Algorithmic Game Theory and Internet Computing

1. Market Equilibrium and Pricing of Goods Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

2. Adam Smith • The Wealth of Nations, 1776. “It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest.” Each participant in a competitive economy is “led by an invisible hand to promote an end which was no part of his intention.”

3. What is Economics? ‘‘Economics is the study of the use of scarce resources which have alternative uses.’’ Lionel Robbins (1898 – 1984)

4. How are scarce resources assigned to alternative uses?

6. How are scarce resources assigned to alternative uses? Prices!

8. How are scarce resources assigned to alternative uses? Prices Parity between demand and supply

9. How are scarce resources assigned to alternative uses? Prices Parity between demand and supplyequilibrium prices

10. Leon Walras, 1874 • Pioneered general equilibrium theory

11. General Equilibrium TheoryOccupied center stage in MathematicalEconomics for over a century Mathematical ratification!

12. Central tenet • Markets should operate at equilibrium

13. Central tenet • Markets should operate at equilibrium i.e., prices s.t. Parity between supply and demand

14. Do markets even admitequilibrium prices?

15. Easy if only one good! Do markets even admitequilibrium prices?

16. Supply-demand curves

17. What if there are multiple goods and multiple buyers with diverse desires and different buying power? Do markets even admitequilibrium prices?

18. Irving Fisher, 1891 • Defined a fundamental market model • Special case of Walras’ model

21. amount ofj Concave utility function (Of buyer i for good j) utility

22. total utility

23. For given prices,find optimal bundle of goods

24. Several buyers with different utility functions and moneys.

25. Several buyers with different utility functions and moneys.Equilibrium prices

26. Several buyers with different utility functions and moneys.Show equilibrium prices exist.

27. 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.

28. First Welfare Theorem • Competitive equilibrium => Pareto optimal allocation of resources • Pareto optimal = impossible to make an agent better off without making some other agent worse off

29. Second Welfare Theorem • Every Pareto optimal allocation of resources comes from a competitive equilibrium (after redistribution of initial endowments).

30. Kenneth Arrow • Nobel Prize, 1972

31. Gerard Debreu • Nobel Prize, 1983

32. Arrow-Debreu Model Agents: buyers/sellers

33. Initial endowment of goods Agents Goods

34. Prices = $25 = $15 = $10 Agents Goods

35. Incomes Agents $50 $60 Goods Prices =$25 =$15 =$10 $40 $40

36. Maximizeutility Agents $50 $60 Goods Prices =$25 =$15 =$10 $40 $40

37. Find prices s.t. market clears Agents $50 $60 Goods Prices =$25 =$15 =$10 $40 Maximize utility $40

38. Arrow-Debreu Model • n agents, k goods

39. Arrow-Debreu Model • n agents, k goods • Each agent has: initial endowment of goods, & a utility function

40. Arrow-Debreu Model • n agents, k goods • Each agent has: initial endowment of goods, & a utility function • Find market clearing prices, i.e., prices s.t. if • Each agent sells all her goods • Buys optimal bundle using this money • No surplus or deficiency of any good

41. Utility function of agent i • Continuous, quasi-concave and satisfying non-satiation. • Given prices and money m, there is a unique utility maximizing bundle.

42. Proof of Arrow-Debreu Theorem • Uses Kakutani’s Fixed Point Theorem. • Deep theorem in topology

43. Proof • Uses Kakutani’s Fixed Point Theorem. • Deep theorem in topology • Will illustrate main idea via Brouwer’s Fixed Point Theorem (buggy proof!!)

44. Brouwer’s Fixed Point Theorem • Let be a non-empty, compact, convex set • Continuous function • Then

45. Brouwer’s Fixed Point Theorem

46. Brouwer’s Fixed Point Theorem

47. Observe: If p is market clearing prices, then so is any scaling of p • Assume w.l.o.g. that sum of prices of k goods is 1. • k-1 dimensional unit simplex

48. Idea of proof • Will define continuous function • If p is not market clearing, f(p) tries to ‘correct’ this. • Therefore fixed points of f must be equilibrium prices.

49. When is p an equilibrium price? • s(j): total supply of good j. • B(i): unique optimal bundle which agent i wants to buy after selling her initial endowment at prices p. • d(j): total demand of good j.

50. When is p an equilibrium price? • s(j): total supply of good j. • B(i): unique optimal bundle which agent i wants to buy after selling her initial endowment at prices p. • d(j): total demand of good j. • For each good j: s(j) = d(j).