 Let us now start discussing the Pareto optimality notion in Poissonian domain. Pareto optimality or Pareto efficiency as we have already referred to earlier is a notion where the we talk about the development of each of the utilities of every player rather than maximizing the sum of or an aggregate measure of something. And also the difference is that in this Pareto optimality we will have to look at the payment component as well while in allocative efficiency we were just looking at the allocation component of the decision problem. So, let us look at the definition first and then we will connect the Pareto optimality with the allocative efficiency. So, what does this say? So, we are going to call a mechanism a mechanism in the Poissonian domain to be Pareto optimal if for every type profile the following thing happens that there does not exist any allocation for a different allocation which is not equal to the allocation of f theta and the set of payments pi 1 to pi n with the additional condition that this payment is at least as much as the payment of the original mechanism. So, if the mechanism is extracting this much amount of payment allocation and the payment that we are going to talk about also at least extracts the same amount of payment then this the utility under that allocation and that payment is at least as much as the utility when you are under this mechanism f comma p. So, if there does not exist any such b and pi then we are going to call that this is Pareto optimal in some sense you cannot really make it better for weakly better for all the players and you can make it strictly better for one player. So, of course this is this inequality is weak for all the players and it this inequality is going to be strict for at least one agent. So, why do we need this additional constraint? Let us look at this. So, if we so, suppose we did not have this constraint then what would have happened? One could have easily found some allocation and could have given sufficient subsidies. So, maybe all this pi i were negative. So, that is the mechanism designer is essentially paying a lot of money to the agents and therefore this inequality would have been very easy to satisfy. So, we do not want that kind of a thing. So, we also in addition to the fact that this inequality should be satisfied we also want that this counter example of this allocation and payment should be such that this payment at least extracts the same amount of money as the original mechanism. So, now we are going to connect this notion of Pareto optimality simply speaking the Pareto optimality is saying that if you are given a specific allocation and a payment rule then there does not exist any other allocation and payments the some of the payments being at least as much as the some of the payments in the original mechanism such that all the players are better off in that new allocation and payment and there is at least one agent who is strictly better off. So, here we are going to compare the Pareto optimality with allocative efficiency we are going to say that if it is Pareto optimal then it implies and it is implied by the fact that it is allocatively efficient. So, this is this is kind of equivalent. So, in this quasi-linear domain we are not really looking at the payments there is a reason for that I mean Pareto optimality and allocative efficiency are one and the same. So, you do not really gain something more by defining Pareto optimality by putting restrictions on the payment rather it is sufficient to look at the allocation alone. So, let us try to prove this we will first prove in the forward direction that is Pareto optimality implies allocative efficiency rather we will prove it in the in the following way that if it is not allocative efficiency. So, that Pareto optimality implies allocative efficiency is equivalent to saying if it is not allocatively efficient then it is not Pareto optimal as well. So, what does not allocative efficiency mean that there exists some allocation which is not the allocation given by this mechanism such that the sum of the valuation is at least as much as the sum of the valuation under that under that allocation rule F. So, remember what was allocative efficiency it was looking at the sum of all these valuations e comma theta l and summing over all the agents and picking that alternative which maximizes the sum. So, this was allocative efficiency. So, it is not allocatively efficient means that there exists some b which actually is larger than this sum and this is going to happen for some theta. Now, because this is larger than that if we just look at the difference between them and define that as delta that is definitely going to be positive by the choice that we have made. Now, we are going to construct so what do we have to show we will have to show Pareto optimality a violation of Pareto optimality. So, we will have to create a count example we already have found some allocation which could be a candidate allocation to form that count example all that that is remaining is to find the payment and this is how we are going to construct that payment. So, let us say this pi i so here it is sum over all the agents we are looking at the individual differences so vi b theta i minus vi of f theta theta i and then we are summing the payment. So, this payment is originally given because that is that is the mechanism itself. So, we are given the mechanism we are just using the same payment here and subtracting out this delta. So, which is guaranteed to be positive by n now why we have chosen this specific form will become clear in one or two steps so now what we can say is because we have our candidate and this we are going to define for every i now this candidate payment and the allocation b is our count example to Pareto optimality then we should be able to show that the utility here which is vi b of theta i minus pi i so just look at the corresponding example so this definition of Pareto optimality if this happens then we are certain that this is not going to be Pareto optimal. So, we write this left hand side minus the right hand side here so minus this right hand side and we see that that is equal to delta over n I mean this is just reorganizing this equality here and because of the fact that this delta is positive strictly positive then this inequality will be positive that means that for every agent i in this new allocation and payment the difference between the utility of that player is getting strictly better off than the allocation and payment under the original mechanism and we are also not violating the payment