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Algorithmic Game Theory
and Internet Computing
Vijay V. Vazirani
Georgia Tech
New (Practical) Complementary Pivot
Algorithms for Market Equilibria
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Leon Walras, 1874
Pioneered general
equilibrium theory
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General Equilibrium Theory
Occupied center stage in Mathematical
Economics for over a century
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Central tenet
Markets should operate at equilibrium
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Central tenet
Markets should operate at equilibrium
i.e., prices s.t.
Parity between supply and demand
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Do markets even admit
equilibrium prices?
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Do markets even admit
equilibrium prices?
Easy if only one good!
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Supply-demand curves
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Do markets even admit
equilibrium prices?
What if there are multiple goods and
multiple buyers with diverse desires
and different buying power?
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Arrow-Debreu Model
n agents and g divisible goods.
Agent i: has initial endowment of goods
and a concave utility function
(yields convexity!)
ÎR+
g
ui : R+
g ® R+
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Arrow-Debreu Model
n agents and g divisible goods.
Agent i: has initial endowment of goods
and a concave utility function
piecewise-linear, concave (PLC)
ÎR+
g
ui : R+
g ® R+
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Agent i comes with
an initial endowment
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At given prices, agent i
sells initial endowment
1p 2p3p
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… and buys optimal bundle
of goods, i.e,
1p 2p3p
max ui (bundle)
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Several agents with own endowments
and utility functions.
Currently, no goods in the market.
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Agents sell endowments
at current prices.
1p 2p3p
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Each agent wants
an optimal bundle.
1p 2p3p
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Equilibrium
Prices p s.t. market clears,
i.e., there is no deficiency or surplus
of any good.
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Arrow-Debreu Theorem, 1954
Celebrated theorem in Mathematical Economics
Established existence of market equilibrium under
very general conditions using a deep theorem from
topology - Kakutani fixed point theorem.
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Kenneth Arrow
Nobel Prize, 1972
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Gerard Debreu
Nobel Prize, 1983
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Arrow-Debreu Theorem, 1954
Celebrated theorem in Mathematical Economics
Established existence of market equilibrium under
very general conditions using a theorem from
topology - Kakutani fixed point theorem.
Highly non-constructive!
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Leon Walras (1774):
Tatonnement process:
Price adjustment process to arrive at equilibrium
Deficient goods: raise prices
Excess goods: lower prices
Inherently algorithmic notion!
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Leon Walras
Tatonnement process:
Price adjustment process to arrive at equilibrium
Deficient goods: raise prices
Excess goods: lower prices
Does it converge to equilibrium?
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GETTING TO ECONOMIC
EQUILIBRIUM: A
PROBLEM AND ITS
HISTORY
For the third International Workshop on Internet and Network Economics
Kenneth J. Arrow
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OUTLINE
I. BEFORE THE FORMULATION OF GENERAL EQUILIBRIUM THEORY
II. PARTIAL EQUILIBRIUM
III. WALRAS, PARETO, AND HICKS
IV. SOCIALISM AND DECENTRALIZATION
V. SAMUELSON AND SUCCESSORS
VI. THE END OF THE PROGRAM
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Part VI: THE END OF THE PROGRAM
A. Scarf’s example
B. Saari-Simon Theorem: For any dynamic system depending on first-order information (z) only, there is a set of excess demand functions for which stability fails. (In fact, theorem is stronger).
C. Uzawa: Existence of general equilibrium is equivalent to fixed-point theorem
D. Assumptions on individual demand functions do not constrain aggregate demand function (Sonnenschein, Debreu, Mantel)
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Centralized algorithms for equilibria
Scarf, Smale, … , 1970s: Nice approaches!
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Centralized algorithms for equilibria
Scarf, Smale, … , 1970s: Nice approaches!
(slow and suffer from numerical instability)
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Theoretical Computer Science
Primal-dual paradigm
Convex programs
Complementary pivot algorithms
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(Complementary) Pivot Algorithms
Dantzig, 1947: Simplex algorithm for LP
Lemke-Howson, 1964: 2-Nash Equilibrium
Eaves, 1975: Equilibrium for Arrow-Debreu
markets under linear utilities
f (x) = cj
j
å xj
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(Complementary) Pivot Algorithms
Very fast in practice (even though
exponential time in worst case).
Work on rational numbers with bounded
denominators, hence no instability issues.
Reveal deep structural properties.
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(Complementary) Pivot Algorithms
Eaves, 1975: Equilibrium for linear
Arrow-Debreu markets
(based on Lemke’s algorithm)
Until very recently, no extension to more
general utility functions!
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(Complementary) Pivot Algorithms
Eaves, 1975: Equilibrium for linear
Arrow-Debreu markets
Until very recently, no extension to more
general utility functions!
Why?
