 It's a great pleasure to be back here. I know many people already. So this talk is about clear competition in market. So market we deal in our everyday life. So this is a vegetable market, stock market, transaction market and more recently online market. So the basic question here is how the price of a good is determined. Why some good is more first year than the other. So let's take an example of a single good. So at higher prices supply is more because more people are ready to produce it. So supply curve is like this. At higher prices there is more supply because more people are ready to produce it for you. But the demand we have in opposite way, at higher prices demand is less because less number of people will be buying this food. And when these two come in, then we call it an equilibrium. So here P star is the equilibrium price of this good and Q star is the demand of this good as well as supply of this good. So at equilibrium demand equals supply. So this was about a single good. But when there are more than one good, so suppose you want to have a dessert and you have two choices as cream and cake. Now the demand of ice cream depends on the price of cake. It is not simple anymore because these two goods are substitutes because like both set your desire of having dessert. And this theory has been studied since century. So August was by Adam Smith in 1700 who was advocate of free market economy. He also coined the term called invisible hand of the market. So what he said that it is not from the benevolence of the beggars that we expect of our dinner but from their regard to their own interest. So he says that when everyone is behaving rationally and selfishly then market automatically operates at an equilibrium. And later in 1874 Yolvaras, who is considered father of general equilibrium theory he was the first one to mathematically formulate a market. And in fact he gave an algorithm to compute market equilibrium which he called tetomide. So algorithm is very natural. It is very simple. You start with some prices and calculate the demand. If the demand of some good is more than its supply then you increase its prices. And if the demand is less than its supply then you decrease the prices. When it will converge then that will be the equilibrium. So he gave this dynamics which will converge at an equilibrium. However he did prove that every market will converge to equilibrium. If not then under what conditions it will converge? Another famous economist of the same time, Irving Fisher he in fact built a hydraulic computer to compute an equilibrium. So this is the 19th century computer to compute an equilibrium. When the water level is stabilized then you get equilibrium. So this was just the beginning of the beautiful theory of market equilibrium. So the plan of this talk is I will mathematically define what are markets and the notion of equilibrium. And then from the series of remarkets of work now we have a clear dichotomy in equilibrium computation. So I will show you that. And then I will show you one algorithmic result. And one hardest result. And then I will briefly talk about some of my other works in markets. So that is the plan. So let's start with markets and equilibrium. So in the market so there are a set of goods and a set of agents. And feel free to ask me question if something is unclear. Each agent brings some goods to market and even prizes of these goods. Agents sell their goods and earn some money. And from the earn money they want to buy a bundle which they prefer the most. And it should be also affordable because then they have the limited amount of money. So this is called optimal bundle. So when they are doing something like this then it also happens that demand of every goods meets its supply. Because at some prices demand of some goods is more than its supply because they prefer something. Then if it also happens that demand meets supply then we call it an equilibrium. So this is the notion of the equilibrium. And for optimal bundle there is a utility function. We can associate the preference of every agent by utility function and optimal bundle is obtained by maximizing this utility function. And it is assumed to be concave because it models law of decreasing marginal returns. So for example you want to have an ice cream for the first scoop you have some happiness. For the second scoop your happiness decreases. And maybe like forth with the scoop your happiness may be negative or you are even ready to pay for not consuming it. So this is the assumptions on utility function of agent. So the agents also bring money or they just bring goods? They bring goods. This is like barter system. It is like exchange market. But any other market like with the money you can convert it to this market. Because money can be another good. There are many ways to do. So this is the more general market model. Most of the market can be converted to this model. And this is given by Aaron the group. Okay let's take a simple example. So suppose there are three goods orange, apple and mango. And there are two agents. This for them one tree and bubbly. And one tree brings like one kilo of apple and one kilo of mango. And bubbly brings like one kilo of orange and one kilo of mango. So this is what they bring to the market. This is their initial bundle. And the utility function is like what they want to consume? Orange and mango in equal proportion. Suppose they want to have a juice for which they want both of them in equal proportion. So that is what they want to consume. And here like bubbly is like apple and mango. So her utility function is she wants to consume apple and mango in equal proportion. Okay. So suppose prices of goods are one, two and one. One rupee for orange, two rupee for apple and so on. So at these prices, earning of one tree is like he has like one kilo of apple and one kilo of mango. So two plus one, three. So three rupees he can earn by selling his initial bundle. And similarly bubbly can earn two rupees by selling an initial bundle of orange and mango. So this is their initial earning at these prices. So what is demand? So at these prices, since he doesn't want to consume apple, so demand of apple is zero. It doesn't want, it doesn't like apple. And what will be the demand of orange and mango because like he has three rupees. And he wants to consume like orange and mango in equal proportion. In the price of orange and mango is one each. So it's like three divided by two. So three by two kilos of orange, he will demand three by two kilos of mango. So here like this is an example. That's why this is a simple utility function but it can be very complicated. And similarly we can compute the demand of bubbly and that will turn out to be like two-third of mango, two-third of apple and zero kilo of orange. So this is the demand of these two agents at these prices. Because at these prices they earn some money and from the earn money they want to maximize their utility