3 (Bargaining Power) Suppose that, in agreement, player 1 always gets 8 times what player 2 gets in the controlled bargaining game (A,0), where

8 is some positive number. Then, the number 8 measures some kind of power of player 1 over player 2, and therefore, the number 8 is defined as the relative bargaining power of player 1.

This definition of bargaining power accords well with that of Chamberlain and Kuhn (1965: p.170), who defined ‘bargaining power as the ability to secure another's agreement on one's own terms.’

Let Z be a vector of all variables, such as the time preference rate of the players, the players' belief that the opportunity of gain will evaporate or be snatched by a third party, the difference in players' skill of negotiation or coalitional strength of members, the difference between the degrees of free rider problem within each coalition and other unaccounted factors in the political environment etc., that influence the bargaining

outcome but do not belong to the choice set of the bargainers. Then, we can infer, on the basis of previous studies referred above, that

8 = 8{Z).

The functional dependence of relative bargaining power of player 1 on Z acknowledges that the relative bargaining power of a player is essentially a dynamic concept. It may change as the exogenous environment changes.

In a given environment Z, a value of 8 equal to unity implies that the two players are equally powerful, neither is favoured against the other, and 8 > 1 implies that player 1 possesses more bargaining power than player 2, and vice versa.

We normalize the measure of bargaining power by defining

© 1

8

1 + 8 , and 0 2 1 1 + 8'

Then, the parameter 0,(Z ) satisfies 0 < 0,(Z ) < 1, and ^ .0 - ( Z ) = 1; and therefore, it can be called the normalized bargaining power of player i.

With this definition of bargaining power, we can proceed on to the extension of Nash's bargaining solution in the presence of unequal bargaining power of the players. The following important theorem in this direction was proved by Roth (1979).9

Theorem 6.2 (Roth, 1979: Theorem 3). For each strictly positive vector 0 , such that ^ . 0 , = 1, there is a unique solution /j. satisfying the axioms of feasibility, invariance and independence, such that /i(A,0) = 0 where A = j x e R2 Xi > 0 ,^ X , < lj. For any bargaining game (91,IT*), n{%Yld) = II e ^K, such that T1 > fl^ and

(n, - nfr(n2 - nj) ! > (n, - nf f (n2 - n'J

for all n e SK such that fl > IT* and IT * IT*.

Proof: See Roth (1979: 16-17), Binmore (1987a: 34-37), and Appendix-6B.

The first part of the theorem states that if the players possess different weights given by the vector 0 , and bargain over a ‘pie’ of size unity, represented by the symmetric bargaining set A with disagreement payoff equal to zero, then player f s share in the unit pie is just 0 ,. This explains why 0- has been defined as the bargaining

power of player i (see also Binmore, 1987a and 1987b). If all players have equal bargaining power, then the theorem implies that each of the players will get an equal share of the pie. In particular, if the set A is defined for a ‘pie’ of size n, that is

A = j x gR2 Xi > 0 , £ X 4. < , and 0 ,'s are suitably normalized so that 0 . = n, then

each player will get one "pie". This is the standard Nash bargaining solution for a symmetric game, which is also called the symmetric Nash bargaining solution. Thus, the theorem 6.1 (Nash's theorem) is a special case of Theorem 6.2 in which the players hold equal bargaining power.10

The second part of the theorem shows that in any bargaining game with predetermined disagreement payoffs, and a compact and convex bargaining set the solution to the bargaining problem that satisfies axiom 1-3 is the one that maximizes the asymmetric Nash product over the bargaining set. The strict inequality implies that the solution is unique. That is, the payoff distribution that maximizes the asymmetric Nash product and the solution that satisfies the "desirable" properties are one and the same. One implies the other.

A note of clarification is warranted here. It may appear that the solution to the bargaining game (SR, n*) that satisfies axioms 1-3 does not necessarily satisfy axiom 5 - the axiom of symmetry. Binmore (1987a), for example, has explicitly stated that the asymmetric Nash solution satisfies axiom 5 if and only if the players have equal bargaining powers. But this would mean that the solution depends on the way players are represented if players do not have equal bargaining powers. If a player's payoffs are now measured along the x-axis, then she will receive a different payoff at the solution of the game than she would obtain had her payoffs were measured along the y-axis. This situation, certainly, is not satisfying.

However, once we isolate the axiom of symmetry from the hangover of identical bargaining power of the players and treat them independently it can be easily seen that the "asymmetric" Nash solution is in fact symmetric. We will show this result as a Corollary to Roth's Theorem. First, to highlight the role of bargaining power in the mathematical description of a bargaining game we state the following definition:

Definition 6.4 (Generalized Bargaining game). A triplet (9t,IT*,0) is defined as