condition because if we just take the sum over all these pi i's you can see that this sum I mean this difference is exactly equal to delta if you are summing over all the agents so once you sum over all these all these agents all that you are and you are summing over this delta over n so this will become exactly equal to delta and that is exactly equal to this part so these two parts will cancel out what will be remaining is the sum over all this pi i so this pi i is exactly equal to sum over pi i so we are also maintaining that payment condition so we found that f is not bad optimal and here is that count example okay so we have proved the forward direction now let us look at the reverse direction and we are going to prove it in a very similar way reverse direction means that if it is allocatively efficient we will have to show that it is bad optimal rather we show that if it is not bad optimal it cannot be allocatively efficient so what does not bad optimal mean that there exists some allocation and this payments such that this payment inequality the payment constraint is made and also this inequality holds for all agents in weak form and then it is strict for some agent j in them now if you just sum over all this inequalities because this inequality is going to be strict for some j so the this weak inequality will become a strict inequality when we are taking the sum over all of them so fair enough now we have we are just looking at the difference between this sum of this valuations under b and the sum of the valuations under f right so we are just transforming it on the right hand side so we can see that this is strictly going to be larger than sum over pi i minus this sum over pi theta i and because of the assumption so because this is not parent efficient so this inequality is also going to get satisfied so we see that this particular term the right hand side of this inequality is actually nonnegative and that implies that this is this at b the sum of the valuations of all the agents is getting strictly better than the sum of the valuation at f of theta and therefore f of theta is f is not an allocatively efficient allocation so that essentially proves both the directions so as soon as we have parent efficiency or parent optimality in this quasi linear domain we we can immediately say that it is allocatively efficient allocatively efficient and vice versa now let us look at the allocative efficient rule so now we don't really have to worry much about the parent optimality because both are the same we can focus on allocative efficiency now what we can see is that this allocative efficient rule is actually implementable so how do we design payments so we have made this point earlier that whenever we say that a specific allocation rule is implementable that means there exist some payments which will implement that in dominant strategies so this rule is the rule which maximizes the sum of the values of all the agents and we will have to just construct the payments and we are going to construct the payment in the following way and this particular structure is named after Groves who has actually given this structure so what we are going to look at is first for this while designing the payment for player i we are defining a function h i which is not a function of theta i the type of player i it is dependent on the types of all the other agents and the second term and from that we are subtracting out the sum of the values of all the agents except agent i under the same outcome so f efficient is essentially the allocatively efficient outcome the allocation in this in this context and we are just looking at that but except agent i now this function h i could be arbitrary so you can pick any any number it can be constant it can be zero whatever it is but it gives you the complete flexibility so what does that mean let us look at this kind of a payment rule using an example so suppose we have single indivisible item allocation and there are four agents here their types are given in the following way the types are given by theta one equal to 10 theta two eight theta three six and theta four is four when they get the object so this type means that this is the value that they get when they get the object zero otherwise and let us also fix this h i theta minus i to be the minimum of theta minus i this is just an arbitrary thing i mean you could have chosen some other numbers the corresponding payments should have been different now if everyone so let us assume for now we will later prove that this is also true that if everybody reports their true type then the values of this h i are going to be as follows and you can you can verify that now the efficient allocation gives this item to agent one why because if you if you give it to because this is single indivisible object if you give it to the to this agent you are maximizing the sum of the values right so that is why it is it is allocatively efficient now you can define the corresponding payments and payments exactly as you we have defined in this case so this is going to be h i theta minus i minus the sum of the valuation of all the other agents when it is when the allocation is efficient so in the efficient allocation only agent one gets this object and nobody else gets this object so therefore the valuation so the sum of the valuation of all the other agents is going to be zero and that agent gets and gets this h i so this h one is exactly equal to four so this is going to be the payment for player one now when you are looking at player two you know that the sum of the valuation so the efficient allocation gives it to the agent one so whose valuation is is going to be 10 so that is the that is the number that we are putting here and this one is the h two and we see that h two is also four so this difference is minus six which means that player two is getting paid six amount of money and similarly you can carry out the rest of the calculation you will see that only player one is paying and other players are getting paid so that is that is an example how if the players were reporting their types truthfully what will happen in the in in this context now we are going to make a stronger claim we are going to claim that this mechanism this gross payment mechanism essentially gross is a class of mechanism because you can just change the h i's and you can get a bunch of payment rules and all of them are essentially members of the gross mechanisms class so all this class of gross mechanisms are dsic now why we why is that in order to prove dsic we will have to satisfy that inequality