Rights on what is Left:
An axiomatic discussion of grandfathering
∗
Jérémy Laurent-Lucchetti† and Justin Leroux‡
Preliminary draft, do not circulate
Abstract
Allocating property rights on an open access resource which has been
freely exploited in the past is often problematic. In practice, involved
agents typically rely on one of two competing principles to determine future allocation. The rst principle, grandfathering favors the status quo
while the other one, historical accountability, is a corrective justice argument. These arguments are usually raised without any precise referential
framework. Our main contribution is to propose a simple framework allowing to (i) dene precisely each argument, (ii) explicit the link between
these arguments and (iii) clarify the methods that they logically imply.
Keywords:
Property rights, distributive justice, tragedy of the
commons, claims problems, axiomatic solutions.
JEL Classication Numbers:
D30, D63, D70, H23, Q2, Q3,
Q56.
∗ We are very grateful for stimulating conversations with Christian Almer, Stefan Ambec,
Olivier Bochet, Sidartha Gordon, Barbara Julien, Pierre Lasserre, Hervé Moulin, Bernard
Sinclair-Desgagné and William Thomson as well as for feedback from participants of the
Montreal Natural Resources and Environmental Economics Workshop and the Social Choice
and Welfare Conference.
† University of Geneva, Geneva School of Economics and Management
‡ HEC Montréal, CIRANO and CRÉ
1
"For to the one who has, more
will be given, and he will have an
abundance."
Matthew, 13:12.
1 Introduction
International negotiations about climate change raise many issues of science,
economics and politics.
They also raise serious issues of distributive justice.
This is due in part to the current consensus that climate change should be managed through emissions reduction targets. The resulting (tense) international
negotiations revolve around the apportionment of a global carbon budget of
sorts.
Fair allocation in this context is controversial because involved agents typically
rely on self-serving arguments, depending on which best accommodates their
situation. Developed countries, which have already emitted much greenhouse
gases, typically argue that they have a legitimate claim to a larger share of the
carbon budget.
Their past emissions were a by-product of a wealth-creation
process. But with development come higher emission needs, which persist even
today. Hence, refusing developed countries a larger share of the carbon budget,
the argument goes, would threaten
ex post
their legitimate right to development
at a time when climate change was not perceived as an issue.
At the very
least, they should not get less than developing countries. Conversely, developing
countries, which have emitted less in the past, claim that they themselves should
obtain a larger share than developed countries because they have beneted less
from it up until now. They also have a right to development and it is their
turn to exercise it.
The rst argument, known as
grandfathering
(see, for example, Ott and Sachs,
2002, or Bovens, 2011), considers an agent's current level of exploitation as a
rm property right over the resource. It requires that past use of the resource
strengthen the claim for future entitlements.
In the pollution case, it trans-
lates as follows: because high-emitting countries were unaware that they were
overusing a commons, it would be unfair to force them to emit less than developing countries. In the climate change context, grandfathering implies that the
share of future emissions of a high emitter should not be lower than those of a
historically low emitter.
2
The competing argument,
historical accountability,
states that agents having
beneted less from a common in the past should obtain a higher share of what
remains. This corrective justice argument considers the resource as a common
property over which all agents have an equally legitimate claim. In the climate
case, this principle is often associated with a general right to development
linked to the future needs of agents who have emitted less in the past and,
as such, are deemed to be less responsible for the accumulated stock of past
emissions (see for example Bovens, 2011, for an exposition of the argument
framed in the climate setting).
It is striking that these arguments are usually raised without any precise referential framework. Consequently, they are usually directly associated with specic
rules for allocating property rights; namely, proportional allocation for grandfathering, and equal sharing for historical accountability. Our main contribution
is to propose a simple framework that allows one to (i) dene precisely each
argument, (ii) make explicit the link between them, and (iii) clarify the sharing
rules that they logically imply.
1
Once dened in a unifying framework, we shall discover that grandfathering
and historical accountability are not incompatible. In fact, taken together, they
characterize the
equal sharing
rule, which splits the remaining resource equally
among all agents, irrespective of past emissions. This rule is totally blind with
respect to past activity and allocates rights over what is left in the most uniform
way possible.
Pushing each argument further allows us to single out two rules to allocate
property rights:
egalitarian
what we call the
rule, respectively.
The
strong grandfathering rule
strong grandfathering rule
and the
strong
allocates emis-
sions rights proportionally to past emissions. In other words, what is casually
construed as the grandfathering rule is actually the
strong grandfathering rule,
which corresponds to an extreme version of the grandfathering argument. By
contrast, the
strong egalitarian rule
is seldom advocated in existing negotiation
even though it is the one equalizing as much as possible the total share of each
agent (summing their initial consumption and their future claim).
