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

4. 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.

5. 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!

6. The new platform for computing

8. Internet Open, distributed platform
with enormous numbers of users interacting
and sharing scarce resources
Rife with situations of conflict of interest
Massive computational power available

9. Internet Open, distributed platform
with enormous numbers of users interacting
and sharing scarce resources
Rife with situations of conflict of interest
Massive computational power available
Algorithmic Game Theory

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

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

12. Utilities derived jointly
: convex + compact
feasible set

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

14. Typically: assume that S contains a point which makes each player happier
Disagreement point =

15. Nash bargaining Captures the main idea that both players
gain if they agree on a solution.
Else, they go back to status quo.
Q: Which solution is the “right” one?

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

19. Thm: Unique soln:
Disagreement point =

20. Nash bargaining Captures the main idea that both players
gain if they agree on a solution.
Else, they go back to status quo.
Link between cooperative and
non-cooperative games

21. Generalizes to n-players
Theorem: Unique solution

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

23. Q: Compute solution combinatorially in polynomial time?

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

25. Convex program giving NB solution

26. Theorem For linear utilities case,
Nash bargaining solution is rational!
Polynomially many bits in size of instance

27. Theorem For linear utilities case,
Nash bargaining solution is rational!
Polynomially many bits in size of instance
Solution can be computed in polynomial time.

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

29. Based on Kelly (1997): Mathematical framework for
understanding TCP congestion control
Jain & V. (2007): Eisenberg-Gale markets

30. Goods = edges Objects = flow paths

31. Given disagreement point, find NB soln.

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

33. Insights into Nash bargaining problems
Chakrabarty, Goel, V., & Wang:
Fairness
Efficiency
Competition monotonicity

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

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

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.

40. Irving Fisher, 1891
Defined a fundamental
market model

41. Fisher’s Model B = n buyers, money m(i) 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,

42. Fisher’s Model B = n buyers, money m(i) 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.

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

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

45. Flexible budget market
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.

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

47. Flexible budget market
Lower bound on utility =
At prices p, must spend to get utility

48. Flexible budget market
Lower bound on utility =
At prices p, must spend to get utility
Define
Find market clearing prices.

49. prices pj Theorem: x and p are optimal primal and dual solutions
iff they are equilibrium allocations and prices.
Also, equilibrium prices are unique.

50. Flexible budget market
Several goods, fixed amount of each good
Several buyers,
with individual utilities and lower bound
Find equilibrium prices and allocations, i.e., prices s.t.,
Each buyer gets an optimal bundle
No deficiency or surplus of any good

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

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

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

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

55. 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

56. Idea of algorithm “primal” variables: allocations
“dual” variables: prices of goods
Approach equilibrium prices from below:
start with very low prices; buyers have surplus money
iteratively keep raising prices,
until outcome is clear.

57. Algorithm figures out if:
A subset of players must get disagreement utility.
There exists Nash bargaining solution
(each player is happier than disagreement utility).

58. Finds dual feasible solution to:

59. Idea of algorithm Iterations: execute primal & dual improvements
Allocations Prices (Money)

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

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

62. 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

63. 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)

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

65. Balanced flow surplus vector: vector of surpluses w.r.t. f.
A max-flow that
minimizes l2 norm of surplus vector.
makes surpluses as equal as possible.

66. Running time? Proof technique of DPSV very difficult –
impossible to carry over to the more
complex algorithm.

67. DPSV proof idea Algorithm reduces l2 norm of surplus vector
by inverse polynomial fraction in each iteration.

68. “Natural” approach Total surplus money = l1 norm of surplus vector
Show: l1 norm of surplus vector reduces by
inverse polynomial fraction in each iteration.

70. Simpler proof l1 norm of surplus vector decreases
by a factor of (1 – 1/n2) in each iteration.
Gives poly run time for more complex algorithm

71. Simpler proof l1 norm of surplus vector decreases
by a factor of (1 – 1/n2) in each iteration.
Gives poly run time for more complex algorithm
Q: Find strongly poly algorithm!!

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

73. Goods = edges Objects = flow paths

74. Given disagreement point, find NB soln.

75. Theorem:
Strongly polynomial, combinatorial algorithm
for single-source multiple-sink case.

76. Other resource allocation problems
k source-sink pairs (directed/undirected)

77. Other resource allocation problems
k source-sink pairs (directed/undirected)
Branchings rooted at sources (agents)
Spanning trees
Network coding

78. Branching construction (for broadcasting)

79. Sources want to obtain rooted branchings

80. Convex program for branching problem
s.t. packing of branchings

81. Other resource allocation problems
k source-sink pairs (directed/undirected)

82. JV: Irrational for 3 source-sink pairs

83. Irrational for 3 source-sink pairs
Dual variables of convex program

84. Other resource allocation problems2
k source-sink pairs (directed/undirected)

85. Max-flow min-cut theorem!

86. Theorem: Strongly polynomial algs for following NB problems:
2 source-sink pairs, undirected (Hu, 1963)
spanning tree (Nash-William & Tutte, 1961)
2 sources branching (Edmonds, JV, 2005)
JV: 3 sources branching: irrational

87. Theorem: Strongly polynomial algs for following NB problems:
2 source-sink pairs, undirected (Hu, 1963)
spanning tree (Nash-William & Tutte, 1961)
2 sources branching (Edmonds, JV, 2005)
JV: 3 sources branching: irrational
No max-min theorems.
Use full power of LP-duality theorem:
2 source-sink pairs, directed
2 sources, network coding

88. Nash Bargaining vs. Market Equilibrium

89. Linear Case of Arrow-Debreu Model
Instead of money, each agent has
an initial endowment of goods.
Find market clearing prices, i.e., prices s.t.
If each agent sells all her goods
Buys optimal bundle using this money
No surplus or deficiency of any good

90. Initial endowment of goods
Agents
Goods

91. Prices
= $ = $ = $10
Agents
Goods

92. Incomes
Agents
$50
$60
Goods
Prices
=$ =$15 =$10
$40
$40

93. Maximize utility
Agents
$50
$60
Goods
Prices
=$ =$15 =$10
$40
$40

94. Find prices s.t. market clears
Agents
$50
$60
Goods
Prices
=$ =$15 =$10
$40
Maximize utility
$40

95. Natural transformation: AD to NB
ci = ui (initial-endowment(i))

96. Initial endowments

97. Utilities

98. AD solution:
gets
NB solution:
gets ~ ½ unit of
gets rest.

99. Open Solve Nash bargaining problems with other
utility functions s.t. solution is still
captured by a convex program!

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

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

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

103. 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

104. 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

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

107. Eisenberg-Gale Program, 1959