1. Algorithmic Game Theory and Internet Computing
Nash Bargaining via
Flexible Budget Markets
Algorithmic Game Theory and Internet Computing
Vijay V. Vazirani

4. The new platform for computing

6. Internet Massive computational power available
Sellers (programs) can negotiate with
individual buyers!

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

8. Internet Massive computational power available
Sellers (programs) can negotiate with
individual buyers!
Algorithmic Game Theory

9. 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.

10. 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!

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

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

13. Utilities derived jointly
: convex + compact
feasible set

14. Disagreement point = status quo utilities

15. Nash bargaining problem = (S, c)
Disagreement point =

16. Nash bargaining
Q: Which solution is the “right” one?

17. Solution must satisfy 4 axioms:
Paretto optimality
Invariance under affine transforms
Symmetry
Independence of irrelevant alternatives

20. Thm: Unique solution satisfying 4 axioms

21. Generalizes to n-players
Theorem: Unique solution

22. Generalizes to n-players
Theorem: Unique solution
(S, c) is feasible if S contains a point that makes
each i strictly happier than ci

23. Bargaining theory studies promise problem
Restrict to instances (S, c) which are feasible.

24. Linear Nash Bargaining (LNB)
Feasible set is a polytope defined by
linear packing constraints
Nash bargaining solution is
optimal solution to convex program:

25. Q: Compute solution combinatorially in polynomial time?

26. Study promise problem? Decision problem reduces to promise problem
Therefore, study decision and search problems.

27. Linear utilities B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players

28. e.g., ci = i’s utility for initial endowment
B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players

29. Convex program giving NB solution

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

31. 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.

32. Resource Allocation Nash Bargaining Problems
Players use “goods” to build “objects”
Player’s utility = number of objects
Bound on amount of goods available

33. Goods = edges Objects = flow paths

34. Given disagreement point, find NB soln.

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

36. Insights into game-theoretic properties of Nash bargaining problems
Chakrabarty, Goel, V. , Wang & Yu:
Efficiency (Price of bargaining)
Fairness
Full competitiveness

37. Linear utilities B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players

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

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

41. General Equilibrium Theory
Crown jewel of mathematical economics for over a century!

42. A central tenet Prices are such that demand equals supply, i.e.,
equilibrium prices.

43. A central tenet Prices are such that demand equals supply, i.e.,
equilibrium prices.
Easy if only one good

44. Supply-demand curves

45. Irving Fisher, 1891
Defined a fundamental
market model

46. Fisher’s 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,

47. Fisher’s 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.

49. General Equilibrium Theory
An almost entirely non-algorithmic theory!

50. Flexible budget market, only difference:
Buyers don’t spend a fixed amount of money.
Instead, they know how much utility they desire.
At any given prices, they spend just enough
money to accrue utility desired.

51. Most cost-effective goods
At prices p, for buyer i: Si =
Define

52. Flexible budget market
Agent i wants utility
At prices p, must spend to get utility

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

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

55. 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

56. Theorem: Nash Bargaining for linear utilities reduces to
Equilibrium for flexible budget markets

57. 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.

58. Primal-Dual Paradigm
Usual framework: LP-duality theory

59. Primal-Dual Paradigm Usual framework: LP-duality theory
Extension to convex programs and
KKT conditions.

62. Yin & Yang

63. Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual paradigm

64. Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual paradigm
Solves Eisenberg-Gale convex program

65. Eisenberg-Gale Program, 1959

66. Eisenberg-Gale Program, 1959
prices pj

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

68. Idea of algorithm primal variables: allocations
dual variables: prices of goods
iterations:
execute primal & dual improvements
Allocations Prices (Money)

69. Flexible budget market
Main differences:
mi ’s change as prices change.
problem is not total.

70. ? Search Decision Infeasible Feasible Allocations Prices (Money)

71. An easier question Given prices p, are they equilibrium prices?
If so, find equilibrium allocations.

72. An easier question Given prices p, are they equilibrium prices?
If so, find equilibrium allocations.
For each i,

73. For each i, most cost-effective goods
p(1)
p(2)
p(3)
p(4)

74. Network N(p) infinite capacities p(1) m(1) p(2) m(2) p(3) m(3) m(4)

75. p: equilibrium prices iff both cuts saturated
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

76. Two important considerations
The price of a good never exceeds
its equilibrium price
Invariant: s is a min-cut

77. Max flow
p(1)
m(1)
p(2)
m(2)
p(3)
m(3)
m(4)
p(4)
p: low prices

78. Two important considerations
The price of a good never exceeds
its equilibrium price
Invariant: s is a min-cut
Rapid progress is made
Balanced flows

79. W.r.t. max-flow f, surplus(i) = m(i) – f(i,t)
Max-flow in N p m i
W.r.t. max-flow f, surplus(i) = m(i) – f(i,t)

80. Balanced flow surplus vector: vector of surpluses w.r.t. f.
A max-flow that
minimizes l2 norm of surplus vector.

81. ? Search Decision Infeasible Feasible Allocations Prices (Money)

82. Balanced flow helps Decision as well!

83. Proof of infeasibility: dual solution to

84. Theorem: Algorithm runs in polynomial time.

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

86. Nonlinear programs with rational solutions!
Open
Nonlinear programs with rational solutions!

87. Nonlinear programs with rational solutions! Solvable combinatorially!!
Open
Nonlinear programs with rational solutions! Solvable combinatorially!!

88. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s

89. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s

90. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
Algorithmic Game Theory (New Millennium):
Rational solutions to nonlinear convex programs

91. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
Algorithmic Game Theory (New Millennium):
Rational solutions to nonlinear convex programs

92. Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
Algorithmic Game Theory (New Millennium):
Rational solutions to nonlinear convex programs
Approximation algorithms for convex programs?!

93. Can Nash bargaining problem for linear utilities case
Open
Can Nash bargaining problem
for linear utilities case
be captured via an LP?