On the Allocation of Punitive Damages– Can punitive damages aid the victims and deter injurers?–
Yasuhiro Ikeda ∗July 10, 2012

Abstract
This article analyzes the eﬀect of punitive damages rate and allocation on the deterrence of injurer’s harmful activity and the incentivesof victim. This article also considers a three-stage game between theplaintiﬀ and defendant, and provides a comparative-static analysis of
the eﬀect of punitive damages and its allocation on players. In this article, we present that the system of decoupling punitive damages could
not deter injurers from harmful activities and not aid the victims

1 Introduction
It is said that punitive damages, on the one hand, could give sanctions themalicious injurer and aid the victim. On the other hand, however, the punitive damages might compel the sound manufacturers to bear the fear of it
and discourage them. Are these stereotyped ideas true at all? We try toshow results diﬀerent from stereotyped ones. That is, the decoupling punitive damages could weaken the victim the manufacturer could make defence
and ﬁght instead of chilling eﬀects under the some conditions. Almost all
the previous literatures do not refer to our indications.
Many legal scholars against the punitive damages point out that it isunreasonable as the public policy violation for the victim to accept more∗

Kumamoto University, Japan1

than the real damages. There are some opinions that some portion of thepunitive damages should go to funds or notional treasury.
Our study presents the eﬀects on the victim and injurer of the allocationof the punitive damages. Especially, following public policy, in our paper, itis obtained that the larger punitive damages rate, the injurer makes defense
more and more, then the victim becomes to hesitate a suit, and ﬁnally theinjurer continues to manufacture a product.
Precisely, we use the game theoretical method in our paper. The simple
model is as follows. There are parties, the victim (plaintiﬀ), and the injurer(defendant; manufacturer). Each party is risk neutral. The manufacturer
produces a product, then obtaining the production payoﬀ. At the sametime, the production gives a harm to the victim.
The time line of the game is as follows. In 0-stage there exists a rule ofthe punitive damages allocating rule. In ﬁrst stage, the manufacturer (defendant) produces a product or not. In second stage, the victim (plaintiﬀ)
ﬁles a suit or not. In third stage, the plaintiﬀ and the defendant dispute incivil court (trial stage). Main results are as follows. In the system of decoupling punitive damages under small share of the punitive damages for thevictim (plaintiﬀ), raising the punitive damages rate compels the manufacturer (defendant) to make defence much more and the plaintiﬀ’s eﬀort lesser
in civil court, and then prevailing for the plaintiﬀ goes down. Therefore, theportion of ﬁling a suit by the victim becomes smaller. Anticipating thissituation, the manufacturer could continue to manufacture a product. The
punitive damages might not aid the victim and deter the injurer if smaller
share for the plaintiﬀ and highly punitive rate for defendant.
This article studies the eﬀects of punitive damages. Diﬀerent from theprevious literatures (Choi and Sanchirico (2004) etc.), this article introduces

2the probability of plaintiﬀ victory as interdependent form, and specify thatthe punitive damages have not necessarily deterrence of defendant’s torts.

2 The Model
The players are the victim (the plaintiﬀ) and the injurer (the defendant,e.g. manufacturer) in this model. Both are risk neutral.The manufacturer
engages a production and gain production payoﬀ β ≥ 0, doing actual harm
R > 0 to the victim.
The time line of this game is as follows. In the zero period, there exist
the allocation rule of punitive damages. In the ﬁrst period, the producerproduce a product or not. Then he does, there is the harm to the victim.
In the second period the plaintiﬀ (victim) bring a suit or not. In the third
period, the plaintiﬀ and the defendant are battle in civil court. The gameis solved by way of the backward induction from the third period.

3 In the Third Stage
The plaintiﬀ and the defendant dispute in civil court on the basis of eacheﬀort. The eﬀort level for the plaintiﬀ in civil court is x ≥ 0, and y ≥ 0 forthe defendant. The cost of the eﬀort level in civil litigation is s(y) = y forthe defendant, c(x) = k + x for the plaintiﬀ. The term k is ﬁxed cost, wherek ∈ (0, k̄]. The population of k stands for various plaintiﬀ. We assume theuniform distribution of k from 0 to the upper bound k̄. Then the probabilityof the plaintiﬀ victory is as follows,
θx



 θx + yp(x, y) =f or x > 0, y > 0(1)


