Consider the effect of an increase in S upon the judgment J(S). In those cases in which S changes from an amount less than D to an amount greater than or equal to D, liability for Cd will shift to the plaintiff, or liability for Cp will shift back to the plaintiff, or both, depending on which costs shift under the offer-of-settlement rule. In all other cases, the increase in S will not affect J(S). Therefore, the expected judgment, denoted E[J(S)], taking the expectation with respect to different possible values of D, is a nonincreasing function of S.
Thus, В is now a function of the settlement amount S proposed in the special offer, because В is a function of E[J(S)]. Let B(S) denote this function, and let S* denote the optimal special offer for the offeror. We can now show the following lemma:

Lemma 1: If one party makes a special offer of settlement, then it can maximize its own payoff by choosing the S that equals the payoff to the plaintiff from the ordinary bargaining game that the parties would play if the offeree were to reject the special offer. That is:
If the offeree rejects S, then the parties immediately begin ordinary bargaining. The Nash bargaining solution, like that described in Proposition 1, results from this ordinary bargaining game. The same proof applies, where the expected judgment E[J(S)] substitutes as a generalization of the expected damages D.
Equality (2) follows from the behavior of the parties as they seek to maximize their payoffs. For example, suppose the plaintiff makes the special offer. Given (4), we know that B’(S) < 0, because E[J(S)] is a noninereasing function of S. The plaintiff knows that if it demands S, it will receive S immediately if S < B(S), because the defendant would accept such a special offer, but B(S) immediately if S > B(S), because the defendant would reject such a special offer. That is, the plaintiff receives min[S, B(S)]. Given that B’(S) < 0, the plaintiff could maximize this payoff, min[S, B(S)], by choosing S such that S = B(S). The plaintiff can improve the terms of a settlement by increasing the demand S, but the plaintiff would choose to do so only as long as the defendant would still agree to the demand. To raise it any higher would not only entail rejection but also reduce the plaintiffs payoff from such a rejection (because it increases the likelihood of unfavorable cost-shifting). Thus, the plaintiff chooses the S that is so large as to be barely acceptable to the defendant.