A Lemke-Type Algorithm for Market Equilibrium under Separable,

Piecewise-Linear Concave Utilities

Ruta Mehta

Indian Institute of Technology – Bombay

Joint work with Jugal Garg, Milind Sohoni and Vijay V. Vazirani

Exchange MarketSeveral agents

Several agents with endowment of goods

Several agents with endowments of goods and different concave utility functions

Given prices, an agent sells his endowment and buys an optimal bundle from the earned money.

1p 2p3p

Parity between demand and supplyequilibrium prices

1p 2p 3p

Do equilibrium prices exist?

1p 2p 3p

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.

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!

Computation

The Linear Case

DPSV (2002) – Flow based algorithm for the Fisher market.

Jain (2004) – Using Ellipsoid method.

Ye (2004) – Interior point method.

Separable Piecewise-Linear Concave (SPLC)

Utility function of an agent is separable for goods.

Amount of good j

Utility

Separable Piecewise-Linear Concave (SPLC)

Utility function of an agent

is separable

Rationality – Devanur and Kannan (2008); Vazirani and Yannakakis (2010).

Amount of good j

Utility

Separable Piecewise-Linear Concave (SPLC)

Utility function of an agent

is separable

Rationality – Devanur and Kannan (2008); Vazirani and Yannakakis (2010).

Devanur and Kannan (2008) – Polynomial time algorithm when number of agents or goods are constant.

Amount of good j

Utility

SPLC – Hardness Results

Chen et al. (2009) – It is PPAD-hard.

Chen and Teng (2009) – Even for the Fisher market it is PPAD-hard.

Vazirani and Yannakakis (2010) It is PPAD-hard for the Fisher market. It is in PPAD for both.

Vazirani and Yannakakis

“The definition of the class PPAD was designed to capture problems that allow for path following algorithms, in the style of the algorithms of Lemke-Howson. It will be interesting to obtain natural, direct path following algorithm for this task (hence leading to a more direct proof of membership in PPAD), which may be useful for computing equilibria in practice.”

Initial Attempts DPSV like flow based algorithm.

Lemke-Howson A classical algorithm for 2-Nash. Proves containment of 2-Nash in PPAD.

Lemke-Howson type algorithm for linear markets by Garg, Mehta and Sohoni (2011).

Extend GMS algorithm.

Linear Case: Eaves (1975) LCP formulation to capture market equilibria. Apply Lemke’s algorithm to find one.

He states: “Also under study are extensions of the overall method to include piecewise linear concave utilities, production, etc., if successful, this avenue could prove important in real economic modeling.”

In 1976 Journal version He demonstrates a Leontief market with only

irrational equilibria, and concludes impossibility of extension.

Our Results Extend Eave’s LCP formulation to SPLC markets. Design a Lemke-type algorithm.

Runs very fast in practice. Direct proof of membership of SPLC markets in PPAD. The number of equilibria is odd (similar to 2-Nash,

Shapley’74). Provide combinatorial interpretation.

Strongly polynomial bound when number of goods or agents is constant.

In case of linear utilities, prices and surplus are monotonic Combinatorial algorithm. Equilibria form a convex polyhedral cone.

Linear Complementarity Problem For LP: Complementary slackness conditions

capture optimality. 2-Nash: Equilibria are characterized through

complementarity conditions.

Given n x n matrix M and n x 1 vector q, find y s.t.

My ≤ q; y ≥ 0

My + v = q; v, y ≥ 0 yTv = 0yT(q – My) =

0

nR

Properties of LCP

yTv = 0 => yivi = 0, for all i.

At a solution, yi=0 or vi=0, for all i.

Trivial if q ≥ 0: Set y = 0, and v = q.

P: My + v = q; v, y ≥ 0

yTv = 0

Properties of LCP

yTv = 0 => yivi = 0, for all i.

At a solution, yi=0 or vi=0, for all i.

There may not exist a solution.

yTv = 0

P: My + v = q; v, y ≥ 0

Properties of LCP

yTv = 0 => yivi = 0, for all i.

At a solution, yi=0 or vi=0, for all i.

If there exists a solution, then there is a vertex of P which is a solution.

yTv = 0

P: My + v = q; v, y ≥ 0

Properties of LCP

Solution set might be disconnected.

There is a possibility of a simplex-like algorithm given a feasible vertex of P.

yTv = 0

P: My + v = q; v, y ≥ 0

Lemke’s Algorithm Add a dimension:

P’: My + v – z = q; v, y, z ≥ 0yTv=0

T = Points in P’ with yTv=0.

Required: A point of T with z=0

Assumption: P’ is non-degenerate.

The set TP’: My + v – z = q; v, y, z ≥ 0

yTv=0

Assumption: P’ is non-degenerate.

n inequalities should be tight at every point.

P’ is n+1-dimensional => T consists of edges and vertices.

The set TP’: My + v – z = q; v, y, z ≥ 0

yTv=0

Assumption: P’ is non-degenerate.

