Market Equilibrium via a Primal-Dual-Type Algorithm

1. Market Equilibrium via a Primal-Dual-Type Algorithm Written By Nikhil R. Devanur, Christos H. Papadimitriou, Amin Saberi, Vijay V. Vazirani Presented by Zhouyan Wang Advised by Prof. Kirk Pruhs November 18, 2003

2. Outline • Introduction • Basic Algorithm: --- Basic Idea --- High Level Idea • Algorithm with Polynomial Time

3. Introduction Definition • Set A --- n Goods with each price pj • Set B --- n’ Buyers with amount ei budget • uij --- the utility derived by i on obtaining a unit amount of goods j • Buyer will use all his money to buy the goods with Max(uij / pj ) since he will get the maximal satisfaction.

4. Introduction Graph Buyers Goods u11 /p1 e1 p1 u1n /pn 1, 2, …,i,… n’ 1, 2, …,j,… n un’1 /p1 en’ un’n /pn pn ei pj : Budget :Price

5. Introduction Problem • The question is: --- How to find the market clearing prices? --- Namely, find the prices for goods such that the buyers will use up all money on hand to buy goods and there are no goods left in the market?

6. Introduction Literature Review • First defined by Irving Fisher in 1891 • No Algorithm with polynomial time has been found • This paper finds such algorithm by assuming uij is a linear function of the amount of goods

7. Basic Algorithm Basic Idea • In the bipartition Graph(A,B), add a source s and sink t • There is an edge from source to every goods. Their capacity is • For every buyer i and goods j, connect them by an edge only if goods j can give buyer i the maximal utility per dollar and its capacity of the edge is infinite • There is an edge from buyer i to sink t and its capacity is ei pj

8. Basic Algorithm New Graph Buyers Goods 1 1 Infinite e1 p1 Source Sink j pj 2, …,i,… en’ pn n’ n ei pj : Budget :Price

9. Basic Algorithm Primal Dual Type Algorithm • In order to find the clear price, solve the max flow problem by: • Keep the price low such that all goods will always be sold out: (s, A U B U t) is the min cut --- Invariant Property. • And then increase the prices gradually such that all buyers have spent all their money.

10. Basic Algorithm High Level Idea Define For Define its money as For Define its money as For Define its neighborhood: Lemma: For given prices p, graph satisfies the Invariant IFF

11. Basic Algorithm High Level Idea,Cont. • Define: • Tight Set: a unique maximal set such that • --- be frozen by the algorithm • 2. Active Sub-graph: • 3. Drop the edges (i,j) with Lemma: If the Invariant holds and is the tight set, then each goods has an edge to certain buyer

12. Basic Algorithm High Level Idea,Cont. • Then gradually raise the prices of all goods in the active graph by the same pace , then: • Event 1: A nonempty set R goes tight. Add it to the tight set and continue the next iteration. • Event 2: An edge (i,j) with becomes an equality edge. Suppose j , remove component (Sl, Tl) into the active subgraph.

13. Basic Algorithm Find Tight Sets Let x* denote the value at which a nonempty set goes tight S* denote the tight set at prices x*p Lemma: For prices (x p) If x <= x*, then (s, A U B U t) is a min-cut If x > x*, then (s, A U B U t) is not a min-cut. Moreover, if (s U A1 U B1, A2 U B2 U t) is a min-cut in G(x p), then S* A1

14. Basic Algorithm Find Tight Sets, Cont. Lemma: x* and S* could be found using n max-flow computations. Proof: Let x = m(B)/m(A). Clearly, x >= x*. If (s, A U B U t) is a min cut in G(x P), then by the above lemma, we get: x = x* and S* = A. Otherwise, let (s U A1 U B1, A2UB2 U t) be the min cut. By the above lemma again, S* A1 A. Therefore, it issufficient to get S* by at most n recursions on the smaller graph (A1, N(A1)).

15. Basic Algorithm Basic Algorithm • Partition the whole algorithm into phases, each phase terminates with the occurrence of Event 1 • Each phase is partitioned into iterations which conclude with a new edge entering the equality subgraph (Event 2) Define M be the total money possessed by the buyers f be the max flow in the graph at prices P M – f is the surplus money with buyers.

16. Basic Algorithm Basic Algorithm, Cont. Let and Lemma 1: Each phase consists of at most n iterations. At the termination of a phase, the prices of the goods in the newly tight set must be rational numbers with denominator Lemma 2: Consider two phases P and P’ , not necessarily consecutive, such that goods j lies in the newly tight sets at the end of P as well as P’. Then the increase in the price of j, going from P to P’ is

17. Basic Algorithm Basic Algorithm, Cont. Lemma 3: After k phases, Corollary 4: The algorithm will end with market clearing price in at most phases, and execute iterations or times of computations.

18. Advanced Algorithm Advanced Algorithm • After the each phase in the basic algorithm, instead of freezing the tight set by prices p, we add one more step: • Let be fixed and ADD to the price of each active subgraph and find a min cut in graph G that maximizes the s-side. Let be the subset of A on the s-side of the min cut. Freeze . • Clearly, and • The algorithm will continue the next phase from the prices p, so Invariant will be maintained.

19. Advanced Algorithm Advanced Algorithm,Cont. • However, for the set of goods in the active subgraph : • Namely,the next phase must increase f by at least . Consider the situation when all goods are frozen and call it as an epoch. • After the 1st epoch, the total surplus is , set and run the next epoch…

20. Advanced Algorithm Advanced Algorithm,Cont. • Consider the 1st epoch that .At the end of this epoch, the surplus , and we will set for the next epoch. • By Lemma 2, the next epoch will be the final epoch and we will get the market clearing prices. Lemma: The number of total phases is: and the total iterations are