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 2

3. First ideas K. Cechlárová, Budapest 2013 3 Adam Smith: An Inquiry into the Nature and Causes of the Wealth of Nations (1776) Francis Ysidro Edgeworth: Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (1881) Marie-Ésprit Léon Walras: Elements of Pure Economics (1874) Manual of Political Economy (1906) Vilfredo Pareto: Manual of Political Economy (1906)

4. Exchange economy set of agents, set of commodities each agent owns a commodity bundle and has preferences over bundles economic equilibrium: pair (prices, redistribution) such that:  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 4 Kenneth Arrow & Gérard Debreu (1954)

5. Example: two agents, two goods K. Cechlárová, Budapest 2013 5 agent 1: agent 2: prices (1,1) prices (1,1) are not equilibrium, as supply  demand

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

7. Economy with indivisible goods K. Cechlárová, Budapest 2013 7 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? already for two agents NP-complete, already for two agents

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

9. K. Cechlárová, Budapest 2013 9 acceptable housesstrict preferences ties trichotomous preferences

10. K. Cechlárová, Budapest 2013 10 a1a1 a2a2 a7a7 a6a6 a4a4 a5a5 a3a3

11. K. Cechlárová, Budapest 2013 11 Lemma. Definition. not equilibrium: a 6 not satisfied a1a1 a2a2 a7a7 a6a6 a4a4 a5a5 a3a3

12. Top Trading Cycles algorithm for Shapley-Scarf model (m=n,  identity) K. Cechlárová, Budapest 2013 12 Step 0. N:=A, round r :=0, p r = n. Step 1. Take an arbitrary agent a 0. Step 2. a 0 points to a most preferred house, in N, its owner is a 1. Agent a 1 points to the most preferred house a 2 in N etc. A cycle C arises. Step 3. r := r +1, p r = p r -1; C r :=C, all houses on C receive price p r, 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

13. Top Trading Cycles algorithm for Shapley-Scarf model (m=n,  identity) K. Cechlárová, Budapest 2013 13 Step 0. N:=A, round r :=0, p r = n. Step 1. Take an arbitrary agent a 0. Step 2. a 0 points to a most preferred house, in N, its owner is a 1. Agent a 1 points to the most preferred house a 2 in N etc. A cycle C arises. Step 3. r := r +1, p r = p r -1; C r :=C, all houses on C receive price p r, N:=N-C. Step 4. If N , go to Step 1, else end. Theorem (Gale 1974).

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

15. K. Cechlárová, Budapest 2013 15 h2h2 h4h4 h1h1 h3h3 a1a1 a4a4 a2a2 a3a3 a5a5 a6a6 p 1 > p 2 a7a7

16. K. Cechlárová, Budapest 2013 16 h2h2 h4h4 h1h1 h3h3 a1a1 a4a4 a2a2 a3a3 a5a5 a6a6 a7a7 Theorem (KC & Schlotter 2010). Definition.

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

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

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

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

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

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

23. Thank you for your attention!