1 / 63

Ruta Mehta

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

hakan
Download Presentation

Ruta Mehta

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Market Equilibrium Ruta Mehta

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

  3. Linear Complementarity Problem

  4. 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.

  5. How to Proceed? No Potential Function!

  6. Lemke’s idea

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

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

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

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

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

  12. Back to markets

  13. Arrow-Debreu Model

  14. amount of goodj Linear utility function utility utility/unit of j

  15. Equilibrium

  16. amount of goodj Linear utility function utility utility/unit of j

  17. Market Clears!

  18. Optimal bundles

  19. Optimal bundles, guaranteed by:

  20. Optimal bundles, guaranteed by:

  21. Optimal bundles, via complementarity

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

  23. Recourse Recall: Equilibrium prices can be scaled.

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

  25. No Secondary Rays Proof on board

  26. Separable Piecewise-Linear Concave Utilities (SPLC)

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

  28. amount ofj Segments of SPLC utility function utility/unit of j Satiated utility

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

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

  31. 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

  32. amount ofj Segments of SPLC utility function utility/unit of j utility

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

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

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

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

  37. LCP captures all market equilibria!

  38. and more …

More Related