function. And utility function can be very complicated. We have just taken some example where they want to consume in this fashion. Okay, so now what is the total demand of apple? So total demand of apple is like zero plus two-third which is two-third. And the total supply is one unit because only one unit, one kilo of apple. So demand is length and supply. And the demand of mango is more than supply because like the total demand is more than two and supply is two. So it is more than supply. So demand is not equal to supply. So this is not an equilibrium price. So these prices are not equilibrium prices. Suppose the prices are one by two. So we can do the same calculation again. And we can see that like they will earn three rupees each and they will demand like this. And here demand matches supply. So this is an equilibrium. So equilibrium means like you demand at the current prices whatever you want to consume which is up whatever and demand with supply. So these two constants should be satisfied. So we get an equilibrium. And here this is an equilibrium. So any question about the notion of equilibrium? So here this was a very simple market. There were like three goods, two agents and the utility function is also very simple. And we were able to start with some prices and in the end we were able to get the equilibrium prices. But why this should be proved for every market and the complicated utility function any concave utility function? And in fact this was the celebrated theorem I around the group from 1954 where they established the existence of the equilibrium using fixed point theorem. So they proved that there always exist an equilibrium given any market and any utility function. But their proof was using fixed point theorem and fixed point theorem is like this you have a square and you have a continuous function from this square to itself. So you have a continuous function. So you take any continuous function from the square to itself there will always exist a point in this square which is fixed by this function. So there is a point here a which is fixed by this function f a equals a So take any continuous function this will always be there. There will be always some point a. So this is what fixed point theorem says. So around the group what they did they formulated the market clear of problem as a fixed point problem and since fixed point problem there always exist a fixed point so it proves the existence of market clear via fixed point theorem. However it doesn't give us any way to compute this point. So it doesn't give us any algorithm any procedure by which we can compute this point. It is just an existential result non-constructive. But we are interested in computing market clear because we want to use it. We want to use it for policy analysis for online markets for many other problems which are which can be collaborated through market equilibrium notion. And computation is very important in the course of this chart he says that due to non-existence of efficient algorithm algorithm for computing equilibrium general equilibrium analysis says remind at a level of affection and mathematical theorizing far removed from its ultimate purpose is a method for the evolution of economic policy. So of course computation is important so one way is to just put a grid on the domain of the function and when we check on every point of the grid whether it gives a fixed point or near to fixed point maybe that is also okay. If this does not give us anything then maybe refine this grid and do the same calculation. In fact this chart uses something similar like this similar like this to find approximate fixed point and using this they were able to get something in market equilibrium. However the algorithm is slow and there are issues of numerical instability. In spite of all these it has been used in practice this algorithm, this chart and this one. So computation of market equilibrium depend on utility functions what you assume about the utility function of IZN it is very much dependent on that so if we assume that the utility function is linear so linear means you have utility from one mango is like if you consume one mango then your utility is one unit let us say for the second mango you get again the one unit of IZN so it was linear similarly you have linear utility from apple here it can be any constant in this example it is like two units of happiness from consuming one apple and for the second apple also you have two units of happiness and so like this you have linear utilities for every good and the total utility is like how much mango utility from mango plus utility from apple plus utility from orange so this is your total utility so here like one times because one is the utility you you get by consuming one unit of apple mango so one times amount of mango plus two times so I am just writing this in mathematical so and there has been a huge amount of work for studying linear utility functions so this is not an exhaustive list so these are just some papers which I had in mind and now we have polynomial and algorithms to compute market equilibrium where every IZN has linear utility function but it does not satisfy diminishing margin of error so it is far away from the real utility function so let's try to generalize this so this was linear utility function so what you do you make it piecewise linear function so now utility function of first okay so from first two units of happiness from the second apple your happiness decreases now you get only one unit of happiness and so on so you have you have met linear into piecewise linear concave concave because like your utility is decreasing so it goes linear till some amount then it decreases the rate of utility and then again it goes linear and then again so it is like piecewise linear concave so you have any assumption that diminishing will be marginal in this sense it will not be diminishing jointly so yeah so this is separable you can talk about so there are utility functions which can satisfy those conditions that it can be joined then it is for non separable okay so this is piecewise linear concave so you have piecewise concave utility from every good and you have piecewise concave from every good and total utility is sum over all these goods so it is separable because the utility is separate for each good so you have a utility from presuming apple from presuming mango separately in our example it was not separable because it is like goods in equal proportion okay so this is the this is the I think natural generalization from linear which satisfy diminishing marginality so it is very it is like of course more realistic but there are no polynomial and algorithm known for this problem even for two segments even the utility has only two segments two linear segments then also this problem is was shown to be intractable okay so non separable utility so this was separable you have separate utility function for each good and in the non separable