Inferences from Litigated Cases
Dan Klerman & Yoon-Ho Alex Lee
Law and Economic Theory Conference
December 7, 2013
MOTIVATING QUESTION
• Take an area of private law.
• Suppose in State A, the legal standard governing
liability is f. In State B, the standard is g, more
plaintiff-friendly. What should we expect to observe in
terms of the rate of plaintiff’s trial victory among
litigated cases, all else equal?
• Alternatively, suppose we observe a higher rate of
plaintiff’s trial victory among litigated cases in State B
with g, as compared to State A, with f. Can we validly
make an inference that g is more plaintiff-friendly than
f, all else equal?
Theoretical Literature on Settlement and Litigation
“50%!”
“Any deviation from 50% is from noise/errors or
asymmetric stakes!”
- Priest & Klein (1984)
“No inferences regarding the legal standard are
possible!”
- Wittman (1985)(interpreting Priest & Klein)
“Any frequency of plaintiff win is possible!”
- Shavell (1996)
Standard Models of Settlement and Litigation
• Divergent Expectations (Priest & Klein (1984))
• Asymmetric Information
– Screening (P’ng (1983), Bebchuk (1984), Shavell (1996))
• Defendant has informational advantage
• Plaintiff has informational advantage
– Signaling (Reinganum & Wilde (1986))
• Defendant has informational advantage
• Plaintiff has informational advantage
OUR AIM: Across all standard classes of models of settlement/
litigation, valid inferences are possible under plausible
assumptions about the distributions of disputes.
Theoretical Literature on Settlement and Litigation
“50%!”
“Any deviation from 50% is from noise/errors or
asymmetric stakes!”
- Priest & Klein (1984)
“No inferences regarding the legal standard are
possible!”
- Wittman (1985)(interpreting Priest & Klein)
“Any frequency of plaintiff win is possible!”
- Shavell (1996)
Asymmetric Information:
The Screening Model
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
• A legal standard is, f(p), a PDF over [0,1], distributing potential Ds
in terms of type p = p’s probability of win. High p for low-quality D.
• Damage = d, respective cost of litigating: CD, Cp > 0.
• D’s p private info, p makes a “take-or-leave” settlement offer at x.
• D’s strategy: Accept if x < pd + CD, reject otherwise.
• p offers x to maximize expected recovery:
𝒙− 𝒄∆
𝒅
𝒑
𝒑𝒅 − 𝒄𝝅
𝒙 − 𝒄∆
𝒇 𝒑 𝒅𝒑 + 𝟏 − 𝑭
𝒅
p’s payoff from Litigating Ds
𝒙
p’s payoff from Settling Ds
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
• A legal standard is, f(p), a PDF over [0,1], distributing potential Ds
“Any
frequency
win is possible!”
– pShavell
(1996)D.
in
terms
of type p =of
p’spprobability
of win. High
for low-quality
• Liability = d, respective cost of litigating: CD, Cp > 0.
Translated: “Give me any probability, and I can
• D’s
p privateainfo,
makes a “take-or-leave”
settlement
offerwin
at x.
construct
PDFp producing
that probability
as p’s
rate.”
• D’s
strategy: Accept if x < pd + CD, reject otherwise.
• p offers x to maximize expected recovery:
Empirical Relevance?: We can’t choose the PDF.
𝒙− 𝒄∆
𝒅
𝒙 − 𝒄∆
Relevant
Question:
𝒄𝝅 𝒇 𝒑 𝒅𝒑
+ standard,
𝟏 − 𝑭 what
Given a𝒑𝒅
PDF−representing
a legal
𝒅
𝒑
condition must a new PDF satisfy in relation to the
original PDF to permit valid natural inferences?
p’s payoff from Litigating Ds
𝒙
p’s payoff from Settling Ds
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
• How should we understand the “more pro-p” standard?
• Given 2 PDFs, f(p) and g(p), what relation describes “more pro-p”?
– Stochastic Dominance: not sufficient
– Monotone Likelihood Ratio Property:
𝒅 𝒈(𝒑)
𝒅𝒑 𝒇(𝒑)
> 𝟎, sufficient
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
Exp. Payoff:
𝒙− 𝒄∆
𝒅
𝒑
𝒑𝒅 − 𝒄𝝅 𝒇 𝒑 𝒅𝒑 + 𝟏 − 𝑭
• FOC over x: 𝟏 − 𝑭
𝒙−𝒄∆
𝒅
=
𝒇(
∗
• Let 𝒑 =
k=
(𝒄𝝅 +𝒄∆ )
:
𝒅
𝒙
𝒙− 𝒄∆
)(𝒄𝝅 +𝒄∆ )
𝒅
𝒅
Hazard rate of f
Threshold D type
𝒙∗−𝒄∆
,
𝒅
𝒙−𝒄∆
𝒅
𝒇(𝒑∗ )
𝟏−𝑭(𝒑∗ )
hf(p)
=
𝟏
.
𝒌
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
• As long as hg(p) < hf(p), 𝒑∗ under f(p) < 𝒑∗ under f(p)
• But if MLRP
𝒅 𝒈(𝒑)
(
𝒅𝒑 𝒇(𝒑)
> 𝟎), we have hg(p) < hf(p).
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
• So MLRP implies 𝒑∗ under f(p) < 𝒑∗ under g(p), with
which we can show p’s win-rate is higher under g(p).
Screening Model (Bebchuk, Shavell): Overview
D has private info, p makes a “take-it-or-leave-it” offer
PROPOSITION 1 (INFERENCES UNDER THE SCREENING MODEL).
