Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
Nash Bargaining via Flexible Budget Markets

• Vijay V. Vazirani
• Georgia Tech

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The new platform for computing
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Internet

• Massive computational power available
• Sellers (programs) can negotiate with
• individual buyers!

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Internet

• Massive computational power available
• Sellers (programs) can negotiate with
• individual buyers!
• Back to bargaining!

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Bargaining and Game Theory

• Nash (1950) First formalization of bargaining.
• von Neumann Morgenstern (1947)
• Theory of Games and Economic Behavior
• Game Theory Studies solution concepts for
• negotiating in situations of conflict of
  interest.

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Bargaining and Game Theory

• Nash (1950) First formalization of bargaining.
• von Neumann Morgenstern (1947)
• Theory of Games and Economic Behavior
• Game Theory Studies solution concepts for
• negotiating in situations of conflict of
  interest.
• Theory of Bargaining Central!

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Nash bargaining

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

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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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Generalizes to n-players

• Theorem Unique solution

(S, c) is feasible if S contains a point that
makes each i
strictly happier than ci
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Bargaining theory studies only instances (S, c)
which are feasible
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Bargaining theory studies only instances (S, c)
which are feasible

• We will need to explicitly test for feasibility

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Linear Nash Bargaining (LNB)

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

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Q Compute solution combinatoriallyin
polynomial time?
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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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ADNB

• B n players with disagreement points, ci
• G g goods, unit amount each
• S utility vectors obtained by distributing
• goods among players

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Convex program for ADNB
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Theorem

• If instance is feasible,
• Nash bargaining solution is rational!
• Polynomially many bits in size of instance

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Theorem

• If instance is feasible,
• Nash bargaining solution is rational!
• Polynomially many bits in size of instance
• Decision and search problems
• can be solved in polynomial time.

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Other NB games
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Given disagreement point, find NB soln.
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• Theorem
• Strongly polynomial, combinatorial algorithm
• for single-source multiple-sink case.
• Solution is again rational.
• Using Megiddo, 1974.

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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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Game-theoretic properties of LNB games --
stress tests

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

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ADNB

• B n players with disagreement points, ci
• G g goods, unit amount each
• S utility vectors obtained by distributing
• goods among players

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Game plan

• Use KKT conditions to
• transform Nash bargaining problem to
• computing the equilibrium in a certain
  market.
• Find equilibrium using primal-dual paradigm.

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Game plan

• Use KKT conditions to
• transform Nash bargaining problem to
• computing the equilibrium in a certain
  market.
• Find equilibrium using primal-dual paradigm.

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Fishers Model

• 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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Flexible budget market,only difference

• Buyers dont spend a fixed amount of money.
• Instead, they have a strict lower bound on
• the utility
  they desire.

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Flexible budget market,only difference

• Buyers dont spend a fixed amount of money.
• Instead, they have a strict lower bound on
• the utility
  they desire.
• Money spent f (utility desired, prices of
  goods)

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Most cost-effective goods

• At prices p, for buyer i
• Define

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Flexible budget market

• Agent i wants utility
• At prices p, must spend
  to get utility

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Flexible budget market

• Agent i wants utility
• At prices p, must spend
  to get utility
• Define
• Find market clearing prices.

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Flexible budget market

• Agent i wants utility
• At prices p, must spend
  to get utility
• Define
• Find market clearing prices -- may not exist!!

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Flexible budget market

• Agent i wants utility
• At prices p, must spend
  to get utility
• Define
• Find market clearing prices -- may not exist!!
• 
  feasible/infeasible

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• Theorem Nash Bargaining for linear utilities
• reduces to
• Equilibrium for flexible budget markets

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• Theorem Nash Bargaining for linear utilities
• reduces to
• Equilibrium for flexible budget markets
• (S(u), c) ? M(u, c)
• (S, c) is feasible iff M is feasible.
• If feasible, x is Nash bargaining solution
• iff x is equilibrium
  allocation.

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An easier question

• Given prices p, are they equilibrium prices?
• If so, find equilibrium allocations.

