### Lecture 15: Bargaining and Cooperative Game

Advanced Microeconomics II

Yosuke YASUDA

Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp

January TBA, 2015

### Cooperative Games

Non-Cooperative (Strategic) Games

examine individual decision making in strategic settings. assume a person decides her action on her own.

does NOT rule out cooperative behaviors. Cooperative (Coalitional) Games

examine group decision making in strategic settings. assume players can agree on their joint action, or can make binding contracts.

simplifies strategic analysis by NOT modeling the negotiation process explicitly.

The two approaches are complements to each other: each of them reflects different kinds of strategic considerations and contributes to our understanding of strategic reasoning.

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### Bargaining Problem

We study two-person bargaining problems from the perspective of cooperative game theory initiated by John Nash. Although its solution concept is called the Nash (bargaining) solution, this is nothing to do with Nash equilibrium.

A bargaining situation is described by a tuple hX, D, %_{1}, %_{2}i:
X is a set of possible agreements: a set of possible
consequences that the two players can jointly achieve.

D ∈ X is the disagreement outcome: the event that occurs if the players fail to agree.

%_{1} and %_{2} are the players’ preference relations over L(X),
the set of lotteries over X.

We assume that D is singleton and X contains an agreement y for which y ≻i D for i = 1, 2.

### Nash Bargaining Solution (1)

Definition 1

A bargaining problem is defined by a pair hU, di where U is the set of pairs (u1(x), u2(x)) for x ∈ X and d = (u1(D), u2(D)).

Given the set of possible agreements, the disagreement outcome, and the players’ preferences, we can construct the set of all utility pairs that can be the outcome of bargaining. Definition 2

A bargaining solution is a function f : (U, d) → U , that assigns a point in U to every bargaining problem hU, di.

A bargaining solution describes the way how the agreement depends on the parameters of the bargaining problem.

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### Nash Bargaining Solution (2)

Definition 3 (N S1)

The utility pair u^{∗}(≥ d) ∈ U is a Nash solution of the bargaining
problem, denoted by f^{N}(U, d), if

(u^{∗}_{1}− d_{1})(u^{∗}_{2}− d_{2}) ≥ (u_{1}− d_{1})(u_{2}− d_{2}) for all u ∈ U. (N)
That is,

f^{N}(U, d) = arg max

(d1^{,d}2)≤(v1^{,v}2)∈U

(v_{1}− d_{1})(v_{2}− d_{2}).

### Nash Bargaining Solution (3)

There is a equivalent definition that has a natural interpretation.
Assume that x^{∗} is “on the table.”

If Player i is willing to object to x^{∗} by proposing an

alternative x, even if she faces the risk that with probability
1 − p the negotiations will break down and end with D,
then Player j is willing to take the analogous risk and reject x
in favor of the agreement x^{∗}.

Definition 4 (N S_{2})

u(x^{∗}) ∈ U is a Nash solution if the following hold: if
p · x ⊕ (1 − p) · D ≻i x^{∗} for some p ∈ [0, 1] and x ∈ X then
p · x^{∗}⊕ (1 − p) · D %_{j} x for j 6= i.

Theorem 5

DefinitionsN S_{1} and N S_{2} are equivalent.

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### Equivalence: Proof

N S_{1} ⇒ N S_{2} Suppose that u_{1}(x^{∗})u_{2}(x^{∗}) ≥ u_{1}(x)u_{2}(x) for all
x ∈ X. Then u_{i}(x^{∗}) > 0 for i = 1, 2. Now, if pu_{i}(x) > u_{i}(x^{∗}) for
some p ∈ [0, 1] then

pu_{i}(x)u_{j}(x^{∗}) > u_{i}(x^{∗})u_{j}(x^{∗}) ≥ u_{i}(x)u_{j}(x),
and hence pu_{j}(x^{∗}) > u_{j}(x).