θ



θ+1f or x = 0, y = 03

This formation means that the probability of plaintiﬀ victory depends oneach eﬀort level. The parameter θ stands for strength weighted for plaintiﬀ’seﬀort relative to defendant’s eﬀort. We assume θ ∈ (0, ∞). If θ ∈ (1, ∞),it means advantageous for plaintiﬀ, and if θ ∈ (0, 1), disadvantageous forplaintiﬀ.
If the plaintiﬀ win, the defendant is imposed the punitive damages αR(α > 1; α is punitive rate), and the plaintiﬀ receives the award R + δ(α −1)R. The parameter δ is the share of the punitive damages for the plaintiﬀ.
The remains of them, (1 − δ)(α − 1)R, goes to the funds or the nationaltreasury. The plaintiﬀ determines its own eﬀort level x, maximizing theexpected damages and the share of punitive damages minus its own eﬀort
cost (referred to the expected payoﬀ, here after), that is, p(x, y)(R + δ(α −1)R)−c(x). The defendant also determines its own eﬀort level y, minimizing
the expected compensatory and punitive damages plus its own eﬀort cost(referred to the expected cost, here after), that is p(x, y)αR + s(y).
We can form the plaintiﬀ’s maximizing problem at the following form.
)][(θx(R + δ(α − 1)R) − x − k .
(2)maxxθx + yThen, we solve this problem, and the plaintiﬀ’s reaction function can beobtained as follows.
x∗ =−y +√(1 + δ(α − 1))Ryθ.
θ(3)Next, we can form the defendant’s minimizing problem at the followingform.

[(miny)θxθx + y]αR + y(4)Similarly, we can obtain the defendant’s reaction function as follows.
y ∗ = −xθ +√αRxθ.

4(5)It turns out that each player’s reaction functions consists of the strategic
substitute and complement part. Then, we can obtain each eﬀort level ofNash equilibrium in third stage as follows.
}{(1 + δ(α − 1))α2 Rθ
(1 + δ(α − 1))2 αRθ,(6){x∗ , y ∗ } =(α + (1 + α + δ(α − 1))θ)2 (α + (1 + α + δ(α − 1))θ)2

3.1 Comparative Statics
We look at the relationship of plaintiﬀ’s eﬀort x∗ and share δ.
∂x∗2(α − 1)(1 + δ(α − 1))α2 Rθ=>0∂δ(α + (1 + α + δ(α − 1))θ)3(7)The relationship of plaintiﬀ’s eﬀort x∗ and punitive damage rate α is asfollows.
∂x∗Rθ(1 + δ(α − 1))((1 − δ)2 θ + α2 δ(1 + δθ) + α(1 − δ)(2δθ − 1))=(8)∂α(α + (1 + α + δ(α − 1))θ)3
The sign of this equation is ambiguous but it can be determined by thecombination of δ, α, and θ.
Lemma 1
The greater share δ of plaintiﬀ is, the larger eﬀort plaintiﬀ makes.
And in the region of the smaller share δ, the greater rate α ofpunitive damages becomes, the lesser eﬀort plaintiﬀ makes.
In turn, we look at the relationship of plaintiﬀ’s eﬀort y ∗ and share δ.

α2 Rθ(α − 1)(α(1 − δθ) − θ(1 − δ))∂y ∗=∂δ(α + (1 + δ(α − 1))θ)3(9)The sign of this equation is ambiguous but it can be determined by thecombination of δ, α, and θ.

5The relationship of the plaintiﬀ’s eﬀort x∗ and punitive damage rate α
is at the following form.
∂y ∗αRθ(2(1 − δ)2 θ + 3α(1 − δ)δθ + α2 δ(1 + δθ))=>0∂α(α + (1 + δ(α − 1))θ)3(10)Lemma 2The bigger share δ of plaintiﬀ, and the more eﬀort the plaintiﬀ
makes, then defendant also makes more eﬀort (does more defence). This eﬀect is indirect one. The greater punitive rate α
the more stakes the plaintiﬀ seek, but the defendant makes eﬀort(defence) more than the plaintiﬀ makes.

3.2 Probability of Plaintiﬀ Victory in the Equilibrium
We can show the probability of plaintiﬀ victory in Nash-equilibrium in thefollowing.
p(x∗ , y ∗ ) =(1 + δ(α − 1))θα + (1 + δ(α − 1))θ(11)

