E¢ cient Incentives from Obligation Law and
the Compensation Principle
PRELIMINARY VERSION!
Urs Schweizer, yUniversity of Bonn
November 15, 2013
Abstract
The compensation principle provides a link between the requirement to compensate for deviations from obligations and generating
incentives for e¢ cient outcomes. The principle leaves much indeterminacy as far as the distribution of costs is concerned that may be
re‡ected by a comparable vagueness of certain provisions from obligation law. The compensation principle would apply if damages were
awarded in line with the di¤erence hypothesis, for one-sided deviations
from obligations at least. To see the compensation principle at work,
di¤erent methods of quantifying damages under uncertain causation
but in line with the di¤erence hypothesis are discussed. Courts, in
contrast, may sacri…ce the compensation goal by concentrating on the
causality requirement instead. A further application concerns contributory negligence. It is shown that reducing damages for contributory
negligence may serve to generate e¢ cient incentives on and o¤ the
equilibrium path..
JEL classi…cation: K12, K13
Keywords: legal obligation, di¤erence hypothesis, compensation
goal, compensation principle, e¢ cient incentives
address: Urs Schweizer, Department of Economics, University of Bonn, Adenauerallee
24, 53113 Bonn, Germany, schweizer@uni-bonn.de
y
Support by the DFG through SFB/TR 15 is gratefully acknowledged.
1
1
Introduction
Legal rules are called e¢ cient if they generate incentives for strategically
acting parties to take decisions leading to a welfare maximizing outcome.
Obligation law provides general rules for contractual and tort relationships, including unjust enrichment and precontractual liability. If a debtor
deviates from a (contractual or legal) obligation, the law o¤ers remedies to
creditors who su¤er from harm caused by the debtor’s deviation. Allegedly,
these remedies aim at compensating creditors.
The compensation principle introduces a link between the legal requirement of compensation and the economic concept of e¢ cient incentives. I
distinguish the unilateral, the bilateral and the multi-lateral compensation
principle according to whether a single party, two parties, or more than two
parties, respectively, act strategically. Aside from that, the outcome may
also be a¤ected by a random move of nature capturing uncertainty.
The unilateral version of the principle establishes the following claim. If
the compensation goal is achieved in the sense that the (passive) creditor
is worse o¤ under no deviation by the debtor from her obligation and if
this obligation maximizes welfare then the debtor has the incentive to meet
her obligation. In spite of its formal simplicity, the compensation principle
turns out to be a convenient analytical tool for establishing the e¢ ciency of
provisions from obligation law.
The compensation goal as required by the compensation principle would
notably be achieved if the creditor were awarded damages in line with the
di¤erence hypothesis. In Germany, this hypothesis is attributed to Friedrich
Mommsen, a legal scholar from the nineteenth century. Accordingly, damages
should account for that part of the harm that was caused by the deviation
from the obligation and should be calculated as the di¤erence of the hypothetical (counterfactual) value of the creditor’s assets if the debtor had met
her obligation and their actual value, given that actually she has not.
In legal theory, the compensation goal and the di¤erence hypothesis are
widely accepted. Yet, as soon as uncertain causation is involved, it becomes
rather demanding to stick to the di¤erence hypothesis and to achieve the
compensation goal in practice.
For a general setting of uncertain causation, I will propose three di¤erent
2
ways of calculating the di¤erence of the hypothetical and actual value of
assets. All three methods ensure that, from the ex ante perspective, the
creditor will be worse o¤ under no deviation by the debtor from her obligation
and, hence, the compensation principle applies to all three of them.
Yet, in cases of uncertain causation, courts may practice a two-step approach instead that puts more weight on the causality requirement than the
compensation goal. In a …rst step, they ask whether or not the debtor’s
deviation has caused any harm at all to the creditor. If this question is denied then the debtor escapes liability. Only if the answer at the …rst step is
a¢ rmative, damages will ever have to be quanti…ed.
In probabilistic terms, no criterion is o¤ered under which conditions exactly the causation requirement should be considered as being ful…lled. Instead, vague notions such as "beyond reasonable doubt" are in use. But
notice, achieving the compensation goal may be at risk whenever recovery is
denied in spite of the debtor’s deviation having caused harm, if not for sure,
though with positive probability. In such cases, the unilateral compensation
principle no longer applies and incentives may well end up being distorted.
The accident model (the traditional workhorse for the economic analysis
of tort law) is contained as a special case in my more general setting of uncertain causation. From the more general perspective, the binary nature of the
accident model turns out to be misleading. The traditional negligence rule,
in particular, proves di¢ cult to justify in terms of the di¤erence hypothesis.
The bilateral compensation principle refers to situations where both the
debtor and the creditor are taking decisions strategically and where each
party may face an obligation of its own. In a bilateral setting, incentives from
legal provisions are called e¢ cient if the Nash equilibrium of the underlying
game leads to an e¢ cient outcome.
The bilateral compensation principle identi…es the following conditions
as su¢ cient for generating e¢ cient incentives. If none of the parties is worse
o¤ under no unilateral deviation by the other party and if the obligation
pro…le consists of an e¢ cient outcome then any Nash equilibrium of the
corresponding game must maximize the (expected) welfare and all of them
(if more than one exists) must be payo¤ equivalent. On top of it and under
the same su¢ cient conditions, if the game is played sequentially, the subgame
perfect equilibrium outcome must be payo¤ equivalent as well.
3
Notice, the above encompassing e¢ ciency result based on the bilateral
compensation principle leaves much indeterminacy as far as transfer payments are concerned. In particular, no reduction of damages claims for
contributory negligence would be needed to ensure e¢ cient equilibrium outcomes. Therefore, reductions of claims may, at best, increase welfare out of
equilibrium or o¤ the equilibrium path.
Various rules of contributory negligence have been examined by the law
and economics literature. Most of them specify the same quantum of damages as long as only one of the parties deviates. In any case, as it turns out,
all these rules compensate parties for unilateral deviations such that the bilateral compensation principle applies to all of them. Therefore, on e¢ ciency
grounds, it seems impossible to elevate one of them. Again, such indeterminacy of the compensation principle may be re‡ected by the vagueness of the
corresponding provisions from obligation law.