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Separable, piecewise-linear concave
utility functions
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Separable utility function
For a single buyer :
Utility from good j , f j : + ® +
Total utility from bundle, f (x) = f j
j
å (xj )
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utility
: piecewise-linear, concave
amount of j
f j
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Arrow-Debreu market under separable,
piecewise-linear concave (SPLC) utilities
Can Eaves’ algorithm be extended to this case?
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Eaves, 1975 Technical Report:
Also under study are extensions of the overall method to
include piecewise-linear utilities, production, etc., if
successful, this avenue could prove important in real
economic modeling.
Eaves, 1976 Journal Paper:
... Now suppose each trader has a piecewise-linear, concave
utility function. Does there exist a rational equilibrium?
Andreu Mas-Colell generated a negative example, using
Leontief utilities. Consequently, one can conclude that
Lemke’s algorithm cannot be used to solve this class of
exchange problems.
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Leontief utility
u x( ) = minx1
a1
,x2
a2
, ... ,xn
an
æ
èçö
ø÷
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Leontief utility: is non-separable!
Utility = min{#bread, 2 #butter}
Only bread or only butter gives 0 utility!
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Devanur & Kannan, 2007,
V. & Yannakakis, 2007:
If all parameters are rational numbers,
there is a rational equilibrium.
Rationality for SPLC Utilities
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Theorem (Garg, Mehta, Sohoni & V., 2012):
Complementary pivot algorithm for Arrow-Debreu
markets under SPLC utility functions.
(based on Lemke’s algorithm)
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Experimental Results
Inputs are drawn uniformly at random.
|A|x|G|x#Seg #Instances Min Iters Avg Iters Max Iters
10 x 5 x 2 1000 55 69.5 91
10 x 5 x 5 1000 130 154.3 197
10 x 10 x 5 100 254 321.9 401
10 x 10 x 10 50 473 515.8 569
15 x 15 x 10 40 775 890.5 986
15 x 15 x 15 5 1203 1261.3 1382
20 x 20 x 5 10 719 764 853
20 x 20 x 10 5 1093 1143.8 1233
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log(total no. of segments)
log
(no
. o
f it
erat
ion
s)
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Linear Complementarity
& Lemke’s Algorithm
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Linear complementarity problem
Generalizes LP
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min cT
x
s.t. Ax ³ b
x ³ 0
max bTz
s.t. AT z£ c
z³ 0
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min cT
x
s.t. Ax ³ b
x ³ 0
max bTz
s.t. AT z£ c
z³ 0
Complementary Slackness:
Let x , z both be feasible.
Then both are optimal iff
"i : zi = 0 or Ax( )i= bi
"j : xj = 0 or AT z( )j= cj
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x, z are both optimal iff
Ax ³ b AT z£ c
x ³ 0 z³ 0
zT
Ax- b( ) = 0 xT
c- AT z( ) = 0
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Let
M =-A 0
0 AT
æ
èç
ö
ø÷ y =
x
z
æ
èç
ö
ø÷ q =
-b
c
æ
èç
ö
ø÷
Then y gives optimal solutions iff
M y £ q
y ³ 0
y. q- M y( ) = 0
square matrix
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Let
M =-A 0
0 AT
æ
èç
ö
ø÷ y =
x
z
æ
èç
ö
ø÷ q =
-b
c
æ
èç
ö
ø÷
Then y gives optimal solutions iff
M y £ q
y ³ 0
y. q- M y( ) = 0
square matrix
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Linear Complementarity Problem
Given n´ n matrix M and vector q find y s.t.
M y £ q
y ³ 0
y. (q- M y) =0
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Linear Complementarity Problem
Given n´ n matrix M and vector q find y s.t.
M y £ q
y ³ 0
y. (q- M y) =0
quadratic
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Linear Complementarity Problem
Given n´ n matrix M and vector q find y s.t.
M y £ q
y ³ 0
y. (q- M y) =0 Clearly, q- M y ³ 0( )i.e., for each i :
yi = 0 or inequality i is satisfied with equality.
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Examples of linear complementarity
LP: complementary slackness
2-Nash: For row player,
either Pr[row i] = 0 or row i is a best response.
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Nonlinear complementarity
Plays a key role in KKT conditions for
convex programs
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Linear Complementarity Problem
Given n´ n matrix M and vector q find y s.t.
M y £ q
y ³ 0
y. (q- M y) =0
i.e., for each i :
yi = 0 or inequality i is satisfied with equality.
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Linear Complementarity Problem
Given n´ n matrix M and vector q find y s.t.
M y £ q
y ³ 0
y. (q- M y) =0
.
Introduce slack variables v
M y + v = q
y ³ 0, v ³ 0 (Since q- M y ³ 0)
y. v = 0
i.e., for each i : yi = 0 or vi = 0.
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M y + v = q
y ³ 0
v ³ 0
y. v=0
Assume polyhedron in 2n defined by
red constraints is non-degenerate.
Solution to LCP satisfies 2n equalities
Þ is a vertex of the polyhedron.
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Possible scheme
Find one vertex of polyhedron
and walk along 1-skeleton, via pivoting,
to a solution.
