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Computational complexity of competitive equilibria in exchange markets

Computational complexity of competitive equilibria in exchange markets. Katar ína Cechlárová P . J. Šafárik University Košic e , Slovakia Budapest, Summer school, 2013. Outline of the talk. brief history of the notion of competitive equilibrium model computation for divisible goods

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Computational complexity of competitive equilibria in exchange markets

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  1. Computational complexity of competitive equilibria in exchange markets Katarína Cechlárová P. J. Šafárik University Košice, Slovakia Budapest, Summer school, 2013

  2. Outline of the talk • brief history of the notion of competitive equilibrium • model computation for divisible goods • indivisible goods – housing market • Top trading cycles algorithm • housing market with duplicated houses • algorithm and complexity • approximate equilibrium and its complexity K. Cechlárová, Budapest 2013

  3. First ideas K. Cechlárová, Budapest 2013 Adam Smith: An Inquiry into the Nature and Causes of the Wealth of Nations (1776) FrancisYsidroEdgeworth: Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (1881) Marie-ÉspritLéonWalras: Elements of Pure Economics (1874) VilfredoPareto: Manual of PoliticalEconomy (1906)

  4. Exchange economy Kenneth Arrow & Gérard Debreu (1954) • set of agents, set of commodities • each agent owns a commoditybundle and has preferences over bundles • economic equilibrium: pair (prices, redistribution) suchthat: • each agent owns the best bundle he can afford given his budget • demand equals supply • if commodities are infinitely divisible and preferences of agents strictly monotone and strictly convex, equilibrium always exists K. Cechlárová, Budapest 2013

  5. Example: two agents, twogoods • agent 1: • agent 2: • prices (1,1) prices (1,1) are not equilibrium, assupplydemand K. Cechlárová, Budapest 2013

  6. Example - continued • agent 1: • agent 2: • prices (1,4) Equilibrium! K. Cechlárová, Budapest 2013

  7. Economy with indivisible goods Equlibrium might not exists! X. Deng, Ch. Papadimitriou, S. Safra (2002): Decision problem: Does an economic equilibrium exist in exchange economy with indivisible commodities and linear utility functions? NP-complete,already for two agents K. Cechlárová, Budapest 2013

  8. Housing market • n agents, each owns one unit of a unique indivisible good – house • preferencesof agent: linear ordering on a subset of houses • Shapley-Scarf economy (1974) • housing market is a model of: • kidney exchange • several Internet based markets K. Cechlárová, Budapest 2013

  9. strict preferences trichotomous preferences ties acceptable houses K. Cechlárová, Budapest 2013

  10. a1 a2 a4 a3 a5 a7 a6 K. Cechlárová, Budapest 2013

  11. Lemma. not equilibrium: a6 not satisfied Definition. a1 a2 a4 a3 a5 a7 a6 K. Cechlárová, Budapest 2013 11

  12. Top Trading Cycles algorithmfor Shapley-Scarf model (m=n, identity) Step 0. N:=A, roundr:=0, pr=n. Step 1.Take an arbitrary agent a0. Step 2.a0pointsto a most preferredhouse, in N, its owner is a1 . Agent a1pointsto the most preferredhouse a2in N etc. A cycle C arises. Step 3. r:=r+1, pr= pr-1; Cr:=C, all houses on C receive price pr, N:=N-C. Step 4.If N , go to Step 1, else end. • Shapley & Scarf (1974):author D. Gale • Abraham, KC, Manlove, Mehlhorn(2004): implementation linear in the size of the market K. Cechlárová, Budapest 2013

  13. Top Trading Cycles algorithmfor Shapley-Scarf model (m=n, identity) Step 0. N:=A, roundr:=0, pr=n. Step 1.Take an arbitrary agent a0. Step 2.a0pointsto a most preferredhouse, in N, its owner is a1 . Agent a1pointsto the most preferredhouse a2in N etc. A cycle C arises. Step 3. r:=r+1, pr= pr-1; Cr:=C, all houses on C receive price pr, N:=N-C. Step 4.If N , go to Step 1, else end. Theorem (Gale 1974). K. Cechlárová, Budapest 2013

  14. Theorem (KC & Fleiner 2008). Theorem (Fekete, Skutella , Woeginger 2003). K. Cechlárová, Budapest 2013

  15. a4 a3 a5 a7 a1 a2 p1 > p2 h2 h1 h3 h4 a6 K. Cechlárová, Budapest 2013

  16. h2 a4 a3 a5 h1 Theorem (KC & Schlotter 2010). a7 a1 a2 h3 h4 a6 Definition. Theorem (KC & Schlotter 2010). K. Cechlárová, Budapest 2013

  17. Definition. Approximating the number of satisfied agents K. Cechlárová, Budapest 2013

  18. Theorem (KC & Jelínková 2011). K. Cechlárová, Budapest 2013

  19. Theorem (KC & Jelínková 2011). K. Cechlárová, Budapest 2013

  20. Theorem (KC & Jelínková 2011). K. Cechlárová, Budapest 2013

  21. Theorem (KC & Jelínková 2011). 2 8 3 9 7 6 1 4 5 K. Cechlárová, Budapest 2013

  22. Theorem (KC & Jelínková 2011). Theorem (KC & Jelínková 2011). K. Cechlárová, Budapest 2013

  23. Thank you for your attention!

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