you have utility from bundle of goods like you want to buy a charger or i packed together you do not have utility from only charger so there are goods for which you want to consume together so these are called complement goods in economics so here like if someone wants to consume mango and apple in equal proportion so there is no utility by consuming only mango no utility by consuming only apple only utility you get when you consume both of them together so it is called non separable and piecewise linear function which is the more general utility function can have any combination combination of separable plus non separable so you can model i think anything any preference of any engine and it is of course it contains separable so it is intractable but it is exact complexity is obtained in these three papers what is the exact complexity so it is intractable in the number of goods and the number of so if you assume that number of goods are constant then it is a problem no it is an open problem if the number of goods are constant then we do not know but if both are not both are general then it is of course it is intractable so these are the three classes i will be talking about linear spnc and pnc so we want to get an algorithm fast algorithm for pnc utility function so that way we can model the real preferences any preferences what would be the case of linear but non separable linear plus non separable so it is also so we show that it comes inside pnc the hardness is same as pnc so the example i took where you want to consume both of the goods together so that is kind of linear plus non separable because there is no diminishing returns so we show that hardness of in fact show the hardness of in this vapour we show the hardness of dead utility function which i will be covering in this talk and so it seems that dead utility function is the most difficult one so that is the most difficult one so i will come to that okay this is the are there cases of in the non separable in any case of i do not know how to put it the x axis is going non linearly so for example the first point is the x axis is 1 ipad 1 tablet second point is 2 ipad 1 tablet so that is also possible i have defined that kind of utility function this is what i am showing so there was no utility function for that so it is for leontia leontia is an economist who defines this function but there is no diminishing returns but if you make them pieces like initial like to fill some amount let us say first charge and first ipad you have some utility from on the second segment maybe you can have same ratio of these groups or maybe different ratio but maybe your red decreases so those can be noted and everything comes in that PLC you can model everything in PLC it should be just concave that is the so we have polynomial time and it is intractable but the intractability of these two problems are different so i will come to that so where they are of course this is more harder than PLC so PLC means fragment objects for example take quarter of an apple or slice if it goes north yeah yeah it is like divisible goods goods are divisible in my talk but goods are divisible infinitely divisible so what is the complexity class we know P and NP so P is the class of problems where solutions can be formed in polynomial time like linear programming you can solve efficiently on your computer in polynomial time and NP is the class of problem where solutions can be verified in polynomial time so it contains hard and hard problems like clerics as well problem, quadratic programming those type of problems we ask here is like these are the decision classes like whether there exist a solution or not so we ask yes and no questions in these classes but in market equilibrium case solution always exist there is no question to us whether there exist an equilibrium or not because by error to do theorem equilibrium always exist so you cannot ask such a question so Meghito in 88 showed that if these problems are NP hard where solution always exist so NP hardness is ruled out for these problems so we want to prove that these are interactable but NP is not the right class to prove the interactability of these problems so Pappadhi Mitra Pricot Pappadhi Pricot in 94 defined a complexity class for PTAD which looks similar to his like substring of his last bit so again I talk like that Pappadhi Mitra Pricot Pappadhi Pricot it stands for polynomial parity argument for directed graphs so he proved that approximate pitch point is PTAD complete so approximate pitch point is like fx-x is within some actual model so finding such a point x is PTAD complete it's hard and the solutions are rational so rational means like you can compute them exactly like Pappadhi Pricot so all the prices are like this all the pitch points are like this but what about the complexity of computing exact pitch point fx-x and the thing is that these two notions like approximate pitch point and the exact pitch point can be very very far away so there may be approximate pitch point sitting somewhere but it may be nowhere near to any exact pitch point so for the exact pitch point at this time we have defined a complexity class called fixed P and here the solutions can be irrational but algebraic so it is like of course exact pitch point contains approximate pitch point so fixed P contents PTAD so these are the two classes defined for like the problem that solutions are directed to exist so let's see where they stand in traditional complexity classes so PTAD is inside NP functional class of NP and fixed P is even like not known to be in NP it may be like harder than NP it is believed to be harder than NP but everything sits inside P space so and fixed P of course contents NP so this is the current status of these complexity classes and now due to a due to a lot of work in computer science we now have a very striking dichotomy in market middle computation which means separable PSE utilities PSE is the most unavailability function so for separable PSE SPSE all equilibrium are rational computing only is PTAD complete there is a practical algorithm in the style of simplex language moves from vertex to vertex and decision version is NP complete because there always exists a solution but if you ask a non-trivial decision problem whether there exist more than one equilibrium this is a non-trivial so those are NP complete problems and for PSE solutions are irrational but algebraic the complexity of finding one is fixed P complete there is no non-trivial algorithm known for this and the decision version is ATR complete which is where ATR stands for extension for every else so these three are our resultant I will now show you some briefly these two results and amazingly we have a similar dichotomy analogous dichotomy in Nash equilibrium computation so I am not covering Nash equilibrium but just to mention that we have analogous dichotomy between computing Nash equilibrium at two player game and more than two player games