Approximate Market Equilibrium for Near Gross Substitutes

Transcription

1 Aroximate Market Equilibrium for Near Gross Substitutes Chinmay Karande College of Comuting, Georgia Tech Nikhil Devanur College of Comuting, Georgia Tech Abstract The roerty of Weak Gross Substitutibility (WGS) of goods in a market has been found to be conducive to efficient algorithms for finding equilibria. In this aer, we give a natural definition of a δ-aroximate WGS roerty, and show that the auction algorithm of [GK04, GKV04] can be extended to give an (ɛ + δ)-aroximate equilibrium for markets with this roerty. 1 Introduction The comutational comlexity of finding a market equilibrium has recieved a lot of interest lately [DPS02, DPSV02, Jai04, DV04, GK04, CSVY06, CMV05] (also see [CPV04] for a survey). A key roerty that has been used in designing some of these algorithms is that of Weak Gross Substitutibility (WGS). A market satisfies WGS if an increase in rice of one good does not lead to a decrease in demand of any other good. Markets with WGS have been well studied in both the economics and the algorithmic game theory literature. It has been shown, for instance, [ABH59] that the tatonnement rocess converges for all markets satisfying WGS. In this aer we extend the definition of WGS to aroximate WGS and design algorithms for the same. We formulate the following alternate definition of WGS: the monetary demand for any good, which is the demand for the good times the rice, is a decreasing function of its rice, given that all the other rices are fixed. Our definition of an aroximate WGS now follows from bounding the increase in the monetary demand for a good with an increase in its rice. 2 Preliminaries In this section, we define the Fisher market model which we refer to throughout this aer. Consider a market with n buyers and m goods. The goods are assumed to be erfectly divisible, and w.l.o.g. a unit amount of each good is available as suly. Each buyer i has an intial endowment e i of money, and utility functions u i : u i (x i ) gives the utility gained by her for having bought (consumed) x i units of good. Given rices P = ( 1,..., m ), a buyer uses her money to buy a bundle of goods X i = (x i1,..., x im ), called the demand vector for buyer i, such that her total utility U i (X i ) = u i(x i ) is maximized, subect to the budget constaint: x i e i. Equilibrium rices are such that the market clears, that is, total demand for every good is equal to the suly: for all goods, i x i = 1. Let v i be dui dx i. We assume that for all i and, U i is non-negative, non-decreasing, differentiable and concave. These constraints translate into v i being non-negative, non-increasing and well defined. From the KKT conditions on the buyers otimization rogram, it follows that v i (x i )/ is equalized over all goods for which x i > 0. So for any otimal bundle of goods X i for buyer i, there exists α such that x i > 0 vi(xi) = α, and x i = 0 vi(xi) α. Definition 2.1 For any ɛ > 0, a rice vector P is an ɛ-aroximate market equilibrium if each buyer can be allocated a bundle X i such that U i (X i ) is at least the otimal utility times (1 ɛ) and 1