Ray: An unbounded edge of T. If y=0 then primary ray, all others are secondary

rays. At a vertex of T

Either z=0 Or ! i s.t. yi=0 and vi=0. Relaxing each gives two

adjacent edges of S.

The set TP’: My + v – z = q; v, y, z ≥ 0

yTv=0

Assumption: P’ is non-degenerate.

Paths and cycles on 1-skeleton of P’.

z=0

z=0

z=0

Lemke’s AlgorithmP’: My + v – z = q; v, y, z ≥ 0

yTv=0

Assumption: P’ is non-degenerate.

Invariant: Remain in T.

Start from the primary ray.

Starting VertexP’: My + v – z = q; v, y, z ≥ 0

yTv=0 Primary Ray:

y=0, z and v change accordingly.

Vertex (v*, y*, z*): y* = 0; i* = argmini qi; z* = |qi*|; vi* = qi + z*;

z=z*

y =

0z=∞

v > 0

vi*=0

The Algorithm Start by tracing the primary ray up to (v*, y*,

z*).

z=z*vi*=0

v > 0, y =

0z=∞

The Algorithm Start by tracing the primary ray up to (v*, y*,

z*). Then relax yi* = 0,

vi*=0yi*=0

v i*>0

v i*=0

y i*>

0

The Algorithm

In general If vi ≥ 0 becomes tight, then relax yi = 0,

And if yi ≥ 0 becomes tight then relax vi = 0.

z=0

vi=0yi=0

vi =0

yi >0

v i>0y i=

0

vi*=0yi*=0

v i*>0

v i*=0

y i*>

0

The Algorithm Start by tracing the primary ray up to (v*, y*,

z*). If vi ≥ 0 becomes tight, then relax yi=0

And if yi ≥ 0 becomes tight then relax vi=0.

vi=0yi=0

vi =0

yi >0

v i>0y i=

0

vi*=0yi*=0

v i*>0

v i*=0

y i*>

0

Properties and Correctness No cycling.

Termination: Either at a vertex with z=0 (the solution), or on an

unbounded edge (a secondary ray).

No need of potential function for termination guarantee.

Exchange Markets A: Set of agents, G: Set of goods

m= |A|, n=|G|.

Agents i with wij endowment of good j utility function :

i nf R R

Separable Piecewise-Linear Concave (SPLC) Utilities

Utility function f i is: Separable – is for jth good, and f i(x) = Piecewise-Linear Concave

Segment k with Slope , and range = b –

a.

: ij

f R R ( )if xj j j

ijku i

jkl

ijf

xj

ijku

a b

Optimal Bundle for Agent i Utility per unit of money: Bang-per-buck

Given prices Sort the segments (j, k) in decreasing order of bpb Partition them by equality – q1,…,qd. Start buying from the first till exhaust all the

money

Suppose the last partition he buys, is qk

q1,…,qk-1 are forced, qk is flexible, qk+1,…,qd are undesired.

ijki

jkj

ubpb

p

p

( )ij jjw p

Forced vs. Flexible/Undesired Let be inverse of the bpb of flexible

partition. If (j, k) is forced then:

Let be the supplementary price s.t.

Complementarity Condition:

i

1, and

ijki i i

jk jk jk ji j

ubpb q l p

p

0ijk 1

ijk

ii j jk

u

p

0 and 0

( ) 0

i i i i i ijk jk jk j jk jk j jk

i i ijk jk jk j

q l p q l p

q l p

Undesired vs. Flexible/Forced If (j, k) is undesired then:

Complementarity Condition:

1, and 0. Therefore 0.

ijki i i

jk jk jki j

ubpb q

p

1 10, and 0

i ijk jki i

jk jki ij jk i j jk i

u uq q

p p

1

0

( ) 0

ijki

jk ij jk i

i i ijk jk i j jk

uq

p

q u p

LCP Formulation

, ,

, ,

0

0, 0, ( ) 0

0, 0, ( ) 0

, 0, 0

0, 0, 0

i ijk j j j jk j

i k i k

i i i i i ijk i j jk jk

i ijk ij j i i

jk jk i j jki i i i i ijk jk j jk jk jk jk

jk ij jj k j j k

j

j

ijk u p q

j q p p p q p

i q w p z q w

q u p

ijk q l p q

z

p

p

l

LCP and Market Equilibria Captures all the market equilibria.

To capture only market equilibria, We need to be zero whenever is zero:

Homogeneous LCP (q=0) Feasible set is a polyhedral cone. Origin is the dummy solution, and the only vertex.

( , , ), 0ijk ji j k p

ijk

jp

Recall: Starting VertexP’: My + v – z = q = 0; v, y, z ≥ 0

yTv=0 Primary Ray:

y=0, z and v changes accordingly.

