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.

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 plaintiffâ€™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).

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