Download Stat155 Game Theory Lecture 22: Nash bargaining. Multiplayer TU

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Stat155
Game Theory
Lecture 22: Nash bargaining.
Multiplayer TU cooperative games.
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms
The Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms
Peter Bartlett
November 15, 2016
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Recall: Cooperative games
Nontransferable utility cooperative games
Feasible payoffs (NTU)
Players can make binding agreements.
(3,6)
Two types:
Transferable utility The players agree what strategies to play and
what additional side payments are to be made.
Nontransferable utility The players choose a joint strategy, but there
are no side payments.
Payoffs
1
2
(5,5)
1
(2,2)
(4,3)
2
(6,2)
(3,6)
3
(1,2)
(5,5)
(4,3)
(1,2)
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(2,2)
(6,2)
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Nash Bargaining Model for NTU games
Nash Bargaining Model for NTU games
Ingredients of a bargaining problem
1
2
Nash’s bargaining axioms
A compact, convex feasible set S ⊂ R2 .
1
A disagreement point d = (d1 , d2 ) ∈ R2 .
2
Think of the disagreement point as the utility that the players get from
walking away and not playing the game.
(And we can assume every x ∈ S has x1 ≥ d1 , x2 ≥ d2 , with strict
inequalities for some x ∈ S.)
Definition
A solution to a bargaining problem is a function F that takes
a feasible set S and a disagreement point d and returns an agreement
point a = (a1 , a2 ) ∈ S.
Pareto optimality: the only feasible payoff vector (v1 , v2 ) with v1 ≥ a1
and v2 ≥ a2 is (v1 , v2 ) = (a1 , a2 ).
Symmetry: If both (x, y ) ∈ S implies (y , x) ∈ S and d1 = d2 then
a1 = a2 .
3
Affine covariance: For any affine transformation
Ψ(x1 , x2 ) = (α1 x1 + β1 , α2 x2 + β2 ) with α1 , α2 > 0 and any S and d,
F (Ψ(S), Ψ(d)) = Ψ(F (S, d)).
4
Independence of irrelevant attributes: For two bargaining problems
(R, d) and (S, d), if R ⊂ S and F (S, d) ∈ R, then
F (R, d) = F (S, d).
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Nash Bargaining Model for NTU games
Nash Bargaining Model for NTU games
Theorem
Nash’s bargaining axioms
Pareto optimality: The agreement point shouldn’t be dominated by
another point for both players. (Criticism: why should one player care
if the agreement point is only dominated for the other player?)
Symmetry: This is about fairness: if nothing distinguishes the players,
the solution should be similarly symmetric.
There is a unique function F satisfying Nash’s bargaining axioms.
It is the function that takes S and d and returns the unique solution to the
optimization problem
max
(x1 ,x2 )
subject to
Affine covariance: Changing the units (or a constant offset) of the
utilities should not affect the outcome of bargaining.
(x1 − d1 )(x2 − d2 )
x1 ≥ d1
x2 ≥ d2
(x1 , x2 ) ∈ S.
Independence of irrelevant attributes: This assumes that all of the
threats the players might make have been accounted for in the
disagreement point.
We call this F the Nash bargaining solution.
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Nash Bargaining Model for NTU games
Nash Bargaining Model for NTU games
Example
Consider a TU game with disagreement point d
and cooperative strategy with total payoff σ.
Then the convex set S is the set of convex
combinations of lines {(aij + p, bij + p) : p ∈ R}.
To maximize (x1 − d1 )(x2 − d2 ) we set
x2 = σ − x1 and choose x1 to maximize
(x1 − d1 )(σ − x1 − d2 )
= (σ − d2 + d1 )x1 −
x12
Proof
(d1,10−d1)
((10−d2+d1)/2,
(10−d1+d2)/2)
(3,6)
(5,5) (10−d2,d2)
The Nash solution is unique. (See text for a slick proof.)
The Nash solution satisfies the axioms:
1
(d1,d2)
2
(4,3)
(1,2)
(2,2)
3
(6,2)
4
− d1 (σ − d2 ).
Pareto optimality: increasing, say, x1 increases (x1 − d1 )(x2 − d2 ).
Symmetry: You can check that it follows from uniqueness of the
solution.
Affine covariance: (α1 x1 + β1 − (α1 d1 + β1 )) = α1 (x1 − d1 )
Independence of irrelevant attributes: A maximizer in S is still a
maximizer in R ⊂ S.
This gives x1 = (σ − d2 + d1 )/2.
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Nash Bargaining Model for NTU games
Nash Bargaining Model for NTU games
Example
Proof Idea
Independence of irrelevant attributes is not completely self-evident.
Any bargaining solution that
satisfies the axioms is the
Nash solution.
1
2
For S and d, if the Nash
solution is a, find the
affine function Ψ so that
Ψ(a) = (1, 1) and
Ψ(d) = (0, 0).
