which are these two actions available to these players and the state of the game was just one one state which we have defined here but if there are multiple states of the game then we define strategy as this mapping and we'll see examples later on in this course so depending on the context games can be represented in represented in many ways and the the most common two representations are called the normal form and extensive form representations the normal form games are typically used for representing games that end in one round so all the players choose their actions or strategies and the game ends right after that extensive form games are kind of sequential games one player makes some move based on that the second player makes some move and then first player makes a move again so imagine a game like chess which is more more succinctly represented in the form of an extensive form game extensive form representation of the game similarly there are repeated games stochastic games and various other forms of games which will not be discussing at least in this course now so game theory is the formal study of of strategic interaction between the players that are called rational and intelligent now what does these two terms mean we call a player to be rational if that player picks the actions to achieve the most desired outcome for that agent right so in this example we have always picked we have always assumed that the player are picking the actions which maximizes their utilities so that is one way of representing rationality and the second part is is called intelligent intelligent agents this is the assumption we make for for the players a player will be called intelligent if she knows the rules of the game perfectly and picks the actions considering that there are other rational and intelligent players in the game so this definition is a little circular in nature but it is purposefully done done in that way so it means that it does not only consider its own payoff maximizing actions it also takes care of what will happen in the game if there were other players who are also thinking in a very similar way like this player and in that case it will pick the actions which is not only just a just their own utility maximizer but also something like a joint maximizer of of certain kind we will discuss this kind of points later on when we discuss equilibrium so equilibrium is a point you can at least imagine that a point in this game a predictable outcome from where none of these players would like to deviate because if they do any deviation unilaterally then that will be not beneficial for them so that is that is intelligence we will see the ramifications of this assumptions of rationality intelligence when we discuss more examples so I mean if you want to make things a little short and concrete intelligence essentially implies that the player has sufficient computational ability to find the optimal action which is which is optimal against other players who are also of the same type it's not only your unilateral so this is the fundamental difference between single agent optimization versus multi-agent optimization so in in classical artificial intelligence you will find techniques which is trying to maximize optimize certain things but in in game theory we will we are dealing with multiple agents and therefore it is not meaningful to talk about an optimization which which has an objective function you are trying to maximize that rather you are trying to find an equilibrium of this game which all this rational and intelligent player will will achieve and that is that is a fundamental difference in in game theory