so we have analogous dichotomy here so that was market sign equilibrium and dichotomy if you have any questions then I can answer otherwise I can move to dichotomy in the Nash equilibrium computation or market equilibrium computation are based on the techniques from linear complementarity problem in LP so let me briefly review these techniques it is little bit technical but it won't take more than 5 minutes so this is telling me okay so the LCP problem is like this we have given a metric sand and a metric cube find bias as it satisfies these three constraints so first constraint is linear inequalities like in LP and the last constraint is like non-linear constraints which is like complementarity constraint if you are familiar with complementarity slackness of LP so it is something like that so you have a complementarity between variable and the constraint so you are pairing them up okay so these two constraints are easy constraints because these are like linear programming the problem is these quality constraints so okay so this is the problem we want to solve so just to simplify writing I will write these complementarity with this inverse m sub i a rows m sub i is the ith row of LP so it captures it generalizes linear programming convex quality programming so you can formulate them as an LCP and there are many other problems from engineering wherever you have complementarity type constraint then the LCP is the natural thing to try okay so here every solution is at a vertex of the polyadron so rationality if the input is rational number then output is also rational there can be disconnected set of solutions so this is unlike LPs and convex programming where the solutions are connected and convex set so here the solutions can be disconnected and even there may not exist one there may not exist any solution and it also captures knapsack type of problem so it is NPR in general so how to find a point in how to solve this LCP so possible method is so you can start from some vertex in the polyadron so this is little bit technical but let me over the interest of those people so you can easily find one vertex it is like solving LP but it is not clear how to pivot from this vertex to another vertex so that you end up on one of the solution vertices so lengthy gave an idea so what is the idea was so we want to solve this LCP so what we do we introduce one new scalar variable so we want to solve this system but we like introduce one more dimension one more variable extra variable and we consider this problem and what happen that you can show that like solutions of the second problem with a new variable equals 0 is same as solutions of 1 so we are interested in finding solutions of 1 which is same as finding solutions of 2 with j equals 0 so this can be easily checked okay so it looks like that maybe you have made the problem more difficult by introducing one more dimension but it helps actually so what happens now let us see how the solutions of 2 look like so they are now set of paths and cycles so paths are like either going infinity or both the direction if it is stopping at a vertex then that vertex has j equals 0 z equals 0 and that can be cycle so in some sense you have connected the solutions in the original problem the solution can be disconnected set of vertices but by introducing one more dimension one more degree of freedom you have connected these solutions and now they look like this set of paths and cycles and we are interested in finding these vertices end point of paths so if we plug in y equals 0 then we we obtain an infinite edge end point of one path and so let me tell you is to extract from that end point so that is easy to find and then follow the path as I said that all the solutions are set of paths and cycles so just follow this path and this is an infinite edge so it will follow this path and if you find j equals 0 then that gives you solution but the problem here is that it may end up on another infinite edge so in that case you have not found the solution so length is infinite is not directed to give you a solution even if there exists one so to prove it it will give you a solution you have to show that there are no res no other infinite edges in that case it has to find one more solution so to use the length is infinite one it shows there are no second edges so it is unlike LP or something because like here you have to first formulate a problem as an LCP but after that also I will not give you anything so you need to do some technical work to prove that it will converge to the solution it does not come free by just the LCP problem so let us come to the market markets with SPLC abilities which is short to be in that category so what we did this problem is an LCP like you might also be doing like you take some problem and may be formulating it as an LP or convex programming so here we formulate it as an LCP and as I said the algorithm does not come free so what we prove that the algorithm will converge to a solution and this is the problem of my time if you assume that the number of edges and number of goods if you assume any one of them to be constant then it is a polynomial algorithm and we cannot hope to hope to get better than this because it is problem in fact and this algorithm also gives existence of equilibrium rationality because all equilibrium prices are rational all number of equilibrium and it also shows fundamental PTA so this just we can be often these as a corollary of our formulation formulation and algorithm okay so there are two challenges here first to formulate the market clear problem as an LCP and then show that the algorithm will converge meaning there are no secondary edges so lets first try to formulate the market as an LCP so we need to like capture two things one is the optimal bundle of edges and the market clearing these are the two difficulties these are the two concerns we need to capture through an LCP so okay so every agent brings so these are the agents these are the goods so every agent brings some amount of goods so lets say w11 denotes the amount of good one agent one brings to the market so it can be zero also it can be so it is non negative number and the total supply is just some of our like wij over all agents so total supply of good j is like some over like whoever brings some over this good among all the agents and that is like total supply of goods in the market and by generally lets assume that it is one total supply of every good is one so SPLC is little bit technical so let me first show you for linear because linear there is only it is easy SPLC is like many segments so lets first try to derive the LCP for linear test so linear is like linear is the utility like this you have like linear function for each good and the total utility is some over all these goods so in this example utility is like one into more times amount of mango and so on so lets say this is the good j and this constant can be anything any non negative