Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
Combinatorial Algorithms for Convex Programs
(Capturing Market Equilibria and Nash
Bargaining Solutions)

• Vijay V. Vazirani
• Georgia Tech

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What is Economics?

• Economics is the study of the use of
• scarce resources which have alternative uses.
• Lionel Robbins
• (1898 1984)

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How are scarce resources assigned to alternative
uses?
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How are scarce resources assigned to alternative
uses?
Prices!
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How are scarce resources assigned to alternative
uses?
Prices
Parity between demand and supply
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How are scarce resources assigned to alternative
uses?
Prices
Parity between demand and supplyequilibrium
prices
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Do markets even admitequilibrium prices?
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General Equilibrium TheoryOccupied center stage
in MathematicalEconomics for over a century
Do markets even admitequilibrium prices?
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Arrow-Debreu Theorem, 1954

• Celebrated theorem in Mathematical Economics
• Established existence of market equilibrium under
• very general conditions using a theorem
  from
• topology - Kakutani fixed point theorem.

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Do markets even admitequilibrium prices?
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Easy if only one good!
Do markets even admitequilibrium prices?
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Supply-demand curves
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What if there are multiple goods and multiple
buyers with diverse desires and different buying
power?
Do markets even admitequilibrium prices?
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Irving Fisher, 1891

• Defined a fundamental
• market model

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linear utilities
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For given prices,find optimal bundle of goods
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Several buyers with different utility functions
and moneys.
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Several buyers with different utility functions
and moneys.Find equilibrium prices.
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Stock prices have reached what looks likea
permanently high plateau
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Stock prices have reached what looks likea
permanently high plateau

• Irving Fisher, October 1929

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Arrow-Debreu Theorem, 1954

• Celebrated theorem in Mathematical Economics
• Established existence of market equilibrium under
• very general conditions using a theorem
  from
• topology - Kakutani fixed point theorem.
• Highly non-constructive!

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General Equilibrium Theory
An almost entirely non-algorithmic theory!
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The new face of computing
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Todays reality

• New markets defined by Internet companies, e.g.,
• Google
• eBay
• Yahoo!
• Amazon
• Massive computing power available.
• Need an inherenltly-algorithmic theory of
• markets and market equilibria.

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Combinatorial Algorithm for Linear Case of
Fishers Model

• Devanur, Papadimitriou, Saberi V., 2002
• Using the primal-dual paradigm

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Combinatorial algorithm

• Conducts an efficient search over
• a discrete space.
• E.g., for LP simplex algorithm
• vs
• ellipsoid algorithm or interior point algorithms.

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Combinatorial algorithm

• Conducts an efficient search over
• a discrete space.
• E.g., for LP simplex algorithm
• vs
• ellipsoid algorithm or interior point algorithms.
• Yields deep insights into structure.

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• No LPs known for capturing equilibrium
  allocations for Fishers model

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• No LPs known for capturing equilibrium
  allocations for Fishers model
• Eisenberg-Gale convex program, 1959

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• No LPs known for capturing equilibrium
  allocations for Fishers model
• Eisenberg-Gale convex program, 1959
• Extended primal-dual paradigm to
• solving a nonlinear convex program

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Linear Fisher Market

• B n buyers, money mi for buyer i
• G g goods, w.l.o.g. unit amount of each good
• utility derived by i
• on obtaining one unit of
  j
• Total utility of i,
• Find market clearing prices.

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Eisenberg-Gale Program, 1959
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Eisenberg-Gale Program, 1959
prices pj
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Why remarkable?

• Equilibrium simultaneously optimizes
• for all agents.
• How is this done via a single objective function?

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Theorem

• If all parameters are rational, Eisenberg-Gale
• convex program has a rational
  solution!
• Polynomially many bits in size of instance

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Theorem

• If all parameters are rational, Eisenberg-Gale
• convex program has a rational
  solution!
• Polynomially many bits in size of instance
• Combinatorial polynomial time algorithm
• for finding it.

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Theorem

• If all parameters are rational, Eisenberg-Gale
• convex program has a rational
  solution!
• Polynomially many bits in size of instance
• Combinatorial polynomial time algorithm
• for finding it.
• 
  Discrete space

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Idea of algorithm

• primal variables allocations
• dual variables prices of goods
• iterations
• execute primal dual improvements

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How are scarce resources assigned to alternative
uses?
Prices
Parity between demand and supply
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Yin Yang
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Nash bargaining game, 1950

• Captures the main idea that both players
• gain if they agree on a solution.
• Else, they go back to status quo.
• Complete information game.

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Example

• Two players, 1 and 2, have vacation homes
• 1 in the mountains
• 2 on the beach
• Consider all possible ways of sharing.

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Utilities derived jointly
convex compact
feasible set
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Disagreement point status quo utilities
Disagreement point
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Nash bargaining problem (S, c)
Disagreement point
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Nash bargaining

• Q Which solution is the right one?

