Algorithmic Game Theory and Internet Computing

1. Dichotomies in Equilibrium Computation: Market Provide a Surprise Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Joint work with Jugal Garg & Ruta Mehta

2. Equilibrium computation • Has its own character, quite distinct from computability of optimization problems

3. Dichotomies Natural equilibrium computation problems exhibit striking

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6. Arrow-Debreu Model • n agents and g divisible goods. • Agent i: has initial endowment of goods and a concave utility function (models satiation!)

7. Arrow-Debreu Model • n agents and g divisible goods. • Agent i: has initial endowment of goods and a concave utility function (piecewise-linear, concave)

8. Agent i comes with an initial endowment

9. At given prices, agent isells initial endowment

10. … and buys optimal bundleof goods, i.e,

11. Several agents with own endowments and utility functions.Currently, no goods in the market.

12. Agents sell endowments at current prices.

13. Each agent wants an optimal bundle.

14. Equilibrium Prices p s.t. market clears, i.e., there is no deficiency or surplus of any good.

15. 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.

16. Kenneth Arrow • Nobel Prize, 1972

17. Gerard Debreu • Nobel Prize, 1983

18. 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!

19. Computability of equilibria Walras, 1874: Tanonnement process Scarf, Debreu-Mantel-Sonnenschein, 1960s: Won’t converge! Scarf, Smale, … , 1970s: Many approaches

20. Open problem (2002) Separable, piecewise-linear, concave (SPLC) utility functions

21. Separable utility function

22. : piecewise-linear, concave utility amount ofj

23. : piecewise-linear, concave utility SPLC utility: Additively separable over goods amount ofj

24. Markets with separable,piecewise-linear, concave utilities • Chen, Dai, Du, Teng, 2009: • PPAD-hardness for Arrow-Debreu model • Chen & Teng, 2009: • PPAD-hardness for Fisher’s model • V. & Yannakakis, 2009: • PPAD-hardness for Fisher’s model

25. Markets with separable,piecewise-linear, concave utilities V. & Yannakakis, 2009: • Membership in PPAD for both models.

26. Markets with separable,piecewise-linear, concave utilities V. & Yannakakis, 2009: • Membership in PPAD for both models. • Indirect proof, using the class FIXP of Etessami & Yannakakis, 2008.

27. Open The definition of PPAD was designed to capture problems that allow for path-following algorithms, in the style of Lemke-Howson and Scarf … It will be interesting to obtain a natural, direct algorithm for this task (hence leading to a more direct proof of membership in PPAD), which may be useful for computing equilibria in practice.

28. Complementary Pivot Algorithms(path-following) Lemke-Howson, 1964: 2-Nash Equilibrium Eaves, 1974: Equilibrium for linear Arrow-Debreu markets (using Lemke’s algorithm)

29. Eaves, 1975 Technical Report: Also under study are extensions of the overall method to include piecewise-linear utilities, production, etc., if successful, this avenue could prove important in real economic modeling. Eaves, 1976 Journal Paper: ... Now suppose each trader has a piecewise-linear, concave utility function. Does there exist a rational equilibrium? Andreu Mas-Colell generated a negative example, using Leontief utilities. Consequently, one can conclude that Lemke’s algorithm cannot be used to solve this class of exchange problems.

30. Leontief utility (complementary goods) • Utility = min{#bread, 2 #butter} • Piecewise-linear, concave: • Equilibrium may be irrational!

31. Leontief utility: is non-separable PLC • Utility = min{#bread, 2 #butter} • Only bread or only butter gives 0 utility

32. Rationality for SPLC Utilities • Devanur & Kannan, 2007, V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under SPLC utilities. • Using flow-based structure from DPSV.

33. Theorem (Garg, Mehta, Sohoni & V., 2012): • Complementary pivot (path-following) algorithm for Arrow-Debreu markets under SPLC utilities. • Membership in PPAD, using Todd, 1976. • Algorithm is simple, no stability issues, & is practical.

34. Experimental Results Inputs are drawn uniformly at random.

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

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37. Eaves, 1975 Technical Report: Also under study are extensions of the overall method to include piecewise-linear utilities, production, etc., if successful, this avenue could prove important in real economic modeling. • Main general class of markets still unresolved.

38. Eaves, 1975 Technical Report: Also under study are extensions of the overall method to include piecewise-linear utilities, production, etc., if successful, this avenue could prove important in real economic modeling. • Main general class of markets still unresolved. • So far, no works exploring rationality …

39. Garg & V., 2013: • SPLC Production: 1 finished good from 1 raw good, e.g., bread from wheat or corn. Total is additive. • LCP and complementary pivot algorithm • PPAD-complete • Rationality • Non-separable PLC Production: From 2 or more raw goods: irrational!

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41. Separable vs non-separable • This dichotomy arises in several places, e.g., concave flows.

43. Leontief-Free Utilitiesand Production!!

45. utility amount of crème brulee

47. utility amount of chocolate cake

48. Utilities of Crème Brulee and Chocolate Cake are not additively separable -- both satiate desire for dessert! • Joint utility should be sub-additive

49. Capture via non-separable utilities! • This non-separability is very different from that of Leontief utilities. • Leontief: goods are complements • Here: goods are substitutes

50. Garg, Mehta & V, 2013: Leontief-free utility functions • Both are subclasses of PLC utilities, with no known generalization to differentiable, concave utilities.