The preceding analysis has assumed that D is distributed about its mean such that Pr(D < D) = ¥t. That is, D < D and D > D are equally likely events. The distribution of D may instead be skewed, however, so that Pr(D < D) ^ ‘A. Suppose we relax our assumption that Pr(D < D) = lh. Lemma 1, which did not turn on this assumption, would still hold. We would, however, need to reconsider the specific conclusions that we derived from the application of that lemma to particular offer-of-settlement rules.
Under one-party offer-of-settlement rules with one-sided cost-shifting, it remains true that a party with an option to make a special offer would always choose to make such an offer. Furthermore, the offeror would still never be better off under a regime in which it could not make a special offer. It would no longer be true, however, that a special offer would always yield an outcome better than D for the offeror.

Under one-party offer-of-settlement rules with two-sided cost-shifting, the optimal special offer would no longer necessarily equal D. Suppose the applicable rule makes a special offer mandatory. Then we can show the following:
Proposition 7: If a one-party offer-of-settlement rule with two-sided cost-shifting requires a party to make a special offer, then:

(a) If Pr(D < D) > xh, then the settlement amount will be S* < D.

(b) If Pr(D < D) < ¥i, then the settlement amount will be S* > D.

Proof: See Appendix.
Cases in which the defendant contests liability present a particularly important category of stewed distributions. The distribution of possible payoffs for the plaintiffs in such cases generally feature a discrete probability mass at D = 0. With some technical modifications, we can extend our model to include payoff distributions with such discrete probability masses. If a defendant that prevails on liability can invoke cost-shifting under an offer-of-settlement rule, then this possibility would stew the distribution relevant for calculating the expected cost-shifting at trial. As suggested by Campbell (1987), the possibility that the defendant could prevail on liability may imply a median strictly below the mean.18 If so, then in these cases Pr(D < D) > 4i. It is also possible, however, that Pr(D < D) < when liability is in dispute.