1. Algorithmic Game Theory and Internet Computing Amin Saberi Stanford University Computation of Competitive Equilibria

2. Outline History Economic theory and equilibria (existence, dynamics, stability) An algorithmic approach: computation, polynomial time computability

3. A bit history Rabbi Samuel ben Meir (12th century, France): 2nd century text: “You shall have inspectors of weights and measures but not inspectors of prices.” Commentary (Aumann): If one seller charges too high a price, then another will undercut him. Adam Smith (1776): Capital flows from low-profit to high- profit industries (demand function implicit?)

4. The beginning of analytical work Standard analysis  demand functions: Cournot (1838)  supply functions: Jenkin (1870)  excess demand: Hicks (1939). Dynamics in 1870’s: Is out-of-equilibrium behavior modeled by demand and supply?

5. Walras, Fisher, Pareto, Hicks Walras [1871, 1874]: first formulator of competitive general equilibrium theory. Recognized need for stability (how to get into equilibrium) His name: tatonnements (gropings).

6. Walras, Fisher, Pareto, Hicks Walras [1871, 1874]: first formulator of competitive general equilibrium theory. Recognized need for stability (how to get into equilibrium) His name: tatonnements (gropings). Fisher (1891): tried to compute the equilibrium prices

7. First computational approach!  Fisher (1891): Hydraulic apparatus for calculating equilibrium

8. Walras, Fisher, Pareto, Hicks Walras [1871, 1874]: tatonnements Pareto (1904): Pointed out that even a simple economy requires a large set of equations to define equilibrium. Argued that market was an effective way to solve large systems of equations, better than an “ordinateur” (his word in the French translation). I believe this is the word now used to translate, “computer.”

9. Walras, Fisher, Pareto, Hicks Walras [1871, 1874]: tatonnements Fisher (1894), Pareto (1904): Markets and computation Hicks (1939): convergence and “Hicksian” condition on the Jacobian of the excess demand functions (the determinants of the minors be positive if of even order and negative if of odd order)

10. Samuelson and successors Samuelson [1944]: Hicksian conditions neither necessary nor sufficient for stability. Metzler [1945]: if off-diagonal elements of Jacobian are non-negative (commodities are gross substitutes), then Hicksian conditions are sufficient. Arrow [1974]: Hicksian conditions were actually equivalent to the statement that the real roots of the Jacobian are negative.

11. Arrow, Debreu and… Arrow-Hurwicz et. al. papers [1977]: Sufficient conditions for stability of Samuelson-Lange system Gross substitution implies that Euclidean norm decreases Will talk about these dynamics in details in the next lecture Arrow-Debreu: existence of equilibrium prices (will show a variation of Debreu’s proof)

12. End of the program? Scarf’s example, Saari-Simon Theorem: For any dynamic system depending on first-order information (z) only, there is a set of excess demand functions for which stability fails. Uzawa: Existence of general equilibrium is equivalent to fixed-point theorem (will show in this lecture) Linear complementarity Programs (LCP) and algorithms: Scarf, Eaves, Cottle…(later in the quarter)

13. Outline History Economic theory and equilibria (existence, dynamics, stability) An algorithmic approach: computation, polynomial time computability

14.  New applications: Internet, Sponsored search, combinatorial auctions  Computation as a lense!  First papers: Megiddo 80’s, DPS 01 prices and ND communication complexity  Lots of new algorithm: convex programs combinatorial algorithms Last 10 years

15.  n buyers, with specified money  m divisible goods (unit amount)  Buyers have CES utility functions: Contains several interesting special cases:  = 1 linear  = 0 Cobb-Douglas  = - 1 Leontief (rate allocation in a network) A CES Market

16.  n buyers, with specified money  m divisible goods (unit amount)  Buyers have CES utility functions: Contains several interesting special cases:  = 1 linear  = 0 Cobb-Douglas  = - 1 Leontief (rate allocation in a network) A CES Market

17.  n buyers, with specified money m i  m divisible goods (unit amount)  Buyers have CES utility functions: Find prices such that  buyers spend all their money  Market clears Market Equilibrium

18.  Buyers’ optimization program:  Global Constraint: Market Equilibrium

19.  The space of feasible allocations is:  How do you aggregate the utility functions U 1, U 2, … U n ? Eisenberg-Gale’s convex program

20.  The space of feasible allocations is:  How do you aggregate the utility functions U 1, U 2, … U n ? First observation: Adding them up is not the answer! Eisenberg-Gale’s convex program

21. Buyer i should not gain (or loose) by  Doubling all u ij s  By splitting himself into two buyers with half of the money Eisenberg-Gale’s convex program

22. Buyer i should not gain (or loose) by  Doubling all u ij s  By splitting himself into two buyers with half of the money  Eisenberg-Gale’s solution: Eisenberg-Gale’s convex program

24.  Optimum dual: Equilibrium prices (also unique)  Gives a poly-time algorithm for computing the equilibrium Eisenberg-Gale’s convex program

25.  Optimum dual: Equilibrium prices (also unique)  Gives a poly-time algorithm for computing the equilibrium  Market is “proportionally” fair for every other allocation achieving Eisenberg-Gale’s convex program

26.  Optimum dual: Equilibrium prices (also unique)  Gives a poly-time algorithm for computing the equilibrium  The program works for all homogenous utility functions, generalized to homothetic KVY 03 (homothetic: U(f(y)) U is homogeneous of degree one and f is a monotone) Eisenberg-Gale’s convex program

27. Application: Congestion Control x1x1 x2x2 x3x3

28. Congestion Control $ $ $ Find the right prices in a Leontief market p 1 = p 2 = 3/2

29.  Primal-dual scheme primal: packet rates at sources dual: congestion measures (shadow prices) A market equilibrium in a distributed setting! Kelly, Low, Doyle, Tan, …. Congestion Control

30. Exchange Economy Agents buy and sell at the same time:

31. Exchange Economy Agents buy and sell at the same time: - 1 -1 0 1 At least as hard as solving Nash Equilibria (CVSY 05) Polynomial-time algorithms known (DPSV 02, J 03, CMK 03, GKV 04,... OPEN!! 

32. Nash = Leontief Use LCP as an intermediate step: Finding the solution of LCP for H > 0 Nash equilibria for a symmetric game H x is equilibrium if:

33. Nash = Leontief Finding the solution of LCP for H > 0 Leontief: H the rate matrix; agent i owns good i x is at equilibrium if:

34. Open Questions Exchange economies with - 1 <  < -1 Markets with indivisible goods  Price equilibria; proportional fair allocation Core of a Game:  LP-based algorithm for transferable payoff  Still open for NTU games

35. Nash = Leontief In Leontief markets, agents consume goods in fixed proportions: Let H > 0 be the utility matrix. Assume agent i owns good i x is an equilibrium if