Then, Some properties can be obtained as follows.
∂p(x∗ , y ∗ )−(1 − δ)θ=≤0∂α(α + (1 + δ(α − 1))θ)2(12)As punitive damages ratio goes up, the plaintiﬀ make eﬀort and defendantalso does eﬀort for defence. Then as the share of δ determines the plaintiﬀ’sposition, the prevailing of the plaintiﬀ goes down or a constant if punitivedamages increases.
α(α − 1)θ∂p(x∗ , y ∗ )=>0∂δ(α + (1 + δ(α − 1))θ)2(13)As the punitive damages rate is increased, the plaintiﬀ’s prevailing goes up
because of the plaintiﬀ’s own share increasing.
64 In the Second Stage
The plaintiﬀ has the incentives to sue as follows on anticipating of the thirdstage.
((1 + δ(α − 1))θα + (1 + δ(α − 1))θ)
(1 + δ(α − 1)R) −(1 + δ(α − 1))2 αRθ−k ≥0(α + (1 + δ(α − 1))θ)2(14)We let T denote the diﬀerence of the ﬁrst and the second term of of left
side of this inequality. We refer T to the gross payoﬀ for the plaintiﬀ. Then
we have the followings.
T ≡(1 + δ(α − 1))3 Rθ2≥ k.
(α + (1 + δ(α − 1))θ)2(15)As the population of the plaintiﬀ is in k ∈ (0, k̄], (i) whenever k̄ ≤ T , allplaintiﬀ sue. (ii) In T ≥ k̄, the plaintiﬀ of k ∈ (0, T ] sue, whereas, them ofk ∈ (T, k̄] leave to sue. (iii) In T ≤ 0, then all the plaintiﬀ does not sue.
Therefore, we look at the comparative analysis. The relationship of α,δ, and T is as follows.
∂T
R(α − 1)(1 + δ(α − 1))2 θ2 ((1 − δ)θ + α(δθ + 3))=≥0∂δ(α + (1 + δ(α − 1))θ)3(16)It is natural that the share of plaintiﬀ goes up, the gross payoﬀ in suitincrease.
∂T
R(1 + δ(α − 1))2 θ2 ((α − 1)θδ 2 + (α + θ + 2)δ − 2)=∂α(α + (1 + δ(α − 1))θ)3∂TR 0,∂αif α Rθδ 2 − (θ + 2)δ + 2
δ(1 + δθ)

Lemma 3On the one hand, if punitive rate α is very small, and plaintiﬀ’s
2
share δ is very small, that is α < θδ −(θ+2)δ+2, raising puniδ(1+δθ)
tive rate δ makes plaintiﬀ7s incentives to ﬁle a suit weaker. On2
the other hand, if α > θδ −(θ+2)δ+2, the greater punitive rate δδ(1+δθ)
becomes, plaintiﬀ’s incentives to sue larger.
7(17)5 In the First Stage
Manufacturer activity gives production payoﬀ β to a manufacturer, whereas
harm R is accompanied by it.
Anticipating second stage, the expected cost for defendant when fractionof plaintiﬀ ﬁling a suit as follows.
Tk̄[((1 + δ(α − 1))δα + (1 + δ(α − 1))θ)αR +(1 + δ(α − 1))α2 Rθ
(α + (1 + α + δ(α − 1))θ)2](18)

By assumption of uniform distribution of k for the plaintiﬀ’s ﬁxed cost, thecondition of a manufacturer (defendant) producing a product as follows.
β≥(1 + δ(α − 1))4 αR2 θ3 ((1 − δ)θ + α(2 + δθ))
≡Hk̄(α + (1 + δ(α − 1))θ)4(19)where, H is expected litigation cost for defendant with a product.
We look at the comparative statics of the relationship of H and α∂HR2 θ3 (1 + δ(α − 1))3=[(1 − δ)3 θ2 + αθ(1 − δ)2 (1 + 4δθ)∂αk̄(α + (1 + δ(α − 1))θ)5+2α3 δ(2 + 3δθ + δ 2 θ2 ) + α2 (1 − δ)(5θ2 δ 2 + 7θδ − 4)](20)

We denote A the second term of the right hand side of the equation above,and present the sign.
∂HQ 0,∂α()if A Q 0 .

We can obtain the following the proposition from the above.
Proposition
If the plaintiﬀ’s share δ of punitive damages is very small andpunitive damages rate α is very small, expected litigation costfor defendant goes down as punitive rate α increase. Then, defendant continues to produce a product with respect to expected
payoﬀ β. If the plaintiﬀ’s share δ is not small, the reverse situation will be obtained.

8(21)6 Conclusion
Under small share δ for the plaintiﬀ of the punitive damages, decoupling
punitive damages system compels defendant (manufacturer) to make a defense, in addition, the plaintiﬀ will be eﬀort lesser in civil litigation, thenprevailing for plaintiﬀ goes down. Then, portion of ﬁling a suit by victims
become smaller. Anticipating these second and third stage, manufacturer(defendant) may continue to produce a product. the system of decoupling
punitive damages could not aid victims and not deter injurer from harmful
activity if smaller share of punitive damages for the plaintiﬀ .

7 Main References
1. Choi, A. and C. W. Sanchirico（2004）“Should plaintiﬀs win whatdefendants lose?–Litigation stakes, litigation eﬀort,and the beneﬁts ofdecoupling,” Journal of Legal Studies, vol.33, pp323-54.
2. Kahan, M. and B. Tuckman (1995), “Special Levies on Punitive Damages: Decoupling, Agency Problems and Litigation Expenditures,” International Review of Law and Economics, vol.15, pp175-185.
3. Polinsky, A.M.,and Y.-K. Che（1991）“Decoupling Liability: Optimalincentives for care and litigation,” Rand Journal of Economics, vol.22,
pp562-70.

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