If the Nash solution is
a = (1, 1) and d = (0, 0),
then the convex hull of S
and its reflection are in
{x1 + x2 ≤ 2}, so any
symmetric, optimal F
returns (1, 1) for this
convex hull, and hence, by
IIA, for S.
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(Karlin and Peres, 2016)
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(Karlin and Peres, 2016)
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Nash Bargaining Model for NTU games: Key points
Outline
Nash’s bargaining axioms
1
Pareto optimality
2
Symmetry
3
Affine covariance
4
Independence of irrelevant attributes
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms
The Nash bargaining solution
Theorem (Nash bargaining solution)
Multi-player transferable utility cooperative games
There is a unique function F satisfying Nash’s bargaining axioms.
It is the function that takes S and d and returns the unique solution to the
optimization problem
max
(x1 ,x2 )
subject to
Characteristic function
Gillies’ core
Shapley’s axioms
(x1 − d1 )(x2 − d2 )
x1 ≥ d1
x2 ≥ d2
(x1 , x2 ) ∈ S.
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Multiplayer TU cooperative games
Multiplayer TU cooperative games
Example
A customer in a marketplace is willing to buy a pair of gloves for $100.
There are three players, two with only left gloves and one with right
gloves, and they need to agree on who sells their glove and how to
split the $100.
It is more complicated than a two-player game:
the players can form coalitions.
Characteristic function
Define a characteristic function: for each subset S of players, v (S) is
the total value that would be available to be split by that subset of
players, no matter what the other players do:
v ({1, 2, 3}) = v ({1, 2}) = v ({1, 3}) = 100,
v ({1}) = v ({2}) = v ({3}) = v ({2, 3}) = v ({}) = 0.
Who holds the power and what’s fair depends on how the different
subsets of players depend on other players and contribute to the
payoff.
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Multiplayer TU cooperative games
Multiplayer TU cooperative games
Gillies’ core
Efficiency
= v ({1, . . . , n}).
P
Stability For all S ⊆ {1, . . . , n}, i∈S ψi (v ) ≥ v (S).
Allocations
Define an allocation function ψ as a map from a characteristic function v
for n players to a vector ψ(v ) ∈ Rn .
This is the payoff that is allocated to the n players.
What properties should an allocation function have?
Pn
i=1 ψi (v )
These seem reasonable properties:
1
The total payoff gets allocated.
2
A coalition gets allocated at least the payoff it can obtain on its own.
(That coalition has the power to act unilaterally.)
(Donald B. Gillies: Canadian-born mathematician, game theorist,
computer scientist. UIUC)
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Multiplayer TU cooperative games
Multiplayer TU cooperative games
Example: any pair for $1
Example: left and right gloves
v ({1, 2}) = v ({1, 3}) = v ({2, 3}) = v ({1, 2, 3}) = 1,
(Write ψi for ψi (v ).)
v ({1}) = v ({2}) = v ({3}) = v ({}) = 0.
3
X
Then
ψi = v ({1, 2, 3}) = 100
i=1
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ψ1 + ψ2 ≥ 100,
ψi = v ({1, 2, 3}) = 1,
i=1
ψ1 + ψ3 ≥ 100.
ψ1 + ψ2 ≥ 1,
One solution: ψ1 = 100.
ψ1 + ψ3 ≥ 1,
ψ2 + ψ3 ≥ 1.
No solutions!
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Multiplayer TU cooperative games
Outline
Example: single gloves for $1, pairs for $10, triples for $100
v ({1}) = v ({2}) = v ({3}) =1,
Two-player nontransferable utility cooperative games
v ({1, 2}) = v ({1, 3}) = v ({2, 3}) = 10,
Bargaining problems
Nash’s bargaining axioms
The Nash bargaining solution
v ({1, 2, 3}) = 100,
Then
Multi-player transferable utility cooperative games
3
X
Characteristic function
Gillies’ core
Shapley’s axioms
ψi = v ({1, 2, 3}) = 100,
i=1
ψ1 ≥ 1, ψ2 ≥ 1, ψ3 ≥ 1,
ψ1 + ψ2 ≥ 10, ψ1 + ψ3 ≥ 10, ψ2 + ψ3 ≥ 10,
ψ1 + ψ2 + ψ3 ≥ 100.
Many solutions!
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Multiplayer TU cooperative games
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Outline
Shapley axioms
Efficiency
Pn
i=1 ψi (v )
= v ({1, . . . , n}).
Two-player nontransferable utility cooperative games
Symmetry If, for all S ⊂ {1, . . . , n} and i, j 6∈ S,
v (S ∪ {i}) = v (S ∪ {j}), then ψi (v ) = ψj (v ).
No freeloaders For all i, if for all S ⊂ {1, . . . , n} v (S ∪ {i}) = v (S), then
ψi (v ) = 0.
Additivity ψi (v + u) = ψi (v ) + ψi (u).
Bargaining problems
Nash’s bargaining axioms
The Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms
Shapley’s Theorem
Shapley’s axioms uniquely determine the allocation ψ.
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