By making a special offer under a one-sided cost-shifting rule, the defendant can also obtain a settlement for less than D, the damages expected at trial. The defendant can do so because a special offer of D under such a rule would give the defendant a threat point to its advantage: if S = D, then each side would be equally likely to bear Cd under the cost-shifting rule, but only the plaintiff could bear Cp. Given the threat of an outcome worse than D for the plaintiff, the plaintiff would be willing to accept a special offer below D.

Example: Consider our numerical example again. In that example, D is uniformly distributed in the interval (60, 140), and therefore:
for 60 < X < 140. Using (6) and our specific values for D, Cd, and Cp to substitute in (5) and solving for S* yields S* = 76. Thus, S* is not only less than the expected damages, D = 100, but also less than the settlement under ordinary bargaining in the absence of a special offer, B(D) = 80.

B. Plaintiff Makes the Special Offer

Suppose now that the plaintiff makes the special offer to settle for an amount S. Consider an offer-of-settlement rule shifting costs only in favor of the plaintiff. Specifically, suppose the defendant would be obliged to pay Cp if the damages D are greater than S. The same intuition set forth above applies by analogy to this case. By the same reasoning used to prove Proposition 2, we can show the following:
Proposition 3: If an offer-of-settlement rule with one-sided cost-shifting permits the plaintiff to make a special offer, then:

(a) The plaintiff will always choose to exercise this option, and the defendant will accept the special offer.
(b) The settlement amount S* will solve:
(c) The settlement amount will be (i) at least the expected settlement amount without the offer-of-settlement rule and (ii) strictly greater than the expected damages D.