Let us now turn to bargaining under an offer-of-settlement rule. We will start by setting forth a general result that will enable us to predict the outcome under any given offer-of-settlement rule. The subsequent analysis will use this basic lemma to derive the settlement outcome under several specific offer-of-settlement rules.
Suppose that one party may make one special offer at time t — 0, immediately before the first round of ordinary bargaining and before the first stage of litigation. First, let us assume that only one party may make such an offer and that this party will indeed mate this offer. Let S denote the amount proposed in this special offer. Later, in Section VI, we will extend our analysis to two-party rules, under which both parties can mate special offers.
The key point to recognize is that if the special offer were rejected, this rejection would not necessarily mean that the case would not be settled; the parties might still mate ordinary settlement offers and thereby reach a settlement. The parties will conduct these settlement negotiations, however, in the shadow of a different expected judgment than they would in the absence of a special offer. Under an offer-of-settlement rule, the special offer will affect the payoffs that the parties can expect if the case ends in a judgment at trial. Specifically, if the special offer S is rejected, then the judgment J may include a cost-shifting element, depending on how D compares with S. Thus, the judgment will be a function of S: let J(S) represent the judgment from trial, including the cost-shifting due under the relevant offer-of-settlement rule given the special offer S. Let В again denote the payoff to plaintiff from the ordinary bargaining game, in this case, the ordinary bargaining game that the parties would play if an offeree rejects a special offer. The outcome В is still a function of the expected judgment, but E[J] is no longer simply D, the expected damages, because J is now a function of S.