Games for Judges: An
Approach to the Design of
Legal Liability Rules
Sergey Knysh, Paul M. Goldbart & Ian Ayres
Department of Physics
University of Illinois
Yale Law School
New Haven, CT
Michigan Law Review,100 (Centenary Volume), 1-79 (2001)
Preprints (2002): Design of Efficient Legal Liability Rules:
I. Continuous Extension of Multistage Rules
II. Comparison of Discrete Vanilla and Exotic Variants
goldbart@uiuc.edu
w3.physics.uiuc.edu/~goldbart
Outline

Property rules & liability rules

Nuisance dispute settings

Options view of liability rules

Optimizing liability rules

Continuum liability rules

Game-theoretic viewpoint

Practical advice for courts
Property rules & liability rules
 Property
 Liability
rules: protect by deterrence
rules: protect by compensation
Property rules & liability rules
 Example:
Abbott breaks
Costello’s arm
 Intentionally?
 focus on taker’s welfare
(a criminal offence)
 traditionally protect via a property rule
 Through negligence?
 focus on takee’s welfare (compensatory damages)
 traditionally protect via a liability rule
Property rules & liability rules
 But
property rules
are not very efficient
 Example: Laurel
steals Hardy’s hat?
 Property rule: Hardy can sue to recover hat (replevin)
 Liability rule: Can also sue for value of hat (trover)
 Advantage? If hat is worth $10 to Laurel, $5 to Hardy?
Property rules & liability rules
 Liability
rules are more efficient
 Example: Fred holds over
in Barney’s apartment
 Barney can sue for trespass
 or to force Fred to rent for
another year
 Goal
of liability rules:
 Add efficiency—by compensating initial entitlement holder
for transfer of entitlement (goes beyond mere deterrence)
Concerns of judges re liability rules
 Traditional
view:
 identity of the initial entitlement holder
 compensation as deterrence
 Modern
view:
 identity of the more efficient chooser
 decouple allocative and distributional concerns
 liability rules: a means by which an imperfectly
informed court can delegate choice to private
litigants thus harnessing their superior information
Central aims
 Focus
on nuisance dispute settings
 Provide
courts with liability rules that are

economically efficient

cheap to implement
Examples of nuisance disputes

A/c noise reduced value of adjacent residence
 Estancias Dallas Corp. v. Shultz (Tex. App. 1973)

Hotel addition obstructed view of adjacent hotel
 Fontainbleu Hotel v. Twenty-Five Twenty-Five
Inc. (Fl. 1959)

Dog-track lights interfered w/ drive-in movie theater
 Amphitheaters, Inc. v. Portland Meadows (Or. 1948)

Pollution from Con Ed plant disrupted new car
preparation business
 Copart Indus. v. Con. Ed. Co. (N.Y. 1977)
Property rules & liability rules

What might courts do in nuisance disputes?
 E.g. Boomer v. Atlantic Cement (N.Y. 1970)
 Resident/Plaintiff (Boomer): discomforted by pollution
 Polluter/Defendant (Atl. Cem.): factory operator

After Calabresi & Melamed (’72):
 Rule 1: nuisance  injunction on Polluter (stop!)
 Property rule
 Rule 2: nuisance  Polluter pays damages to continue
 Liability rule
 Rule 3: not a nuisance  Polluter continues
 Property rule
 Rule 4: nuisance?  Resident pays Polluter damages to stop
 Liability rule
Options: Calls and Puts

Call option
 choice of whether or not to pay a nonnegotiated amount to purchase entitlement
 choice of forcing seller to sell (be paid)

Put option
 choice of whether or not to be paid a nonnegotiated amount to sell entitlement
 choice of forcing buyer to buy (pay)
Liability rules as options (Morris ’93)

Call: seller forced to sell

Put: buyer forced to buy
 Rule 2: Polluter can pay damages to continue
 Liability rule: initial entitlement to Res; call option to Pol
 Rule 4: Resident can pay damages to stop Polluter
 Liability rule: initial entitlement to Pol; call option to Res
 Rule 5: Polluter can require damages & stop
 Liability rule: initial entitlement to Pol; put option to Pol
 Rule 6: Resident can require dam’s & allow Polluter
 Liability rule: initial entitlement to Res; put option to Res

