Suppose, however, that (f believes) c has bargaining power of 0.8, while all other parties (including f) have equal bargaining power so that, in coalition II.1, the total bargaining power is 1. In coalition II.1 then, the payoff schedule would be ya = yb = yd = yf = 1.25, yc = 20. In coalition I.2, by contrast, the payoffs implied by equal bargaining power would be ye = yf = yg = 3. Thus, if f were to shift from I.2 to II.1, he (believes he) would be worse off. Accordingly, he would refuse to shift and (supposing f reasons in this way) coalition structure I is stable.
This is an instance of what I have elsewhere (McCain, 2013, pp. 94, 122) called the extended core. Should this somewhat relaxed concept of stability be chosen rather than the conventional concept of domination discussed in the previous subsection? There is, of course, no mathematically conclusive answer to this question. Only plausible arguments can be given. (Compare, for example, von Neumann and Morgenstern, 1944/2004, p. 506.) In that framework, the answer may depend on our understanding of bargaining power and its determinants. There is no widely received theory here! If we think of bargaining power as intrinsic to the person - superior ability to dissemble and bluff, for example, or optimism (as in Brandenburger and Stuart, 2007, p. 542) - then the extended core may seem to be the appropriate concept. If we think of bargaining power as something that arises from the coalition and its situation as a whole, then we might doubt that one person’s bargaining power could be so great as to prevent the formation of a coalition that would otherwise be mutually beneficial.
Recall that bargaining power arises from threats, in that the other agents make concessions to prevent the carrying out of the threat. This will depend on the credibility of the threat. In coalition II.1, for example, c might threaten to withdraw, reducing all payoffs to 1. In a noncooperative framework, this threat would not be credible, but here we are assuming ideal rationality, and ideally rational agents will make such commitments if they expect to be better off as a result. But the commitment that makes c better off in this case will be a commitment NOT to withdraw. That is c, if rational, will make commitments that will voluntarily reduce his bargaining power so as to make the coalition II.1 possible.
On the basis of this plausible argument, the rest of this section will assume that a stable coalition structure will be one that is undominated, using the concept of domination as adapted in the previous subsection to FCS games. While this formulation does not support sweeping theorems on the uniqueness of pay schedules and similar matters, it does allow us to recover and somewhat extend some familiar ideas from neoclassical economics.