Cooperative Game Models
10.1 Bargaining Problems
10.1.1 The Nash Bargaining Solution
We start with the definition of a two-person bargaining problem.1 Definition 10.1 A two-personbargaining problemis a pair .S; d/, where
.i/ S R2is a convex, closed and bounded set,2
.ii/ dD .d1; d2/2 S such that there is some point x D .x1; x2/2 S with x1 > d1
and x2> d2.
Sis thefeasible setand d is thedisagreement point. The interpretation of a bargaining problem .S; d/ is as follows. The two players bargain over the feasible outcomes in S. If they reach an agreement xD .x1; x2/2 S, then player 1 receives utility x1and player 2 receives utility x2. If they do not reach an agreement, then the game ends in the disagreement point d, yielding utility d1
to player 1 and d2to player 2. This is an interpretation, and the actual bargaining procedure is not spelled out.
For the example in Sect.1.3.5, the feasible set and the disagreement point are given by
SD fx 2 R2j 0 x1; x2 1; x2p
1 x1g; d1D d2D 0 :
See also Fig.1.7. In general, a bargaining problem may look as in Fig.10.1. The set of all such bargaining problems is denoted by B.
Fig. 10.1 A two-person bargaining problem
d
S
1We restrict attention here to two-person bargaining problems. For n-person bargaining problems and, more generally, NTU-games, see the Notes section at the end of the chapter and Chap.21.
2A subset of Rkis convex if with each pair of points in the set also the line segment connecting these points is in the set. A set is closed if it contains its boundary or, equivalently, if for every sequence of points in the set that converges to a point that limit point is also in the set. It is bounded if there is a number M > 0 such that jxij M for all points x in the set and all coordinates i.
10.1 Bargaining Problems 173
d (c)
S T
F (S, d )
F (T, d ) d
T S
F (T, d ) = F (S, d )
(d) d
S
P (S ) F (S, d )
(a)
S
d
(b)
Fig. 10.2 Illustration of the four conditions (‘axioms’) determining the Nash bargaining solution—cf. Theorem10.2. In (a) the Pareto optimal subset of S is the thick black curve. The bargaining problem .S; d/ in (b) is symmetric, and symmetry of F means that F should assign a point on the thick black line segment. In (c), which illustrates scale covariance, we took d to be the origin, and T results from S by multiplying all first coordinates by 2: then scale covariance implies that F1.T; d/ D 2F1.S; d/. The independence of irrelevant alternatives axiom is illustrated in (d)
We consider the following question: for any given bargaining problem .S; d/, what is a good compromise? We answer this question by looking for a map F W B ! R2 which assigns a feasible point to every bargaining problem, i.e., satisfies F.S; d/ 2 S for every .S; d/ 2 B. Such a map is called a (two-person)bargaining solution. According to Nash (1950), a bargaining solution should satisfy four conditions, namely: Pareto optimality, symmetry, scale covariance, and independence of irrelevant alternatives. We discuss each of these conditions in detail. The conditions are illustrated in Fig.10.2a–d.
For a bargaining problem .S; d/ 2 B, the Pareto optimal points of S are those where the utility of no player can be increased without decreasing the utility of the other player. Formally,
P.S/D fx 2 S j for all y 2 S with y1 x1, y2 x2, we have yD xg is thePareto optimal (sub)set of S. The bargaining solution F isPareto optimal if F.S; d/ 2 P.S/ for all .S; d/ 2 B. Hence, a Pareto optimal bargaining solution assigns a Pareto optimal point to each bargaining problem. See Fig.10.2a for an illustration.
174 10 Cooperative Game Models
A bargaining problem .S; d/2 B issymmetricif d1 D d2and if S is symmetric with respect to the 45ı-line through d, i.e., if
SD f.x2; x1/2 R2j .x1; x2/2 Sg :
In a symmetric bargaining problem there is no way to distinguish between the players other than by the arbitrary choice of axes. A bargaining solution is symmetricif F1.S; d/D F2.S; d/for each symmetric bargaining problem .S; d/ 2 B. Hence, a symmetric bargaining solution assigns the same utility to each player in a symmetric bargaining problem. See Fig.10.2b.
Observe that, for a symmetric bargaining problem .S; d/, Pareto optimality and symmetry of F would completely determine the solution point F.S; d/, since there is a unique symmetric Pareto optimal point in S.
The condition of scale covariance says that a bargaining solution should not depend on the choice of the origin or on a positive multiplicative factor in the utilities. For instance, in the wine division problem in Sect.1.3.5, it should not matter if the utility functions were Nu1.˛/ D a1˛C b1 and Nu2.˛/ D a2p
˛C b2, where a1; a2; b1; b2 2 R with a1; a2 > 0. Saying that this should not matter means that the final outcome of the bargaining problem, the division of the wine, should not depend on this. One can think ofNu1;Nu2 expressing the same preferences about wine as u1; u2in different units.3Formally, a bargaining solution F isscale covariant if for all .S; d/2 B and all a1; a2; b1; b22 R with a1; a2> 0we have:
F
f.a1x1C b1; a2x2C b2/2 R2j .x1; x2/2 Sg; .a1d1C b1; a2d2C b2/ D .a1F1.S; d/C b1; a2F2.S; d/C b2/ : For a simple case, this condition is illustrated in Fig.10.2c.