Who pays? Who chooses who pays?
Realizations
 Rule 2: Entitlement to Resident; call to Polluter
 Boomer v. Atlantic Cement (N.Y. 1970)
 Rule 4: Entitlement to Polluter; call to Resident
 Spur Indus., Inc. v. Del E. Webb Dev. Co. (Ariz. 1972)
 Rule 6: Entitlement to Resident; put to Resident
 Thelma builds an
encroaching wall
on Louise’s land;
Louise can sue Thelma
to remove the wall
or to force Thelma to buy the encroached land permanently
Basic ingredients for analyzing
liability rules

Imperfectly informed court

Explore classes of rules

Which to use? Ex post efficiency as criterion
 Vanilla rules
 Exotic rules
 dual-chooser rules (vetos, higher-order,…)

Emerging guidelines for courts
Imperfectly informed courts
value of asset to defendant
D is higher
valuer
VD  VP
VD
In any given instance

Plaintiff P

Defendant D

j.p.d.


P is higher valuer
value of asset to plaintiff
VP
P alone knows her
valuation, VP
D alone knows his,
VD
P, D & court know
j.p.d. f (VP ,VD )
i.e. joint probability
distrib. of valuations
(possibly correlated)
Imperfectly informed courts: Property rule
value of asset to defendant
D is higher
valuer
mean values
VD
V
P
, VD

What might the court do?


P is higher valuer
value of asset to plaintiff

VP
compute means w.r.t.
f : VP & VD
allocate asset to higher
mean valuer (here: D)
What can the court do to
promote better efficiency?
Imperfectly informed courts: Liability rule
One stage / Call flavour


The court could…

allocate asset to P

give call option to D

set damages
(C,VD  C )
if D exercises
VD
C
How would D respond?


Resp. profits
(to P, to D)?
by exercising option
when worth it to him
How to set damages?

to elicit a response
that maximizes the
expected total value
(VP ,0)

if D doesn’t
VP
So… where to arrange the bar?
Imperfectly informed courts: Liability rule
One stage / Call flavour

Upshot…


optimally efficient
(i.e. best mean total
profit) if D “pivots” at
nonchooser’s mean
Resp. profits
to P & D ?
(C,VD  C )
if D exercises
VD
(rational) D does this
if damages are also at
nonchooser’s mean
C
VP  (VD  C )
 (VD  C )
(VP ,0)
if D doesn’t
VP
Imperfectly informed courts: Liability rule
One stage / Put flavour



allocate asset to P

give put option to P

set damages
if P does put
Resp. profits
(to P, to D)?
VD
C
How would P respond?


(C,VD  C )
The court could…
by exercising option
when worth it to her
(VP ,0)
if P doesn’t put
How to set damages?

to elicit a response
that maximizes the
expected total value
VP

So… where to arrange the bar?
Imperfectly informed courts: Liability rule
One stage / Put flavour

(C,VD  C )
Upshot…
 optimally
efficient
(i.e. best mean total
profit) if P “pivots” at
nonchooser’s mean
 (rational)
P does this
if damages are also at
nonchooser’s mean
C
VD  (VP  C )
 (VP  C )
if P does put
Resp. profits
(to P, to D)?
VD
(VP ,0)
if P doesn’t put
VP
Imperfectly informed courts: Veto rule


(VP ,0)
The court could…

allocate asset to P

give call option to D
at damages C

give call option to P
at same damages
(C,VD  C )
VD
D takes but P
doesn’t take
back
D takes and P
takes back
Transfer to D can be
vetoed by D or P
(VP ,0)
Resp. profits
(to P, to D)?

D doesn’t take
VP
So… where to arrange the bars?
Imperfectly informed courts: Veto rule


(VP ,0)
Upshot…

efficient for D to pivot
at damages C

efficient for P, too
(C,VD  C )
VD
D takes but P
doesn’t take
back
D takes and P
takes back
Optimal damages obey
straightforward formula
VD  C  (VD  C )  (VP  C )
 C  VP  (C  VP )  (VD  C )
(VP ,0)
D doesn’t take
VP
“Phase diagram” for rules
…roughly
SCR: select more volatile valuer as chooser
DCR: select lower mean
valuer as vetoee
SCR v. DCR: select
SCR if diff. in var’s
exceeds diff. in means

Simplest setting…
 uniform
rectangular j.p.d., 1/t is the golden ratio
P
 , range 2 ; D mean , range 2
mean
 rules:

property (Pr), one-stage liability (Li), veto (Ve)
Multi-stage liability rule: Call flavour

The court could give…


call option to
D at damages C D1

call-back option to
P at damages C P1

call-back option to
D at damages C D2

call-back option to
P at damages C P2


asset to P
VD
(CD2 ,VD  CD2 )
(VP  CP2 , CP2 )
(CD1 ,VD  CD1 )
(VP  CP1 , CP1 )
(VP ,0)
VP
…..
How to choose the
damages?
Resp. profits
(to P, to D)?