number lets call this uj so everything in blue is like given to you its the input uj is the rate of utility from good j and the amount it consumes lets say denotes by xj so total utility is like some over all these goods uj times xj uj times xj is the utility from good j and you just sum over all the goods so that is the total utility lj times xj so the utility functions, linear utility functions are given by these numbers so how much what is the rate of utility by consuming per unit of good j so we have uij these are the number 0 to use it can be 0 if you don't like to consume it but it goes linear from the first unit uij and second unit uij so this is the linear test so lets write market clearing constraint we need to make sure that market clears this is the one constraint we need to capture through LCD so lets introduce variable pj for price of good j and qj for money spent on goods j by agile type because agile type may be buying some goods lets say qj is the money agile type is giving to good j so for each good j what we need to make sure demand equal supply we need to capture market clearing we need to capture market clearing through LCD so we need to capture demand equal supply for every good j what is the demand of good j demand is like qij is the money agile type is giving to good j qij upon pj is the quantity agile type is taking agile type is consuming of good j so qij upon pj is the amount you are consuming agile type and it just come over all the agile types so this is the total demand of good j and that should be equals to its supply which we assume to be one so this is the constraint we will put in our LCD so here like it summation does not depend on j so it is a linear term for the agile type we need to capture spending equals earning because you are not allowed to spend more than what you have so spending is like qij money you are giving to good j you just sum over all the goods so it is like summation qij and that should be equal to earning wij is the amount of good j you brought to the market pj is the money you can earn by selling this amount of good and you just sum over all the goods so this is also a linear constraint so market carrying for market carrying we have linear constraint so we can capture market carrying using these variables and these linear constraints so now comes to the optimal one we want to make sure everybody buys the most preferred one which maximizes the utility function so for that uij upon pj what is this value uij is the rate of utility by consuming one unit of good j and divided by its price is the utility you get by spending one rupee of your money it is like utility per rupee it is called bank per buck and at optimal bundle so you will see what are the prices of goods and you will just compute uij upon pj that will give you what is the bank per buck from consuming this good and if you want to maximize the utility function your utility, total utility so you will only spend your money on the maximum bank per buck goods where your rate of utility is maximum that will give you an optimal bundle so if you are spending on some good uij is greater than zero then we want to make sure that uij upon pj that is the bank per buck from this good should be maximum among all goods so this is the constraint we need to so let's denote this maximum by one over land i so this is like how we calculate the let's denote this value by one over land i and then more we want that uij upon pj this is the bank per buck of goods should be at most the maximum bank per buck so this is a linear constraint but over land i because if you simplify this then it becomes linear so that was the reason why we want uij upon pj so what we what we are saying that the bank per buck from goods should be at most the maximum bank per buck this is like this is by the definition of one over land i and what we want that if you are uij greater than zero so this is a non linear constraint what we are saying here if uij is greater than zero uij is positive then this should be equality then this goods should be uij upon pj equals one over land i so these two constraints satisfy our optimal fundamental constraint and by simplifying this this is like this this is linear constraint and these are the complementarity constraint which are hard and in our notation it becomes like this so there can be multiple goods which have the same bank per buck because uij upon pj can be many goods and you are indifferent in terms of any of these goods you can consume any one of these so market algorithm can give you any good you should be okay with that so from the formulation of uij yes yes so that maximum is kind of unique for it's not unique but the value is unique so this is a unique value and there can be many goods which satisfy those are the different j's okay this is like or maximization is just dependent on i no no this depends on all the goods k and people so this depends on i yes yes every agent will have different so uij is one over land i yes so the j part these are maximized there could be multiple j's which have qij greater than zero but all of them will have qij over pj yes that we ensure we want to ensure that actually and these concerns that we ensure that that here like what we are saying qij greater than zero then it should be quality because otherwise how will you get multiplication of these two equals zero maximum time for product is not how much if two things give you the maximum time for product yes you are indifferent market can give you any all of those will appear as solutions yes so it will be a convex solution in fact for linear data it has convex solution very good point so mentioned without loss of generating the assume supply to be one unit obviously yes what you do if you are given any supply so it is like redefining the utility function so you have to the ratio of the supply if you assume an equilibrium each unit is one so that will correspond to a particular utility function transformation of a particular utility function but initially i have to choose between the space of different utility functions here the utility function is fixed if i am going to scale my utility functions i don't know which one to choose so give me your market so your market may have more than one supply more or less than one supply so what i will do i will convert it to another market where like i will make supply of these two to be one and i will modify uij i will solve for this market and then i will come this solution i can get the solution here whatever is getting i did here right so that is assuming i know the supply the ratio of the supply is in the equilibrium no the supply is fixed supply is part of it so this is the part of it so it is like the issue is the supply and the determinant the price it is the supply at the utility function and then the output is the equilibrium price is at the allocation supply is given supply is given yeah so this is like once the supply is given then you can do so