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Solution must satisfy 4 axioms

• Paretto optimality
• Invariance under affine transforms
• Symmetry
• Independence of irrelevant alternatives

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Thm Unique solution satisfying 4 axioms
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Generalizes to n-players

• Theorem Unique solution

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Linear Nash Bargaining (LNB)

• Feasible set is a polytope defined by
• linear constraints
• Nash bargaining solution is
• optimal solution to convex program

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Q Compute solution combinatoriallyin
polynomial time?
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Game-theoretic properties of LNB games

• Chakrabarty, Goel, V. , Wang Yu, 2008
• Fairness
• Efficiency (Price of bargaining)
• Monotonicity

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Insights into markets

• V., 2005 spending constraint utilities
• 
  (Adwords market)
• Megiddo V., 2007 continuity properties
• V. Yannakakis, 2009 piecewise-linear,
• 
  concave utilities
• Nisan, 2009 Googles auction for TV ads

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How should they exchange their goods?
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State as a Nash bargaining game
S utility vectors obtained by distributing
goods among players
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Special case linear utility functions
S utility vectors obtained by distributing
goods among players
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Convex program for ADNB
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Theorem (V., 2008)

• If all parameters are rational,
• solution to ADNB is rational!
• Polynomially many bits in size of instance

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Theorem (V., 2008)

• If all parameters are rational,
• solution to ADNB is rational!
• Polynomially many bits in size of instance
• Combinatorial polynomial time algorithm
• for finding it.

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Flexible budget markets

• Natural variant of linear Fisher markets
• ADNB flexible budget
  markets
• Primal-dual algorithm for finding an
• equilibrium

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How is primal-dual paradigm adapted to
nonlinear setting?
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Fundamental difference betweenLPs and convex
programs

• Complementary slackness conditions
• involve primal or dual variables, not
  both.
• KKT conditions involve primal and dual
• variables simultaneously.

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KKT conditions
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KKT conditions
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Primal-dual algorithms so far(i.e., LP-based)

• Raise dual variables greedily. (Lot of effort
  spent
• on designing more sophisticated dual
  processes.)

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Primal-dual algorithms so far

• Raise dual variables greedily. (Lot of effort
  spent
• on designing more sophisticated dual
  processes.)
• Only exception Edmonds, 1965 algorithm
• for max weight
  matching.

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Primal-dual algorithms so far

• Raise dual variables greedily. (Lot of effort
  spent
• on designing more sophisticated dual
  processes.)
• Only exception Edmonds, 1965 algorithm
• for max weight
  matching.
• Otherwise primal objects go tight and loose.
• Difficult to account for these reversals --
• in the running time.

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Our algorithm

• Dual variables (prices) are raised greedily
• Yet, primal objects go tight and loose
• Because of enhanced KKT conditions

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Our algorithm

• Dual variables (prices) are raised greedily
• Yet, primal objects go tight and loose
• Because of enhanced KKT conditions
• New algorithmic ideas needed!

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Nonlinear programs with rational solutions!
Open
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Nonlinear programs with rational
solutions!Solvable combinatorially!!
Open
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Primal-Dual Paradigm

• Combinatorial Optimization (1960s 70s)
• Integral optimal solutions to LPs

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Exact Algorithms for Cornerstone
Problems in P

• Matching (general graph)
• Network flow
• Shortest paths
• Minimum spanning tree
• Minimum branching

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Primal-Dual Paradigm

• Combinatorial Optimization (1960s 70s)
• Integral optimal solutions to LPs
• Approximation Algorithms (1990s)
• Near-optimal integral solutions to LPs

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Primal-Dual Paradigm

• Combinatorial Optimization (1960s 70s)
• Integral optimal solutions to LPs
• Approximation Algorithms (1990s)
• Near-optimal integral solutions to LPs

WGMV 1992
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Approximation Algorithms

• set cover facility
  location
• Steiner tree k-median
• Steiner network multicut
• k-MST feedback
  vertex set
• scheduling . . .

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Primal-Dual Paradigm

• Combinatorial Optimization (1960s 70s)
• Integral optimal solutions to LPs
• Approximation Algorithms (1990s)
• Near-optimal integral solutions to LPs
• Algorithmic Game Theory (New Millennium)
• Rational solutions to nonlinear convex
  programs

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Primal-Dual Paradigm

• Combinatorial Optimization (1960s 70s)
• Integral optimal solutions to LPs
• Approximation Algorithms (1990s)
• Near-optimal integral solutions to LPs
• Algorithmic Game Theory (New Millennium)
• Rational solutions to nonlinear convex
  programs
• Approximation algorithms for convex programs?!

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• Goel V., 2009
• ADNB with piecewise-linear, concave utilities

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Convex program for ADNB
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Eisenberg-Gale Program, 1959
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Common generalization
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Common generalization

• Is it meaningful?
• Can it be solved via a combinatorial,
• polynomial time algorithm?

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Common generalization

• Is it meaningful? Nonsymmetric ADNB
• Kalai, 1975 Nonsymmetric bargaining games
• wi clout of player i.

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Common generalization

• Is it meaningful? Nonsymmetric ADNB
• Kalai, 1975 Nonsymmetric bargaining games
• wi clout of player i.
• Algorithm

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Open

• Can Fishers linear case
• or ADNB
• be captured via an LP?