The multi-lateral compensation principle, …nally, refers to cases with more
than two strategically acting parties. If each party, by meeting his or her
obligation, will never be worse o¤ as compared with the situation where
all other parties meet theirs and if the obligation pro…le maximizes welfare
then all parties meeting their obligations constitutes a Nash equilibrium that
remains robust even against coordinated deviations by coalitions of players. This version of the principle becomes relevant, e.g., in cases of multiple
debtors or creditors. The multi-lateral compensation principle too remains
indeterminate as far as distributional issues are concerned.
Most of the related literature takes a common law perspective. Cooter
(1985) has identi…ed unifying principles behind tort and contract law. Moreover, he has pointed out that, in a bilateral choice setting, exact compensation
of both parties may not be feasible, a fact, he refers to as the compensation
paradox.
The present paper refers to a legal system where unifying principles need
not be detected as they are already explicitly codi…ed by obligation law. To
ensure compensation for deviations from obligations, courts rely on the difference hypothesis. Compensation for one-sided di¤erences turns out to be
su¢ cient for generating e¢ cient incentives for both parties (provided that
the obligation pro…le is e¢ cient). Cooter’s compensation paradox remains
a phenomenon o¤ the equilibrium only. The present paper’s compensation
4
principle exploits, in fact, the link between the compensation requirement
for unilateral deviations and e¢ cient incentives. Based on the legal requirement of compensation, the principle provides a convenient economic tool for
establishing the e¢ ciency of provisions from obligation law in a unifying way.
Based on the work of Rea (1987), Grady (1988) and Kornhauser and
Revesz (1991), the textbook by Miceli (2008) nicely summarizes earlier …ndings on liability under sequential moves. Miceli is looking for rules that
would ensure e¢ cient incentives, o¤ and on the equilibrium path, for the
second moving party without distorting the incentives of the …rst moving
party. Yet, the di¤erent combinations of rules he examines fail quite generally to satisfy all of the desirable properties. Based on this observation,
Miceli sees a con‡ict between second best incentives of the second moving
party (i.e. e¢ cient incentives o¤ the equilibrium path) and e¢ cient incentives for the party that moves …rst. The present paper, in contrast, shows
that a strict adherence to the appropriate di¤erence hypothesis would satisfy
all the properties, which Miceli considers as desirable.
With respect to uncertain causation, the present paper extends my earlier …ndings (Schweizer (2009)) beyond the binary accident model. For sake
of completeness, …nally, let me also refer to Schweizer (2005) on the economic analysis of obligation law. While, in that paper, I had focussed on the
mathematical saddle point property, the present paper takes a more legal perspective by relying on the compensation goal and the di¤erence hypothesis
instead.
The paper is organized as follows. Section 2 introduces the unilateral
version of the compensation principle. Section 3 spells out the di¤erence hypothesis in a setting of certain causation. The di¤erence hypothesis ensures
that the compensation principle applies. For a general setting of uncertain
causation, section 4 introduces three di¤erent methods of quantifying damages in line with the di¤erence hypothesis. Furthermore, this setting o¤ers a
new perspective on the binary accident model.
Section 5 establishes the bilateral version of the compensation principle.
In a setting of certain causation, section 6 proposes a rule based on the difference hypothesis that would ensure e¢ cient incentives on and o¤ the equilibrium path without distorting incentives of the …rst moving party. Section
7 introduces a version of the bilateral accident model capturing uncertain
5
causation. Lawyers consider causality as an indispensable requirement for
liability of a party. In a setting of bilateral interaction, the requirement of
causality may actually raise deep philosophical concerns. From the e¢ ciency
perspective, however, such concerns are of lesser relevance as follows from the
bilateral compensation principle, which entirely rests on one-sided deviations
only.
Section 8 provides a brief account of the multi-lateral compensation principle. Section 9 concludes.
2
The unilateral compensation principle
The unilateral compensation principle applies to models with general features as follows. The only strategically acting party A, also referred to as
debtor/injurer/she, takes a decision a from a given set A of alternatives.
Party B, also referred to as creditor/victim/he, remains passive but is a¤ected
by party A’s decision. Let w(a) denote the (expected) collective welfare as a
function of party A’s decision.
Party A has the obligation to choose some given ao 2 A. If she deviates
by choosing a instead then, depending on the legal regime in place, she may
owe a transfer payment (damages) to party B. Party A’s (expected) payo¤
net of transfer payments amounts to (a; ao ) and party B’s (expected) payo¤
including transfer payments to (a; ao ).
If party A meets her obligation then these payo¤s add up to welfare, i.e.
(ao ; ao ) + (ao ; ao ) = w(ao )
is assumed to hold. If A deviates by choosing a 6= ao then welfare is assumed
to be an upper bound such that
(a; ao ) + (a; ao )
w(a)
is assumed to hold for all a 2 A. The di¤erence (if any) may be lost to
transaction costs.
The unilateral compensation principle applies to any such situation and
provides the following e¢ ciency criterion. Suppose party A’s obligation ao
maximizes welfare w(a) and the compensation goal is achieved in the sense
that party B can never be worse o¤ if party A deviates from ao . Then
6
meeting the obligation maximizes party A’s payo¤. Moreover, if more than
one decision maximizes party A’s objective function then all of them have
to be e¢ cient and payo¤ equivalent. The following proposition provides a
rigorous account of this principle.
Proposition 1 (unilateral compensation principle) If party A’s obligation is
e¢ cient, i.e. ao 2 arg maxa2A w(a); and if the compensation goal is met in
the sense that
(ao ; ao )
(a; ao )
holds for any deviation ao , then
ao 2 arg max (a; ao )
arg max w(a)
a2A
a2A
most hold. Moreover, if (a; ao ) = (ao ; ao ) then
(a; ao ) = (ao ; ao ).
