Market Equilibrium Ruta Mehta

LCP and Lemke’s Scheme Linear case – Eaves (1975) SPLC case – Garg, M., Sohoni, Vazirani (2012)

Linear Complementarity Problem    

Examples of linear complementarity LP: complementary slackness either a primal inequality is satisfied with equality or corresponding dual variable = 0. KKT conditions for QP 2-Nash: For row player, either Pr[row i] = 0 or row i is a best response.

 

How to Proceed? No Potential Function!

Lemke’s idea

 

 

           

 

Follow the path starting with the primary ray Lemke’s Scheme Follow the path starting with the primary ray  

Follow the path starting with the primary ray Lemke’s Scheme Follow the path starting with the primary ray                 Complementary Pivot

Does not guarantee a solution Lemke’s Scheme Follow the path starting with the primary ray               Does not guarantee a solution

Follow the path starting with the primary ray Lemke’s Scheme Follow the path starting with the primary ray No Secondary rays                  

Paths pair-up rest of the solutions No secondary ray   Paths pair-up rest of the solutions      

Back to markets

Arrow-Debreu Model  

Linear utility function amount of good j utility   utility/unit of j

Equilibrium    

Linear utility function amount of good j   utility utility/unit of j  

   

  Market Clears!

Optimal bundles  

Optimal bundles, guaranteed by:

Optimal bundles, guaranteed by:

Optimal bundles, via complementarity

LCP (Eaves, 1975)   All zeros is a solution!

Recall: Equilibrium prices can be scaled. Recourse Recall: Equilibrium prices can be scaled.

Resulting LCP   Theorem: Resulting LCP captures exactly the set of market equilibria.

   

No Secondary Rays Proof on board

Separable Piecewise-Linear Concave Utilities (SPLC)  

Segments of SPLC utility function amount of j utility/unit of j Non-satiated utility

Segments of SPLC utility function amount of j utility/unit of j Satiated utility

In general, equilibrium may not exist. Vazirani & Yannakakis: Deciding this is NP-hard.

A weak sufficient condition Consider graph G on A, with Maxfield, 1997: If G is strongly connected, then the market has an equilibrium.

Assuming Strong Connectivity Chen et al. (2009): PPAD-hard VY (2009): In PPAD, Rationality GMSV (2012): LCP, No secondary rays Computation, existence, oddness, containment in PPAD

Segments of SPLC utility function amount of j utility/unit of j utility

Bang-per-buck of segments w.r.t. p

Optimal bundle for i w.r.t. prices p Sort all his segments by decreasing bpb. Partition by equality: Start allocating until money runs out.

Forced, flexible and undesirable partitions Flexible: last allocated partition Forced: all partitions before flexible Undesirable: all partitions after flexible

Forced, flexible and undesirable partitions Forced: all segments fully allocated Flexible: remaining money spent on any segments Undesirable: no segments allocated

LCP captures all market equilibria!

and more …

and more … Pick an equilibrium, set variables.

and more … prices = 0 Set accordingly. Pick an equilibrium, set variables. prices = 0 Set accordingly.

Recall: Equilibrium prices can be scaled. Recourse Recall: Equilibrium prices can be scaled.

Recall: Equilibrium prices can be scaled. Recourse Recall: Equilibrium prices can be scaled. Theorem: Resulting LCP captures exactly the set of market equilibria.

LCP for SPLC utilities

Recourse Will add z variable to apply Lemke’s scheme. Theorem: Resulting LCP captures exactly the set of market equilibria. Will add z variable to apply Lemke’s scheme.

Theorem: Assuming strong connectivity, no secondary rays exist. Using all complementarity conditions! Corollary 1: The path starting with the primary ray must end with z = 0, i.e., an equilibrium. Alternative proof of existence of equilibrium, assuming strong connectivity!

Corollary 2: Alternative proof of containment in PPAD. Using Todd, 1976. Corollary 3: The number of equilibria is odd, up to scaling. (Because rest of equilibria appear at ends of paths.)

Experimental Results Inputs are drawn uniformly at random. |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 5 130 154.3 197 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 1093 1143.8 1233

log(total no. of segments) log(no. of iterations) log(total no. of segments)

Combinatorial interpretation Eaves, 1976 Journal paper: That the algorithm can be interpreted as a ‘global market adjustment mechanism’ might be interesting to explore. Linear case: z = maximum surplus of an agent, Decreases monotonically. Hence: Equilibrium prices unique if non-degenerate, o.w., form a convex set. Open: For SPLC case.

Many questions on SPLC case Explore structural properties, e.g., index, degree, stability -- similar to 2-Nash Smoothed analysis Example that takes exponential time Find rest of equilibria Combinatorial interpretation and how algorithm overcomes hurdle

Can 2-Nash be solved using Lemke?