When p is screening, the probability that p will prevail in
litigated cases is strictly higher under a “more proplaintiff” legal standard, as characterized by monotone
likelihood ratio property.
PDF Families over [0,1] Exhibiting MLRP:
Over [0,1]: uniform, beta, rising triangle, falling triangle
Truncated: normal, exponential, binomial, Poisson
PROPOSITION 1A (INFERENCES UNDER THE SCREENING MODEL).
Same result when D is screening.
Asymmetric Information:
The Signaling Model
Signaling Model (Reinganum & Wilde): Overview
p has private info, p makes a “take-it-or-leave-it” offer
• p knows type. Type is probability that plaintiff will prevail at
trial
• p makes a “take-it-or-leave-it” offer, s(p), that increases with
type
• D rejects offer with probability, p(s), that increases with the
offer and is largely independent of the distribution of disputes
• So high probability plaintiffs are disproportionately represented
among litigated cases
• Pro-plaintiff shift in law
– increases proportion of high probability plaintiffs in
population (e.g. in pool of litigated + settled cases)
– increases proportion of high probability plaintiffs in litigated
subset
– increases observed probability of plaintiff success
Signaling Model (Reinganum & Wilde): Overview
p has private info, p makes a “take-it-or-leave-it” offer
PROPOSITION 2 (INFERENCES UNDER THE SIGNALING MODEL).
When p has the informational advantage, the probability
that p will prevail in litigated cases is strictly higher
under a more pro-plaintiff legal standard, as
characterized by monotone likelihood ratio property.
PROPOSITION 2A (INFERENCES UNDER THE SIGNALING MODEL).
Same result when D has the informational advantage.
The Priest-Klein Model
Priest-Klein Model: Overview
DISTRIBUTION OF ALL DISPUTES (SETTLED OR LITIGATED)
p WINS
(BLUE)
p WINS
(BLUE)
DEGREE
OF D
FAULT
PRO-p STANDARD
Distributions of litigated disputes
if parties make moderate errors
Distributions of litigated disputes
if parties make small errors
DEGREE
OF D
FAULT
PRO-D STANDARD
Priest-Klein Model: Overview
PROPOSITION 3 (INFERENCES UNDER THE PRIEST-KLEIN MODEL).
Under the Priest-Klein model, if the distribution of
disputes has a log concave CDF, then p’s win-rate among
litigated cases increases as the decision standard
becomes more pro-p.
PDFs with Log-Concave CDFs:
normal, generalized normal, skew
normal, exponential, logistic, Laplace,
chi, beta, gamma, log-normal, Weibull…
Priest-Klein Model
• As legal standard becomes more pro-D, p’s win-rate decreases
• Effect varies with standard deviation of prediction error
• Paper presents evidence that standard deviation is large
Why We Disagree with P&K I
• Priest & Klein understood that plaintiff trial win
rates would vary with the legal standard unless
prediction errors were very small (e.g. σ=0.1)
• They estimated prediction errors from trial rates
based on simulations
– Lower prediction errors produce lower trial rates
– Exact effect depends on (C-S)/J = (cost of trial –
cost of settlement)/judgment
• If (C-S)/J = 0.33, then 2% trial rate implies very
small prediction errors (σ=0.1)
• If (C-S)/J ≥ 0.66, then 2% trial rate implies
prediction errors large enough to make
plaintiff trial win rates vary significantly with
legal standard
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Why We Disagree with P&K II
• Priest & Klein assumed (C-S)/J=0.33, because 33% is
standard contingent fee percentage
• That is wrong for two reasons
– 1) (C-S)/J = (Cπ-S π)/J + (CΔ-S Δ)/J
• So, at best, contingent fee measures (Cπ-S π)/J
• (C-S)/J ≈ 2 (Cπ-S π)/J = 0.66
– 2) Under simple contingent fee, lawyer gets
paid same percentage whether case settles or
goes to trial
• C/J = S/J which implies (C-S)/J = 0
• Can’t estimate (C-S)/J from contingent fee
• RAND (1986) estimates (C-S)/J = 0.75
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Extensions
• Effect of different decisionmakers
– Republican versus Democratic judges
– Male versus female judges
– 6 or 12 person jury
• Whether factor affects trial outcome
– Race or gender of plaintiff
– In-state or out-of-state defendant
– Law firm quality
• Effect of change in composition of cases
– Business cycle induces stronger or weaker plaintiffs
to sue (Siegelman & Donohue 1995)
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Caveats
• Assumes that distribution of underlying behavior doesn’t
change
– Not usually true
– Exceptions
• Retroactive legal change
• Uninformed defendants
• Analysis of judicial biases or case factors, when cases
randomly assigned
– Advice to empiricists
• Worry less about settlement selection
• Worry more about changes in behavior
• Distribution of disputes (litigated & settled)
– Monotone likelihood ratio property for asymmetric
information
– Logconcave for Priest-Klein
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Conclusions
• Selection effects are real
• But, under all standard settlement models, change in legal
standard, under plausible assumptions, will lead to predictable
changes in plaintiff trial win rate
– Bebchuk screening model
– Reinganum-Wilde signaling model
– Priest-Klein divergent expectations model
• Pro-plaintiff change in law will lead to increase in plaintiff trial
win rate
• So may be able to draw valid inferences from litigated cases
– Measure legal change
– Measure biases of decision makers
– Identify factors affecting outcomes
• Good news for empiricists
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