Proof. For any decision a 2 A, it follows from the assumptions made
that
(a; ao )
w(a)
(a; ao )
w(ao )
(ao ; ao ) = (ao ; ao )
must hold which means that ao maximizes party A’s objective function indeed.
If her objective functions attains a further maximum at some other decision a then (a; ao ) = (ao ; ao ) and, hence,
w(ao ) = (ao ; ao ) + (ao ; ao )
(a; ao ) + (a; ao )
w(a)
must hold. Yet, since ao is assumed to maximize welfare, all the above
inequalities must be binding such that, in particular, the equalities w(ao ) =
w(a) and (ao ; ao ) = (a; ao ) must both be ful…lled. This means that a also
maximizes welfare and party B is equally well o¤ under a as under ao .
While the formal proof of the unilateral compensation principle is straightforward, the principle proves useful nonetheless to establish the e¢ ciency of
incentives in a unifying way and for a great variety of applications from
obligation law.
7
3
The di¤erence hypothesis under certain causation
As a …rst application, I show that the compensation principle applies if,
under a negligence regime, damages are awarded in line with the di¤erence
hypothesis. Party A still decides on a 2 A. Decision a comes with costs c(a)
to be borne by party A. In the present section, the value of party B’s assets is
assumed to be a deterministic function v(a) of party A’s decision a. Welfare
amounts to w(a) = v(a)
c(a).
Suppose party A faces the obligation to choose ao but actually chooses
a instead. The di¤erence v(ao )
v(a) of the hypothetical value v(ao ) and
the actual value v(a) of B’s assets captures the loss, which party B su¤ers
from party A’s deviation. But notice, a loss only occurs if this di¤erence
is positive. Otherwise party B rather enjoys an enrichment caused by the
deviation which, however, party B may keep for free as compensation for
(unjust) enrichment is not considered by the present paper. Without such
compensation, damages in line with the di¤erence hypothesis amount to
d(a; ao ) = max [v(ao )
v(a); 0]
for short.
Under such damages, the compensation goal is achieved in the sense that
(a; ao ) = v(a) + d(a; ao )
v(a) + [v(ao )
v(a)] = v(ao ) = (ao ; ao )
holds with respect to party B’s payo¤ (a; ao ) (including damages) for any
deviation a and, hence, the compensation principle applies. As a consequence, if party B receives damages in line with the di¤erence hypothesis
and if party A’s obligation maximizes welfare then she has the incentive to
meet her obligation. Moreover, any other decision that maximizes her objective function is e¢ cient as well and leads to a payo¤ equivalent outcome. All
these claims immediately follow from the compensation principle established
in the previous section.
The di¤erence hypothesis is a concept derived from obligation law. E¢ cient incentives, in contrast, re‡ect an economic desideratum. The compensation principle provides a link between the two perspectives.
8
4
The di¤erence hypothesis under uncertain
causation
Party A still takes a decision a 2 A and bears the costs c(a) that come
with it. The decision still a¤ects the value of party B’s assets though in an
uncertain way. Uncertainty is captured by a random move ! which nature
draws from outcome space
0
. Event
occurs with exogenously given
probability ( 0 ).
Party B’s assets can attain values from a (for simplicity, …nite) list fv0 ; v1 ; :::; vn g
which is assumed ranked as v0 > v1 > ::: > vn . If party A takes the decision
a and nature draws ! then party B’s assets will be of value V (a; !).
Party A has the obligation to choose ao but suppose she deviates and
chooses a instead. Courts can observe this deviation. Moreover, to begin
with, let me assume that courts can even observe the true move ! of nature.
Then, in line with the di¤erence hypothesis and as in the previous section,
courts would award damages
D(a; !; ao ) = max [V (ao ; !)
V (a; !); 0]
to party B (compensation for enrichment is still ruled out). The (ex post)
payo¤ of party B including damages then amounts to
V (a; !) + D(a; !; ao )
V (ao ; !)
and can never be lower than B’s payo¤ V (ao ; !) if A had met her obligation.
Since this inequality holds for all moves ! of nature it must hold, a fortiori,
for B’s expected payo¤s
(a; ao ) = E [V (a; !)] + d(a; ao )
E [V (ao ; !)] = (ao ; ao )
where expectations are taken with respect to the given distribution of the
random move ! of nature and where d(a; ao ) = E[D(a; !; ao )] denotes expected damages awards.
As a consequence, the compensation principle applies and, hence, if party
A’s obligation maximizes the expected welfare
w(a) = E [V (a; !)]
c(a)
then A has the incentive to meet her obligation.
9
So far, I have assumed that courts can observe the true move of nature
which essentially means that they know the hypothetical value of B’s assets
for sure even if this value is of counterfactual nature. Such an assumption
seems quite restrictive. To relax it, let me now assume that courts can only
observe the actual value vi of party B’s assets but not the hypothetical one
(interim perspective). This informational setting is re‡ected by the partition
=
of the outcome space
0 (a)
where
[
1 (a)
[ ::: [
n (a)
= f! 2
i (a)
: V (a; !) = vi g denotes the
observable event that, at the given deviation a, the actual value of party
B’s assets is vi . This event
i (a)
occurs with ex ante probability qi (a) =
f i (a)g.
In the following, I discuss three di¤erent methods of quantifying dam-
ages, all arguably in line with the di¤erence hypothesis. As a …rst method,
courts may award damages as above but, since the move of nature remains
uncertain, in (interim) expected terms. More precisely, according to the …rst
method, damages equal to the expected value
dIi (a; ao ) = E [D(a; !;
of D(a; !;
o
o
)j
i (a)]
) conditional on the observed event
i (a)
that party B’s assets
under the actual decision a are of value vi would be awarded.