that we have already defined and let us do it in step by step we are just focusing on player i and we are arbitrarily picking this player i so therefore this is without loss of generality suppose its own true type is theta i and other players are reporting theta minus i tilde so if this agent reports its type truthfully then the outcome becomes a and if it misreports to some theta i prime then the outcome becomes b so by definition we know that because this is allocatively efficient this v i a i theta i plus this thing so we are just writing the sum of the valuations of all the agents at a when this agent is reporting is its type truthfully and we are just decomposing it there is a purpose why we are doing this decomposition because this particular decomposition will show up later in our proof and we'll use this inequality and this is going to be at least as much as any other allocation in particular if you put b as well this will also be larger than that because a is essentially the allocatively efficient outcome this so whenever we are looking at this type so this type is the true type theta i when its true type is theta i and you are trying to maximize it you will get the allocation a as your outcome b might be an efficient allocation when you are looking at treating as if the player i is true type is theta i prime and not theta i so when its true type is theta i a maximizes this sum over all the other alternatives in particular the alternative b as well okay so let's save this equation one and we'll reuse it now what do we really need to show so utility of player i when he reports theta i is at least as much as the utility when he reports theta i prime so let us write down the left hand side of the dsic inequality so this is the left hand side so we have the the the rule is nothing but f efficient the the mechanism is f efficient and p one up to p n this is the mechanism and under this mechanism what is the the utility when player i is reporting is type truthfully that is reporting is its type to be theta i and the other players are reporting their types to be theta minus i tilde that is given by this so player i is reporting theta i and others are reporting theta minus i tilde similarly the payment is calculated according to that and we are evaluating player i is evaluating this utility when its true type is theta i that is its true type and therefore this inequality should hold when its type is theta i so let us write this down writing down the the payment function so we have all we already know that the payment is given by this roves payment rule so we can just expand that out by each i theta minus i tilde and the other part is the the sum of the valuation of all the agents except agent i at the same efficient allocation so now there is a reason why we have chosen this structure of of a specific form so now we know that this this is nothing but a because the that is the allocation when agent i is reporting its type to be theta i others are reporting theta minus i tilde and that is the same here as well so we can club these two things together this and that and then we can write that this is going to be the sum of the valuations of all the agents right so sum of the valuation at the efficient allocation and because of that fact that this is essentially maximizing the sum so what is the definition of a efficient just do this exercise yourself so if i theta i and theta minus i tilde what is it it is actually the arg max over the summation of all these v is a of theta i so you can imagine that this you can just write so instead of writing it in this way it is better to write it in the following way so we just isolate this theta agent i because its type is theta i for all the other agents so sum over v j's where it is a and theta minus so theta j so this is agent j theta j tilde's j is not equal to all so if you look at this sum here it is actually maximizing this sum and because this is the efficient thing then we can actually write it this to be greater than equal to so whenever we are plugging the same value inside this a we are going to get this this sum to be greater than equal to the same sum vi of theta i plus the sum over all these things for every other for every other b which is not equal to a and that is exactly what we have written here this is the inequality one and we are just now using the same fact that this is because this is greater than equal to any other alternative in particular when we are looking at this alternative b which is nothing but when a player i is reporting its type to be theta i prime and yeah so that's that's it we can actually write this the same inequality here this will this will get satisfied think about it I mean this will be clear because there is a little bit of notation involved it might get get a little confusing in the very beginning but remember the fact that agent i is true type is theta i and this is exactly where we are trying to maximize it so everywhere you will look at that what is happening when we are trying to maximize this we are looking at agent i is true type is theta i and that is why it is the the arg max on that profile on that profile of theta's and therefore this is going to be greater than equal to this inequality for all b's for all b's which are not equal to a and in particular this inequality will hold whenever we are speaking a very specific outcome which is theta i when agent i is reporting theta i prime and theta minus i others are reporting theta minus i tilde and we are running the same efficient allocation this could be a different alternative a different allocation but for that allocation this inequality should hold and that is that is essentially the end of the proof now we can club these two things together so notice that this is completely independent of the type reported by player i so this will not change and this two things together we can write it as the payment of agent i when it is reporting its type to be theta i prime and others are reporting theta minus i tilde so together you can write that this right hand side is nothing but when he is misreporting when the agent i is misreporting to theta i prime this is the inflator that it gets and because of this inequality is getting satisfied we can say that this mechanism f efficient is implemented by this gross payment rule so actually this gross mechanism class of gross mechanism is actually dominant strategy incentive compatible that is because of the fact that we have shown this for any arbitrary i so you can repeat this argument for any i and it this inequality will hold for all i in it