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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

- Celebrated theorem in Mathematical Economics
- Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

- 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!

The Linear Case

- DPSV (2002) – Flow based algorithm for the Fisher market.
- Jain (2004) – Using Ellipsoid method.
- Ye (2004) – Interior point method.

- Utility function of an agent
is separable for goods.

Utility

Amount of good j

- Utility function of an agent
is separable

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

Utility

Amount of good j

- 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

- 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.

“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.”

- 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.

- 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.

- 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.

- 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

- 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

- 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

- 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

- 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

- 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.

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.

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.

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

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.

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*

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

z=∞

v > 0, y = 0

z=z*

vi*=0

- 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

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

- 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

- 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.

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

- 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

- 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.

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

- Complementarity Condition:

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

- 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.

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*

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

p2=1

p2

p2

p1=1

0

0

p1

p1

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

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

yTv = 0

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.

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

- 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.

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).

P’: My + - zd= q; y, v, z ≥ 0;b ≥ 0

yTv = 0

Assumption: Market satisfies Strong Connectivity

- and accordingly

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.

- 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.

- 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.

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

- Market mechanism interpretation

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

- 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

- 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