B. Basic Models of Settlement Bargaining
10.1 Perfect Information
We start with the perfect-information version. In keeping with the earlier discussion, the following model emphasizes final outcomes and suppresses bargaining detail. While the analysis below may seem like analytical overkill, it will allow us to structure the problem for later, more complex, discussions in this Part and in Part C.
The players are P and D; further, assume that the level of damages, d, is commonly known and that D is fully liable for these damages. Court costs are kP and kD, and each player is responsible for his own court costs. Each player has an individual action he can take that assures him a particular payoff. P can stop negotiating and go to court; thus the payoff to P from trial is πPt = d - kP (note the superscript t for trial) and the payoff to D from trial is πDt = d + kD. Under the assumption that d - kP > 0, P has a credible threat to go to trial if negotiations fail. For the purposes of most of this paper (and most of the literature) we assume this condition to hold (the issue of it failing will be discussed in Section 16.1). By default, D can “assure” himself of the same payoff by stopping negotiations, since P will then presumably proceed to trial; no other payoff for D is guaranteed via his individual action. The pair (πDt, πPt) is called the threat or disagreement point for the bargaining game (recall that D’s payoff is an expenditure).
What might they agree on? One way to capture the essence of the negotiation is to imagine both players on either side of a table, and that they actually place money on the table (abusing the
card-game story from earlier, this is an “ante”) in anticipation of finding a way of allocating it. This means that D places d + kD on the table and P places kP on the table. The maximum at stake is the sum of the available resources, d + kP + kD, and therefore any outcome (which here is an allocation of the available resources) that does not exceed this amount is a possible settlement.
P’s payoff, πP, is his bargaining outcome allocation (bP) minus his ante (that is, πP = bP - kP).
D’s payoff, πD, is his ante minus his bargaining outcome allocation (bD); thus D’s cost (loss) is πD
= d + kD - bD. Since the joint bargaining outcome (bP + bD) cannot exceed the total resources to be allocated (the money on the table), bP + bD# d + kD + kP, or equivalently, P’s gain cannot exceed D’s loss: πP = bP - kP # d + kD - bD = πD. Thus, in payoff terms, the outcome of the overall bargaining game must satisfy: 1) πP #πD; 2) πP $πPt and 3) πD#πDt. That is, whatever is the outcome of bargaining, it must be feasible and individually rational. In bargaining outcome terms this can be written as: 1N) bP+ bD# d + kD + kP; 2N) bP$ d and 3N) bD$ 0. For the diagrams to come, we restate (2N) as bP - d $ 0.
Figure 1(a) illustrates the settlement possibilities. The horizontal axis indicates the net gain (-πD) or net loss (that is total expenditure, πD) to D. The vertical axis indicates the net gain (πP) or net loss (-πP) to P. The sloping line graphs points satisfying πP = πD while the region to the left of it involves allocations such that πP < πD. The best that D could possibly achieve is to recover his ante d + kD and get kP, too; this is indicated at the right-hand lower-end of the line at the point (kP, -kP), meaning D has a net gain of kP and P has a net loss of kP. At the other extreme (the upper- left end point) is the outcome wherein P gets all of d + kP + kD, meaning P has a net gain of d + kD which is D’s net loss.
P’s gain (πP)
D’s gain (−πD) d + kD
-(d + kD)
d - kP
-(d - kP) kP -kP disagreement
point
Figure 1: Settlement Under Perfect Information
(a) (b)
settlement frontier
bP - d
bD kP + kD
kP + kD settlement frontier adjusted for disagreement point
NBS
Note also that points below the line represent inefficient allocations: this is what is meant by “money left on the table.” The disagreement point (-(d + kD), d - kP) draws attention (observe the thin lines) to a portion of the feasible set that contains allocations that satisfy inequalities (2) and (3) above: every allocation in this little triangle is individually rational for both players. The placement of the disagreement point reflects the assertion that there is something to bargain over;
if the disagreement point were above the line πP = πD then trial is unavoidable, since there would be no settlements that jointly satisfy (2) and (3) above. This triangular region, satisfying inequalities (1), (2) and (3), is the settlement set (or bargaining set) and the endpoints of the portion of the line πP = πD that is in the settlement set are called the concession limits; between the concession limits (and including them) are all the efficient bargaining outcomes under settlement, called the settlement frontier. Another way that this frontier is referred to is that if there is something to bargain over (i.e., the disagreement point is to the left of the line), then the portion of the line between the
concession limits provides the bargaining range or the settlement range, which means that a range of offers by one party to the other can be deduced that will make at least one party better off, and no party worse off, than the disagreement point.
Figure 1(b) illustrates the settlement set as bargaining outcomes, found by subtracting the disagreement point from everything in the settlement set, so now the disagreement point is the origin. Doing this helps adjust the region of interest for asymmetries in the threats that P and D can employ. This leaves any remaining asymmetries in power or information in the resulting diagram.
In the case at hand, the only power difference might appear in the difference between the costs of going to trial; other power differences such as differences in risk preferences or patience will be discussed in Section 17.1, while informational differences will be discussed in the sections on asymmetric information in Parts B and C.
Notice that, in view of (2N) and (3N), the vertical (non-negative) axis is labeled bP - d while the horizontal (non-negative) axis is labeled bD. Moreover, since aggregate trial costs determine the frontier in Figure 1(b), the bargaining problem here is symmetric. The Nash Bargaining Solution (NBS) applies in either picture, but its prediction is particularly obvious in Figure 1(b): recalling the discussion in Section 8.2, requiring the solution to be efficient (axiom 2) and that, when the problem is symmetric the solution is, too (axiom 4), means that splitting the saved court costs is the NBS in Figure 1(b). Thus, to find the NBS in Figure 1(a), we add the disagreement point back into the solution from Figure 1(b). Therefore, under the NBS, P’s payoff is πP = d - kP + (kP + kD)/2 = d + (kD - kP)/2. D’s net position is -(d + kD) + (kP + kD)/2. In other words, D’s payoff (his expenditure, πD) is πD = d + (kD - kP)/2. The result that players should “split the difference” relative to the disagreement point is always the NBS solution for any bargaining game with payoffs in monetary
terms.
Observe that if court costs are the same, then at the NBS P and D simply transfer the liability d (that is, πP = πD = d + 0/2 = d). If D’s court costs exceed P’s, P receives more from the settlement than the actual damages, reflecting his somewhat stronger relative bargaining position embodied in his threat with respect to the surplus that P and D can jointly generate by not going to court. A similar argument holds for P in the weaker position, with higher costs of going to court: he settles for less than d, since kD - kP < 0.