Proof. The proof is essentially the same as that of Spier (1989, 1992) for closely related models. The proposition is established by induction starting with a game of length T = 2.
Verification Step: T = 2
Begin with the plaintiff’s strategy: decision rules for accepting and rejecting values of the first-period settlement proposal S1. Three cases must be considered. First, if
the expected net present value of the continuation game following rejection exceeds the value of settling at the first-period proposal S1, the plaintiff’s optimal strategy
must be to reject S1. Second, if the expected net present value of the continuation
game is exceeded by the value of settling at S1, the plaintiff’s optimal strategy must
be to accept S1. The remainder of the plaintiff’s equilibrium strategy concerns the
Let S2∗(S1) denote the equilibrium settlement proposal that the defendant would
make in the second period of the game if the first-period proposal S1 were rejected.
A plaintiff is indifferent between accepting and rejecting the first-period settlement proposal when Up(S1) = max{Up(S2∗(S1)), Wp(x)}: that is, when settlement at S1
yields the same payoff as the better of settlement in the continuation game, or receipt of a trial verdict. Similar to the assumption made in proving Proposition 1, break ties by assuming a plaintiff indifferent between settlement and trial chooses to settle in some period.
The tie-breaking assumption compels first-period settlement when indifference concerns only settlement at S1 or rejection in favor of an eventual trial verdict: i.e.
when Up(S1) = Wp(x) > Up(S2∗(S1)). When indifference concerns settlement across
multiple periods, i.e. Up(S1) = Up(S2∗(S1)) ≥ Wp(x), the plaintiff’s preferences re-
quire settlement in some period, but do not alone specify which period. Timing of acceptance in this aspect of the plaintiff’s equilibrium strategy is dictated by the PBE concept, which restricts the plaintiff’s strategy to prescribe a pattern of settlement for which the defendant’s strategy is sequentially rational.50 This point will be revisited
after the interior-solution to the defendant’s equilibrium strategy is derived.
For the defendant’s strategy, begin by considering two potential boundary solu- tions: (i) a solution in which no types of plaintiff settle, and (ii) a solution in which all types of plaintiff settle. There exist no equilibria of the first type, as a positive measure of plaintiff types must settle in every equilibria. This result was established for a single-period game in Proposition 1, and generalizes to a two-period game be- cause the continuation game reached in the second period is itself just a single-period game and therefore characterized by Proposition 1.
50This is intuitively comparable to a randomized equilibrium, where randomization is only possible
as a consequence of the randomizing player’s indifference between actions, and where behavior of the randomizing player is defined by the need to make the opposing player’s actions optimal.
Next consider a solution in which all types of plaintiff eventually settle. By the first proposition of Lemma 1, no equilibrium proposal can be more valuable to a plaintiff than S1∗: Up(S1∗) ≥ . . . ≥ Up(ST∗) in any equilibrium. Thus, if all plaintiff
types eventually settle, it must be that S1∗ is as valuable as the expected net present value of a trial verdict to the highest-type plaintiff: Up(S1B) = Wp(x).51 Substituting
terms and solving for S1B yields the equilibrium settlement proposal for the boundary solution in which all types of plaintiff settle:
S1B = δ2(πx − kp) − δcp. (9)
All types of plaintiff immediately accept S1B under Assumption 3, so the value of the boundary solution to the defendant is Ud(S1B).52
With the boundary solution out of the way, consider the interior solution where some but not all types of plaintiff settle. A convenient way to construct the equilib- rium is by working backwards from the continuation game following rejection of S1.
Since some types of plaintiff never settle by assumption, the second-period continua- tion game is necessarily reached with positive probability in an interior equilibrium.
Assumption 2 restricts the plaintiff’s strategy to be monotone in type, so in any interior solution there exists some cutoff type x2(S1) under which types of plaintiff
accept S1and above which types of plaintiff reject S1. Note also that since Assumption
4 specifies the population distribution of plaintiff types ρ(x) = f (x) to be uniform on [x, x], existence of a cutoff type means the distribution of plaintiff types remaining in the second-period continuation game, ρ(x|S1) = f (x|x > x2(S1)), is simply the
uniform distribution with support [x2(S1), x].
51All plaintiff types would also settle for U
p(S1∗) > Wp(x), but this would not be an equilibrium
as the defendant could profitably deviate in the direction of a lower first-period proposal.
52Assumption 3 is a refinement ruling out unintuitive boundary equilibria where, e.g., all types of
Optimal play in the continuation game can be expressed as a function of S1 alone.
