Computational complexity of competitive equilibria in exchange - PowerPoint PPT Presentation

Computational complexity of competitive equilibria in exchange markets Katarína Cechlárová P. J. Š afárik University Ko š ice, Slovakia Budapest, Summer school, 2013

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

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

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

Example: two agents, two goods 1 1 ω ( 2 , 1 ); u ( x , x ) x x n agent 1: = = 1 2 1 2 2 2 ω ( 1 , 0 ); u ( x , x ) x n agent 2: = = 1 2 2 n prices (1,1) 3 5 ⎛ ⎞ i x , ∑ = ⎜ ⎟ 2 2 x x ⎝ ⎠ 2 2 3 3 ⎛ ⎞ 2 = i 1 ( ) x 0 , 1 ( ) x , 3 , 1 ∑ = ⎜ ⎟ ω = 2 2 ⎝ ⎠ prices (1,1) are not equilibrium, as x x supply ≠ demand 1 1 5 K. Cechlárová, Budapest 2013

Example - continued 1 1 ω ( 2 , 1 ); u ( x , x ) x x n agent 1: = = 1 2 1 2 2 2 ω ( 1 , 0 ); u ( x , x ) x n agent 2: = = 1 2 2 n prices (1,4) i ( ) x 3 , 1 ∑ = i ( ) ω 3 , 1 ∑ x x = 2 2 1 3 ⎛ ⎞ ⎛ ⎞ 2 x 0 , 1 x 3 , = ⎜ ⎟ = ⎜ ⎟ 4 4 ⎝ ⎠ Equilibrium! ⎝ ⎠ x x 1 1 6 K. Cechlárová, Budapest 2013

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 7 K. Cechlárová, Budapest 2013

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

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

a 1 a 2 a 4 a 3 a 5 a 7 a 6 10 K. Cechlárová, Budapest 2013

Definition. Lemma. a 1 a 2 a 4 a 3 a 5 a 7 not equilibrium: a 6 a 6 not satisfied 11 11 K. Cechlárová, Budapest 2013

Top Trading Cycles algorithm for Shapley-Scarf model (m=n, ω identity) 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. n Shapley & Scarf (1974): author D. Gale n Abraham, KC, Manlove, Mehlhorn (2004): implementation linear in the size of the market 12 K. Cechlárová, Budapest 2013

Top Trading Cycles algorithm for Shapley-Scarf model (m=n, ω identity) 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). 13 K. Cechlárová, Budapest 2013

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

h 2 a 4 a 3 a 5 h 1 a 7 a 1 a 2 h 4 h 3 a 6 p 1 > p 2 15 K. Cechlárová, Budapest 2013

Definition. Theorem (KC & Schlotter 2010). h 2 a 4 a 3 a 5 h 1 Theorem (KC & Schlotter 2010). a 7 a 1 a 2 h 3 h 4 a 6 16 K. Cechlárová, Budapest 2013

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

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

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

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

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

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

Thank you for your attention!