2 Progressivity
Finally, we discuss the "progressivity" of these allocation rules.
1 Our
simple framework is inspired by the rationing and surplus sharing literature.
The
interested reader can read Moulin (2002) and Thomson (2003) for comprehensive surveys, as
well as Young (1987) and Young (1990) for applications to the taxation problem or Livne
(1986) for a related model of Nash Bargaining with changes in the disagreement point.
2 By
a progressive rule, we mean one which gives a higher quantity of what is remaining
3
is a key concept underlying many debates about the establishment of property
rights market and our framework allows for a formal treatment of its implications. It happens that the
rule
strong grandfathering rule
and the
strong egalitarian
are, respectively, the most progressive and the less regressive methods that
can be justied by our properties.
ciated with
grandfathering
Strong grandfathering
is traditionally asso-
while it is the most regressive method compatible
Symmetrically, the equal sharing method is traditionally
historical accountability principle but, surprisingly, it is the
most regressive method meeting this principle. The strong egalitarian method
is the one which aims to equalize as much as possible the total shares among
with the argument.
associated with the
all agents within the bounds imposed by our properties and thus should be the
one associated with the egalitarian goal.
2 Property rights as a means to achieving eciency
When several entities independently exploit a common resource in a free access
regime, failure to account for the externalities one's activity imposes on others
often leads to a
tragedy of the commons
(Hardin, 1968), meaning the over-
exploitation of the resource. Examples of such
common-pool resources,
in the
language of Ostrom (1990), include sheries and clean water, which are actual
resources.
Other examples, like clean air and the atmospheric concentration
of greenhouse gases (GHGs), are not resources
per se
but are aspects of the
environment that are altered by waste that human activity releases.
ideas, we shall focus on the climate change problem throughout.
To x
In addition
to being an area where the issue of grandfathering has become contentious in
international negotiations about climate change (Lange et al., 2010), it has also
been the object of much recent attention in the philosophy literature (Caney,
2009 and 2011; Bovens, 2011; Knight, 2013).
One way of addressing the tragedy of the commons is to limit access to the
resource by setting quotas.
This ensures that resource exploitation is main-
tained at the desired level. If the issued quotas take the form of exploitation
rights, making these rights tradeable allows them to end up in the hands of the
exploiters who value them the most, thus achieving economic eciency.
to agent
i,
relatively to agent
j,
if agent
i
has exploited less the resource than
regressive rule is conversely dened (as in Young, 1990).
4
j
before. A
This economic instrument is silent regarding the initial allocation of rights,
however. Any allocation will do on eciency grounds, but the question of their
distribution remains. Although some economic agents may end up exploiting
less than they are allowed, and hence selling some of their rights, the fact that
these have nancial valuedue to the existence of a marketbenets even those
agents who choose not to exercise all of their rights to exploit.
When applied to climate policy, the 'resource' to manage is actually the absorption capacity of the atmosphere, meaning its ability to handle the waste that
human activity generates in the form of greenhouse gases (GHGs). As a result,
the property rights not quite rights on the exploitation of a 'resource' but rights
for polluters to emit given amounts of GHGs in the atmosphere. Such emissions
trading practices are also known as 'cap-and-trade', with the aggregate 'cap' set
to meet an emissions 'target', based on the notion of a 'carbon budget' (i.e., the
amount of GHGs we can still aord to emit).
Other ways of mitigating emissions exist, such as carbon taxation. There, the
State retains its 'property right' on emissions, but lets emitters pollute in exchange for a fee. More 'polycentric' approaches have also been proposed. For
example, according to Ostrom (2014), procedures that take place at multiple
scales and levels of government can prevent the free-rider problem while at the
same time allowing for more experimentation. Nevertheless, we shall focus on
cap-and-trade, because emissions trading is an economic instrument that is increasingly used in practice (Tietenberg, 2003), but chiey because the notion of
apportioning a global carbon budget has been central to international climate
negotiations for the past 20 years. Indeed, it has been the case since the Kyoto
protocol adopted in 1997 introduced the principle of 'common but dierentiated
responsibilities' to justify the dierential treatment of countries.
3 How to allocate rights
The method traditionally used in international negotiations over commons problem is to allocate property rights free of charge, based upon
historic use.
Fair
allocation in this context is a controversial issue because involved agents typically rely on self-serving arguments to guide the initial allocation: agents which
have already highly exploited the resource claim that they should not be punished for there past exploitation, while agents which have exploited less claim
that they should obtain a larger share of what is remaining because they have
5
beneted less from it. The rst argument is traditionally named "grandfathering" and the second "historical accountability".
To evaluate more precisely each argument and their compatibility we propose the following simple model.
