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

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

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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**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.**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. Utility Amount of good j**Separable Piecewise-Linear Concave (SPLC)**• 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)**• 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**• 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 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**• 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**• 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. 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. • 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**• 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**• 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 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 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 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**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**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**• Start by tracing the primary ray up to (v*, y*, z*). z=∞ v > 0, y = 0 z=z* vi*=0**The Algorithm**• 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**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**• 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**• 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**• 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**• 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 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**• 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 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 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**• If u is a solution then so is αu, α≥ 0. • Impose p ≥ 1. p2=1 p2 p2 p1=1 0 0 p1 p1**Non-Homogeneous LCP**• Starting vertex:and the rest are zero. • End point of the primary ray.**Non-Homogeneous LCP**• Let yand v= [s, t, r,a] then in short My + - zd= q; y, v, z ≥ 0;b ≥ 0 yTv = 0**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.**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.