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A Lemke-Type Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave UtilitiesPowerPoint Presentation

A Lemke-Type Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities

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### A Lemke-Type Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities

Ruta Mehta

Indian Institute of Technology – Bombay

Joint work with JugalGarg, MilindSohoniand Vijay V. Vazirani

Exchange Market Separable, Piecewise-Linear Concave UtilitiesSeveral agents

Several agents Separable, Piecewise-Linear Concave Utilitieswith endowment of goods

Several agents with endowments of goods Separable, Piecewise-Linear Concave Utilitiesand different concave utility functions

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

Parity between demand and supply optimal bundle from the earned money.equilibrium prices

Do equilibrium prices exist? optimal bundle from the earned money.

Arrow-Debreu Theorem, 1954 optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

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) optimal bundle from the earned money.

- Utility function of an agent
is separable for goods.

Utility

Amount of good j

Separable Piecewise-Linear Concave (SPLC) optimal bundle from the earned money.

- Utility function of an agent
is separable

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

Utility

Amount of good j

Separable Piecewise-Linear Concave (SPLC) optimal bundle from the earned money.

- 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.

Utility

Amount of good j

SPLC – Hardness Results optimal bundle from the earned money.

- 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 optimal bundle from the earned money. 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 optimal bundle from the earned money.

- 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) optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

- Extend Eave’s LCP formulation to SPLC markets.
- Design a Lemke-type algorithm.
- Runs very fast in practice.
- Direct proof of membershipof 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 optimal bundle from the earned money.

- For LP: Complementary slackness conditions capture optimality.
- 2-Nash: Equilibria are characterized through complementarity conditions.
- Given n x n matrix Mand n x 1 vector q, find ys.t.

My ≤ q; y ≥ 0

My + v = q; v, y ≥ 0

yT(q – My) = 0

yTv = 0

Properties of LCP optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

- yTv = 0 => yivi= 0, for all i.
- At a solution, yi=0 or vi=0, for all i.
- There may not exist a solution.

P: My + v = q; v, y ≥ 0

yTv = 0

Properties of LCP optimal bundle from the earned money.

- 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.

P: My + v = q; v, y ≥ 0

yTv = 0

Properties of LCP optimal bundle from the earned money.

- Solution set might be disconnected.
- There is a possibility of a simplex-like algorithm given a feasible vertex of P.

P: My + v = q; v, y ≥ 0

yTv = 0

Lemke’s Algorithm optimal bundle from the earned money.

- Add a dimension:
P’: My + v – z = q; v, y, z ≥ 0

yTv=0

- T=Points in P’ with yTv=0.
- Required: A point of T with z=0
Assumption: P’ is non-degenerate.

The set optimal bundle from the earned money.T

P’: 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 optimal bundle from the earned money.T

P’: 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 ! is.t. yi=0 and vi=0. Relaxing each gives two adjacent edges of S.

The set optimal bundle from the earned money.T

P’: 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 Algorithm optimal bundle from the earned money.

P’: 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 Vertex optimal bundle from the earned money.

P’: My + v – z = q; v, y, z ≥ 0

yTv=0

- Primary Ray:
- y=0, z and vchange accordingly.

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

v > 0

vi*=0

z=∞

y = 0

z=z*

The Algorithm optimal bundle from the earned money.

- Start by tracing the primary ray up to (v*, y*, z*).

z=∞

v > 0, y = 0

z=z*

vi*=0

The Algorithm optimal bundle from the earned money.

- Start by tracing the primary ray up to (v*, y*, z*).
- Then relax yi* = 0,

vi*=0

yi*>0

vi*>0

vi*=0

yi*=0

The Algorithm optimal bundle from the earned money.

In general

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

z=0

yi=0

yi>0

vi>0

vi=0

vi*=0

yi*>0

vi*>0

vi*=0

yi*=0

vi=0

yi=0

The Algorithm optimal bundle from the earned money.

- 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.

yi=0

yi>0

vi>0

vi=0

vi*=0

yi*>0

vi*>0

vi*=0

yi*=0

vi=0

yi=0

Properties and Correctness optimal bundle from the earned money.

- No cycling.
- Termination:
- Either at a vertex with z=0 (the solution), or on an unbounded edge (asecondary ray).

- No need of potential function for termination guarantee.

Exchange Markets optimal bundle from the earned money.

- A: Set of agents, G:Set of goods
- m= |A|, n=|G|.
- Agents i with
- wijendowment of good j
- utility function

Separable Piecewise-Linear Concave (SPLC) Utilities optimal bundle from the earned money.

- Utility function fi is:
- Separable – is for jth good, and fi(x) =
- Piecewise-Linear Concave
Segment k with Slope , and range = b – a.

a

b

Optimal Bundle for Agent optimal bundle from the earned money.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.

Forced vs. Flexible/Undesired optimal bundle from the earned money.

- Let be inverse of the bpb of flexible partition.
- If (j, k) is forced then:
Let be the supplementary price s.t.

- Complementarity Condition:

Undesired vs. Flexible/Forced optimal bundle from the earned money.

- If (j, k) is undesired then:
- Complementarity Condition:

LCP Formulation optimal bundle from the earned money.

LCP and Market optimal bundle from the earned money.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.

Recall: Starting optimal bundle from the earned money.Vertex

P’: 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* = argminiqi; z* = |qi*| = 0; vi* = qi + z* = 0;

The origin

v > 0

vi*=0

z=∞

y = 0

z=z*

Non-Homogeneous LCP optimal bundle from the earned money. Impose p ≥ 1.

- If u is a solution then so is αu, α≥ 0.

p2=1

p2

p2

p1=1

0

0

p1

p1

Non-Homogeneous LCP optimal bundle from the earned money.

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

Non-Homogeneous LCP optimal bundle from the earned money.

- Let yand v= [s, t, r,a] then in short
My + - zd= q; y, v, z ≥ 0;b ≥ 0

yTv = 0

Lemke-Type Algorithm optimal bundle from the earned money.

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.

Detour – Strong Connectivity optimal bundle from the earned money.

Strong Connectivity (Maxfield’97) optimal bundle from the earned money.

- G = Graph with agents as nodes.
- Edges
G is Strongly Connected.

Strong Connectivity optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

Lemke-Type Algorithm optimal bundle from the earned money.

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).

Lemke-type Algorithm optimal bundle from the earned money.

P’: My + - zd= q; y, v, z ≥ 0;b ≥ 0

yTv = 0

Assumption: Market satisfies Strong Connectivity

- and accordingly

Correctness optimal bundle from the earned money.

Assumption: 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.

Consequences optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

- 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 optimal bundle from the earned money. The maximum surplus monotonically decreases, and prices monotonically increase. Unique equilibrium if the input is non-degenerate. In general, equilibria form a polyhedral cone.

- Eaves (1975) – “That the algorithm can be interpreted as a `global market adjustment mechanism' might be interesting to explore.”

- Market mechanism interpretation

Experimental Results optimal bundle from the earned money.

- Inputs are drawn uniformly at random.
- from [0, 1], from [0, 1/#seg], and from [0, 1]

What Next? optimal bundle from the earned money.

- 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 optimal bundle from the earned money.

Properties of LCP optimal bundle from the earned money.

- 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.

P: My + v = q; v, y ≥ 0

yTv = 0

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