(rms, individuals or countries).
t,
by
where
xi ≥ 0
t ∈ R+ .
of a resource.
Consider a society of
i ∈ N
x = (xi )iN
has already exploited a quantity
be the prole of these quantities.
r ≡ t−
P
N
= (t, x) ∈ R+ × R+ \ {0} |r ≡ t − N xi ≥ 0 the
We dene what remains of the resource as
R
N
relevant to our problem. A
where
N
ranges over
rule
agents
The initial stock of this resource is denoted
Each agent
Let
N = {1, ..., n}
P
N
xi
and denote by
set of proles that are
is a function dened on the union of all
RN 's,
N, which associates with each N ∈ N and each (t, x) ∈ RN
a point in
RN
+
vector
for
(t, x).
3.1
The grandfathering view
whose coordinates add up to
r.
We denote by
grandfathering "
The rst argument, known as "
s (t, x)
this
awards
in the literature (see for exam-
ple Ott and Sachs, 2002 or Bovens, 2011), considers agent's current level of
exploitation as a rm property right over the resource. This argument requires
that future shares should be in line with their past levels. This argument can
be tracked as far as Locke (Second Treatise of Government) who argues that
an agent could claim to ownership of a common resource by working on this
resource (by "mixing" his labor with the resource). This argument is further
developed in the "entitlement theory of justice" of Robert Nozick (1974).
In
the pollution case, it translates as follows : because the high-emitting countries
were unaware that they were overusing a commonsemissions is the by-product
of a wealth creation processit would be unfair to ask them to restrict their
3
use to a lower total share than other agents.
In our setup,
grandfathering
Axiom. [Grandfathering,
translates as follows:
GF]
For each N ∈ N, each (t, x) ∈ RN , and each
pair {i, j} ⊆ N ,
xi ≤ xj =⇒ si (t, x) ≤ sj (t, x) .
3 Bovens
(2011) states the argument as follows: we would, at least to some extent, respect
dierential investments made, especially the investments made at the time when these were
morally unproblematic.
6
Grandfathering
establishes that if an agent has exploited a resource more than
an other, he should not obtain a lower amount of the remaining resource.
It
clearly rewards early consumption by granting rights to future consumption.
3.2
The historical accountability view
historical accountability ", states
The competing argument, traditionally named "
that agents whom have beneted less from a common in the past should obtain a
higher amount of what remains. This corrective justice argument considers the
resource as a common property over which all agents have an equally legitimate
claim. In the word of Bovens (2011), "
on the ground that:
historical accountability " can be justied
"latecomers or future generations will object that they
never had the opportunity to [benet from the resource] and are disenfranchised
now due to no fault of their own." In the pollution case, this principle is often
associated with a general "right to development" linked to the future needs of
agents who have emitted less in the past and, as such, are deemed to be less
responsible for the accumulated stock of past emissions.
In our simple model, this argument translates as follows
Axiom. [Historical accountability, HA] For each N
∈ N, each (t, x) ∈ RN ,
and each pair {i, j} ⊆ N ,
xi ≤ xj =⇒ si (t, x) ≥ sj (t, x) .
Historical accountability states that if an agent has exploited a resource less than
an other, then he should not obtain a lower share of the remaining resource. This
argument protects low early consumers from receiving too little a share of the
remaining resources.
4 Grandfathering or historical accountability? Or
both?
We now discuss the dierent methods implied by the arguments above and their
extensions.
7
4.1
Both views are compatible
grandfathering and historwhere the remaining resource
It is noteworthy that the only method meeting both
ical accountability
equal sharing
is the
method,
is slitted equally among all agents, irrespective of their past use of the resource.
Dene the
equal sharing
rule, which awards the remaining resource in an egal-
itarian fashion as follows:
esi (N, r, x) = r/n
for all
i ∈ N.
This rule satises
both GF and HA. Likewise, it is immediate to see that GF and HA together
characterize this method.
The equal sharing method is traditionally associated with the
ability
historical account-
principle but, surprisingly, it is the most regressive method meeting the
property. It is indeed totally blind with respect to past emissions and allocate
rights over what is left in the most uniform way possible.
