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Algorithmic Game Theoryand Internet Computing
Amin Saberi
Stanford University
Computation of Competitive Equilibria
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Outline
History
Economic theory and equilibria (existence, dynamics, stability)
An algorithmic approach: computation, polynomial time computability
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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?)
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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?
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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).
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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
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First computational approach!
Fisher (1891): Hydraulic apparatus for calculating equilibrium
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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.”
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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)
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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.
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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)
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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)
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Outline
History
Economic theory and equilibria (existence, dynamics, stability)
An algorithmic approach: computation, polynomial time computability
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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
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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
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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
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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
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Buyers’ optimization program:
Global Constraint:
Market Equilibrium
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The space of feasible allocations is:
How do you aggregate the utility functions U1, U2, … Un ?
Eisenberg-Gale’s convex program
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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
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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
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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
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Eisenberg-Gale’s convex program
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Optimum dual: Equilibrium prices (also unique)
Gives a poly-time algorithm for computing the equilibrium
Eisenberg-Gale’s convex program
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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
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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
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Application: Congestion Control
;3
2;
3
1
Maximize
321
321
xxx
xxx
x1
x2x3
121 xx 131 xx
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Congestion Control
21 pp$
$$
321 Maximize xxx Find the right prices in a Leontief market
p1 = p2 = 3/2
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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
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Exchange Economy
Agents buy and sell at the same time:
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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!!
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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:
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Nash = Leontief
Finding the solution of LCP for H > 0
Leontief: H the rate matrix; agent i owns good ix is at equilibrium if:
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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
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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