Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
The Logic of Bargaining
Dongmo Zhang
Intelligent Systems Lab
University of Western Sydney
Australia
Thematic Trimester on Game Theory @ IRIT, France
6 July 2015
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Bargaining Theory
“A two-person bargaining situation involves two individuals
who have the opportunity to collaborate for mutual benefit in
more than one way.”
[John Nash, 1950]
“Under such a definition, nearly all human interaction can be
seen as bargaining of one form or another.”
[Ken Binmore et al. 1992]
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Multi-agent systems
A multi-agent system (MAS) is a system composed of multiple
interacting intelligent agents, each of which is:
Autonomous: acts on its own.
Self-interested: directs its activity towards achieving its goals.
Decentralized: no designated controlling agent.
“Research in multi-agent systems is concerned with the study,
behaviour, and construction of a collection of possibly pre-existing
autonomous agents that interact with each other and their
environments.”
[Katia P. Sycara, 1998]
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Outline
1
Motivation
2
Game-theoretic solutions
3
Logical model
4
Solution construction
5
Axiomatic system
6
Continuous domain
7
Conclusion
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Bargaining problems
Bargaining problem: Pie Devision.
Devision range: x ∈ [0, 1], y = 1 − x
Utility of player 1: u1 (x)
Utility of player 2: u2 (y )
Bargaining game: (S, d), where S ⊆ <2 & d ∈ S
Bargaining solution: f (S, d) ∈ S
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Game-theoretic bargaining solutions
Nash’s bargaining solution (NBS):
the maximizer of the product of
utilities.
Kalai-Smorodinsky’s Solution (KSS):
the maximizer of the points in S on
the segment connecting d and
a(S,d).
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Axiomatization of Bargaining Solutions
A bargaining solution is the NBS iff
it satisfies:
A bargaining solution is the KSS iff
it satisfies:
Pareto-optimality
Pareto-optimality
Symmetry
Symmetry
Scale invariance
Scale invariance
Independence of irrelevant
alternatives.
Restricted Monotonicity
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Split a pie
Example
Two players bargain over the split of a pie.
u1 (x) = x
u2 (y ) = y 2
S = {(x, y 2 ) : 0 ≤ x ≤ 1 and y = 1 − x}, d = (0, 0).
Nash’s prediction: (33.3, 66.7)
Kalai-Smorodinsky’s prediction:
(38.2, 61.8)
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
The “numbers” illusion
“Can this prediction be tested as in the sciences?”
“The use of numbers, even if analytically convenient, obscures the
meaning of the model and creates the illusion that it can produce
quantitative results.”
“I am not convinced that Nash’s theory has done more than clarify
the logic of one consideration which influences bargaining
outcomes. I can not see how this consideration will
comprehensively explain real-life bargaining results.”
“Were game theorists to use a more natural language to specify the
model, the solution would become clearer and more meaningful.”
[Ariel Rubinstein, 2000]
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Example: Political negotiation
Two political parties in the Parliament bargain over a government
rescue plan in response to the 2008 financial crisis. Proposals from
the parties:
Inject funds into struggling financial institutions
Rescue car makers
Relieve homeowners of heavy house mortgage
Sponsor job training and job creation.
Increase taxes
Obviously each party has their benefits from different industries
therefore has preference on different rescue plans. However,
representing the preference for each item in numbers can be a hard
job for each party.
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Logical solution: a possibility
Represent bargaining terms in logic:
bank: fund financial institutions;
car : rescue car makers;
house: help house mortgagors;
training : create training opportunities;
incTax: increase taxes.
Represent constraints in logic:
¬(bank ∧ house): mortgagees and mortgagors shouldn’t be
both funded.
(car ∧ bank) → incTax: it is impossible to rescue both car
industry and financial institutions without increasing taxes.
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Representation of bargaining problems: demands
Bargainers’ demands:
Party A wants to inject almost all funds into the major banks
but a small amount for job training. Tax increase is never a
policy of party A.
Party B insists on funding car makers and individual
homeowners.
Both parties know that there is no need to support both sides
of house mortgage. Also the government budget does not
allow to rescue both car industry and financial institutions
unless increase taxes.
A: {bank, training , ¬incTax, ¬(bank ∧ house), (car ∧ bank) → incTax}
B: {car , house,¬(bank ∧ house), (car ∧ bank) → incTax}
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Representation of bargaining problems: conflicts
A: {bank, training , ¬incTax, ¬(bank ∧ house), (car ∧ bank) → incTax}
B: {car , house,¬(bank ∧ house), (car ∧ bank) → incTax}
Conflicts between the negotiation parties:
Conflict 1
{bank, house, ¬(bank ∧ house)} ` ⊥
Conflict 2
{bank, car , ¬incTax, (car ∧ bank) → incTax} ` ⊥
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Representation of Preferences
Each party ranks their bargaining items in total pre-order,
representing the firmness the party insists the items (the higher the
firmer).