4.2
A strong view of grandfathering
We can push the
grandfathering
argument by extending it to groups of user,
which is crucial when considering groups of countries negotiating over a resource
(such as the European Union in Climate Negotiation for example):
Axiom. [Grouped
Grandfathering, GGF]
For each N ∈ N, each (t, x) ∈
RN , and each pair {N 0 , N 00 } ⊆ N ,
X
xi ≤
i∈N 0
X
xi =⇒
i∈N 00
X
si (t, x) ≤
i∈N 0
X
si (t, x) .
i∈N 00
Grandfathering rewards early consumption by granting rights to future consumption, grouped grandfathering extends this right to groups of users: a group
of users who used the resource more than another should not have less of the
remaining resource.
consistency and continuity, grouped grandstrong grandfathering method, also known as the
Along with two technical properties,
fathering
characterizes the
"proportional" method (see Chambers and Thomson, 2002, for the original
4 Formally, the
proof ).
strong grandfathering
method which we denote
sgf
is
dened as follows:
4 The
formal description of consistency and continuity are found in the appendix. These
two properties are purely technical, mostly devoid of ethical content, and we thus relegate
their exposition for the interested reader in appendix 1. The proofs of all theorems are also
in the appendix.
8
Denition 1. [Strong Grandfathering, SGF] For each N
R
N
, and each
∈ N, each (t, x) ∈
i ∈ N,
sgfi (t, x) =
!
xi
P
j∈N
"
× r = xi × P
xj
#
t
j∈N
xj
−1 .
Theorem 1. A sharing rule satises GGF, CONS and CONT if and only if it
is the
strong grandfathering
Strong grandfathering
method.
extends the original allocation in a linear fashion toward
the remaining amount to be shared. It thus asserts that the inequality in the
original claims should be extended to the rest of the resource. It is noteworthy
that this method is traditionally associated one-to-one with the grandfathering
argument (See for example Rose et al., 1998) while it is the most regressive
method meeting the
grandfathering
property. It would be indeed very dicult
to nd support for a rule extending the rights to use the resource above what
is allocated by the
strong grandfathering.
experiment to assess this argument:
We can use the following thought
consider that the remaining resource is
exactly equal to the resource which has been consumed already.
grandfathering
The
strong
method exactly replicates the original shares to the remaining
resource. It thus extends agent's current level of exploitation to the remaining
resource in the strictest way possible, negating any corrective justice concerns.
4.3
A strong view of historical accountability
We now turn toward a strong interpretation of the
historical accountability argu-
enough-and-as-good condition. We
historical accountability and consistency
ment, that we call
show that this property,
along with
characterizes the strong
egalitarian method (also known as "uniform gains" method). The enough-andas-good property is an interpretation of the "enough-and-as-good" Lockean conditionwhich Nozick dubs "the Lockean Proviso"stating that though every
appropriation of property is a diminution of another's rights to it, it is acceptable as long as it does not make anyone worse o than they would have been
without it. In our context,
enough-and-as-good
considers that an agent should
not obtain more than the quantity she would have obtained if all agents had
exploited the resource similarly as her.
9
Axiom. [Enough-and-as-good,
EAAG]
For each N ∈ N, each (t, x) ∈ RN ,
and each i ∈ N ,
si (t, x) ≤ si t, xi ,
(1)
where xi = (xi , xi , ..., xi ) ∈ RN
+.
EAAG conveys the idea that exploiting a common access resource (or polluting)
creates an externality to others. In order to take into account this externality,
every agent makes the following thought experiment: what would be my allocation over what is left, given a resource stock and a method, if every other agent
had precisely exploited the same as myself ? This property is especially binding
5
for agents which have already beneted more from the resource.
This property, along with
itarian
historical accountability, characterizes the strong egal-
method which equalizes as much as possible the total share of each agent
(summing their initial extraction and their future claim).
Denition 2. [Strong Egalitarian, SE] For each N
and each
each
(t, x) ∈ RN ,
(λ − xi )+ = r
(2)
∈ N,
i ∈ N,
sei (t, x) = (λ − xi )+ ,
with
λ
s.t.
X
N
Theorem 2. A sharing rule satises HA, EEAG and CONS if and only if it is
the Strong Egalitarian rule.
The
strong egalitarian method aims at equalizing the total shares (xi +si (t, x)) i∈N
among all agents. Intuitively, the method increases the shares of those with the
smallest claim as long as possible, until all shares are equalized.
Then it in-
creases all shares equally.
As we will discuss in the next section, the
strong egalitarian
method is the
method that erases as much as possible the original inequality in the distribution and should be thus considered the symmetric of
strong grandfathering
in
discussions on property rights allocation. This is the most progressive method
that can be justied by our properties and should be the one associated with
an egalitarian objective.
5 For
a discussion on such protective criteria, see Herrero and Villar (2001) and Yeh (2004).
These papers discuss of properties which insure a preferential treatments to small claims in
the rationing problem.
10
4.4
A discussion on progressivity
These discussion above allows us to draw clear conclusions with respect to the
"progressivity" of both methods.
