THE TERMS OF SETTLEMENT: BARGAINING WITH AND WITHOUT SPECIAL OFFERS 4

Suppose instead that the defendant makes the special offer. By similar reasoning, the defendant would have to pay max[S, B(S)] and could minimize the plaintiffs payoff by choosing S = B(S). Thus, (2) holds whether the plaintiff or the defendant mates the special offer.13 Finally, (3) follows from (2) and (4).
Remark: Figure 1 illustrates why S = B(S) is the optimal offer for either party to make.

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The function B(S) represents the plaintiffs payoff if the offeree rejects the special offer and the parties instead settle through ordinary bargaining. B(S) is a nonincreasing function of S, because a larger S increases the plaintiffs expected litigation costs under an offer-of-settlement rule. The 45-degree line represents S, which is the plaintiffs payoff if the offeree accepts the special offer S.
If the plaintiff mates the special offer S, then the defendant would choose S if S < B(S) and B(S) otherwise. Thus, the plaintiffs payoff as a function of S would be min[S, B(S)], represented by the lower envelope of S and B(S). The plaintiff would maximize this function, min[S, B(S)], by choosing the S identified by the intersection S = B(S). Similarly, if the defendant mates the special offer S, then the plaintiff would choose S if S > B(S) and B(S) otherwise. Thus, the plaintiff’s payoff as a function of S would be max[S, B(S)], represented by the upper envelope of S and B(S). The defendant would minimize this function, max[S, B(S)], by choosing the S identified by the intersection S = B(S).