 In the quasi-linear domain, the social choice function can be actually decomposed into two components. So, you have already defined that this social choice function maps each of this the Cartesian product of all the type sets or the type profiles into an outcome. Now, outcome has two components we have already discussed in the allocation component and the payment component. The allocation comes from the set A and payment comes from the set R to the n, one for each of these players. And now what we can do is we can actually decompose this f into two parts. So, one function which maps this theta into A and the other component which maps this theta into R to the n. And we are going to call each of those things as the first component as the allocation rule and that is what we will be denoting with this lowercase f. So, what does that do? So, if each of these players report their types, it will take an allocation decision which will come from this A. So, to give an example, each of these players suppose are saying how much the value is specific object, then the indivisible item allocation just gives this item to one of these agents that is one allocation. Now, each of these types are denoted by lowercase theta i. So, therefore f of theta 1 to theta n is essentially an alternative which is living in this set A. And the second component, this component here is denoted by P. So, this is essentially a collection of functions. So, let us look at the ith component of this vector. So, the ith function, so this is a vector of functions because we will have to define the payment for each of these players. So, for player i, it is defined in the following way that if they report their types in this way, each of these players, then it will pick one real number. And notice that this real number can be either positive or negative. So, which means that in the remember our the definition of our utility, our utility had this form that vi of A comma theta i minus pi i. So, essentially this pi i is nothing but the real number value of that. So, we can as well replace that by pi of theta 1 to theta n. So, that is both these things are the same. Now, if this pi is positive, that means this agent is asked to pay. So, you purchase an object and you also ask to pay certain amount of money. So, therefore, there is a difference. But if the pi itself is negative, that means the agent is getting paid. So, not only it gets allocated, also gets some amount of compensation money. And those setups are also admissible under this model. So, as before this pi 1 to pi theta 1 to theta n is essentially one payment, real number which is living in this set R. Let us look at some of these allocation rules to understand what it means. So, one of the very straightforward and not so interesting allocation rule is a constant allocation rule. No matter what about the agents are choosing, you are going to choose this allocation A and you are completely agnostic about the types of the agents. So, that is a constant rule not very interesting one. The second rule that we have already discussed to a certain extent, this is known as the dictatorial rule. So, suppose we identify a specific player and call that player the dictator and we are going to pick the alternative or the allocation which maximizes its valuation and we do not look at the types of any other player. So, here even though this type, this type vector has lots of components. So, theta D is the the dictator's type and theta minus D is the type of all the other players apart from the dictator. But this allocation function, this rule F does not look at anybody else's, it is just picking that allocation which maximizes the valuation of this player. So, that is a dictatorial rule. A mechanism or allocation rule could be something which is known as the allocatively efficient rule and also sometimes called the utilitarian rule. So, what does that mean? So, suppose we look at the the sum of the values of all the agents. So, different people, so if you pick a specific alternative, let us say building a bridge or building a park, building a museum that gives different valuation for different players. So, there may be there are lots of people in the population who values environment more, lots of people who values transportation more and so on. So, we are taking the sum of the values of all the agents and picking that allocation which maximizes that sum. So, if we do that then what we are going to call that allocation is what is known as the allocatively efficient. So, we are taking the sum, picking that alternative that maximizes the sum. Just to keep a note here that this is different from parity efficiency even though sometimes will be a little loose in saying what kind of efficiency it is and we will just say that in the context of quasi-linear mechanism design we will just say that it is an efficient rule. What we mean actually is the allocatively efficient rule not a Pareto efficient rule. So, Pareto efficiency is a property that is defined for the outcome which also considers the payment. So, this is only looking at the valuation component and we will see a connection very soon how we can connect this allocatively efficient rule and the Pareto efficient rule. In some sense you can think of Pareto efficiency it is making a kind of vector to be better off. So, for each of these agents the infinity should be better off and there should be some agent for whom it is strictly better off. While allocatively efficient efficiency is looking at an aggregated matrix taking the sum and picking the alternative that maximizes that sum. There could be some cases where some people might be getting less valuation than other people but yes this is how it is. Allocative efficiency just maximizes the sum. So, the fourth example of an allocation rule is what is known as the affine maximizer rule. This is just a generalization of the previous rule. So, here what we were doing is we are just taking the simple sum of all the players. We are not distinguishing between different agents in some sense you are being fair to all the agents in some equal way. But suppose we do not do that we become a little unequal and put certain weights on each of these agents. So, let us say we put some weights lambda for each of these players take that weighted sum of their valuations and then translate that weighted sum with a function which is a function of that alternative or the allocation. So, that is the reason this is called affine maximizer because this is now an affine sum. We are taking the weighted combination of all these valuations of all the players and then you are translating with respect to this allocation giving a selective biases to certain allocation and you are picking that alternative that maximizes this affine sum. Of course, to define an affine maximizer we will first have to come up with all this lambda i's that will be fixed and also this kappa which is a function which translates in this the allocations and we are also going to assume that this lambda i's are not all zero. So, they are non-negative quantities but not all of them are equal to zero. So, then the type of mechanism that