6
Progressivity is a key concept underlying
many debates about the establishment of property rights market and our framework allows for a formal treatment of its implications. It happens that the two
methods we single out
strong grandfathering
and
strong egalitarian are,
re-
spectively, the most progressive and the less regressive methods that can be
justied by our properties.
strong grandfathering rule
grandfathering. More regressive
The
is the method traditionally associated with
rules exist, in principle. In fact, the most re-
gressive rule allowed by GF is the one that awards all to the heaviest consumer:
ordering the agents in decreasing order of past consumption,
y = (r, 0, ..., 0). So far as we have described
if x1 = x2 . Allocating more to agent 1 that to
this most regressive rule, awards
it, the rule is not well-dened
x1 > x2 ≥ ... ≥ xn ,
mr
agent 2 would be arbitrary. On the other hand, if we modiy it so as to treat
equals equally, and hence awarding
y mr
=


r
|arg maxi {x1 ,...,xn }|
0
where
||
if
xi ∈ maxi {x1 , ..., xn }
,
(3)
otherwise
is the cardinality operator, the rule violates CONT.
[Forthcoming graph describing this rule, as well as the two ranges of GF and
HA.]
One could certainly cook up families of continuous rules that treat equals equally
and that are more regressive than
sgf
(see, e.g., Expression (25) in Appendix
C), but it is hard to see this additional regressivity as anything but arbitrary. In
particular, there does not seem to be a compelling argument why any such rule
would be more justied than another within the same family. More importantly
for our discussion, it is hard to see why any such rule would be more justied
than
sgf .
In other words,
sgf
is arguably the most regressive rule one can
reasonably propose.
By contrast, the most progressive method compatible with
equal sharing
6 By
method. Symmetrically The
equal sharing
grandfathering
is the
method is traditionally
a "progressive" method, we mean one which gives a higher quantity of what is remain-
ing to agent
i,
relatively to agent
j,
if agent
i
has exploited less the resource than
and conversely for a "regressive" method (as in Young, 1990).
11
j
before,
x2 , z2
y2
t
sgf
es
se
x2
y1
x
x1
x1 + x2
Figure 1: Awards paths for the
degree ray from
x.
x. sgf
x1 , z1
t
es, se,
and
sgf
rules.
es
is located on the 45-
is located on the ray from the origin that passes through
The awards path for
seg
awards nothing to the agent having exploited the
most until the other agent has caught up; it then coincides with the 45-degree
ray from the origin.
associated with the
historical accountability
principle but, surprisingly, it is
the most regressive method meeting this principle: it negates the role of past
strong egalitarian
total shares among
consumption when allocating rights over future resources. The
method is the one aiming to equalize as much as possible the
all agents within the bounds imposed by our properties. It should be the one
associated with the egalitarian goal in international negotiations.
5 Concluding remarks
Allocating property rights over an open access resource that has been freely exploited in the past is often problematic. The most common method for the applications discussed here is allocating property rights free of charge, based upon
12
historic use.
Fair allocation in this context is a very controversial issue because
involved agents typically rely on two dierent principles to determine the allocation:
grandfathering
and
historical accountability.
We construct a simple model
to show that these two positions are in fact compatible and characterize the
equal sharing
method. Pushing both arguments toward stronger extensions, we
show that they characterizealong with technical propertiestwo well-known
and the
strong egalitarian
The equal sharing method is traditionally associated with the
historical account-
methods in the literature the
strong grandfathering
methods.
ability
principle but, surprisingly, it is the most regressive method meeting the
strong grandfathering rule is the method traditionally associated with grandfathering but it is the most regressive method
meeting the property. Finally we highlight that the strong egalitarian method
is the rule which aims to equalize as much as possible the total shares among
property. Symmetrically, The
all agents. Thus, this method should be the one associated with the egalitarian
goal if one considers the family of methods we propose.
For the important problem of allocating
CO2
emissions, many observers have
urged that in an international agreement, emissions rights should be allocated by
reference to population (see for example, Posner and Sunstein, 2008). It should
be noted that our model encompasses this situation: we can simply dene the
society
N
being the
as being composed of individuals and the prole of emissions
per capita
x
as
prole of emissions (i.e. the ratio of emissions of a country
over its population).
13
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[23] Young, H.P. 1990."Progressive taxation and equal sacrice,"
nomic Review, 80(1), 253-266.
15
American Eco-
A Two technical properties: Consistency and Continuity
The rst technical axiom is
consistency,
which ensure that the awards be non
ambiguous. Consistency requires that the sharing rule yields identical awards
whether it is applied to a subset of agents or to the entire population. If two ways
of looking at the problem led to two dierent awards vectors, the application of
the rule would be subject to disagreement, which would defeat the purpose of
resorting to a sharing rule to settle disputes.
