Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
Computation of Competitive Equilibria

• Amin Saberi
• Stanford University

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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 1870s 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 Debreus proof)

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End of the program?

• Scarfs 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
  algorithmsScarf, 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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Last 10 years

• New applications Internet, Sponsored search,
  combinatorial auctions
• Computation as a lense!
• First papers Megiddo 80s, DPS 01prices and ND
  communication complexity
• Lots of new algorithm convex programs
  combinatorial
  algorithms

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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)

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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)

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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

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Market Equilibrium

• Buyers optimization program
• Global Constraint

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Eisenberg-Gales convex program

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

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Eisenberg-Gales 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!

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Eisenberg-Gales 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

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Eisenberg-Gales 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-Gales solution

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Eisenberg-Gales convex program
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Eisenberg-Gales convex program

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

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Eisenberg-Gales 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

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Eisenberg-Gales 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)

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Application Congestion Control
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Congestion Control
Find the right prices in a Leontief market p1
p2 3/2
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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, .

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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

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Nash Leontief
Leontief H the rate matrix agent i owns good i
x is at equilibrium if
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Open Questions

• Exchange economies with -1 lt ? lt -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 gt 0 be the utility matrix. Assume agent i
  owns good i
• x is an equilibrium if