To calculate such damages dIi (a; ao ) explicitly, the probability of the
subevent
o
ij (a; a )
= f! 2
i (a)
: V (ao ; !) = vj g
that the hypothetical value is vj must be known (for j = 0; :::; n) from which
the corresponding conditional probability
f
qij (a; ao ) =
o
ij (a; a )g
f i (a)g
can be derived. The expected value of damages conditional on the actual
value of party B’s assets being vi amounts to
dIi (a; ao ) =
n
X
qij (a; ao ) max[vj
j=0
vi ; 0] =
i
X
qij (a; ao ) (vj
vi ):
j=0
From the ex ante perspective, party B’s expected damages awards amount
to
dI (a; ao ) =
n
X
qi (a) dIi (a; ao ) = E[D(a; !; ao )] = d(a; ao )
i=0
10
and, under method I, are the same as if the move of nature were observable.
Therefore, the compensation goal is still achieved and the compensation principle still applies if damages were awarded according to method I.
Loosely speaking, method I speci…es the expected value of the di¤erence
as the appropriate quantum of damages. More in line with court practice,
however, may be to award the di¤erence of the expected hypothetical and
the actual value of party B’s assets. Probability theory o¤ers the following
two options. Either the expected value
Hi (a; ao ) = E [V (ao ; !) j
conditional on the observed event
o
i (a)
o
i (a)] =
n
X
qij (a; ao ) vj
j=0
or the unconditionally expected value
H(a ) = E [V (a ; !)] =
n
X
qi (a) Hi (a; ao )
i=0
of party B’s assets may serve as the hypothetical (counterfactual) value of
B’s assets such that damages in spirit of the di¤erence hypothesis
o
o
dII
i (a; a ) = max [Hi (a; a )
o
o
vi ; 0] or dIII
i (a ) = max [H(a )
vi ; 0]
would be awarded under methods II and III, respectively.
Notice, at the lowest value possible, i.e. i = n; methods I and II specify
the same quantum
o
o
dIn (a; ao ) = dII
n (a; a ) = Hn (a; a )
vn :
At the highest value, i.e. i = 0, party B cannot have su¤ered any harm at
all and, for that reason,
o
II
o
dI0 (a; ao ) = dII
0 (a; a ) = d0 (a; a ) = 0
holds for all three methods. At intermediate values 0 < i < n, however, the
methods may quantify damages di¤erently.
Party A decides on a ex ante and faces damages claims amounting, in
expected terms, to
II
o
d (a; a ) =
n
X
qi (a)
o
dII
i (a; a )
and d
i=0
III
o
(a; a ) =
n
X
i=0
under method II and III, respectively.
11
o
qi (a) dIII
i (a )
The following proposition shows that, from the ex ante perspective, all
three methods achieve the compensation goal and, hence, the compensation
principle applies under all three of them. On e¢ ciency grounds, none would
outperform the other ones.
Proposition 2 Suppose, if party A, by choosing a, has deviated from her
obligation ao and if the actual value of party B’s assets is vi ; party B can reo
cover damages dm
i (a; a ) according to one of the thee methods m 2 fI; II; IIIg.
Then party B will be worse o¤ under no such deviation and, hence, the compensation principle applies under all three methods.
Proof. For method I, the claim has already been established. Under
method II, expected awards amount to
dII (ao ) =
n
X
qi (a) max [Hi (a; ao )
vi ; 0]
i=0
n
X
o
qi (a) Hi (a; a )
i=0
n
X
qi (a) vi = E[V (ao ; !)]
E[V (a; !)]
i=0
whereas, under method III, they amount to
III
d
o
(a ) =
n
X
i=0
n
X
qi (a) max [H(ao )
qi (a) H(ao )
i=0
vi ; 0]
n
X
qi (a) vi = E[V (ao ; !)]
E[V (a; !)]:
i=0
Therefore, the compensation goal is achieved under methods II and III as
well.
Method III, in particular, may come close to what courts (in Germany at
least) actually do at the second step of a two-step approach. As a …rst step,
the question is asked whether party A’s deviation has caused any harm at
all (haftungsbegründende Kausalität). If this question is denied then party A
will not be held liable. Only if the answer from the …rst step is yes damages
must ever be quanti…ed.
At the second step, the unconditionally expected value H(ao ) may well
serve as the hypothetical value of party B’s assets from which the actual
value must be deducted, well in line with method III above. To defend this
claim, think of an used car that was damaged in an accident and suppose
12
the accident would have been avoided for sure if the injurer had met her
obligation. Most likely, courts would then award the average market value
of an undamaged used car with the same general features. In fact, lists
providing market values of used cars are widely available such that courts do
not even have to calculate averages on their own. Such lists provide, in fact,
unconditionally expected values, well in line with method III above.
But remember, as a …rst step, courts examine the causality requirement.
Even in a setting of uncertainty fully speci…ed as above, it remains unclear
under what conditions exactly this requirement should be considered as being ful…lled. Yet, denying liability at the …rst step puts the compensation
goal at risk. As a consequence, the compensation principle need no longer
apply and incentives may well end up being distorted. Separating causality
from quantifying damages proves arti…cial indeed. Since all three methods
proposed above take causality into account, the …rst stage seems super‡uous
anyhow.
In the literature, attention has been paid almost exclusively to a binary
version of the above setting where party B’s assets can attain only two values
v0 and v1 = v0
L. The loss L of …xed size is assumed to occur with
probability "(a) as a function of party A’s decision a: The binary version of
the setting is referred to as the unilateral accident model. To conclude this
section, let me spell out the three methods for this special case.
The accident model requires partitioning the outcome space
two events
0 (a)
in the event
of no accident and
1 (a)
1 (a)
into the
of an accident, respectively. Only
of an accident does party B su¤er from any harm at
all. Therefore, only for this event
1 (a),
which occurs with probability "(a),
damages have ever to be quanti…ed.
To implement methods I and II, the accident event
partitioned into the two subevents
10 (a)
and
11 (a)
1 (a)
must be further
where the true hypo-
thetical value of B’s assets would be v0 and v1 , respectively. The probability
q10 (a; ao ) = f
10 (a)g=
f
1 (a)g
is the probability that party A’s deviation
a 6= ao has caused the accident, conditional on the event that, at deviation a,
the accident has actually occurred. The expected value of damages according
to method I amounts to dI1 (a; ao ) = q10 (a; ao ) L.