Axiom. [Consistency,
CONS]
For each N ∈ N, each (t, x) ∈ RN , and each
N 0 ⊂ N , if y ≡ s (t, x), then


X
yN 0 = s t −
(xi + yi ) , xN 0  .
i∈N \N 0
The second technical axiom is
continuity, which requires that the rule does not
respond wildly to small changes in initial consumption:
Axiom. [Continuity,
N
CONT] For each N ∈ N, each (t, x) ∈ R , and for
+∞
each sequence tk , xk k=1 of elements of RN such that tk = t for all k ∈ N,
lim xk = x =⇒
k→+∞
lim s tk , xk = s (t, x) .
k→+∞
B Proofs of theorems
B.1
Strong grandfathering
Proof. If.
Let
N ∈N
and
N 0 , N 00 ⊆ N ,
such that
P
i∈N 0
xi ≤
P
follows immediately from the expression of the proportional shares,
xi × r/
sgf
P
j∈N
xj
i∈N 00
xi .
It
sgfi (t, x) =
, that the proportional rule satises GGF. The fact that
satises CONT is similarly immediate.To check that the proportional rule
16
satises CONS, let
T ⊂N


X
sgfi t −
i∈T
and compute the share of an agent
:

X
(xi + sgf i (t, xN )) , xT  = sgfi t −
i∈N \T
xj
1+ P
j∈N
j∈N \T

!
r
, xT 
xj
(4)

X
= sgfi t −
t
xj
P
j∈N
j∈N \T

!
, xT 
xj
(5)
= sgfi
P
xj
P
xj
j∈N
t
j∈N
!
P
j∈N \T
−t
P
j∈N
xj
!
xj
(6)
P
= sgfi
j∈T
xj
j∈N
xj
tP
= xi × P
1
j∈T
!
, xT
(7)
 P

X
x
j
j∈T
× t P
−
xj 
xj
j∈N xj
j∈T
(8)
!
t
= xi
P
j∈N
xj
−1
(9)
= sgfi (t, x) .
Only if.
(10)
The proof is adapted from that of Theorem 1 established in Chambers
and Thomson (2002) for claims problems. We actually show that
sgf
is the only
rule that satises CONS, CONT and Equal Treatment of Equal Groups (which
7 Let
is implied by GGF).
N
(t, x) ∈ R
such that there exist
of generality, suppose
Let
α > 0
s be a rule satisfying these properties.
a > 0, γ ∈ N
x1 = maxi∈N xi ,
N
for which
Let
x = aγ .
which of course implies
N ∈N
and
Without loss
γ1 = maxi∈N γi .
be the ratio at which the strong grandfathering rule rewards past
consumption:
sgf (t, x) = αx.
We now consider the augmented problem obtained by addingγ1 agents, who
each previously consumed
t+(1 + α) aγ1 .
and
xl = a
7 Equal
P
i∈T 0
a
and increasing the total available resource to
We denote by
for all
l ∈ N̄ .
N̄
the set of new agents, with
N ∩ N̄ = ∅, |N̄ | = γ1
Hence, the augmented problem writes
Treatment of Equal Groups:
P
i∈T
si (t, x).
17
xi =
P
i∈T 0
xi
t0 =
=⇒
P
(t0 , x0 ),
i∈T
with
si (t, x) =
!
, xT
N 0 = N ∪ N̄ , r0 = t0 −
0
0
i∈N 0 xi , and x
N̄
0
0
0
ones in R . By GGF, yl = yl0 for all l, l
for all
Let
P
= x + aeN̄ ,
∈
eN̄
where
N̄ .8 Dene
ᾱ > 0
is the vector of
such that
yl0 = ᾱa
l ∈ N̄ .
j ∈N
and consider a subset
0
agents. By GGF, yj =
0
get that yi = ᾱxi for all
N j ⊆ N̄
made up of
0 9
Hence,
i∈N j yi .
P
yj0
γj
of the newly added
= γj (ᾱa) = ᾱxj .
Similarly, we
i ∈ N.
Awards must sum to the remaining resource:
!
X
yi0
X
0
+ γ1 ᾱa = t −
i∈N
xi + γ1 a
(11)
i∈N
!
ᾱ
X
xi + γ1 ᾱa = t + (1 + α) aγ1 −
i∈N
ᾱ
xi + γ1 a
(12)
i∈N
!
X
X
xi + γ1 a
!
=
X
t−
xi
+ αaγ1
(13)
i∈N
i∈N
!
ᾱ
X
xi + γ1 a
X
=α
i∈N
xi + αaγ1 ,
(14)
i∈N
where the last step comes from the fact that the strong grandfathering rule also
awards all of the remaining resource. It follows from Equation (14) that
so that
0
0
0
y = sgf (t , x ).
