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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

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

- 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?)

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

- 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

- Fisher (1891): Hydraulic apparatus for calculating equilibrium

- 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 [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 [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-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)

- 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)

- 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

- 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)

- 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)

- 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

- Buyers’ optimization program:
- Global Constraint:

- The space of feasible allocations is:
- How do you aggregate the utility functions U1, U2, … Un ?

- 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!

Buyer i should not gain (or loose) by

- Doubling all uij s
- By splitting himself into two buyers with half of the money

Buyer i should not gain (or loose) by

- Doubling all uij s
- By splitting himself into two buyers with half of the money

- Optimum dual: Equilibrium prices (also unique)
- Gives a poly-time algorithm for computing the equilibrium

- Optimum dual: Equilibrium prices (also unique)
- Gives a poly-time algorithm for computing the equilibrium
- Market is “proportionally” fairfor every other allocation achieving

- 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)

x1

x2

x3

Application: Congestion Control

$

$

$

Congestion Control

Find the right prices in a Leontief market

p1 = p2 = 3/2

Congestion Control

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

Kelly, Low, Doyle, Tan, ….

Agents buy and sell at the same time:

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:

Use LCP as an intermediate step:

Finding the solution of LCP for H > 0

Leontief: H the rate matrix; agent i owns good i

x is at equilibrium if:

- 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

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