Computational complexity of competitive equilibria in exchange markets

Katarína CechlárováP. J. Šafárik UniversityKošice, Slovakia

Budapest, Summer school, 2013

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 complexityapproximate equilibrium and its complexity

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First ideas

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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)

Vilfredo Pareto: Manual of Manual of Political Economy (1906)Political Economy (1906)

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

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Kenneth Arrow & Gérard Debreu (1954)

Example: two agents, two goods

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agent 1: agent 2: prices (1,1)

212111 ),();1,2( xxxxu ω

22122 ),();0,1( xxxu ω

1x

2x

23,

231x

1x

2x 1,02 x

prices (1,1) are not equilibrium, as supply demand

25,

23ix

1,3i

Example - continued

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agent 1: agent 2: prices (1,4)

212111 ),();1,2( xxxxu ω

22122 ),();0,1( xxxu ω

1x

2x

43,31x

1x

2x

41,02x

Equilibrium!

1,3ix

1,3iω

Economy with indivisible goods

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Equlibrium might not exists!Equlibrium might not exists!

X. Deng, Ch. Papadimitriou, S. Safra (2002):

Decision problem: Does an economic equilibrium exist iDoes an economic equilibrium exist inn exchange economy with indivisible exchange economy with indivisible commodities and linear utility functionscommodities and linear utility functions? ?

NP-complete, already for two agentsalready for two agents

Housing market

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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

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acceptable houses

strict preferences

ties

trichotomous preferences

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a1 a2

a7

a6

a4

a5

a3

K. Cechlárová, Budapest 2013 1111

Lemma.

Definition.

not equilibrium: a6 not satisfied

a1 a2

a7

a6

a4

a5

a3

Top Trading Cycles algoritTop Trading Cycles algorithhmm for for Shapley-Scarf model (m=n,Shapley-Scarf model (m=n, identity) identity)

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Step 0. N:=A, round r:=0, pr=n.Step 1. Take an arbitrary agent a0.Step 2. a0 points to a most preferred house, in N, its owner is a1 . Agent a1 points to the most preferred house a2 in 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

Top Trading Cycles algoritTop Trading Cycles algorithhmm for for Shapley-Scarf model (m=n,Shapley-Scarf model (m=n, identity) identity)

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Step 0. N:=A, round r:=0, pr=n.Step 1. Take an arbitrary agent a0.Step 2. a0 points to a most preferred house, in N, its owner is a1 . Agent a1 points to the most preferred house a2 in 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).

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Theorem (Fekete, Skutella , Woeginger 2003).

Theorem (KC & Fleiner 2008).

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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).

Definition.

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Approximating the number of satisfied agents

Definition.

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Theorem (KC & Jelínková 2011).

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Theorem (KC & Jelínková 2011).

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Theorem (KC & Jelínková 2011).

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Theorem (KC & Jelínková 2011).

1

23

45

6 7

89

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Theorem (KC & Jelínková 2011).

Theorem (KC & Jelínková 2011).

Thank you for your attention!