So… where to
arrange the bars?
Multi-stage liability rule: Call flavour




Find pivots that max.
expected total value
Set damages to elicit
these responses
Strategic over-bidding
(pivots smaller than
damages)
Note: shrinking areas
of inefficiency
VD
(CD2 ,VD  CD2 )
(VP  CP2 , CP2 )
(CD1 ,VD  CD1 )
(VP  CP1 , CP1 )
(VP ,0)
VP
Infinite-stage liability rule: Call flavour

Discrete pivots  pivot functions
VP( k )  vP (q) , VD( k )  vD (q)

Optimal pivot functions
 converge to diagonal
 perfect efficiency
VD
vP (q)  vD (q)

Reparametrization  can take
vP (q)  vD (q)  q

Discrete damages
 damages functions
CP( k )  cP (q) , CD( k )  cD (q)
VP
Infinite-stage liability rule: Call flavour


Rational D & P pivot
optimally, provided
damages functions obey

simple ODE’s

simple BC’s
Solvable (to quadratures)
for any j.p.d. (including
arbitrary correlations)
dcP
 P (q)
  ( q ) (c P  c D  q )
dq
dcD
 D (q)
  ( q ) (c P  c D  q )
dq
cP (qmin )  0
cP (qmax )  cD (qmax )  qmax
 P (q)  (VP  q)  (VD  q)
 D (q)  (VD  q)  (VP  q)
 (q)   (VD  q)  (VP  q)
Infinite-stage liability rule: Call flavour

General distributions…

what aspects of j.p.d feature?

simple geometry if j.p.d. uniform
(even if shape gives corr’s)
 P ( s)  (VP  s)  (VD  s )
 D ( s)  (VD  s )  (VP  s )
 ( s)   (VD  s)  (VP  s)

Illustrative example

uniform triangular (correlated)

readily solvable
Infinite-stage liability rule: Call flavour
How to use the results


Court computes parametric
damages curve
And requires D & P to “bid”


bids separate call-exercise
from non-exercise ranges
Asset goes to higher bidder


cD (q)
for damages projected from
lower bid (to higher bidder’s
axis)
Asset ends up in hands of
higher valuer
cP (q)
Infinite-stage liability rule: Call example

j.p.d. a uniform
“2 x 1” rectangle
(no correlations)
VD
VP

Use? Litigants issue bids

Asset goes to higher bidder

At damages set by lower bid
Game-theoretic aspects

SD  SP
SD
Introduce generalized
damages functions
CD (S P , S D )

And strategy functions
 what
P & D bid, given
their private valuations
CP (S P , S D )
S P (VP )
SP
S D (VD )

P’s expected profit
VP  CP ( S P , S D )  ( S P  S D )
 C D ( S P , S D ) ( S D  S P )

D’s expected profit
C P ( S P , S D ) ( S P  S D )
 VD  C D ( S P , S D )  ( S D  S P )
Game-theoretic aspects

Seek Nash Equilibrium
(my strategy is optimal
for me if you fix yours,
& vice versa)



P' s expected profit   0
 S P (VP )

D' s expected profit   0
 S D (VD )
Make simple choice for
damages functions
CD ( S P , S D )  CD ( S P )
Demand that the N-E
strategies be “revealing”
S P (VP )  VP
CP ( S P , S D )  CP ( S D )
S D (VD )  VD

Upshot: previous call-option damages conditions

Court has made it so that it pays not to lie!
Emerging guidelines

Explore classes of liability rules
 single-chooser, veto, continuous
 decouple allocational & distributional concerns

Property rules
 give entitlement to (estimated) higher valuer
 suggested as a general practice, but…

Liability rules
 do better—by harnessing private information
 continuous versions can be fully efficient
 go to end-stage, limit transaction costs