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Combinatorial Algorithms for Market Equilibria Vijay V. Vazirani

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Arrow-Debreu Theorem: Equilibria exist.

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Do markets operate at equilibria?

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Arrow-Debreu Theorem: Equilibria exist. Do markets operate at equilibria? Can equilibria be computed efficiently?

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Arrow-Debreu is highly non-constructive

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“Invisible hand” of the market: Adam Smith Scarf, 1973: approximate fixed point algs. Convex programs: Fisher: Eisenberg & Gale, 1957 Arrow-Debreu: Newman and Primak, 1992

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Used for deciding tax policies, price of new products etc. New markets on the Internet

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Algorithmic Game Theory Use powerful techniques from modern algorithmic theory and notions from game theory to address issues raised by Internet. Combinatorial algorithms for finding market equilibria.

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Two Fundamental Models Fisher’s model Arrow-Debreu model, also known as exchange model

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Combinatorial Algorithms Primal-dual schema based algorithms Devanur, Papadimitriou, Saberi & V., 2002 Combinatorial algorithm for Fisher’s model Auction-based algorithms Garg & Kapoor, 2004 Approximation algorithms.

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Approximation Find prices s.t. all goods clear Each buyer get goods providing at least optimal utility.

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Primal-Dual Schema Highly successful algorithm design technique from exact and approximation algorithms

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Exact Algorithms for Cornerstone Problems in P: Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

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Approximation Algorithms set cover facility location Steiner tree k-median Steiner network multicut k-MST feedback vertex set scheduling...

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Main new idea Previous: problems captured via linear programs DPSV algorithm: problem captured via a nonlinear convex program

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Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i,

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Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i, Find prices s.t. market clears

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Eisenberg-Gale Program, 1959

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DPSV Algorithm “primal” variables: allocations of goods “dual” variables: prices algorithm: primal & dual improvements Allocations Prices

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Buyer i’s optimization program: Global Constraint: Market Equilibrium

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People Goods $100 $60 $20 $140

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Prices and utilities $100 $60 $20 $140 $20 $40 $10 $60 10 20 4 2 utilities

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Bang per buck $100 $60 $20 $140 $20 $40 $10 $60 10 20 4 2 10/20 20/40 4/10 2/60

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Bang per buck Utility of $1 worth of goods Buyers will only buy goods providing maximum bang per buck

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Equality subgraph $100 $60 $20 $140 $20 $40 $10 $60 10 20 4 2 10/20 20/40 4/10 2/60

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Equality subgraph $100 $60 $20 $140 $20 $40 $10 $60 Most desirable goods for each buyer

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Any goods sold in equality subgraph make agents happiest How do we maximize sales in equality subgraph?

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Any goods sold in equality subgraph make agents happiest How do we maximize sales in equality subgraph? Use max-flow!

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Max flow 100 60 20 140 20 40 10 60 infinite capacities

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Idea of Algorithm Invariant: source edges form min-cut (agents have surplus) Iterations: gradually raise prices, decrease surplus Terminate: when surplus = 0, i.e., sink edges also form a min-cut

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Ensuring Invariant initially Set each price to 1/n Assume buyers’ money integral

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How to raise prices? Ensure equality edges retained i j l

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How to raise prices? Ensure equality edges retained i j l Raise prices proportionately

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100 60 20 140 20x 40x 10x 60x initialize: x = 1 x

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100 60 20 140 20x 40x 10x 60x x = 2: another min-cut x>2: Invariant violated

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100 60 20 140 40x 80x 20 120 active frozen reinitialize: x = 1

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100 60 20 140 50 100 20 120 active frozen x = 1.25

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100 60 20 140 50 100 20 120

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100 60 20 140 50 100 20 120 unfreeze

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100 60 20 140 50x 100x 20x 120x x = 1, x

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m buyers goods

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m p buyers goods ensure Invariant

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m p buyers goods equality subgraph ensure Invariant

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m pxpx x = 1, x

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} { S

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} { S freeze S tight set

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} { S prices in S are market clearing

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x = 1, x S active frozen pxpx

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x = 1, x S active frozen pxpx

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x = 1, x S active frozen pxpx

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new edge enters equality subgraph S active frozen

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unfreeze component active frozen

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All goods frozen => terminate (market clears)

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All goods frozen => terminate (market clears) When does a new set go tight? Solve as parametric cut problem

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Termination Prices in S* have denominators Terminates in max-flows.

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Polynomial time? Problem: very little price increase between freezings

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Polynomial time? Problem: very little price increase between freezings Solution: work with buyers having large surplus

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Max flow 100 60 20 140 20 40 10 60

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100 60 20 140 20 40 10 60 20 0 10 60 40 0 Max flow

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surplus(i) = m(i) – f(i) 100 60 20 140 20 40 10 60 20 0 10 60 40 0 60 20 70

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surplus(i) = m(i) – f(i) 100 60 20 140 20 40 10 60 20 0 10 60 40 0 60 20 70 Surplus vector = (40, 60, 20, 70)

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Balanced flow A max-flow that minimizes l 2 norm of surplus vector tries to make surpluses as equal as possible

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Algorithm Compute balanced flow

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active frozen Active subgraph: Buyers with maximum surplus

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active frozen x = 1, x pxpx

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active frozen new edge enters equality subgraph

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active frozen Unfreeze buyers having residual path to active subgraph

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active frozen Unfreeze buyers having residual path to active subgraph Do they have large surplus?

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f: balanced flow R(f): residual graph Theorem: If R(f) has a path from i to j then surplus(i) > surplus(j)

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active frozen New set tight

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active frozen New set tight: freeze

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Theorem: After each freezing, l 2 norm of surplus vector drops by (1 - 1/n 2 ) factor. Two reasons: total surplus decreases flow becomes more balanced

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Idea of Algorithm algorithm: primal & dual improvements measure of progress: l 2 -norm of surplus vector Allocations Prices

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Polynomial time Theorem: max-flow computations suffice.

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Weak gross substitutability Increasing price of one good cannot decrease demand for another good.

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Weak gross substitutability Increasing price of one good cannot decrease demand for another good. => never need to decrease prices (dual variables).

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Weak gross substitutability Increasing price of one good cannot decrease demand for another good. => never need to decrease prices (dual variables). Almost all primal-dual algs work this way.

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Arrow-Debreu Model Approximate equilibrium algorithms: Jain, Mahdian & Saberi, 2003: Use DPSV as black box. Devanur & V., 2003: More efficient, by opening DPSV.

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Garg & Kapoor, 2004 Auction-based algorithm Start with very low prices Keep increasing price of good that is in demand B has excess money. Favorite good: g Currently at price p and owned by B’ B outbids B’

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Outbid

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Auction-based algorithm Go in rounds: In each round, total surplus decreases by factor Hence iterations suffice, total money M= total money

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Arrow-Debreu Model Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money

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Arrow-Debreu Model Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money Any good with price >1 is fully sold

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Arrow-Debreu Model Start with all prices 1 Allocate money to agents (initial endowment) Perform outbid and update agents’ money Any good with price >1 is fully sold Eventually every good will have price >1

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Garg, Kapoor & V., 2004: Auction-based algorithms for additively separable concave utilities satisfying weak gross substitutability

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Kapoor, Mehta & V., 2005: Auction-based algorithm for a (restricted) production model

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Q: Distributed algorithm for equilibria? Appropriate model? Primal-dual schema operates via local improvements

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