Methods II and III make use of a hypothetical value of party B’s assets,
13
namely the conditionally expected value
H1 (a; ao ) = q10 (a) v0 + q11 (a) v1
and the unconditionally expected value
H(ao ) = [1
"(ao )] v0 + "(ao ) v1 ,
respectively.
Method III leads, in the binary accident model, to the quantum
o
o
dIII
1 (a ) = H(a )
v1 = [1
"(ao )] L
which party A owes to B, in case an accident has occurred. The traditional
negligence rule, in contrast, is widely claimed to quantify L as damages instead. Hence the two methods lead to the same quantum only if the accident
were avoided for sure (i.e. "(ao ) = 0) whenever party A has met her obligation. Otherwise, method III and the traditional negligence rule quantify
damages di¤erently.
Such a discrepancy seems surprising and rather disturbing. In fact, the
binary accident model proves misleading as, in the general setting, no parameter exists that would correspond to the L of the binary version. Moreover,
what plausibly re‡ects court practice in the general setting does not lead to
damages amounting to L in the binary setting under method III (unless
"(ao ) = 0).
As a theoretical possibility, methods I and II would also quantify damages
o
o
dI1 (a; ao ) = dII
1 (a; a ) = L provided that the conditional probability q10 ( ; a )
of party A’s deviation being causal for the accident to occur were equal to
one. This condition would mean that, given the accident has occurred under
A’s deviation, it would have been avoided for sure if A had met her obligation.
In principle, this condition may be met even if the unconditional probability
remains positive, i.e. "(ao ) > 0. But if it does then, for any move of nature,
for which the accident occurs if the obligation has been met, the accident
would have been avoided under the deviation. As such a situation seems
entirely implausible, it can be ruled out for practical purposes. In this sense,
the following proposition has been established.
Proposition 3 In the accident model as the binary version of the more general setting of uncertain causation, none of the three methods would specify
14
L as the appropriate quantum of damages unless the accident were avoided
for sure if party A had met her obligation.
For decades, the accident model has served as the workhorse of the economic analysis of tort law. Within the accident model, since decades, the
negligence rule has been assumed to specify L as the appropriate quantum of
damages if the injurer was at fault. As long as causation remains uncertain,
however, such a speci…cation can hardly be reconciled with the di¤erence
hypothesis as the present section has shown.
In Schweizer (2009), I have shown that proportional liability as examined
by Shavell (1985) may well be in line with quantifying damages according to
method I and II which are the same in the accident model.
5
The bilateral compensation principle
The bilateral compensation principle concerns situations where two parties
A and B act strategically and where both face obligations. Party A chooses
her decision a from the set A of alternatives whereas party B chooses b from
the set B. At decision pro…le (a; b) 2 A
B, the (expected) welfare amounts
to w(a; b).
Party A has the obligation to choose ao , party B to choose bo . If they
deviate, transfer payment in one or the other direction (and possibly other
remedies) may be due. Let (a; b; ao ; bo ) and
(a; b; ao ; bo ) denote the (ex-
pected) payo¤ including transfer payments of party A and party B, respectively, if they have actually chosen the decision pro…le (a; b). By assumption,
the sum of these payo¤s is equal to welfare if both meet their obligation, i.e.
(ao ; bo ; ao ; bo ) + (ao ; bo ; ao ; bo ) = w(ao ; bo )
whereas the sum may be lower, if they deviate, i.e.
(a; b; ao ; bo ) + (a; b; ao ; bo )
holds for any decision pro…le (a; b) from A
w(a; b)
B. The di¤erence if any may
be lost again to transaction costs.
The following proposition will be referred to as bilateral compensation
principle. It establishes that, if the obligation pro…le (ao ; bo ) maximizes welfare w(a; b) and if no party is worse o¤ under no unilateral deviation from the
15
obligation pro…le by the other party, the obligation pro…le is a Nash equilibrium of the game with payo¤ functions
and
. Moreover, if several Nash
equilibria exist all must be payo¤ equivalent.
Proposition 4 (bilateral compensation principle) If the obligation pro…le is
e¢ cient, i.e. (ao ; bo ) 2 arg max(a;b)2A
B
w(a; b) and if the compensation goal
for unilateral deviations is achieved in the sense that
(ao ; bo ; ao ; bo )
(ao ; b; ao ; bo ) and
(ao ; bo ; ao ; bo )
(a; bo ; ao ; bo )
hold for any unilateral deviations b 6= bo and a 6= ao , respectively, then (ao ; bo )
is a Nash equilibrium of the game with payo¤ functions
and
. Moreover,
any other Nash equilibrium (xN ; bN ) (if more than one exists) must be payo¤
equivalent and e¢ cient as well.
Proof. For better readability, I omit writing the obligation pro…le as an
argument. Then, for any deviation a 6= ao it follows that
(a; bo )
w(a; bo )
(a; bo )
w(ao ; bo )
(ao ; bo ) = (ao ; bo )
must hold, which means that ao is a best response of party A to party B’s
meeting his obligation bo . For the same reason, bo must be a best response
of party B to party A’s meeting her obligation. Therefore, the obligation
pro…le (ao ; bo ) is a Nash equilibrium indeed.
For any other Nash equilibrium (aN ; bN ), it follows that
(aN ; bN )
(ao ; bN )
(ao ; bo )
(aN ; bN )
(aN ; bo )
(ao ; bo )
as well as
and, hence,
w(ao ; bo ) = (ao ; bo ) + (ao ; bo )
(aN ; bN ) + (aN ; bN )
w(aN ; bN )
must hold. Yet, since (ao ; bo ) maximizes welfare, all of the above inequalities
must be binding such that, in particular, w(ao ; bo ) = w(aN ; bN ), (aN ; bN ) =
(ao ; bo ) and
(aN ; bN ) = (ao ; bo ) must all hold. This fully establishes the
proposition.