now we can so using market training and optimal model consent we have an LCP coding here when this was given by Eats in 1975 and he also said that also under study he also studied piecewise engineering if it is a production when it is successful this then you could prove important in real economic modeling so he made this program open and we kind of explained his algorithm to separable piecewise engineering when there are more segments okay so so let me note go into the proof details so now you have segments so you have pieces like for each of it you have many pieces linear there was only one red now you have different red and every so he had like let's say so every segment has two parameters one is the red of the utility you derive on this segment and one is the major amount you can consume on this segment the length of the segment and we want to put that QIJK the money spent on this segment should not be more than LJK times Pichai because this is just coming from the input LJK is the major amount you can consume on this segment and LJK times Pichai is the money you are spending on this segment so this is easy constraint so the difficulty here is like let me just tell you about the difficulty difficulty in capturing optimal model market training is okay market training you can easily by extending linear case but in optimal model we need to make sure that if you are allocating some money on the second segment then the first segment should be fully allocated because you want to consume like first you want to consume first segment and the second segment so we want to capture these things as an entity some like complementarity and linear constraints so this is the difficulty and for this we introduce new price supplement variables for each segment and huge complementarity carefully so this is the and there are some other problems which but this is the most challenging one to get an LCP formulation so we get an LCP formulation for SPLC and we show that solutions of this LCP is same in market so we got the LCP formulation and now what about the algorithm so we can apply that algorithm LJK algorithm which I discussed but it does not converge to a solution it may not converge to a solution it may converge on another rate so we prove that there are no such rates in the polyadrin so whenever you will start from the Y1 to 0 it will converge to a solution so this is a highly technical part of the part to prove that it will converge to a solution and so it set us open future of years from 1975 it also set us open future of Bajrani and Yarnaka case by showing membership in PTAD and it is it is strongly polynomial so it is polynomial if you assume that the number of goods and number of agents are constant and it is fast in practice so we have implemented this algorithm and ran on randomly generated instances and we found that it works very fast linear in the number of variables like some people from CMU they have some network problem where they have to price the cases of the network they are using our algorithm maybe they can model the other utility function as a separate piece and in fact this is only for randomly generated instances but we want to apply it to more practical scenarios where you can model the utility function and you want to price something so those kind of problems can be we can there you can use this kind of this algorithm the problem is intractable but the algorithm works very fast which people is that this is stopped in 2012 with Ruta Mehta and then we extend this algorithm to include production also so I have not talked about market with production so then there are production firms they can take some raw material from the market and produce some good and then they want to maximize their profit so if you include production also then we extend this algorithm to production and later we extend to non-separative utility function also which we call Lyondipri so PNC is like harder so we could extend these things to Lyondipri which contains many nice utility functions which are which cannot be captured by SCNC where you allow non-separative okay so in the production case the produce group are not coming back to this market no no produce group has to be consumed so then the supply is not fixed yes yes they are supply is not fixed then it is like whatever initial content agents are bringing whatever is the production is happening equals agents are consuming directly or the firms are consuming it so it is strictly a more general problem that is the difficulty that is the difficulty so so any more questions so next I move to hardness results so what we show that Lyondiparkets are not Lyondipri is like undefined so markets can do arithmetic that is how we show it so Lyondipunction is like this where you want to consume goods in fixed proportion like our example of mango and apple you have juice for mango and apple together then you want to buy them together or charger or iPad or like sandwich if you want to consume a sandwich like you need two slices of bread and one slice of cheese so this is the this is the Lyondipunction which is called Lyondipunction so we relate the computing market equilibrium in Lyondipung where the agents Lyondipunction are Lyondip to multivariate polynomials so here the problem is like multivariate polynomials so multivariate polynomials means there are variables and there are many set of these polynomials and the problem is to find a simultaneous zero of these polynomials and this is known to be notoriously hard p-space problem in fact it was not even decidable before 67 so last we said this problem is decidable then KNE in 1882 that it is decidable p-space so this is very hard problem so over real over real yes so here like you want X over real so what we show we take such a system of polynomials so you have such a system of polynomials so we convert these polynomials into a market so we take polynomials we construct a market where each agent utility function is Lyondipunction very simple each agent wants some goods in equal proportion and the variable here variable in the multivariate polynomials is like is good here so that will be captured through prize of goods so for each variable there will be a good and like the value of this variable will be captured to prize of this good and we want to make sure that we show that solutions of the set of multivariate polynomials are same as the clear of prices of the market so this is the reduction we get will briefly tell you about the reduction so so the basic building block of the multivariate polynomials are these two functions you have a linear function where like these B, C and D are constant so this is a linear polynomial and a quadratic polynomial and using these two you can build any polynomial so what we do we want to capture these variables to prize of some good and we want to make sure that in equilibrium these prices it enforces these relations what we want to it enforces these relations either linear or