The continuation game is a one-period game to which Proposition 1 applies, simplified by the assumption that potential damages are uniform on [x2(S1), x]. Substituting
the distributional specification into the interior solution to Proposition 1 yields the optimal continuation game settlement proposal as a function of S1:
S2∗(S1) = δ(πx2(S1) + kd). (10)
The cutoff type at which the equilibrium continuation-game proposal S2∗(S1) is just
rejected, x3(S1), is also given by Proposition 1 with x2(S1) substituting for x:
x3(S1) = π−1(δ−1S2∗(S1) + kp). (11)
Finally, the value of x2(S1) can be derived from optimal play in the continuation
game. Note that since some but not all types of plaintiff settle in an interior equilib- rium, both propositions of Lemma 1 apply to the sequence of equilibrium settlement proposals: i.e. Up(S1∗) ≥ Up(S2∗) and Up(S1∗) ≤ Up(S2∗). Combining the Lemma 1 im-
plication that Up(S1) = Up(S2∗(S1)) ⇐⇒ S2∗(S1) = δ−1S1+ cp with the specification
of S2∗(S1) in equation (10) allows x2(S1) to be solved in terms of S1:
x2(S1) = π−1(δ−2S1 + δ−1cp− kd) (12)
The above terms can be used to represent the defendant’s problem as a function of S1 alone. As expressed in equation (13), the defendant chooses a value of S1 in order
to maximize the sum of (i) the value of settlement at S1, weighted by the measure of
plaintiff types that accept S1, (ii) the value of settlement at S2∗(S1), weighted by the
present value of a trial verdict conditional on the plaintiff being a type that rejects both S1 and S2∗(S1), weighted by the measure of types that reject both proposals:
Vd(S1) = P [x ≤ x2(S1)] Ud(S1) + P [x2(S1) < x ≤ x3(S1)] Ud(S2∗(S1)) + P [x > x3(S1)] E [Wd(x)|x > x3(S1)] (13) = x2(S1) − x x − x [−S1− cd] +x3(S1) − x2(S1) x − x [−δS ∗ 2(S1) − cd− δcd] +x − x3(S1) x − x −δ2 πx + x3(S1) 2 − kd − cd− δcd . (14)
Equation (14) follows from (13) by definition of the uniform distribution and expan- sion of defendant valuation terms.
It is possible but tedious to mechanically derive the FOC for maximization of equation (14) by simply taking the derivative of every term with respect to S1. An
easier approach invokes the envelope theorem with respect to S2∗(S1) and x3(S1),
capitalizing on the definition of S2∗(S1) as optimal behavior in the continuation game
following rejection of S1.53 Either way, the FOC provides a simple expression for the
interior-solution equilibrium proposal S1I:
S1I = δ2(πx + kd) + δcd. (15)
53Abusing notation, let V
d(S1, S2) represent the defendant’s objective function over simultaneous
choice of both S1 and S2, and let S2∗(S1) be the argmax of Vd(S1, S2) with respect to S2. If
Vd(S1) = Vd(S1, S2∗(S1)), the envelope theorem provides a simplifying result:
dVd(S1) dS1 = ∂Vd(S1, S2) ∂S1 S 2=S∗2(S1).
To complete the interior solution, it remains to establish the plaintiff’s equilibrium strategy for the timing of settlement in cases where the plaintiff prefers settlement to trial, but is indifferent between accepting any of multiple settlement proposals: i.e. Up(S1I) = Up(S2∗(S1I)) ≥ Wp(x). Evaluating the definition of x2(S1) in equation (12)
at SI
1 provides the upper bound on types that accept S1I in an interior equilibrium:
x2 = x + π−1δ−1(cp+ cd). (16)
The equilibrium strategy of a plaintiff of type x ≤ x2 is accordingly to accept SI 1.
The strategy of plaintiff types x > x2 is to reject SI
1, either in favor of subsequent
settlement for the equally preferred second-period proposal Up(S2∗(S1I)), or in favor of
an eventual trial verdict.54
Note again that the timing of settlement is not a result of plaintiff preferences—in fact, it is premised on the plaintiff’s indifference between settlement in either period. Equilibrium rules for settlement timing tailor the support of plaintiff types remaining in each period so that satisfaction of Lemma 1, requiring Up(S1I) = Up(S2∗(S1I)), is a
natural consequence of sequentially rational play by the defendant.
Similar to the one-period game, the equilibrium first-period settlement proposal in the two-period game depends on parameter values. When Vd(S1I) ≥ Ud(S1B), the
defendant prefers the interior solution—balancing the marginal benefit of a lower settlement proposal against the marginal cost of bargaining and more frequent trial outcomes—and accordingly proposes S1∗ = SI
1. When Vd(S1I) < Ud(S1B), bargaining
and trial costs are sufficiently high that the defendant can do no better than to recoup costs by settling with all types of plaintiff and so proposes S1∗ = SB
1 .