16
The bilateral compensation principle easily extends to subgame perfect
equilibria of sequential move games. Suppose party A moves …rst by choosing
a from A before party B, after having observed A’s move a; chooses b from
B. In subgame perfect equilibrium, for any a, party B reacts with a best
response
N
: A ! B such that
N
(a) 2 arg max (a; b)
b2B
holds for all a. At the …rst stage, party A anticipates the best response
N
of party B and, hence, party A chooses
aN 2 arg max (a;
N
a2A
(a)):
Then, under the same assumptions as made in the previous proposition,
it can be shown that the subgame perfect equilibrium must also be payo¤
equivalent to the Nash equilibrium under simultaneous moves.
Proposition 5 If the obligation pro…le is e¢ cient, i.e. (ao ; bo ) 2 arg max(a;b)2A
B
w(a; b)
and if the compensation goal for unilateral deviations is achieved in the sense
that
(ao ; bo ; ao ; bo )
(ao ; b; ao ; bo ) and
(ao ; bo ; ao ; bo )
(a; bo ; ao ; bo )
hold for any unilateral deviations b 6= bo and a 6= ao , respectively, then
(aN ;
N
(aN ); ao ; bo ) = (ao ; bo ; ao ; bo ) and
(aN ;
N
(aN ); ao ; bo ) = (ao ; bo ; ao ; bo )
must hold for any subgame perfect equilibrium aN and
game with payo¤ functions
and
N
.
: A ! B of the
Proof. For better readability, I omit writing the obligation pro…le as an
argument. Since
N
is a best response it follows that
(aN ;
N
(aN ))
(aN ; bo )
(ao ; bo )
must hold. Moreover,
(aN ;
N
(aN ))
(ao ;
N
(ao ))
(ao ; bo )
and, hence,
w(aN ;
N
(aN ))
(aN ;
N
(aN )) + (aN ; (aN ))
(ao ; bo ) + (ao ; bo ) = w(ao ; bo )
17
must hold. Yet, since the obligation pro…le maximizes welfare, all of the
above inequalities must be binding. Therefore, in particular,
w(aN ;
N
(aN )) = w(ao ; bo )
as well as
(aN ;
N
(aN )) = (ao ; bo ) and (aN ;
N
(aN )) = (ao ; bo )
must be ful…lled. This establishes the proposition.
Since the e¢ ciency claim of the bilateral compensation principle rests
on compensation for unilateral deviations only, much indeterminacy remains
with respect to quantifying damages under two-sided deviations. The ample
scope left may be used to generate e¢ cient incentives even o¤ the equilibrium
path as will be shown in the next section.
6
Contributory negligence under certain causation
To focus on incentives o¤ the equilibrium path, I consider the following setting under certain causation. Party A decides a 2 A and bears costs c(a) as
a function of her decision a. If party B chooses b 2 B then the value of B’s
assets net of the costs for b (which B must bear) is a function v(a; b) of the
decision pro…le (a; b). Welfare amounts to w(a; b) = v(a; b)
c(a).
Since both parties take decisions, both may face obligations. To begin
with, let me assume that A and B have the obligation to choose ao and bo ,
respectively, but that they have chosen the decision pro…le (a; b) instead. If
contributory negligence were not taken into account, damages in line with
the di¤erence hypothesis would amount to
d(a; b; ao ; bo ) = max[v(ao ; b)
v(a; b); 0]
from which the payo¤ functions
(a; b; ao ; bo ) =
c(a)
d(a; b; ao ; bo )
and
(a; b; ao ; bo ) = v(a; b) + d(a; b; ao ; bo )
18
of party A and B, respectively, emerge.
It follows immediately that the compensation goal of both parties with
respect to unilateral deviations of the other party will be achieved such that
the bilateral compensation principle applies. The following proposition is an
immediate corollary of proposition 4 and, for that reason, needs no proof of
its own.
Proposition 6 Suppose the obligation pro…le (ao ; bo ) maximizes welfare w(a; b)
and the game with payo¤ functions
moves …rst). If aN and
this game then
(aN ;
N
N
and
is played sequentially (party A
: A ! B form a subgame perfect equilibrium of
(aN )) = (ao ; bo ) and
holds and the equilibrium path (aN ;
N
N
(aN ;
(aN )) = (ao ; bo )
(aN )) maximizes welfare.
Notice the above e¢ ciency result holds in spite of the fact that, so far,
contributory negligence by the creditor has not been taken into account.
Obligation law, however, may actually reduce damages. The German civil
code (§ 254 BGB), e.g., requires that "where fault on the part of the injured
person contributes to the occurrence of the damage, ... the extent of compensation to be paid depend on the circumstances, in particular to what extent
the damage is caused mainly by one or the other party. This also applies if
the fault of the injured person is ... failing to ... reduce the damage."
To illustrate such mitigation of damages, let me assume again that parties
choose sequentially, …rst party A and then party B. Suppose party A deviates
from her obligation ao by choosing a instead. Then party B has the obligation
to minimize losses which, according to the di¤erence hypothesis, would be
equivalent to maximize the net value v(a; b) of his assets. Let
V (a) = max v(a; b)
b2B
denote this maximum value as a function of party A’s actual decision a.
If party B deviates from his obligation to minimize losses, nonetheless, he
is awarded damages amounting to
D(a; ao ) = max[V (ao )
19
V (a); 0]
only. As these damages do not depend on B’s actual decision b, his objective
function
v(a; b) + D(a; ao ) = w(a; b) + c(a) + D(a; ao )
di¤ers from welfare w(a; b) by a factor only independent of B’s decision b. As
a consequence, party B has the incentive to react, on and o¤ the equilibrium
path, with a socially best response to any deviation a of party A from her
obligation ao .
If party B reacts in this way his payo¤ amounts to
(a; ao ) = V (a) + D(a; ao )
V (ao ) =
(ao ; ao )
and cannot attain lower values than if party A had met her obligation. Party
A’s payo¤ function is
(a; ao ) =
c(a)
D(a; ao )
in this regime of contributory negligence.