quadratic so any system of polynomials can be converted into these two set of polynomial okay so just to how we do it so addition is simple so what is this you have three goals you want to make sure the prize of first good is addition of second and third good what we want to make sure in the market so that is easy that looks natural also because you want to buy two goods together like is the complementary good like cheese and bread and if you bundle these markets and create another good cheese sandwich so prize of cheese sandwich is equals to prize of cheese plus prize of bread so doing addition is like it is a natural thing and it is simple so here like suppose there are two agents they go and she creates an apple and orange and he wants to consume apple and orange and she wants to consume mango they are like this will enforce this linear linear relation then only he will reach to equilibrium because she does not want to consume mango any of these these things and she does not want to consume mango but suppose there is the one more one difficulty here is that these polynomials there are two common variables so you have system of polynomials and there are common variables in different different polynomials so if we write market for one polynomial so there will be a variable for some prize there will be some prize variable for some variable here in the same prize will be happening in the different market so what we do we create a market for each polynomial but now they are interfering in different markets it is like pavai market, vashi market like that and we want to make sure that they maintain the same prize because it is the same variable so that is the difficulty so here is like if there is another market where she has mango then this relation may not be satisfied because she will interfere with the with the market getting concerned in this some market I hope I am here if you want to ensure that there are no extra supply you take a polynomial and convert it to a market and take another polynomial and convert it to another market and then take all of them together all the markets together and give as a reduction so for that we introduce a notion of close of market where there are no flaws in different different markets so we create a market here where there is no flaw from this market to another market even though when we will consider all of them together it satisfies this property that is the difficulty and of course multiplication is very very difficult how you make sure that the prize of some good is like is like prize of multiplication of prize of two other goods this does not look natural also how to make sure that in the market so this is very challenging we have many gadgets which make sure that we can get a market which can do these kind of multiplication is this exact or is this approximate what are these things here so these are gadgets so here you have to create a market some gadgets and goods so here what is that input to this gadget is like good g1 whose quantity is 1 and the prize is p and we want output from this market is like prize is 1 and quantity is p we are like converting supply into prizes so if you want to multiply two prizes you have to convert one prize into supply at some point because then only this quantity someone will consume and that person will pay like this quantity times its prizes so we get multiplication of these prizes so this is what you call production no no this is not production here like you have to create a market inside it like some number of gadgets and some number of goods which take this goods whose prize is p and quantity is 1 and consume among themselves something and then leave to the other market so this is a small market part of a big market leave to the market another good g2 whose prize is 1 and the quantity p so its like converting reversing these like supply and prize to prizes and this is like combined which takes two goods and combine the prize of two goods this has split up its like opposite of this and we put them in this format because we need to also make sure that close this property that there is no flow outside this market so that makes it more difficult what does this mean this means you are solving multiple so what we do we take column when we get a market we take another column and get a market and then we combine all these together and form a big market and the difficulty is that common variables in different column so here like common goods so how to make sure that these like supply constant supply and demand constant no interfering between these two some market so for this we introduce we have to translate the prize we have to translate the prize because finally we need to take them as one market we just add all the asides all the goods and that is one market okay so I think I am running out of time so where we can discuss so we have to market system of multiple qualifications then we get hard days for that some of my other works so recently I have done like some work in like these are many of my questions in marketing so we have complexity of approximate marketing so this is like very research work and some dynamic questions kind of a dynamics like I mentioned some dynamics in the beginning you start with some prizes then increase the prize the demand is more than supply so the question is like how fast it will converge so in computer science we study like it will converge in following by time or what is the rate of convergence so we prove the past convergence which improves upon the previous convergence these are many other problems also and then I have done some learning utility function because utility in my talk we assume that they are given to you but like it is not given to you then you have to learn because these are the private values so what is the quite complexity of learning these utility functions because you offer like what we do you offer some prizes and get the demand and can you interpret something about something from this demand so if you need to you need to give multiple set of prizes and multiple set of demands and finally you have to come up with oh this person has this ability so those kind of quite complexity lower bound and upper bound how many queries you require we are like this I think master has like piecewise we are here for capabilities so we are like he only has not satisfied increasing marginal returns it is like you scan the input bundle your utility scales so I define in you till some amount it is some rate after that the rate decreases like this and you can change the ratio also and then I study algorithmic and harness results for that so I proved that like if the ratio remains same then it is polynomial time if the ratio is different and we are like ok let me know so this is the what I am currently doing so one is the smooth analysis of complementary pivot algorithm so this is the framework we have given by Esfilman at time for simplex algorithm so simplex algorithm runs very fast