Computational complexity
of competitive equilibria
in exchange markets
Katarína Cechlárová
P. J. Šafárik University
Košice, 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 complexity
 approximate equilibrium and its complexity
K. Cechlárová, Budapest 2013
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First ideas

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
Political Economy (1906)
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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)
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Example: two agents, two goods



agent 1: ω  (2,1);
2
agent 2: ω  (1,0);
prices (1,1)
1
u1 ( x1 , x2 )  x1 x2
2
u ( x1 , x2 )  x2
3 5
 x   2 , 2 
i
x2
3 3
x  , 
2 2
1
x2
x 2  0,1
  3,1
i
prices (1,1) are not
equilibrium, as
x1 demand
supply
x1
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Example - continued



agent 1: ω  (2,1);
2
agent 2: ω  (1,0);
prices (1,4)
1
x2
 3
x   3, 
 4
1
u1 ( x1 , x2 )  x1 x2
2
u ( x1 , x2 )  x2
 x  3,1
i
x2
 1
x   0, 
 4
2
x1
i
ω
  3,1
Equilibrium!
x1
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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
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Housing market




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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strict preferences
trichotomous
preferences
ties
acceptable houses
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a1
a2
a4
a3
a7
a5
a6
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Definition.
Lemma.
not equilibrium:
a6 not satisfied
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a1
a2
a4
a3
a7
a5
a6
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Top Trading Cycles algorithm for
Shapley-Scarf model (m=n, identity)
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
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Top Trading Cycles algorithm for
Shapley-Scarf model (m=n, identity)
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
a4
a3
a5
h1
a1
a2
h4
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a7
a6
p1 > p2
h3
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Definition.
Theorem (KC & Schlotter 2010).
h2
a4
a3
a5
h1
Theorem (KC & Schlotter 2010).
a1
a2
h4
a7
a6
h3
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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).
2
3
9
7
6
1
8
4
5
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Theorem (KC & Jelínková 2011).
Theorem (KC & Jelínková 2011).
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Thank you for your
attention!