The final condition is regarded as the most controversial one. Consider a bargaining problem .S; d/ with solution outcome z D F.S; d/ 2 S. In a sense, zcan be regarded as the best compromise in S according to F. Now consider a smaller bargaining problem .T; d/ with T S and z 2 T. Since z was the best compromise in S, it is should certainly be regarded as the best compromise in T: z is available in T and every point of T is also available in S. Thus, we should conclude that F.T; d/ D z D F.S; d/. As a less abstract example, suppose that in the wine division problem the wine is split fifty-fifty, with utilities .1=2;p
1=2/. Suppose now that no player wants to drink more than 3=4 liter of wine: more wine does not increase utility. In that case, the new feasible set is
T D fx 2 R2j 0 x1 3=4; 0 x2p
3=4; x2p
1 x1g :
3The usual assumption is that the utility functions are expected utility functions, which uniquely represent preferences up to choice of origin and scale.
10.1 Bargaining Problems 175
According to the argument above, the wine should still be split fifty-fifty: T S and .1=2;p
1=2/ 2 T. This may seem reasonable but it is not hard to change the example in such a way that the argument is, at the least, debatable. For instance,suppose that player 1 still wants to drink as much as possible but player 2 does not want to drink more than 1/2 L. In that case, the feasible set becomes
T0D fx 2 R2j 0 x1 1; 0 x2p
1=2; x2p
1 x1g ;
and we would still split the wine fifty-fifty. In this case player 2 would obtain his maximal feasible utility, and .1=2;p
1=2/ no longer seems a reasonable compromise since only player 1 makes a concession.
Formally, a bargaining solution F isindependent of irrelevant alternatives if for all .S; d/; .T; d/2 B with T S and F.S; d/ 2 T, we have F.T; d/ D F.S; d/. See Fig.10.2d for an illustration.
The theorem below says that these four conditions determine a unique bargaining solution FNash, defined as follows. For .S; d/ 2 B, FNash.S; d/is equal to the unique point z2 S with zi difor iD 1; 2 and such that
.z1 d1/.z2 d2/ .x1 d1/.x2 d2/for all x2 S with xi di, iD 1; 2 : The solution FNash is called theNash bargaining solution. The result of Nash is as follows.
Theorem 10.2 The Nash bargaining solution FNashis the unique bargaining solu-tion which is Pareto optimal, symmetric, scale covariant, and independent of irrelevant alternatives.
For a proof of this theorem and the fact that FNashis well defined—i.e., the point z above exists and is unique—see Chap.21.
Example 10.3 In this example we illustrate the role of the conditions in Theo-rem10.2. In fact, we show how the proof of this theorem (cf. Chap.21) works in an example. Consider the bargaining problem .S; d/, where S D f.x1; x2/2 R2 j 0 x1 2; 0 x2 4 x21g and d D .0; 1/. The Nash bargaining solution outcome is obtained by solving the problem max0x12 x1.3 x21/, which yields the point .1; 3/. Alternatively, consider the bargaining problem .S0; d0/, obtained by subtracting 1 from the second coordinates of the points in S, including d, yielding
S0D f.x1; x2/2 R2j 0 x1 2; 1 x2 3 x21g; d0D .0; 0/ : Next, consider the bargaining problem .S00; d00/, obtained from .S0; d0/by dividing all second coordinates by 2, yielding
S00D f.x1; x2/2 R2j 0 x1 2; 1
2 x2 3 2
1
2x21g; d00D .0; 0/ :
176 10 Cooperative Game Models
Fig. 10.3 Illustrating Example10.3
S T
(0, 0)
(1, 1)
( 1, 1) (3, 1)
( 1, 3)
The Pareto optimal boundary of S00 is described by the function f .x1/D 32 12x21 for 0 x1 2. At x1 D 1 the derivative of this function is equal to 1, so that the straight line through the point .1; 1/ with slope 1 is tangential to S00, i.e., the set S00 is below this line. Now consider the bargaining problem .T; .0; 0// with T the triangle and inside with vertices . 1; 1/, .3; 1/, and . 1; 3/; see Fig.10.3.
The bargaining problem .T; .0; 0// is symmetric, so that by symmetry and Pareto optimality of the Nash bargaining solution, the outcome is the point .1; 1/. This point is also in S00, and moreover, S00 is a subset of T, so that by independence of irrelevant alternatives the point .1; 1/ is also the Nash bargaining solution outcome of .S00; .0; 0//. By scale covariance, this implies that the Nash bargaining solution outcome of .S0; .0; 0//is the point .1; 2/. Again by scale covariance, we obtain that the Nash bargaining solution outcome of .S; .0; 1// is the point .1; 3/. We have reached this result by using only the properties of the Nash bargaining solution in Theorem10.2, and not the formula. Observe that the result is in accordance with
what we established by direct computation.