Vertex (v*, y*, z*): y* = 0; i* = argmini qi; z* = |qi*| = 0; vi* = qi + z* = 0;

The origin

z=z*

y =

0z=∞

v > 0

vi*=0

Non-Homogeneous LCP If u is a solution then so is αu, α ≥ 0. Impose p ≥ 1.

p1

p2

0 p1

p1=1

p2=1

p2

0

Non-Homogeneous LCP

Starting vertex: and the rest are zero. End point of the primary ray.

arg max iji j

z w

,

,

1, , 0, 0

,

1,

, 0, 0

, 0, 0

, , 0, 0

ijk

i i i i i i ijk i j jk jk jk jk jk jk

i i i i i i i ijk jk j jk jk jk jk jk j

j j j j j ji k

ijk ij j i ij i i i i

j k j

j

k

jik

j q p t p

ijk u p r q r q r

ijk q l

t p t

i q w p z

p

s w s s

ijk

a l a a

1, 0i ij jk jkp b b

Non-Homogeneous LCP

Let y and v = [s, t, r, a] then in short

My + - zd = q; y, v, z ≥ 0; b ≥ 0

yTv = 0

,

,

1, , 0, 0

,

1,

, 0, 0

, 0, 0

, , 0, 0

ijk

i i i i i i ijk i j jk jk jk jk jk jk

i i i i i i i ijk jk j jk jk jk jk jk j

j j j j j ji k

ijk ij j i ij i i i i

j k j

j

k

jik

j q p t p

ijk u p r q r q r

ijk q l

t p t

i q w p z

p

s w s s

ijk

a l a a

1, 0i ij jk jkp b b

[ , , , ]p q v

b

Lemke-Type Algorithm

P’: My + - zd = q; y, v, z ≥ 0; b ≥ 0

yTv = 0

A solution with z=0 maps to an equilibrium.

does not participate in complementarity

condition.

If a becomes tight, then the algorithm

gets stuck.

v

b

0ijkb

0ijkb

Detour – Strong Connectivity

Strong Connectivity (Maxfield’97) G = Graph with agents as nodes. Edges

G is Strongly Connected.

ijf

Strong Connectivity Weakest known condition for the existence of

market equilibrium (Maxfield’97).

Assumed by Vazirani and Yanakkakis for the PPAD proof.

It also implies that the market is not reducible. Reduction is an evidence that equilibrium does not

exist.

Secondary ray => Reduction => Evidence of no market equilibrium.

Back to The Algorithm

Lemke-Type Algorithm

P’: My + - zd = q; y, v, z ≥ 0; b ≥ 0

yTv = 0

does not participate in complementarity

condition.

If a becomes tight, then the algorithm

gets stuck.

This is expected otherwise NP = Co-NP

Since checking existence is NP-hard in general

(VY).

v

b

0ijkb

0ijkb

Lemke-type Algorithm

P’: My + - zd = q; y, v, z ≥ 0; b ≥ 0

yTv = 0Assumption: Market satisfies Strong

Connectivity

and accordingly

v

b

0ijk jp 0 i

jk jp

1i ijk j jkp b i i

jk j jkp b

CorrectnessAssumption: Market satisfies Strong Connectivity

If ∆ is sufficiently large (polynomial sized), then never becomes tight.

Secondary rays are non-existent Since a secondary ray => equilibrium does not

exist.

Algorithm terminates with a market equilibrium.

0ijkb

Consequences Obtained a path following algorithm.

Runs very fast in practice.

Proves the membership of SPLC case in PPAD using Todd’s result on orientating complementary pivot

path

Start the algorithm from an equilibrium by leaving z=0, it reaches another equilibrium Since secondary rays are non-existent. Pairs up equilibria => The number of equilibria is

odd.

Combinatorial Interpretation Prices are initialized to 1. Goods with price more than 1 are fully sold.

Only agents with maximum surplus are in the market z captures the maximum surplus.

Allocation configuration does not repeat. Strongly polynomial bound when number of

agents or goods are constant.

The Linear Case Eaves (1975) – “That the algorithm can be

interpreted as a `global market adjustment mechanism' might be interesting to explore.”

The maximum surplus monotonically decreases, and prices monotonically increase. Market mechanism interpretation

Unique equilibrium if the input is non-degenerate.

In general, equilibria form a polyhedral cone.

Experimental Results Inputs are drawn uniformly at random.

from [0, 1], from [0, 1/#seg], and from [0, 1]

|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

ijku i

jkl ijw

What Next? SPLC case:

Analyze how the obtained equilibrium different. Combinatorial algorithm. Explore structural properties like index, degree,

stability similar to 2-Nash. Extension to markets with production.

Rational convex program for the linear case.

Thank You

Properties of LCP

yTv = 0 => yivi = 0, for all i.

At a solution, yi=0 or vi=0, for all i => n inequalities tight.

P is non-degenerate => every solution is a vertex of P. Since P is an n–dimensional polyhedron.

yTv = 0

P: My + v = q; v, y ≥ 0