54Types of plaintiff that reject SI
1 divide into two classes. A plaintiff of type x2 < x ≤ x3(S1I)
rejects SI
1, but subsequently accepts the equally preferred second-period proposal, S2∗(S1I). A plaintiff
of type x > x3(SI
Since Vd(S1) is continuous at the boundary solution, S1B, the interior solution is
preferred identically when the interior proposal is less than the boundary proposal: Vd(S1I) ≥ Ud(S1B) ⇐⇒ S1I ≤ S1B. This allows S
∗
1 to be expressed parsimoniously:
S1∗ = min{S1I, S1B}. (17)
It is simple to verify by substitution and simplification that all terms defined in solving the game of length T = 2 adhere to the general definitions provided in Proposition 2.
Inductive Step
Inductive logic is used to demonstrate the interior solution to a general multi-period game. Though grouped under the heading of the inductive step for narrative con- venience, all other aspects of the equilibrium can be derived without induction. To fit the framework of an inductive proof, these aspects of the equilibrium (including much of the plaintiff’s strategy and the boundary solution where all types of plaintiff settle) are established directly for a game of length T + 1.
Begin with the plaintiff’s equilibrium strategy in a game of length T + 1. The first proposition of Lemma 1 establishes that the plaintiff must weakly prefer S2∗ to every subsequent equilibrium settlement proposal: Up(S2∗) ≥ . . . ≥ Up(ST +1∗ ). The value of
rejecting S1 in a game of length T + 1, max{Up(S2∗(S1)), . . . , Up(ST +1∗ (S1)), Wp(x)}, is
thus equivalent to max{Up(S2∗(S1)), Wp(x)}. But this last expression for the value of
continuation is exactly the expression that was used in deriving the plaintiff’s strategy in the two-period game of the verification step. Aside from the timing of settlement in an interior solution, the plaintiff’s strategy is thus exactly the set of rules derived in the verification step for a game of length T = 2.
Turning to the defendant’s strategy, begin by considering two potential boundary solutions: (i) a solution in which no types of plaintiff settle, and (ii) a solution in which all types of plaintiff settle. There exist no equilibria of the first type, since every equilibrium involves settlement with a positive measure of plaintiff types. This result was established for a single-period game in Proposition 1, and generalizes to any multi-period game as the continuation game reached in the final period is itself just a single-period game and therefore characterized by Proposition 1.
A solution of the second type does exist, where the defendant makes a settlement proposal accepted by all types of plaintiff. By the first proposition of Lemma 1, S1 must be weakly preferred to every subsequent equilibrium settlement proposal:
Up(S1∗) ≥ . . . ≥ Up(ST +1∗ ). Thus, if all plaintiff types eventually settle, it must be that
S1∗ is as valuable as the expected net present value of a trial verdict to the highest- type plaintiff: Up(S1B) = Wp(x). Substituting terms and solving yields the equilibrium
settlement proposal for the boundary solution in which all types of plaintiff settle:
S1B = δT +1(πx − kp) − cp T
X
i=1
δi. (18)
All types of plaintiff immediately accept SB
1 under Assumption 3, so the value of the
boundary solution to the defendant is Ud(S1B).55
Next consider the interior solution where some but not all types of plaintiff settle. Derivation of the defendant’s equilibrium strategy is based on inductive reasoning. Suppose the equilibrium asserted in Proposition 2 holds for a game of length T : specifically, assume that for a game of length T , the interior solution involves a first-
period settlement proposal of S1I = δT(πx + kd) + cd T −1 X i=1 δi, (19)
which is accepted by a plaintiff of type
x ≤ x + π−1δ−T +1(cp+ cd). (20)
The following shows this solution then also holds for a game of length T + 1.
As in the verification step, the easiest way to construct the interior solution is by working backwards from the continuation game following rejection of S1. Since some
types of plaintiff reject every equilibrium proposal by assumption, the continuation game is necessarily reached with positive probability in an interior equilibrium.
As reasoned in the verification step, the Assumption 2 requirement that the plaintiff’s strategy be monotone in type means there must exist some cutoff type x2(S1) under which types of plaintiff accept S1 and above which types of plaintiff
reject S1. With plaintiff types distributed uniform in the population, the distribu-
tion of plaintiff types remaining in the continuation game following rejection of S1,
ρ(x|S1) = f (x|x > x2(S1)), is accordingly uniform on support [x2(S1), x].
Optimal play in the continuation game can be expressed as a function of S1 alone.
