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.

6. How to Proceed?
No Potential Function!

7. Lemke’s idea

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

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

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

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

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

17. Back to markets

18. Arrow-Debreu Model

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

20. Equilibrium

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

23. Market Clears!

24. Optimal bundles

25. Optimal bundles, guaranteed by:

26. Optimal bundles, guaranteed by:

27. Optimal bundles, via complementarity

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

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

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

32. No Secondary Rays
Proof on board

33. Separable Piecewise-Linear Concave Utilities (SPLC)

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

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

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

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

38. 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

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

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

43. 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.

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

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

49. LCP captures all market equilibria!

50. and more …

51. and more …
Pick an equilibrium,
set variables.

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

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

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

55. LCP for SPLC utilities

56. 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.

57. 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!

58. 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.)

59. 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

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

61. 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.

62. 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

63. Can 2-Nash be solved using Lemke?