By CONS,
yN =
0
yN
ᾱ = α,
= sgf (t, x) .
a
Finally, consider the case where there exist no
and
γ
as specied in the be-
ginning of the proof. Any such problem can be approximated by a sequence of
problems in which
Remark.
a
and
γ
do exist. The result then follows from CONT.
Because Group Historical Accountability (GHA) would also imply
ETEG, and because
sgf
does not even satisfy HA, no rule satises GHA, CONS
and CONT.
B.2
Strong egalitarian
Proof. If.
It is immediate from the denition of
increasing in her past consumption. Hence,
se
satises EAAG, notice that
se
se
se that an agent's share is non-
satises HA. Now, to show that
partitions the population
N ∈N
into
N =
8 Actually, only Equal Treatment of Equals is needed, whereby x = x =⇒ y = y .
i
j
i
j
9 In fact, only Equal Treatment of Equals Groups is needed.
18
N+ ∪ N0 ,
where
N+ = {i ∈ N |sei (t, x) > 0}
N0 = {i ∈ N |sei (t, x) = 0}.
and
This partition allows us to rewrite the denition of
X
(λ − xi )
= t−
X
⇐⇒
λ
such that
xi ≥ t/n.
t−
=
Because
xj > t/n
Expression (16), the fact that
(15)
P
N0
xi
(16)
|N+ |
i ∈ N+ ,
for all
λ > t/n
t−
⇐⇒
xi
xi < t/n for all i ∈ N+ .
We will show by contradiction that
In turn, this implies that
as follows:
N
N+
i ∈ N+
λ
Suppose there exists
it must be that
j ∈ N0 .
λ > xi ≥ t/n.
However, according to
rewrites as follows:
P
N0
xi
|N+ |
X
nt − n
xi
t
n
(17)
> |N+ |t
(18)
>
N0
⇐⇒
|N0 |t
>
n
X
xi
(19)
N0
t
n
⇐⇒
a contradiction with the fact that
xj > t/n
>
1 X
xi ,
N0
(20)
N0
for all
j ∈ N0 .
xi < t/n
sei ti , xi =
Therefore,
i ∈ N+ . To show that se satises EAAG, observe that
n λ − xi + , with λi ≥ 0 such that n λi − xi + = (t − nxi )+ .10 Thus, for
i
any i ∈ N such that xi ≥ t/n, we have sei t, x
= 0 = sei (t, x). Also,
for all
i
i ∈ N such that xi < t/n, we have that λi = t/n > λ, so
sei t, xi = λi − xi > (λ − xi )+ = sei (t, x) . Hence, se satises EAAG.
for any
Finally, the fact that
se
satises CONS follows from the denition of
deed, considering the reduced problem after the departure of a coalition
10 Note
some
that although we have assumed that
P
i ∈ N.
19
N
xi ≥ t ,
it may be the case that
that
λ.
In-
T,
the
nxi > t
for
associated parameter, which we denote
X
(λ0 − xi )+ = r −
λ0
X
is dened by:
sei (t, x)
T
N \T
=r−
X
(λ − xi )+
T
=
X
(λ − xi )+ −
X
N
=
(λ − xi )+
T
X
(λ − xi )+ ,
N \T
where the third equality follows from the denition of
ness of
λ,
it must be that
0
λ = λ.
Only if.
λ.
Owing to the unique-
se
We actually show that
only rule that satises HA, EAAG, and CONS. Let
s
is the
be a rule satisfying these
properties. Without loss of generality, suppose agents are ranked in increasing
order of past exploitation:
Case 1.
t,
xn ≤ t/n.
Suppose
(t − nxi ) /n.
The proof considers two cases:
By EAAG and ETE (implied by HA),
However, because all of the resource must be allocated,
so that
yi =
Case 2.
x1 ≤ x2 ≤ ... ≤ xn .
t
− xi = sei (t, x)
n
Suppose now that
T = {k, .., n} ⊂ N .
sej (t, x), ∀j ∈ T .
xk−1 ≤ t/n < xk
for some
t−
(xi + si (t, x)) , xN \T
≤
t−
i∈T
yi ≤
(xi + yi ) =
(21)
k ∈ {2, n}.11
T
is zero:
By EAAG and ETE (implied by HA), for all
!
si
i
∀i ∈ N.
By EAAG, the share of any agent in
X
P
Pn
j=k
xj
yj = 0 =
i ∈ N \T :
− xi .
k−1
Dene
(22)
Moreover, because all of the resource must be allocated,
!!
X
xi + si
t−
X
(xi + si (t, x)) , xN \T
=t−
i∈T
i∈N \T
n
X
xj ,
(23)
j=k
so that
!
si
t−
X
(xi + si (t, x)) , xN \T
=
i∈T
11 Note
that the case
t/n < x1
t−
Pn
j=k
k−1
xj
− xi
cannot happen because we assumed
20
∀i ∈ N \T.