As a consequence, the unilateral compensation principle applies if the
corresponding welfare function W (a) is speci…ed as
W (a) = V (a)
c(a) = max w(a; b)
b2B
and if party A’s obligation ao maximizes this welfare function W (a): Under
these provisions, ao must also maximize A’s payo¤ function. Moreover, if
more than one decision maximizes A’s payo¤ function, all of them must
maximize welfare as well. These claims follow directly from the unilateral
compensation principle and, for that reason, require no further proof. The
following proposition summarizes these …ndings.
Proposition 7 Suppose party A has the obligation to choose ao and this
obligation maximizes welfare W (a). Moreover, if party A deviates by choosing
a instead then party B receives damages D(a; ao ) = max[V (ao )
V (a); 0] ,
no matter whether or not B meets his obligation to mitigate losses. Then
ao 2 arg max (a; ao )
a2A
must hold.
20
arg max W (a)
a2A
While the subgame perfect equilibrium outcome will already be e¢ cient
as soon as the compensation goal for unilateral deviations relative to an
e¢ cient obligation pro…le is achieved, the regime of contributory negligence as
examined above allows to generate e¢ cient incentives even o¤ the equilibrium
path.
7
Contributory negligence under uncertain causation
To discuss contributory negligence in a setting with uncertain causation, I
consider the following version of the bilateral accident model. The choice
variables x 2 X and y 2 Y of party A and B, respectively, directly coincide
with precaution costs, such that the choice sets X and Y can be interpreted
as subsets of the real line, endowed with the usual order relation. Uncertainty
is captured by a random move ! 2
e:X
of nature. The function
! f0; 1g
Y
describes the interaction between the two parties and nature. If the pro…le
(x; y; !) is chosen then e(x; y; !) = 0 means that no accident occurs whereas
if e(x; y; !) = 1 then an accident has occurred, leading to a loss of …xed size
L in terms of the value of party B’s assets.
From this function e, the accident probability
"(x; y) = f! 2
: e(x; y; !) = 1g
can be derived. The existing literature, in fact, directly postulates the accident probability as a function "(x; y) of the pro…le (x; y) of precaution spendings, decreasing in both arguments. The present paper, instead, introduces
a random move of nature explicitly to take causality into account.
Suppose party A and B face the obligation not to spend less than xo and
y o , respectively, on precaution. Moreover, to begin with, suppose courts can
observe the move of nature. If party A meets her obligation she will never
be held liable, i.e. zero damages D(xo ; y; !) = 0 are awarded for any move
of nature ! and independent of party B’s actual precaution spending y if A
meets her obligation.
21
If, however, party A unilaterally deviates by actually spending less, x <
xo , on precaution then party B can recover L provided that party A’s deviation has caused the accident. Taking causality into account, damages under
unilateral x < xo deviations by party A amount to
D(x; y o ; !) = max[e(x; y o ; !)
e(xo ; y o ; !); 0] L:
Party B’s ex post payo¤ including damages
yo
e(x; y o ; !) L + D(x; y o ; !)
yo
e(xo ; y o ; !) L
can never be lower than if party A had met her obligation of spending xo on
precaution. Since this is true for any move of nature, the compensation goal
for unilateral deviations of party A will, a fortiori, be achieved from the ex
ante perspective.
Party A’s payo¤ under any unilateral deviation of party B is even constant
xo as party A, by meeting her obligation, escapes liability.
and equal to
As a consequence, the bilateral compensation principle applies. Therefore, if the obligation pro…le (xo ; y o ) maximizes expected welfare w(x; y) =
x
y
"(x; y) L then any Nash equilibrium of the game with payo¤
functions
(x; y) =
x
E[D(x; y; !)] and (x; y) =
y
"(y) L + E[D(x; y; !)]
must maximize welfare. If more than one such equilibrium exists, all are
payo¤ equivalent. Notice, this e¢ ciency result holds independently of how
exactly damages D(x; y; !) would be quanti…ed if both parties, by choosing
x < xo and y < y o ; deviate at the same time.
Nonetheless, quantifying damages under two-sided deviations is of much
concern from the legal perspective. No party should ever be held liable unless
her or his deviation has caused harm. Yet, under two-sided deviations, the
causality requirement raises new conceptual di¢ culties. Suppose an accident
has actually occurred, i.e. e(x; y; !) = 1 (otherwise there would be no need to
quantify damages). Then four (hypothetical) subcases can be distinguished:
(1) e(x; y o ; !) = 1 and e(xo ; y; !) = 0
(2) e(x; y o ; !) = 0 and e(xo ; y; !) = 1
(3) e(x; y o ; !) = 0 and e(xo ; y; !) = 0
(4) e(x; y o ; !) = 1 and e(xo ; y; !) = 1
22
No dispute on causality should arise in the …rst two cases. In case (1), the
accident would have occurred even if party B had met his obligation and, for
that reason, party A’s deviation must have caused the accident such that she
owes damages D(x; y; !) = L to party B. In case (2), for similar reasons, it
must have been party B’s deviation that has caused the accident and, hence,
party A will not be held liable, i.e. D(x; y; !) = 0 in case (2).
In case (3), no single deviation would have caused the accident on its own
and, hence, it must be the joint deviation that has caused it. In this case, the
two parties may be held jointly and severally liable, for which the German
civil code (§ 426 BGB) o¤ers the following provision: "The joint and several
debtors are obliged in equal proportions in relation to one another unless
otherwise determined". If shared in equal proportions party B could recover
D(x; y; !) = L=2 in case (3).
In case (4), …nally, each deviation would have been su¢ cient to cause
the accident on its own. Since the statement party A has caused the accident cannot be denied, she may be held fully liable. Alternatively, case (4)
may also qualify as joint and several liability such that the corresponding
provisions of obligation law would apply again.
Such ambiguities of the causality requirement may be re‡ected by the
indeterminacy left by the bilateral compensation principle.
In the law and economics literature, mainly referring to U.S. common law,
rules of contributory negligence such as the following ones are being discussed.
