We will now apply the general principle set forth in Lemma 1 to derive the settlements that emerge under particular offer-of-settlement rules. Offer-of-settlement rules may provide for cost-shifting only in one direction, or they may provide for two-sided cost-shifting. In this section, we assume that the offer-of-settlement rule would only shift costs in favor of the party making the special offer. For example, suppose the defendant makes a special offer S. Under an offer-of-settlement rule shifting costs only in favor of the defendant, the plaintiff would be obliged to pay Cd to the defendant if the damages D are less than or equal to S.

A. Defendant Makes the Special Offer
First, we will consider a one-sided cost-shifting rule that allows the defendant to make one special offer. We will show that whether the rule requires such an offer or, like Rule 68, merely permits one, the outcome will be the same. Subsequently we also address the case in which the plaintiff may make the only special offer.
Proposition 2: If an offer-of-settlement rule with one-sided cost-shifting permits the defendant to make a special offer, then:

(a) The defendant will always choose to exercise this option, and the plaintiff will accept the special offer.

(b) The settlement amount S* will solve:
(c) The settlement amount will be (i) no greater than the expected settlement amount without the offer-of-settlement rule and (ii) strictly less than the expected damages D.
Proof: See Appendix.
Remark: One-sided cost-shifting can never hurt an offeror who invokes an offer-of-settlement rule: the offeror might win reimbursement but would never have to reimburse the offeree. Thus, given the option of making a special offer under a one-sided cost-shifting rule, the defendant would always choose to make such an offer. Therefore, when cost-shifting is onesided, it makes no difference whether the rule requires or merely permits the defendant to make a special offer. Furthermore, because the defendant cannot harm its bargaining position by making a special offer under such an offer-of-settlement rule, the settlement it can obtain by invoking the rule will never be less favorable to the defendant than the expected settlement in the absence of such a rule.