The continuation game is just a game of length T , and so is characterized by the proposed equilibrium under the assumption that the proposition holds for a game of length T . Substituting x2(S1) for x in the interior solution given by equation (19)
yields the optimal continuation game settlement proposal as a function of S1:
S2∗(S1) = δT(πx2(S1) + kd) + cd T −1
X
i=1
The cutoff type at which the continuation game proposal S2∗(S1) is just rejected,
x3(S1), is given by inequality (20) with x2(S1) substituting for x:
x3(S1) = x2(S1) + π−1δ−T +1(cp+ cd). (22)
Finally, the Lemma 1 restriction that Up(S1) = Up(S2∗(S1)) ⇐⇒ S2∗(S1) = δ−1S1+ cp
in an interior equilibrium, combined with the specification of S2∗(S1) in equation (21),
allows x2(S1) to be solved in terms of S1:
x2(S1) = π−1(δ−2S1+ δ−1cp− kd). (23)
The defendant’s problem for a game of length T +1 is an intuitive generalization of the two-period problem described in the verification step. Without formal definition, let St∗(S1) denote the equilibrium settlement proposal made in period t > 2, and
let xt(S1) be the lower bound on plaintiff types remaining under equilibrium play
in period t > 3. As expressed in equation (24), the defendant chooses a value of S1 in order to maximize the sum of (i) the value of settlement at S1, weighted by
the measure of plaintiff types that accept S1, (ii) the value of settlement at St∗(S1),
weighted by the measure of plaintiff types that reject all prior proposals but accept St∗(S1) for all t ∈ {2, . . . , T + 1}, and (iii) the expected net present value of a trial
verdict conditional on the plaintiff being a type that rejects all equilibrium proposals, weighted by the measure of types that reject all proposals:
Vd(S1) = P [x ≤ x2(S1)] Ud(S1) + P [x2(S1) < x ≤ x3(S1)] Ud(S2∗(S1)) + . . . + P [xT +1(S1) < x ≤ xT +2(S1)] Ud(ST +1∗ (S1)) + P [x > xT +2(S1)] E [Wd(x)|x > xT +2(S1)] (24) = x2(S1) − x x − x (−S1− cd) +x3(S1) − x2(S1) x − x (−δS ∗ 2(S1) − cd− δcd) + . . . +xT +2(S1) − xT +1(S1) x − x −δ TS∗ T +1(S1) − cd T +1 X i=1 δi−1 ! +x − xT +2(S1) x − x −δ T +1 πx + xT +2(S1) 2 − kd − cd T +1 X i=1 δi−1 ! . (25)
Equation (25) follows from (24) by substitution and expansion of terms.
In deriving the FOC for maximization of equation (25), informality in the defini- tion of St∗(S1) for t > 2 and xt(S1) for t > 3 is circumscribed by application of the
envelope theorem. These terms are defined by optimal behavior in nested continua- tion games and so can be treated as constants when taking the derivative with respect to S1; they are subsequently absent at the point of evaluation.56 The resulting FOC
56The reasoning is analogous to that of note 53. Define the T -vector ~S
−1 = [S2, . . . , ST +1] and
let Vd(S1, ~S−1) be the defendant’s objective function over simultaneous choice of S1 and ~S−1, with
~
S−1∗ (S1) the argmax of Vd(S1, ~S−1) with respect to ~S−1. If Vd(S1) = Vd(S1, ~S−1∗ (S1)), the envelope
theorem provides a simplifying result: dVd(S1) dS1 = ∂Vd(S1, ~S−1) ∂S1 S~−1= ~S∗ −1(S1).
Since each xt(S1) for t > 3 is by definition a function of a St−1∗ (S1) for t ≥ 2, the former are also
provides a simple expression for the interior-solution equilibrium proposal S1I: S1I = δT +1(πx + kd) + cd T X i=1 δi. (26)
Evaluating the definition of x2(S1) in equation (23) at S1I provides the upper bound
on plaintiff types that accept S1I in an interior equilibrium:
x2 = x + π−1δ−1(cp+ cd). (27)
The equilibrium strategy for plaintiff types x ≤ x2 is accordingly to accept SI 1. The
strategy for plaintiff types x > x2 is to reject S1I, either in favor of subsequent settle- ment for the equally preferred second-period proposal Up(S2∗(S1I)), or in favor of an
eventual trial verdict.
As discussed in the verification step, the interior solution is preferred identically when the interior proposal is less than the boundary proposal: Vd(S1I) ≥ Ud(S1B) ⇐⇒
SI
1 ≤ S1B. This allows S ∗
1 to be expressed parsimoniously:
S1∗ = min{S1I, S1B}. (28)
Exactly the hypothesized solution for a game of length T + 1, the solution in equation (28) completes the inductive proof.