If both parties have violated their obligations by spending less than xo and
y o , respectively, then party B cannot recover under the negligence rule with
a defense of contributory negligence. Under comparative negligence, party A
would be required to pay some fraction 0
(x; y; !; xo ; y o )
1 of the total
loss only, i.e.
D(x; y; !) = (x; y; !; xo ; y o ) L
if both parties have deviated from their obligations. Typically, this fraction
is assumed decreasing in party A’s precaution spendings x but increasing in
the other party’s spending y.
Which of these (or of still further) quanti…cations would actually apply
remains di¢ cult to tell even if the move of nature happens to be observable. It
might even turn out that, for some moves of natures, the negligence rule with
a defense of contributory negligence while, for others, the rule of comparative
23
negligence would apply. The traditional accident model, however, which just
speci…es the accident probability "(x; y) as a function of the two parties’
precaution spendings cannot take such a possibility into account.
On e¢ ciency grounds, the exact quanti…cation of damages under twosided deviations does not matter as long as the obligation pro…le maximizes
the expected welfare. This insight is an immediate consequence of the bilateral compensation principle.
8
The multi-lateral compensation principle
The multi-lateral compensation principle concerns situations as follows. A
…nite set N = f1; :::; ng of parties strategically decide on the pro…le a =
(a1 ; :::; an ) from A = A1
An . The (expected) welfare under pro…le a
:::
amounts to w(a).
Party i, in fact, decides on ai 2 Ai while facing obligation aoi . Let ao =
(ao1 ; :::; aon ) denote the obligation pro…le and
i (a; a
o
) party i’s payo¤ if pro…le
a is actually chosen. These payo¤s include transfer payments that may be
due under deviations from the obligation pro…le.
For the multi-lateral compensation principle to apply, the obligation pro…le ao must maximize welfare w(a) and the compensation goal must be
achieved in the sense that
i (a
o
; ao )
o
o
i (ai ; a i ; a )
holds for any party i and any deviation a
i
by the other parties.
Moreover, it is required that welfare w(ao ) results whenever all parties
meet their obligation, i.e.
n
X
i (a
o
; ao ) = w(ao )
i=1
whereas welfare w(a) is an upper bound it they deviate by choosing pro…le
a instead, i.e.
n
X
i (a; a
o
)
w(a)
i=1
is assumed to hold for all such deviations.
If all the above assumptions are met then the obligation pro…le ao must
be a Nash equilibrium of the game with payo¤ functions
24
i (a; a
o
) and all
Nash equilibria (if more than one exists) must be payo¤ equivalent. Moreover, these Nash equilibria turn out to be robust even against coordinated
deviations by any coalition of players.
If coalition C
N of players is able of fully coordinating their decisions,
the coalition aims at maximizing their coalitional payo¤ function
C (aC ; a C ; a
o
)=
X
i (aC ; a C ; a
o
)
i2C
over their coalitional decision pro…le aC 2
Y
Ai .
i2C
Under the premises of the multi-lateral compensation principle, the obligation pro…le ao = (aoC ; aoN nC ) turns out to be a Nash equilibrium of any of
the two-party game with payo¤ functions
C
all parties from N nC meet their obligations
aoN nC
and
N nC .
then aoC
In particular, if
remains to be a
coordinated best response by the coalition C.
The proof of this claim immediately follows from the bilateral compensation principle. Observe, in contrast to situations of oligopoly where coordination within a cartel allows (if such coordination is not prohibited by
law) to increase the cartel’s pro…t, under the premises of the multi-lateral
compensation principle, coordination within a coalition cannot increase the
coalitional payo¤.
The multi-lateral compensation principle leaves much indeterminacy as
far as distributional issues are concerned. Such indeterminacy is matched
by the vagueness of provisions from obligation law on how multiple debtors
should share liability.
9
Conclusion
From the legal perspective, compensation of the creditor is considered as
the primary goal of obligation law whereas the economic analysis sees the
e¢ ciency of incentives as the more important goal. Some texts suggest a
con‡ict between the two perspectives. Cooter (1985), e.g., even talks of a
compensation paradox.
The present paper, in contrast, propagates the compensation principle
as a tool to establish the e¢ ciency of provisions from obligation law. While
exact compensation in a bilateral setting may be beyond reach indeed, ef25
…cient incentives require compensation for unilateral deviations only. Any
regime that awards damages in line with the di¤erence hypothesis achieves
the compensation goal for unilateral deviations and, according to the compensation principle, provides e¢ cient incentives provided that the obligation
pro…le maximizes welfare.
Achieving the compensation goal is a su¢ cient but not a necessary condition for e¢ cient incentives. The present paper refers to a whole bundle of
known e¢ ciency results where this su¢ cient condition is met and where the
compensation principle establishes e¢ ciency from a unifying perspective. No
doubt, many more applications are around where the argument also works
even if the existing literature has relied instead on the methodologically more
demanding …rst order approach.
There exist, however, e¢ ciency results where the …rst order approach
works but the compensation principle does not. But such e¢ ciency results
are of a less robust nature as they require more restrictive assumptions.
If uncertain causation is involved, courts put more weight on the requirement that the debtor’s deviation has actually caused harm than to honour
the compensation goal. The present paper gives priority to the compensation
requirement instead, not for justice reasons but because e¢ cient incentives
have been shown to result if the compensation goal is achieved from the ex
ante perspective at least.
10
References
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Kornhauser, L.A. and Revesz, R. (1991), "Sequential Decisions by
a Single Tortfeasor", Journal of Legal Studies, 20, 363-380.
Miceli, T.J. (2008), Economics of the Law (2nd edition), Palo Alto.
Rea, S.A. (1987), "The Economics of Comparative Negligence", International Review of Law and Economics, 7, 149-162.
Schweizer, U. (2005), "Law and Economics of Obligations”, International Review of Law and Economics, 25, 209-228.
26
Schweizer, U. (2009), "Legal Damages for Losses of Chances", International Review of Law and Economics, 29, 153-160.
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27