2 the market clears exactly. Definition 2.2 A market satisfies WGS if for two rice vectors P and P, such that = for all goods k and k < k, the demand of good at P is at least its demand at P. [GKV04] gave the following equivalent condition for WGS, when the utilities are searable. Lemma 2.3 ([GKV04]) A market satsfies WGS if and only if the function x i v i (x i ) is nondecreasing for all i and. 3 Extending WGS 3.1 An alternate definition of WGS Let the monetary demand for a good be its demand times its rice. We motivate our alternate definition of WGS by analysing how the monetary demand for a good changes as its rice goes u. Consider a buyer i who has an otimal bundle of goods (x i1,..., x im ). For any good such that x i > 0, vi(xi) = α. Now suose rice of good k is driven u to k > k, rest of the rices being unchanged. In this case, clearly the current allocation no more reresents an otimal bundle for the buyer. We now describe a way for the buyer to adust her allocation in order to attain the new otimum. This is done in two stages. In the first stage, she sells some of good k to equalize the marginal rate of utility of good k with that of other goods. Let x ik be such that v ik(x ik ) = α k. The difference between the values of the new and original holdings is x ik k x ik k. This is the amount of money she will have to ay as a result of the increase in rice. From Lemma 2.3, x i x i = 1 α (x iv i (x i) x i v i (x i )) 0 This means that she has some money left over at the end of the first stage. In the second stage, she slits the left over money among all goods in such a way that vi(xi) remains the same for all goods with x i > 0. Two things are worth noting: the monetary demand for good k and the value of α both decrease as a result. This leads us to an alternate formulation of WGS. Lemma 3.1 A market satisfies WGS if for two rice vectors P and P, such that = for all goods k and k < k, the monetary demand for good k at P is smaller than that at P. Proof Suose that the market is WGS. Let the demand at P and P be X and X resectively. Then WGS imlies that x x for all k. Therefore x x for all k. Since x = x, it follows that k x k kx k. Now assume that k x k kx k. We need to rove that x x. We can ignore those for which x = 0. Again, since x = x, there exists some k such that x x, and in turn x x. Therefore α = vi(x ) x > 0, x x. vi(x) = α. Hence for all k such that Elasticity of demand: This formulation is related to the Price Elasticity of Demand (see [SN92] for details) defined as % change in demand E d = (1) % change in rice Elasticity is a widely used measure of resonsiveness of the demand to change in rices. alternate definition of WGS simly states that E d > 1. Our 2

3 3.2 Aroximate-WGS utility functions We have seen how WGS can be interreted from the demand ersective as well as the revenue ersective. Extending the revenue interretation from Lemma 3.1, we say that a market satisfies δ-aroximate WGS if increasing the rice of a good does not cause its monetary demand to increase by more than a factor of (1 + δ). Definition 3.2 For any δ 0, a market satisfies δ-aroximate WGS if for two rice vectors P and P, such that = for all goods k and k < k, the monetary demand for good k at P is at most (1 + δ) times that at P. In the next section, we will rove that this definition allows us to design efficient aroximation algorithms for these markets. Henceforth we will refer to definition 3.2 as δ-aroximate weak gross substitutability. The above definition gives the following necessary condition on a δ-aroximate WGS market. Lemma 3.3 If a market satisfies δ-aroximate WGS, then i,, x > x xv i (x) x v i (x ) (1 + δ) Proof Let P be a rice vector so that the demand of buyer i for good at these rices is x. Increase to get a rice vector P so that the demand of buyer i for good at P is x. Let α = vi(x) and α = vi(x ). Then as in the rooof of Lemma 3.1, α α. And by definition, x x(1 + δ), which imlies x v i(x ) α that x v i (x ) xv i (x)(1 + δ). 4 Auction Algorithm xvi(x)(1+δ) α. Combining the two inequalities, we get In this section we show that by slightly modifying the auction algorithm in [GKV04], we can comute an (ɛ+δ)-aroximate equilibrium for a market exhibiting δ-aroximate WGS roerty. This result shows that WGS is not a hard threshold: Markets do not suddenly become intractable if they slightly violate the WGS roerty. We will use the auction algorithm of [GKV04] as the starting oint. This algorithm comutes ɛ-aroximate equilibrium when the market satisfies WGS. An outline of the auction algorithm is as follows: Ascending rices: Prices start out at suitably low values and are raised in multilicative stes. At any stage, some buyers may have an allocation of good at rice, where as others may have bought the same good at rice (1+ɛ). Buyer i s holding of good at rice is denoted by h i and that at rice /(1 + ɛ) is denoted by y i. Total allocation of good to buyer i is x i = h i + y i. Decreasing surlus: A buyer s surlus is the money she hasn t sent. It is denoted by r i = e i h i y i /(1 + ɛ). Each buyer exhausts her surlus by buying goods at rice from others whose allocation of good is at rice /(1 + ɛ). If no other buyer has the good at lower rice, the rice is raised from to (1 + ɛ). Finally due to rising rices, total surlus of all buyers r = i r i aroaches zero. Buy-back: Suose buyer i had x i amount of good with bang er buck α i = vi(xi). Suose now that other cometing buyers raise the rice to, reducing x i to x i. If v i(x i ) α i, then before buying other desirable goods, buyer i, buys back some of good, until the rate of utility gain er dollar for good returns to α i. 3

