Computational complexity of competitive equilibria in exchange - - PowerPoint PPT Presentation
Computational complexity of competitive equilibria in exchange markets Katarna Cechlrov P. J. afrik University Ko ice, Slovakia Budapest, Summer school, 2013 Outline of the talk n brief history of the notion of competitive
Budapest, Summer school, 2013
n brief history of the notion of competitive
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
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n Adam Smith: An Inquiry into
the Nature and Causes of the Wealth of Nations (1776)
n Francis Ysidro
Psychics: An Essay on the Application
(1881)
n Marie-Ésprit Léon
Economics (1874)
n Vilfredo Pareto: Manual of
Political Economy (1906)
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 exists
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Kenneth Arrow & Gérard Debreu (1954)
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n agent 1: n agent 2: n prices (1,1) 2 1 2 1 1 1
2 2 1 2 2
1
x
2
x ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 2 3 , 2 3
1
x
1
x
2
x
1 ,
2 =
x
prices (1,1) are not equilibrium, as supply ≠demand
i
i
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n agent 1: n agent 2: n prices (1,4) 2 1 2 1 1 1
2 2 1 2 2
1
x
2
x ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 4 3 , 3
1
x
1
x
2
x ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 4 1 ,
2
x
i
i
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(2002): Decision problem: Does an economic equilibrium exist in exchange economy with indivisible commodities and linear utility functions? NP-complete, already for two agents
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n n agents, each owns one unit of
n preferences of agent: linear
n Shapley-Scarf economy (1974) n housing market is a model of:
¨ kidney exchange ¨ several Internet based markets
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acceptable houses strict preferences ties trichotomous preferences
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a1 a2 a7 a6 a4 a5 a3
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not equilibrium: a6 not satisfied
a1 a2 a7 a6 a4 a5 a3
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n Shapley & Scarf (1974): author D. Gale n Abraham, KC, Manlove, Mehlhorn (2004): implementation
linear in the size of the market
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h2 h4 h1 h3 a1 a4 a2 a3 a5 a6
p1 > p2
a7
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h2 h4 h1 h3 a1 a4 a2 a3 a5 a6 a7
Theorem (KC & Schlotter 2010). Theorem (KC & Schlotter 2010).
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