In this section, we allow up to two special offers, Sp and Sd, by the plaintiff and by the defendant, respectively. Specifically, first one party may or must make a special offer. If the first party does not mate a special offer, or if the second party rejects the special offer, then the second party may or must make a special offer. Assume that the second party can reject the first special offer and mate a second special offer immediately. If neither party accepts a special offer, then the parties immediately enter the first round of ordinary bargaining.
Given one or both of these special offers, the offer-of-settlement rule then provides for possible cost-shifting if the parties litigate to judgment. If D > Sp, then the defendant reimburses the plaintiff for its litigation costs. If D ^ Sd, then the plaintiff reimburses the defendant for its litigation costs. If Sd < D < Sp, then the plaintiff pays the defendant’s costs if and only if the plaintiff’s special offer triggers two-sided cost-shifting, and the defendant pays the plaintiff’s costs if and only if the defendant’s special offer triggers two-sided cost-shifting.
We will refer to all offer-of-settlement rules fitting the description set forth above as “two party” rules. Let St denote the first special offer, and if the second party rejects that offer and mates a second special offer, let Sj denote the second special offer. That is, let S; represent the special offer by party i, whether party i is the plaintiff or the defendant, and let S;* denote the optimal special offer for party i.
The structure we have described includes many possible two-party rules, which vary with respect to whether:
(1) the plaintiff or the defendant is the first party to move,
(2) the rule requires or merely permits the first party to make a special offer,
(3) the first special offer triggers one-sided or two-sided cost-shifting,
(4) the rule requires or merely permits the second party to make a special offer, and
(5) the second special offer triggers one-sided or two-sided cost-shifting.
Thus, there are thirty-two possible two-party rules depending upon the five binary choices listed above. As it turns out, however, none of these choices affect the outcome. All these two-party rules prove to be equivalent and lead to a settlement for the amount D:
Proposition 6: Under any two-party offer-of-settlement rule, the settlement will be S,* = D.
Proof: See Appendix.
Remark: The basic intuition for this result is as follows. Once a party can invoke the offer-of-settlement rule, it can do at least as well as it can under a settlement for the amount D. If both sides can invoke the rule, then the outcome must be a settlement for the amount D.