Party A
¬(bank ∧ house)
(car ∧ bank) → incTax
¬inTax
bank
training
Party B
¬(bank ∧ house)
(car ∧ bank) → incTax
car
house
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Logical model of bargaining
Bargaining game: G = ((X1 , ≤1 ), · · · , (Xn , ≤n )).
Bargaining solution: f (G ) = (C1 , · · · , Cn ), where Ci ⊆ Xi .
S
Agreement: A(G ) =
fi (B)
i∈N
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Solution construction: the simultaneous concession
solution (SCS)
¬(bank ∧ house)
(car ∧ bank) → incTax
¬inTax
bank
training
¬(bank ∧ house)
(car ∧ bank) → incTax
car
house
The solution is {¬(bank ∧ house), (car ∧ bank) → incTax,
¬inTax}, meaning that nothing is agreed.
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Dummy demands
Party A
¬(bank ∧ house)
(car ∧ bank) → incTax
¬inTax
bank
training
Party B
¬(bank ∧ house)
(car ∧ bank) → incTax
car
house
coffee
The solution is {¬(bank ∧ house), (car ∧ bank) → incTax,
¬inTax,car}, meaning that rescuing car makers has been agreed.
Key idea
We can use dummy demands to represent intentional delays, which
plays a similar role as non-linearity of utility functions.
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Axiomatic system
The logic solution is characterized by the following axioms:
1
Consistency:
S
fi (G ) is consistent.
i∈N
2
Comprehensiveness:
For each i, fi (G ) is comprehensive.
3
Collective Rationality:
If G is non-conflictive, then fi (G ) = Xi for all i.
4
Disagreement:
If G represents a disagreement situation, then fi (G ) = ∅ for
all i.
5
Contraction independence:
If G 0 @max G , then f (G ) = f (G 0 ) unless G is non-conflictive.
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Solution characterisation
Theorem
A logical bargaining solution is the simultaneous concession
solution if and only if it satisfies
Consistency
Comprehensiveness
Collective Rationality
Disagreement
Contraction independence
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Continuous domain
Pie devision problem
Devision range: x1 ∈ [0, 1], x2 = 1 − x1
Utility of player 1: u1 (x1 )
Utility of player 2: u2 (x2 )
For each i = 1, 2, let Pi (l) represent the following proposition:
xi ≥ ui−1 ((1 −
l
l
)ui (0) + ui (1))
Li
Li
i.e., evenly divide player i’s utility into Li pieces. It means
that each time, a player gives up equally amount of utility.
Let C represent the constraints that rule out invalid pie
division.
X 1 = C ∪ {P1 (1), · · · , P1 (L1 )}
X 2 = C ∪ {P2 (1), · · · , P2 (L1 )}
(1)
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Continuous domain
Round Player A’s demand Player B’s demand Agreement
0
x1 ≥ 100
x2 ≥ 100.0
×
1
x1 ≥ 90
x2 ≥ 94.9
×
2
x1 ≥ 80
x2 ≥ 89.4
×
3
x1 ≥ 70
x2 ≥ 83.7
×
4
x1 ≥ 60
x2 ≥ 77.5
×
5
x1 ≥ 50
x2 ≥ 70.7
×
6
x1 ≥ 40
x2 ≥ 63.2
×
√
7
x1 ≥ 30
x2 ≥ 54.8
where L1 = L2 = 10. Both the Nash solution (33.3, 66.7) and the
Kalai-Smorodinsky solution (38.2, 61.8) belong to the range. When
LA = LB = 100, the solution ranges become x1 ∈ [38, 38.4] and
x2 ∈ [61.6, 62], which means that it approaches to
Kalai-Smorondinsky solution.
Motivation Game-theoretic solutions Logical model Solution construction Axiomatic system Continuous domain Conclusion
Conclusion
A logical theory for multi-issue, n-person, raw domain
bargaining.
An elegant axiomatic system: simple and intuitive.
The proposed solution approaches to Kalai-Smorondinsky
solution for continuous domain.
Bargaining reasoning:
Identify conflicts.
Modeling risk.
Representation of bargaining problems
Logical language + ordering > utility function
The logical theory of bargaining is not a rival but a
complementary of game-theoretic bargaining theory.