Given that the parties are risk-neutral, the defendant seeks to minimize its expected costs (its litigation costs plus any payment to the plaintiff), and the plaintiff seeks to maximize its expected payoff (any payment from the defendant minus its litigation costs). Assume that the parties have no mechanism (such as a repeat player might develop by cultivating a reputation for intransigence) that would enable them to bind themselves to a particular bargaining strategy. That is, neither party can commit credibly to a strategy of intransigence, which would enable it to obtain a larger fraction of the gains from settlement. Thus, each party would accept an offer if and only if it were unable to improve its expected payoff by rejecting the offer instead.
For simplicity, assume that an offeror under any applicable offer-of-settlement rule makes a special offer at time t = 0 before the first stage of the litigation process, before the parties incur any litigation costs, and before the first round of ordinary bargaining. We can extend the analysis to include the case in which special offers can be made at later points in time.10 Given the opportunity to mate a special offer, the offeror would choose to mate an offer if and only if it were unable to improve its expected payoff by doing otherwise.

The structure of the bargaining game described above, including n, D, f(D), and cd and cp‘ for i = 1, …, n, as well as any applicable offer-of-settlement rule, is common knowledge to the participants. This assumption ensures that settlement occurs with certainty and always occurs before the parties incur any litigation costs.11 The assumption of perfect information allows us to focus on the issue that we are interested in studying: the effect of various offer-of-settlement rules on settlement amounts.
It will prove useful to refer to a more specific example throughout our analysis. For this purpose, suppose that Cp = 60, Cd = 20, and D is uniformly distributed in the interval (60, 140), so that D = 100. We will use this numerical example to illustrate our results below.