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Algorithmic Game Theory and Internet Computing

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### Algorithmic Game Theoryand Internet Computing

Computation of Competitive Equilibria

Amin Saberi

Stanford University

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

Last 10 years

- 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

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
- 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)

Market Equilibrium

- 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:

Eisenberg-Gale’s convex program

- 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 Eisenberg-Gale’s solution:

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

- 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)

- Primal-dual scheme primal: packet rates at sources dual: congestion measures (shadow prices)
A market equilibrium in a distributed setting!

Kelly, Low, Doyle, Tan, ….

Exchange Economy

Agents buy and sell at the same time:

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!!

Nash equilibria for a symmetric game H

Finding the solution of LCP for H > 0

x is equilibrium if:

Nash = LeontiefUse LCP as an intermediate step:

Finding the solution of LCP for H > 0

Nash = LeontiefLeontief: H the rate matrix; agent i owns good i

x 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

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