type of allocation rule that we get is known as the affine maximizer rule. You can think of so, still we are in the domain of utilitarian rules, allocation rules that we are looking at. So, maybe we are just taking the sum or you are taking unweighted sum. But what if you do something like a egalitarian, so a more equalized treatment of all the agents. So, how do you, what do I mean by this egalitarian treatment? Suppose we look at the valuation of a specific agent at its type and for a specific allocation A. Now, we look at for that particular allocation fix that allocation here and look at what is the worst valuation in the whole population who is getting worse off if we pick this alternative, this allocation and then pick that allocation which maximizes that minimum value. So, in some sense we are actually giving a push to the lowest possible valuation in the whole population and we are maximizing the minimum of the valuation in this in the whole population. If we do that, that kind of an allocation rule is known as the maximin allocation rule which is also called the egalitarian. So, you will see in some literature the term egalitarian being used or utilitarian is being used. So, egalitarian means that this is actually looking at the maximin while utilitarian is the sum of the valuations. Now, that is that where a few examples about the allocation rule. Let us now look at the payment rules. So, certain properties that we want from our payment rule when we are running this kind of mechanism. The first thing is that even though we are charging we are using payments transferable utilities. It should not be the case that the mechanism in order to satisfy certain properties run into a deficit that is now the mechanism designer or a person who is actually selling an object has to pump in money to run that mechanism. So, that condition is called no deficit that means the sum of these payments of all these agents should be non-negative. That means at the end of the day there should not be any money that will be needed to run this mechanism rather the mechanism designer might gain some amount of money or the transfer should be exactly equalized that someone is getting paid and someone is paying money and the sum is exactly equal to 0. Now, the second property is that of no subsidy. This is the property where the payments are always non-negative for all the agents. So, this is the on an aggregate level this is non-negative here it is non-negative for every agent. Of course, no subsidy will imply it is non-deficit but yes no subsidy is a more stricter condition. So, we are asking every agent to either pay or not pay at all but there would not be any situation where the mechanism will pay a specific agent. And finally, the other property that we will be looking at later on is what is known as budget balance. So, this is when this no deficit condition is is met with the equality which means that the sum of the payment is essentially getting redistributed among these agents someone is paying a certain amount of money and someone is getting paid but the sum of the payments is exactly equal to 0. So, this is desirable in certain settings because you do not really want to earn money or maybe gain a lot of money and we will see that this could be a problem in certain mechanisms. You do not want a lot of money to be taken away just to run that mechanism rather it will be very desirable that in order to run the mechanism if certain make a certain party certain of agents are paying money which is being redistributed among the other agents that will be very desirable. Let us now recall the definition of incentive compatibility that we defined long ago in the very beginning of this section the mechanism design section. So, in we are going to reiterate the same definition in the context of Quasillinia preferences and in the context of mechanisms with transfers. So, we know that the mechanism is a tuple of this allocation and payment f comma p and we are going to call that mechanism dominant strategy incentive compatible if the following thing happens we have already seen an equivalent thing. So, in the previous definition of dominant strategy incentive compatibility we have just written it in the terms of the utilities. Now, we are going to write it explicitly in terms of the valuation and the payment. So, what are the two components? The first component is the allocation decision which is given by this f and the second decision is this payment decision which is given by this p. Now, what can happen is that player i suppose player i's type two type is theta i and it can report theta i or misreport something else which is theta i prime. Now, this dominant strategy incentive compatibility would ensure that when it reports its type theta i truthfully then this left hand side is the utility that that agent gets and the other agents can pick any type. So, it need not necessarily be their true types it could be true types or some false types whichever they want to choose. But under the same condition that the other players are choosing the same types if player i changes its type report to theta i prime then that utility is not going to be better than the utility that it gets when it reports its truthfully. So, and this inequality should get satisfied for all theta minus i tilde's so theta minus i tilde for all the other agents and also for all theta i and theta i prime. So, if this holds then we are going to call this mechanism to be dominant strategy incentive compatible. So, in other words in one liner definition would be that dsic means that truth telling is a weekly dominant strategy equilibrium. We have seen this in the previous case here we are just writing it in the context of mechanisms with transfers. And we are also going to use some terminology and it is good to know that because we will be using this kind of a terminology later on in this course. We are going to focus mostly on the allocation part. So, in the in the quasi linear domain we are often very much interested just in the allocation. So, the allocation whether that is that is maximizing the sum of the valuation in some sense that is the social welfare maximizer or not that is that is a objective that we have and payment serves as a mechanism or as a artifact of essentially making that happen. So, it is not that we are jointly deciding of course the mechanism decides the allocation and payment jointly. But the question that will be often asking that whether we can implement and this is the term that we will be using whether we can implement a function this allocation function f using some payment rule in dominant strategies or is f implementable in dominant strategies by some payment rule. So, as if the payment rule is not that important as long as we can find some payment rule and we can implement that f that payment will implement f in dominant strategies will be happy our primary objective is to satisfy f and not the whole combination of f and d. This will make more sense when we discuss more examples but in some cases will directly refer to f as the social choice function. So, even though in our context we have used f capital f