algorithmic game theory and internet computing

Algorithmic Game Theoryand Internet Computing

Amin Saberi

Stanford University

Computation of Competitive Equilibria

Outline History

Economic theory and equilibria (existence, dynamics, stability)

An algorithmic approach: computation, polynomial time computability

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?)

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?

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).

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

First computational approach! Fisher (1891): Hydraulic apparatus for calculating

equilibrium

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.”

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)

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.

Arrow, Debreu and… Arrow-Hurwicz et. al. papers [1977]: Sufficient

conditions for stability of Samuelson-Lange systemGross 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)

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)

Outline History

Economic theory and equilibria (existence, dynamics, stability)

An algorithmic approach: computation, polynomial time computability

New applications: Internet, Sponsored search, combinatorial auctions

Computation as a lense!

First papers: Megiddo 80’s, DPS 01prices and ND communication complexity

Lots of new algorithm: convex programs combinatorial algorithms

Last 10 years

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

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

n buyers, with specified money mi m divisible goods (unit amount) Buyers have CES utility functions:

Find prices such that buyers spend all their money Market clears

Market Equilibrium

Buyers’ optimization program:

Global Constraint:

Market Equilibrium

The space of feasible allocations is:

How do you aggregate the utility functions U1, U2, … Un ?

Eisenberg-Gale’s convex program

The space of feasible allocations is:

How do you aggregate the utility functions U1, U2, … Un ?

First observation: Adding them up is not the answer!

Eisenberg-Gale’s convex program

Buyer i should not gain (or loose) by Doubling all uij s By splitting himself into two buyers with half of the

money

Eisenberg-Gale’s convex program

Buyer i should not gain (or loose) by Doubling all uij s By splitting himself into two buyers with half of the

money

Eisenberg-Gale’s solution:

Eisenberg-Gale’s convex program

Eisenberg-Gale’s convex program

Optimum dual: Equilibrium prices (also unique)

Gives a poly-time algorithm for computing the equilibrium

Eisenberg-Gale’s convex program

Optimum dual: Equilibrium prices (also unique)

Gives a poly-time algorithm for computing the equilibrium

Market is “proportionally” fairfor every other allocation achieving

Eisenberg-Gale’s convex program

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

Application: Congestion Control

;32;

31

Maximize

321

321

xxx

xxx

x1

x2x3

121 xx 131 xx

Congestion Control

21 pp$

$$

321 Maximize xxx Find the right prices in a Leontief market

p1 = p2 = 3/2

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

Exchange EconomyAgents buy and sell at the same time:

Exchange EconomyAgents 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!!

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:

Nash = Leontief

Finding the solution of LCP for H > 0

Leontief: H the rate matrix; agent i owns good ix is at equilibrium if:

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

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