Both parties share the same expectation regarding the outcome at trial. Let us assume at first that the defendant concedes liability and the parties dispute only the amount of damages that the defendant should pay the plaintiff. Let D represent the damages that the court would award to the plaintiff at trial. From the perspective of the parties, D will be a random variable. Let D denote the expected value of D, where D > 0. Assume that D is distributed according to a continuous probability density function f(D). Assume also that the probability that D exceeds its mean equals the probability that it does not: that is, assume that Pr(D < D) = Pr(D > D) = Vi. We will later extend our analysis to include skewed distributions, for which Pr(D < D) 9* Vi, as well as disputes over liability.
Let J represent the total amount that a judgment at trial would require the defendant to pay to the plaintiff. Under an offer-of-settlement rule, this judgment may include a cost-shifting element in addition to damages: J may include a positive amount in order to reimburse the plaintiff for its litigation costs, or J may include a negative amount in order to reimburse the defendant for its litigation costs. The amount J will be uncertain because D is a random variable and cost-shifting under an offer-of-settlement rule will turn on the value of D. Even in the absence of any offer-of-settlement rule, J will be a random variable, because then J = D.
The parties may make ordinary offers to settle out of court without invoking any offer-of-settlement rules. As is customary in the bargaining theory literature, assume that the parties would make ordinary offers in a sequence of bargaining rounds. In particular, assume that before each stage of the litigation process, there would be a bargaining round in which one party would make an ordinary offer and the other party would either accept or reject the offer, the identity of the party making the offer would be determined randomly at the start of the round, and each party would be equally likely to be the offeror. If the offeree rejects the offer, then the parties would begin the next stage of the litigation process, in which the parties would incur another fraction of their litigation costs, and the parties would enter another round of bargaining.