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Possible scheme
Find one vertex of polyhedron
and walk along 1-skeleton, via pivoting,
to a solution.
But in which direction is the solution?
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Lemke’s idea
Add a new dimension:
M y + v - z1 = q
y ³ 0
v ³ 0
z ³ 0
y. v= 0
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Lemke’s idea
Add a new dimension:
M y + v - z1 = q
y ³ 0
v ³ 0
z ³ 0
y. v=0
Note: Easy to get a solution to augmented LCP:
Pick y = 0, z large and v = q + z1 . Then v³ 0.
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Lemke’s idea
Add a new dimension:
M y + v - z1 = q
y ³ 0
v ³ 0
z ³ 0
y. v= 0
Want: solution of augmented LCP with z= 0.
Will be solution of original LCP!
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S: set of solutions to augmented LCP,
each satisfies 2n equalities.
Polyhedron is in 2n+1 space.
Hence, S is a subset of 1-skeleton, i.e.,
consist of edges and vertices.
Every solution is fully labeled, i.e.,
"i : yi =0 or vi =0
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Vertices of polyhedron lying in S
Two possibilities:
1). Has a double label, i.e.,
Only 2 ways of relaxing double label, hence
this vertex has exactly 2 edges of S incident.
$i : yi =0 and vi =0
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Vertices of polyhedron lying in S
Two possibilities:
2). Has z = 0
Only 1 way of relaxing z = 0, hence
this vertex has exactly 1 edge of S incident.
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Vertices of polyhedron lying in S
Two possibilities:
2). Has z = 0
Only 1 way of relaxing z = 0, hence
this vertex has exactly 1 edge of S incident.
This is a solution to original LCP!
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Hence S consists of paths and cycles!
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z= 0
z= 0z= 0
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ray: unbounded edge of S.
principal ray: each point has y = 0.
secondary ray: rest of the rays.
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Lemke’s idea
Add a new dimension:
M y + v - z1 = q
y ³ 0
v ³ 0
z ³ 0
y. v=0
Note: Easy to get a solution to augmented LCP:
Pick y = 0, z large and v = q + z1 . Then v³ 0.
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z¯
principal
ray
y = 0
yi = 0
vi = 0
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z¯
principal
ray
y = 0
yi = 0
vi = 0 yk = 0
vk = 0
yi
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z¯
principal
ray
y = 0
yi = 0
vi = 0 yk = 0
vk = 0
yi yk
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principal
ray
y = 0
yi = 0
vi = 0
Path lies in S!
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z= 0
principal
ray
y = 0
yi = 0
vi = 0
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y = 0
principal
ray secondary
ray
yi = 0
vi = 0
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Problem with Lemke’s algorithm
No recourse if path starting with
primary ray ends in a secondary ray!
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Problem with Lemke’s algorithm
No recourse if path starting with
primary ray ends in a secondary ray!
We show that for each of our LCPs,
associated polyhedron has no secondary rays!
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Dramatic change!
Polyhedron of
original LPC: no clue where solution is.
augmented LCP: know a path
leading to solution!
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Theorem (Garg, Mehta, Sohoni & V., 2012):
1). Derive LCP whose solutions
correspond to equilibria.
2). Polyhedron of LCP has no secondary rays.
Corollary: The number of equilibria is odd,
up to scaling.
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z= 0
z= 0z= 0
z= 0z= 0
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Theorem (Garg, Mehta, Sohoni & V., 2012):
1). Derive LCP whose solutions
correspond to equilibria.
2). Polyhedron of LCP has no secondary rays.
3). If no. of goods or agents is a constant,
polyn. vertices of polyhedron are solutions
=> strongly polynomial algorithm
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Derive LCP (assume linear utilities)
Market clearing
Every good fully sold
Every agent spends all his money
Optimal bundles
Every agent gets a utility maximizing bundle
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Model
Utility of agent i: uij
j
å xij
Initial endowment of agent i: wij , j ÎG
W.l.o.g. assume 1 unit of each good
in the market.
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Variables
pj : price of good j
qij : amount of money
spent by i on j
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Guaranteeing optimal bundles
Agent i spends only on
bang-per-buck of i
Si = argmax j
uij
pj
ìíï
îï
üýï
þï
= max j
uij
pj
ìíï
îï
üýï
þï
=(1
li
at equilibrium)
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Optimal bundles, guaranteed by:
"j : uij
pj
£1
li
qij > 0 Þuij
pj
=1
li
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Optimal bundles, guaranteed by:
"j : uij
pj
£1
li
qij > 0 Þuij
pj
=1
li
qij > 0 oruij
pj
=1
li
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Optimal bundles, via complementarity
"i : "j : uijli £ pj
qij (uijli - pj )= 0
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LCP for linear utilities
"j : qij
i
å £ pj comp pj
"i : wij
j
å pj £ qij
j
å comp li
"i, j : uijli £ pj comp qij
& non-negativity for pj , qij , li