in practice but in the worst case they are exposure time algorithm so they proved that a smooth complexity of simplex is polynomial time complementary pivot algorithm because like worst case it will be exposure but we have run we have many algorithms complementary pivot algorithm all of them perform very fast in practice on the randomly generated instance so there has to be some reason for that so that will explain that why they are fast in practice there are many open problems in pt80 like there are many gaps in market the problems are not to be polynomial time and not to be hard also so they are lying between polynomial time and hardness strongly polynomial algorithm for lp linear programming so strongly polynomial means like number of iteration depends only on the size of the matrix like number of inequalities and number of variables of course and the last thing like more interesting market algorithms for other setting I want to apply these algorithms for other setting like more link pricing like how to present problems and other setting because like market is like and same with those people are anyway using on the ones of network problems thanks for the attention and thanks for the work electronic market is typically a dynamic pricing business dynamic pricing like expected yeah yes yes yes yes yes I think there is a theory but I think not from algorithmic side but there are models for that any of these solutions there which are new in fact so these are very recent ones very recent ones we have not applied to these settings but like it will be very interesting to to see if we can get something because the market model is very general you can have good time and operation perishable perishable good yes I think TCS like the classical CS community has not had these problems these are like very recent but last five years in the real world markets people making distributed there is no central authority doing this so is there any practical use for something with all the this is like this is centralized this is like doing centrally there is another dynamic what distributed where you are like changing the market parameters there is none of that there is nothing like that how the price is like that so so I think this is used in policy analysis like US government or they use for policy analysis how to price something where to put the taxes what to do just to understand how the equilibrium changes and more recently we have because like market gives efficient and fair solution fair allocation so more recently like if you want to allocate some items to some agents where you want to maximize the minimum values and minimum happiness then how to allocate them so these problems are NPR problems in general but using market algorithms now you can compute them and then you can round them because they give like and then many potatoes and should you like you want to assign jobs to the processor like how to do that so that like everybody is like having some you want to maximize the minimum minimum time so everywhere like these problems are formulated with market problems wherever you are like you want to price something it's like can you give an inclusion of the way formulated the constraint that the first segment should have been upstream will be more second segment yes yes you need to use complementarity we use one more variable it will be somehow it's not easy to I think but it's like one more variable and we put them in such a way using complementarity but if it is greater than 0 then it should be equality so that finally it comes out to be like if the k-1 segment is like if you have some clone k-1 segment and k-1 all the 1 to k-1 are fully available so using complementarity we can do it there's some more variables you need to yes I have triggered in one of those one of the locations and suppose I get the information about the entire market and I use an algorithm where you have to get an equilibrium the distributed market may arrive at a different position but till this computation give me any kind of commercial advantage knowing that there is this there is this solution is it very helpful yes I think we don't have anything better than this there is no other model so this is the only thing we have so maybe it may not give you something very nice but this will give you something so we need to model the problem maybe separately maybe it will be reduced to this model most of the time it reduces like finance market everything reduces to this market model so the solutions may not be unique but are they close by nothing like that is also even like that is one more problem there exists one model that will be many and it will be very far away very far away the same problem is in the game so police analysis so traditionally it has been used in police analysis like how to put text size or how to design these things how to put people in some particular direction do you want to add for some behavior it's like mechanism design you can use these to analyzing the change in the player it will give you a more important problem than yes yes not bad but very nice equilibrium condition that is that approximately how many people supply on a market equilibrium condition may not always happen so then that has to be modeled does that change the so you allow some play in both directions so in realistic setting there might be one of the or four days then demand will be supplied I think if it is not then I think it won't matter because it will be very small compared to the supply than used volume you are considering but if it is multiplicative some fraction like 10% of the supply may be may not may be but supposing you give a value for the producer I am making money but I have a value for folding it then I consume right is that or even the other way around where you have a smaller demand then supply you go to maximize the demand as close as possible like LXC pricing for making sure that the receipts are there right there could be another constraint minimization of the supply yeah I think we have but I think demand is for supply I think fine then you are anyway if you are not selling some item you are throwing it away maybe you will not like so it is like of course there is always some relaxation market of that is clear because it will automatically make it move production model will allow for some you know modeling carryover production model will allow carryover but today's production and tomorrow's production may be different so then you can have that carryover so the utility for tomorrow's so yes you can put that so the markets of production where they have time you can allow that a good production very good so this is a very fundamental movement like you have to generalize these things for a specific subject so in the network example we should talk about yes or you said that people have written it so each communication link is a separate good or something like that it will price that no question I have a question I am too bad