which takes this capital theta and gives the outcome which is a Cartesian product of allocation as well as this payments but sometimes only the this is the terminology that we will find in literature as well that theta to a that itself is called the social choice function and we will be calling that whether this social choice function f is implementable in dominant strategies or not. So, that is just a kind of a terminology some nomenclature that we will be we will be using. So, yeah so it will make sense when we discuss examples. So, what actually needs to be satisfied for a dominant strategy incentive compatible mechanism. So, let us say our mechanism is f comma b and suppose we have two players and their types are typesets are same theta either their types have a high type or a low type and therefore, the function the allocation function is a Cartesian product mapping from that Cartesian product of these two typesets to the set of allocations. So, what should happen you can so this is just writing down the same expression. So, this inequality for each of these cases I am just doing one of them to to just give you a feeling if you do it yourself then you will be able to understand what is going on if you just do not look at the this inequalities and try to write down what is the DSIC conditions that will reinforce the understanding of what dominant strategy incentive compatibility is. So, we are looking at player one first. So, the valuation of that player one when his true type this player one's true type is theta h and it is reporting type it is typed with theta h. So, therefore, the payment function also has the same thing and the other player player two is reporting any type it need not be its true type this should be at least as much as when he is reporting his type to be theta l. So, its true type is theta h it is reporting misreporting to theta l and then this is the utility that that agent should get and this inequality is saying that reporting truthfully is better than not reporting it truthfully and this inequality should hold for all theta twos. Now, we only have theta l and theta h. So, we are just expanding it out. Similarly, when player one has a type which is its true type is low type then it should report it truthfully. So, report theta l truthfully and that utility should be more than the utility and he is misreporting even if it is true type is theta l it is misreporting to theta h. We will see certain implications of this when we talk about options but yeah. So, this thing should also hold and they should hold for all theta twos. So, this is only for the inequalities the same set of inequalities for player one. Similarly, you can write it for player two and I do not want to spend time on that. So, let us look at one specific property of the payment rule that implements an allocation rule. So, suppose we have a mechanism with transfer which is given by f comma p the allocation rule and the payment rule this is incentive compatible. So, because it is incentive compatible we already have this p i s. Now, let us define some other payment rule. So, a payment rule which is just adding another term. So, we look at the original payment. So, notice that this is a function of theta i and theta minus i the reported types of all these players and now for player i I am just forgetting its own type I am just adding a function which is function of theta minus i that is its sort of constant from the point of view of player i. So, whatever the other players are choosing as their type that determines that h i function and we are just taking the sum here. Now, the question is the way we have so we have created a new set of payment functions. So, if we look at this mechanism f comma q instead of p now we have q is that mechanism dominant strategy incentive compatible? The answer is yes and why is that you can just write it down. I mean you just write the conditions explicitly. So, all that we have here is this h i theta minus i tilde if you write down the original expression and that h i theta minus i tilde is unchanged on both sides of this inequality. So, essentially this part will get cancelled out and all that is left that you are left with is the same inequality and because this f and p together was dominant strategy incentive compatible this inequality will always be true and for that reason you can always add this kind of a function h i theta minus i to a payment scheme which is already known to implement the allocation rule f without the loss of dominant strategy incentive compatibility. So, the summary is that if we can find a payment that implements an allocation rule then there exist uncountably many because h i could be arbitrary you can pick any kind of h i and many payments that can that can implement that the same allocation rule. The converse question so here it is just saying that if you have a payment you have many payments but the question is when can we say that the payments that implement f differ only by this factor h i. So, in some sense we are not only identifying a specific payment we are identifying a specific payment up to the change of that payment with h i theta minus i. So, you can find and so there does not exist any other kind of payment which will actually implement it. So, if there exist any other payment that can only be written as a factor an additive factor of h i theta minus i of the original payment. So, let me just give the second implication of this incentive compatibility on payment and leave that the proof as an exercise. So, suppose there are two type profiles so one is theta so theta is nothing but theta i theta minus i as we have defined earlier and there is another type profile which is theta tilde where only player i's type is changing to theta i tilde other player's types are still theta minus i and let us assume that this even though player i has changed its type so its type has changed to theta i tilde the social choice outcome or the allocation outcome did not change. So, there could be some many situations you can think of it if player i's type changes from theta i to theta i tilde it is not changing sufficiently to change the outcome altogether. So, if the outcome the allocation outcome is the same and if p implements this function this allocation function in dominant strategies then what we are going to say we are going to conclude the claim is that the payment should also be the same and this is not very difficult to show you can just write down the inequalities for DSIC under these two conditions and use the fact that the allocation is the same so that the valuations essentially cancel out from both ends and use it when when you are using the theta i to be the true type of player i then what is the implication of the DSIC condition and when it is theta i tilde so player i's true type is theta i tilde and it is misreporting to theta i let us write the corresponding condition and the in the previous case when it is true type is theta i and it is misreporting to theta i tilde what is the what is the corresponding condition inequality for the DSIC and you will find that this is pi theta and pi theta tilde both will be equal