N S_{2} ⇒ N S_{1} Suppose that Player i prefers x to x^{∗} and

u_{i}(x^{∗})
ui(x) ^{<}

uj(x)

uj(x^{∗})^{(⇔ u}^{i}^{(x}

∗_{)u}

j^{(x}^{∗}^{) < u}i^{(x)u}j(x)). Then there exists
0 < p < 1 such that ^{u}_{u}^{i}^{(x}^{∗}^{)}

i(x) < p < _{u}^{u}^{j}^{(x)}

j(x^{∗}) ^{so that u}^{i}^{(x}

∗_{) < pu}
i(x)
and u_{j}(x) > pu_{j}(x^{∗}), contradicting the definition of x^{∗}. Hence

ui(x^{∗})
ui(x) ^{≥}

u_{j}(x)

uj(x^{∗}) ^{so that u}^{i}^{(x}

∗_{)u}

j(x^{∗}) ≥ ui(x)uj(x).

The simplicity of the characterization (N) is attractive and accounts for the widespread application of the Nash solution.

### Axiomatic Approach (1)

A beauty of the Nash solution is that it is uniquely characterized by the following four simple axioms (properties).

INV (Invariance to Equivalent Utility Representations) Suppose
that the bargaining problem hU^{′}, d^{′}i is obtained from hU, di by the
transformations v^{′}_{i} = α_{i}v_{i}+ β_{i} for i = 1, 2, where α_{i}> 0 for
i = 1, 2. Then, for i = 1, 2,

fi(U^{′}, d^{′}) = αifi(U, d) + βi.

SYM (Symmetry) If the bargaining problem hU, di is symmetric,
i.e., d_{1} = d_{2} and (v_{1}, v_{2}) ∈ U if and only if (v_{2}, v_{1}) ∈ U , then

f_{1}(U, d) = f_{2}(U, d).

IIA (Independence of Irrelevant Alternatives) If hU, di and
hU^{′}, di are bargaining problems with U ⊆ U^{′} and f (U^{′}, d) ∈ U ,
then f (U^{′}, d) = f (U, d).

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### Axiomatic Approach (2)

PAR (Pareto Efficiency) Suppose hU, di is a bargaining problem
with v, v^{′}∈ U and v^{′}_{i}> vi for i = 1, 2. Then f (U, d) 6= v.

The axioms SYM and PAR restrict the behavior of the solution on single bargaining problems, while INV and IIA require the solution to exhibit some consistency across bargaining problems.

Theorem 6

The Nash solutionf^{N}(U, d) is the only bargaining solution that
satisfies INV, SYM, IIA, and PAR.

✞

✝

☎

Rm For the proof, see for example the chapter 2 of Osborne✆ and Rubinstein, Bargaining and Markets, 1990.

### Sketch of the Proof

1 _{f}^{N}(U, d) always exists and is unique.

2 _{f}^{N}(U, d) satisfies all four axioms.

3 No other solution, denoted f (U, d), satisfies all four axioms.

How to Prove Step 3

1 _{Let f}^{N}(U, d) = z and hU^{′}, 0i be the bargaining problem
obtained from hU, di by affine transformation such that
f^{N}(U^{′}, 0) = (1/2, 1/2).

2 Since both f^{N} and f satisfy INV, f^{N}(U, d) 6= f (U, d) if and
only if f^{N}(U^{′}, 0) 6= f (U^{′}, 0).

3 SYM, IIA, and PAR necessarily imply f (U^{′}, 0) = (1/2, 1/2)
and thereby f^{N}(U^{′}, 0) = f (U^{′}, 0), a contradiction.

✞

✝

☎

Rm See Figure 2.1 in Osborne and Rubinstein (1990).✆

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### Nash Program

The idea of relating axiomatic solutions to equilibria of strategic models (noncooperative games) was suggested by Nash (1953) and is now known as the “Nash program”.

Nash’s axiomatic model achieves great generality by avoiding any specification of the bargaining process; the solution defined by the axioms is unique, and its simple form is highly tractable, facilitating application.

However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’s axioms is in doubt.

The characteristics of such a strategic model clarify the range of situations in which the axioms are reasonable.