P
N
xi ≤ t.
(24)
By CONS,
yN \T = sN \T t −
P
i∈T
(xi + si (t, x)) , xN \T = seN \T (t, x),
yield-
ing the result.
C Independence checks
For completeness, we provide the independence checks for theorems 1 and 2.
Theorem 1: GGF, CONS, and CONT to characterize
•
Rules other than
es
•
and
sgf .
sgf
satisfy CONS and CONT but not GGF, including
sgf
satisfy GGF and CONS but not CONT. We con-
se.
Rules other than
struct one for completeness as follows.
aS > 0
and
γS ∈ N
S
for any subset
xS = aS γS
Whenever
S ⊆ N,
for some
the rule coincides with
sgf .
Whenever it is not the case, consider the sequence that converges to
whose elements can all be written in
the smallest
a(k)
possible. For any
aγ -form, x(k) = a(k) γ (k)
ε > 0,
dene
sgfi t, x(k) − sgfi (t, x) < ε
and assign
si (t, x) = sgfi t, x(k)
kε ∈ N
for all
k∈N
x
, with
such that
k ≥ kε ,
ε will
P
P
minS,S 0 ⊂N
i∈S yi −
i∈S 0 yi .
. Choosing a low enough value of
ε<
guarantee that GGF is satised. For example,
1
n
CONS follows from the fact that removing any subset of agents does not
make the reduced consumption expressible in
aγ -form.
Clearly, the rule
does not satisfy CONT.
•
sgf
Rules other than
satisfy GGF and CONT but not CONS. Consider
for example the following variant of
xī ≥
P
j6=ī
xj ≡ x̄N ,
sgf .
Denote
j
with
∆ = xī −x̄N
(x−ī , x̄N )
j
and
−ī
λ is the solution of λ
(25)
for
N
P
j6=ī
sgf
j 6= ī,
sgfj (t, (x−ī , x̄N )) = 2∆,
the past consumption prole where
consumption. Whenever it is not the case, apply
deviates from
If
assign

y = sgf (t, (x , x̄ )) + 2∆
N
ī
ī
−ī
y = (1 − λ) sgf (t, (x , x̄ ))
where
ī ∈ arg maxi∈N xi .
x̄N
sgf .
replaces agent
ī's
In words, the rule
if the highest consumer has consumed more than all the
other agents combined. It then siphons resources from these agents to the
21
highest consumer in a proportional way. It satises CONT for the same
sgf
reasons that
satises CONT, and because
It satises GGF because
coincide with
sgf ,
sgf
λ
is continuous at
∆ = 0.
satises GGF. Whenever the rule does not
the fact that
xī > x̄N
prevents the agents other than
the largest consumer from exceeding his past consumption by combining
theirs. Also, among these agents, GGF cannot be violated because the rule
is proportional to a (virtual) solution vector of
sgf .
However, the rule does
not satisfy CONS: a counterexample is a prole where
xn >
P
j6=n
xj
and
xn−1 >
the rule now treats agent
P
j<n−1
n−1
xj .
x1 < ... < xn
with
After the departure of agent
n,
as the 'very high consumer' rather than
proportionally to the other remaining agents.
Theorem 2: HA, EAAG and CONS to characterize
•
Rules other than
es.
•
satisfy HA and CONS but not EAAG: for example,
We give a counterexample to show thates does not satisfy EAAG.
Consider
yet
se
se.
N = {1, 2}, x = (1, 2)
and
t = 4.
EAAG recommends
y2 = 0,
eg (t, x) = 0.5.
Rules other than
se satisfy HA and EAAG but not CONS: counterexample
forthcoming.
•
Rules other than
se
satisfy EEAG and CONS but not HA: for example,
consider the following variant of the uniform gains method that favors the
agent 1:
with
λ≥0
such that

y = (2λ − x )
1
1 +
y = (λ − x )
for j 6= 1,
j
j +
P
(2λ − x1 )+ + N \{1} (λ − xj )+ = r.
(26)
This rule does
not satisfy HA; in fact, it does not even satisfy ETE.
Equal sharing method
consider
N ≥3
recommends
Notice that
and a prole
y1 = y2 + y3 ,
x
es
such that
does not satisfy GGF. Indeed,
x1 = 2
and
x2 = x3 = 1.
which is obviously violated by
es3 = r/n.
22
GGF
es: es1 = es2 =
The sharing rule
es
is violated for any
true for all
i∈N
i∈N
if
EAAG either.
P
xi > N xj /n.
does not satisfy
such that
xi = xj
for all
i, j ∈ N .
23
Indeed, inequality (1)
Hence, it can only be