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### Lecture 15: Bargaining and Cooperative Game

Yosuke YASUDA

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

January TBA, 2015

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### 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, %2i: 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.

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### 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 fN(U, d), if

(u1− d1)(u2− d2) ≥ (u1− d1)(u2− d2) for all u ∈ U. (N) That is,

fN(U, d) = arg max

(d1,d2)≤(v1,v2)∈U

(v1− d1)(v2− d2).

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### 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 S2)

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 S1 and N S2 are equivalent.

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### Equivalence: Proof

N S1 ⇒ N S2 Suppose that u1(x)u2(x) ≥ u1(x)u2(x) for all x ∈ X. Then ui(x) > 0 for i = 1, 2. Now, if pui(x) > ui(x) for some p ∈ [0, 1] then

pui(x)uj(x) > ui(x)uj(x) ≥ ui(x)uj(x), and hence puj(x) > uj(x).

N S2 ⇒ N S1 Suppose that Player i prefers x to x and

ui(x) ui(x) <

uj(x)

uj(x)(⇔ ui(x

)u

j(x) < ui(x)uj(x)). Then there exists 0 < p < 1 such that uui(x)

i(x) < p < uuj(x)

j(x) so that ui(x

) < pu i(x) and uj(x) > puj(x), contradicting the definition of x. Hence

ui(x) ui(x)

uj(x)

uj(x) so that ui(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.

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### 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, di is obtained from hU, di by the transformations vi = αivi+ βi for i = 1, 2, where αi> 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., d1 = d2 and (v1, v2) ∈ U if and only if (v2, v1) ∈ U , then

f1(U, d) = f2(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 vi> 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 solutionfN(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.

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### Sketch of the Proof

1 fN(U, d) always exists and is unique.

2 fN(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 fN(U, d) = z and hU, 0i be the bargaining problem obtained from hU, di by affine transformation such that fN(U, 0) = (1/2, 1/2).

2 Since both fN and f satisfy INV, fN(U, d) 6= f (U, d) if and only if fN(U, 0) 6= f (U, 0).

3 SYM, IIA, and PAR necessarily imply f (U, 0) = (1/2, 1/2) and thereby fN(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.

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