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Algorithms for the Linear Case, and Beyond …. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. Irving Fisher, 1891. Defined a fundamental market model Special case of Walras’ model.

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## Algorithmic Game Theory and Internet Computing

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**Algorithms for the Linear Case,**and Beyond … Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech**Irving Fisher, 1891**• Defined a fundamental market model • Special case of Walras’ model**Several buyers with different utility functions and**moneys.Find equilibrium prices!!**Linear Fisher Market**• Assume: • Buyer i’s total utility, • mi: money of buyer i. • One unit of each good j. • Findmarket clearing prices!**Eisenberg-Gale Program, 1959**prices pj**Convex programs thatcapture market equilibria**• Underly all “efficient” markets (so far). • Rational convex programs for many markets! • Algorithms: • combinatorial, for rational (primal-dual) • continuous (ellipsoid/interior point)**Combinatorial algorithms for rational convex programs**Natural extension of field of combinatorial optimization!**Combinatorial algorithms for rational convex programs**Natural extension of field of combinatorial optimization! Central aspect of C.O. – efficient algorithms for solving integral LP’s, e.g. matching, flow.**Combinatorial Algorithm for Linear Case of Fisher’s Model**• Devanur, Papadimitriou, Saberi & V., 2002 By extending the primal-dual paradigm to the setting of convex programs & KKT conditions**Combinatorial algorithms**Yield deep structural insights. Preferable for applications.**Auction for Google’s TV ads**N. Nisan et. al, 2009: • Used market equilibrium based approach. • Combinatorial algorithms for linear case provided “inspiration”.**Primal-Dual Paradigm**• Highly successful algorithm design technique from exact and approximation algorithms**Exact Algorithms for Cornerstone Problems**in P: • Matching (general graph) • Network flow • Shortest paths • Minimum spanning tree • Minimum branching**Approximation Algorithms**set cover facility location Steiner tree k-median Steiner network multicut k-MST feedback vertex set scheduling . . .**An easier question**• Given prices p, are they equilibrium prices? • If so, find equilibrium allocations.**An easier question**• Given prices p, are they equilibrium prices? • If so, find equilibrium allocations. • Equilibrium prices are unique!**Bang-per-buck**• At prices p, buyer i’s most desirable goods, Si = • Any goods from Siworth m(i) constitute i’s optimal bundle**For each buyer, most desirable goods, i.e.**m(1) p(1) m(2) p(2) m(3) p(3) m(4) p(4)**Network N(p)**p(1) m(1) p(2) m(2) t s p(3) m(3) m(4) p(4) infinite capacities**Max flow in N(p)**p(1) m(1) p(2) m(2) p(3) m(3) m(4) p(4) p: equilibrium prices iff both cuts saturated**Idea of algorithm**• “primal” variables: allocations • “dual” variables: prices of goods • Approach equilibrium prices from below: • start with very low prices; buyers have surplus money • iteratively keep raising prices and decreasing surplus**An important consideration**• The price of a good never exceeds its equilibrium price • Invariant: s is a min-cut**Invariant: s is a min-cut in N(p)**p(1) m(1) p(2) m(2) s p(3) m(3) m(4) p(4) p: low prices**AllocationsPrices**Idea of algorithm • Iterations: execute primal & dual improvements**Fundamental difference betweenLP’s and convex programs**• Complementary slackness conditions: involve primal or dual variables, not both. • KKT conditions: involve primal and dual variables simultaneously.**Primal-dual algorithms so far(i.e., LP-based)**• Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.)**Primal-dual algorithms so far**• Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.) • Only exception: Edmonds, 1965: algorithm for max weight matching.**Primal-dual algorithms so far**• Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.) • Only exception: Edmonds, 1965: algorithm for max weight matching. • Otherwise primal objects go tight and loose. Difficult to account for these reversals -- in the running time.**Our algorithm**• Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions**Our algorithm**• Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions • New algorithmic ideas needed!**Key Algorithmic Idea**• Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions • Balanced Flows: For limiting no. of such events**Max-flow in N**p m s t i W.r.t. a max-flow f, surplus(i) = m(i) – f(i,t)**Max-flow in N**p m i surplus vector = vector of surpluses w.r.t. f**Obvious potential function**• Total surplus money = l1 norm of surplus vector • Reduce l1 norm of surplus vector by inverse polynomial fraction in each iteration**Balanced flow**• A max-flow that minimizes l2 norm of surplus vector. • Makes surpluses as equal as possible.**Balanced flow**• A max-flow that minimizes l2 norm of surplus vector. • Makes surpluses as equal as possible. • All balanced flows have same surplus vector.**Our algorithm**• Reduces l2 norm of surplus vector by inverse polynomial fraction in each iteration.**Property 1**• f: max-flow in N. • R: residual graph w.r.t. f. • If surplus (i) < surplus(j) then there is no pathfrom i to j in R.**Property 1**R: i j surplus(i) < surplus(j)**Property 1**R: i j surplus(i) < surplus(j)**Property 1**R: i j Circulation gives a more balanced flow.**Property 1**• Theorem: A max-flow is balanced iff it satisfies Property 1.

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