4 Near-Otimality: ɛ-aroximate otimality of the artial bundle of goods is maintained for each buyer throughout the algorithm. Therefore, when the total surlus tends to zero, the current rice and allocation vectors reresent ɛ-aroximate equilibrium. We require only slight modification in the auction algorithm, in order to get it working for δ-aroximate WGS markets. The seudocode of the modified algorithm can be found in figure 1. The modification to original algorithm from [GKV04] aears in algorithm main. Figure 1: Modified auction algorithm rocedure initialize Set initial rices as:, = v 1(a )e 1/( av1(a)). Allocate entire quantity of all the goods to buyer 1. Consequently, we have: 1. : α 1 = ( av1(a))/e1 2. r 1 = 0 end rocedure initialize algorithm main initialize while i : r i > ɛe i while (r i > 0) and ( : (1 + δ)α i < v i(x i)) if k : y k > 0 then outbid(i, k,, (1 + δ)α i) else raise rice() while (r i > 0) and ( : α i < v i(x i)) if k : y k > 0 then outbid(i, k,, α i) else raise rice() = arg max q α iq if k : y k > 0 outbid(i, k,, α i/(1 + ɛ)) α i = v i(x i)/ else raise rice() end algorithm main rocedure raise rice() i : y i = h i; h i = 0 = (1 + ɛ) end rocedure raise rice rocedure outbid(i, k,, α) Transfer good from buyer k to buyer i until one of the following events haen: 1. Buyer i s surlus reduces to zero. 2. Buyer k has no more of good left. 3. Value of v i(x i)/ dros to α. end rocedure outbid We have modified buy back ste mentioned above, and slit it into two rounds: (Let X i and P denote the allocation and rice vectors at the begining of each round) 1. First round: For each good such that vi(x i ) > (1 + δ)α i, buyer i buys it back to an amount x i such that (1 + δ)α i = vi(x i ). (As oosed to buy back until α i = vi(x i ) in the original algorithm). If buyer i has to raise rice of good in this rocess, may be strictly higher than. As we shall see, this ensures that the buyer does not send more on any good than she originally had when the value of α i was set. Therefore, the buy-back is ossible for all goods. 4

