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 Market Several 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.

Parity between demand and supply equilibrium prices

Do equilibrium prices exist?

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

Properties of LCP  y T v = 0 => y i v i = 0, for all i.  At a solution, y i =0 or v i =0, for all i.  Trivial if q ≥ 0: Set y = 0, and v = q. P : My + v = q; v, y ≥ 0 y T v = 0

Properties of LCP  yTv = 0 => y i v i = 0, for all i.  At a solution, y i =0 or v i =0, for all i.  There may not exist a solution. y T v = 0 P : My + v = q; v, y ≥ 0

Properties of LCP  yTv = 0 => y i v i = 0, for all i.  At a solution, y i =0 or v i =0, for all i.  If there exists a solution, then there is a vertex of P which is a solution. y T v = 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. y T v = 0 P : My + v = q; v, y ≥ 0

Lemke’s Algorithm  Add a dimension: P’ : My + v – z = q; v, y, z ≥ 0 y T v=0  T = Points in P’ with y T v=0.  Required: A point of T with z=0 Assumption: P’ is non-degenerate.

The set T P’ : My + v – z = q; v, y, z ≥ 0 y T v=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 T P’ : My + v – z = q; v, y, z ≥ 0 y T v=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. y i =0 and v i =0. Relaxing each gives two adjacent edges of S.

The set T P’ : My + v – z = q; v, y, z ≥ 0 y T v=0 Assumption: P’ is non-degenerate. Paths and cycles on 1 -skeleton of P’. z=0

Lemke’s Algorithm P’ : My + v – z = q; v, y, z ≥ 0 y T v=0 Assumption: P’ is non-degenerate.  Invariant: Remain in T.  Start from the primary ray.

Starting Vertex P’ : My + v – z = q; v, y, z ≥ 0 y T v=0  Primary Ray:  y=0, z and v change accordingly.  Vertex (v*, y*, z*): y* = 0; i* = argmin i q i ; z* = |q i* |; v i * = q i + z*; z=z* y = 0 z=∞ v > 0 v i* =0

The Algorithm  Start by tracing the primary ray up to (v*, y*, z*). z=z* v i* =0 v > 0, y = 0 z=∞

The Algorithm  Start by tracing the primary ray up to (v*, y*, z*).  Then relax y i* = 0, v i* =0 y i* =0 v i* >0 v i* =0 y i* >0

The Algorithm In general  If v i ≥ 0 becomes tight, then relax y i = 0,  And if y i ≥ 0 becomes tight then relax v i = 0. z=0 v i =0 y i =0 v i =0 y i >0 v i >0 y i =0 v i* =0 y i* =0 v i* >0 v i* =0 y i* >0

The Algorithm  Start by tracing the primary ray up to (v*, y*, z*).  If v i ≥ 0 becomes tight, then relax y i =0  And if y i ≥ 0 becomes tight then relax v i =0. v i =0 y i =0 v i =0 y i >0 v i >0 y i =0 v i* =0 y i* =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  w ij endowment of good j  utility function

Separable Piecewise-Linear Concave (SPLC) Utilities  Utility function f i is:  Separable – is for j th good, and f i ( x ) =  Piecewise-Linear Concave Segment k with Slope, and range = b – a. ab

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 – q 1,…,q d.  Start buying from the first till exhaust all the money  Suppose the last partition he buys, is q k  q 1,…,q k-1 are forced, q k is flexible, q k+1,…,q d are undesired.

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:

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

LCP Formulation

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.

Recall: Starting Vertex P’ : My + v – z = q = 0; v, y, z ≥ 0 y T v=0  Primary Ray:  y=0, z and v changes accordingly.  Vertex (v*, y*, z*): y* = 0; i* = argmin i q i ; z* = |q i* | = 0; v i * = q i + z* = 0; The origin z=z* y = 0 z=∞ v > 0 v i* =0

Non-Homogeneous LCP  If u is a solution then so is αu, α ≥ 0.  Impose p ≥ 1. p1p1 p2p2 0 p1p1 p 1 =1 p 2 =1 p2p2 0

Non-Homogeneous LCP  Starting vertex: and the rest are zero.  End point of the primary ray.

Non-Homogeneous LCP  Let y and v = [s, t, r, a] then in short My + - zd = q; y, v, z ≥ 0; b ≥ 0 y T v = 0

Lemke-Type Algorithm P’ : My + - zd = q; y, v, z ≥ 0; b ≥ 0 y T v = 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

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

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 y T v = 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 P’ : My + - zd = q; y, v, z ≥ 0; b ≥ 0 y T v = 0 Assumption: Market satisfies Strong Connectivity  and accordingly

Correctness 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  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#InstancesMin ItersAvg ItersMax Iters 10 x 5 x x 5 x x 10 x x 10 x x 15 x x 15 x x 20 x x 20 x

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 => y i v i = 0, for all i.  At a solution, y i =0 or v i =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. y T v = 0 P : My + v = q; v, y ≥ 0