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An easier question

• Assume market is feasible.
• Given prices p, are they equilibrium prices?
• If so, find equilibrium allocations.
• For each i,

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For each i, most cost-effective goods

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Network N(p)
p(1)
m(1)
p(2)
m(2)
t
s
p(3)
m(3)
m(4)
p(4)
infinite capacities
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Max flow in N(p)
p(1)
m(1)
p(2)
m(2)
p(3)
m(3)
m(4)
p(4)
p equilibrium prices iff both cuts saturated
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An important consideration

• The price of a good never exceeds
• its equilibrium price
• Invariant s is a min-cut

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Invariant s is a min-cut in N(p)
p(1)
m(1)
p(2)
m(2)
s
p(3)
m(3)
m(4)
p(4)
p low prices
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Initialization

• Assume money of each buyer is 1 -- this
• is a linear Fisher market!
• Find equilibrium prices, p.
• Clearly, N(p) satisfies the Invariant.

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Flexible budget market

• New difficulties
• mi s change as prices change.
• need to decide if market is feasible.

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Network N(p)
N - N
J
I
N(I, J)
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Raise prices in J

• p . x, for each p in J
• initialize x 1
• raise x
• money increases
• resulting network N(xp)

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Raise prices in J

• p . x, for each p in J
• initialize x 1
• raise x
• money increases
• resulting network N(xp)
• How does surplus change?

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1-surplus surplus -1

• bi in -1, infinity)
• Lemma f is max-flow in N(p) gt
• x.f is max-flow in N(xp) and bi(x) x bi

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1-surplus surplus -1

• bi in -1, infinity)
• Lemma f is max-flow in N(p) gt
• x.f is max-flow in N(xp) and bi(x) x bi
• Buyer i is good iff bi lt 0
• Surplus 1 x bi as x

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If market is feasible,

• Can find p s.t. all buyers good
• If so, can raise prices until all surplus vanishes

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Search
Infeasible
Feasible
?
Decision
Allocations Prices (Money)
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KKT conditions
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Relaxed KKT conditions
-bi
-bi
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Potential function

• Algorithm uses balanced flows
• Reduces potential by an inverse polynomial factor
  
• in each phase (strongly polynomial time).

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Search
Infeasible
Feasible
?
Decision
Allocations Prices (Money)
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If market is infeasible,

• Can find p s.t.

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Proof of infeasibility dual solution to
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Search
Infeasible
Feasible
?
Decision
Allocations Prices (Money)
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Decision

• Best viewed as a tug-of-war
• between 2 teams of buyers
• good and rest!

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FEASIBLE
INFEASIBLE
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Decision

• After each iteration
• buyers change sides
• changes
• Terminate if
• All buyers Good gt market is feasible
• gt market is
  infeasible.

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Decision

• Need to reduce prices (dual variables)!
• Second only to Edmonds matching algorithm
• Need balanced flows!
• Need potential function using l2 norm!

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Potential function

• Reduces potential by an inverse polynomial factor
  
• in each phase (strongly polynomial time).

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• Theorem Algorithm runs in polynomial time.

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• Theorem Algorithm runs in polynomial time.
• Q Find strongly polynomial algorithm!

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Subclasses of LNB

• LNB2 Restrict LNB to 2-person games
• Theorem All games in LNB2 are rational
• and solvable in polynomial
  time.

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Given disagreement point, find NB soln.
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UNB and SNB

• Uniform Utility Nash bargaining games
• Coefficients in packing constraints are 0/1
• Submodular Utility Nash bargaining games
• Function defining r.h.s. is submodular
• Theorem Each game in SNB is rational and
• is solvable in strongly polynomial
  time.

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• Theorem
• Strongly polynomial, combinatorial algorithm
• for single-source multiple-sink case.
• Solution is again rational.

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Properties studied (CGVWY, 2009)

• Full fairness
• Max-min and min-max fair in set of Pareto opt.
  sols
• Price of bargaining
• Efficient if price of bargaining 1
• Strong monotonicity
• Increase ci gt vj cannot increase
• Population monotonicity
• Play with subset gt utility can only increase

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• Theorem For each of these properties
• A game in UNB has it iff it is in
  SNB.
• i.e., each of these properties characterizes
• SNB in UNB.

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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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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
• 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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Open

• Can linear Fisher markets
• or ADNB
• be captured via an LP?

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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

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

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Common generalization

• Can it be solved via a combinatorial,
• polynomial time algorithm?
  OPEN!
• Is it meaningful? Nonsymmetric ADNB
• Kalai, 1975 Nonsymmetric bargaining games
• wi clout of player i.