5 2. Second round: For each good such that vi(x i ) > α i, buyer i buys it back to an amount x i such that α i = vi(x i ), until she has surlus money left. This round is identical to the buy-back ste in the original algorithm. Lemma 4.1 First buy-back round finishes with each good considered having α i = vi(x i ) (1+δ). Proof Consider a situation when buyer i has x i amount of good when her turn arrives to send her surlus. Let be the current rice and α i < vi(x i ) (1+δ). Let x i be the amount and be the rice such that x i x i < x i AND > AND α i = vi(x i ) (1+δ) where x i and is the endowment and rice of good resectively when α i was set, i.e. α i = vi(xi). Then using roerty of the market from lemma 3.3, we get: v i (x i ) (1 + δ) = α i = v i(x i ) x i v i(x i ) (1 + δ)x i x i x i The above equation certifies that the value of good held by buyer i at the end of the first buy-back round is at most the value of her holding at original rice. When cometing buyers reduced x i to x i, the value was returned to buyer i in dollars. The above equation says that she can safely buy back uto x i of good, using u only the surlus value returned to her for good. The same is true for all such goods, hence buyer i can buy all goods considered in the first buy-back round uto x i such that α i = vi(x i ) (1+δ). Correctness and convergence of the algorithm can be roved along the lines of [GKV04], by showing that the algorithm maintains following invariants at the start of each iteration of the outer while loo in rocedure main: I1:, i x i = a I4: i,, x i > 0 (1+ɛ)vi(xi) α i I2: i, x i e i I5:, does not fall I3: i,, r i = 0 α i vi(xi) (1+δ) I6: r does not increase Invariant I1 says that all goods are fully sold at any stage. Invariant I2 conveys the fact the buyers never exceed their budget the initial endowment. Invariants I3 and I4 together guarantee the otimality of the bundle each buyer has after she has exhausted her surlus. Only I3 is different from the earlier auction algorithms. By invariants I3 and I4, as well as the manner in which α i s are modified in the algorithm, we have the following at the termination: i, For each buyer i and goods and k Above constraints imly that all the bang-er-buck values for a buyer (1 + ɛ)v i (x i ) α i v i(x i ) (1 + δ) α i (1 + ɛ)α ik ( ) vi(x i) are within a factor (1 + ɛ) 2 (1 + δ) of each other at termination. Therefore, the modified algorithm terminates with (ɛ+δ)-aroximate equilibrium, ignoring the higher order terms. The analysis of the running time is similar to that in [GKV04]: Lemma 4.2 If bidding is organized in rounds, i.e. if each buyer is chosen once in a round to exhaust his surlus in rocedure main, the total unsent surlus money r = i r i decreases by a factor of (1 + ɛ). 5

6 5 Oen Problems Our result holds for searable utility functions. Clearly, definition 3.2 makes sense in the nonsearable setting as well. An imortant oen roblem therefore is to devise an algorithm that finds aroximate equilibrium for non-searable utility markets. Alternatively, it will be interesting to see if other algorithms that solve WGS markets extend to aroximate-wgs markets. References [ABH59] K. Arrow, H. Block, and L. Hurwicz. On the stability of the cometitive equilibrium: II. Econometrica, 27(1):82 109, [CMV05] B. Codenotti, B. McCune, and K. Varadaraan. Market equilibrium via the excess demand function. In Proceedings of ACM Symosium on Theory of Comuting, [CPV04] B. Codenotti, S. Pemmarau, and K. Varadaraan. Algorithms column: The comutation of market equilibria. In ACM SIGACT News 35(4), [CSVY06] B. Codenotti, A. Saberi, K. Varadaraan, and Y. Ye. Leontief economies encode nonzero sum two-layer games. In Proceedings of ACM Symosium on Discrete Algorithms, [DPS02] Xiaotie Deng, Christos Paadimitriou, and Shmuel Safra. On the comlexity of equilibria. In Proceedings of ACM Symosium on Theory of Comuting, [DPSV02] N. R. Devanur, C. H. Paadimitriou, A. Saberi, and V. V. Vazirani. Market equilibrium via a rimal-dual-tye algorithm. In Proceedings of IEEE Annual Symosium on Foundations of Comuter Science, Journal version available at htt://wwwstatic.cc.gatech.edu/ nikhil/ubs/ad/market-full.s. [DV04] [GK04] [GKV04] [Jai04] [SN92] N. R. Devanur and V. V. Vazirani. The sending constraint model for market equilibrium: Algorithmic, existence and uniqueness results. In Proceedings of 36th STOC, R. Garg and S. Kaoor. Auction algorithms for market equilibrium. In Proceedings of 36th STOC, R. Garg, S. Kaoor, and V. V. Vazirani. An auction-based market equilbrium algorithm for the searable gross substitutibility case. In Proceedings, APPROX, K. Jain. A olynomial time algorithm for comuting the Arrow-Debreu market equilibrium for linear utilities. In In Proc. of the IEEE Annual Symosium on Foundations of Comuter Science FOCS, P.A. Samuelson and W.D. Nordhaus. Economics. McGraw-Hill, fourteenth edition,