Efficient and Fair Allocation of Disputed
Properties
Biung-Ghi Ju
Seoul National University
and
Juan D. Moreno-Ternero
Universidad Pablo de Olavide
CORE, Université catholique de Louvain
Francqui Conference; Brussels, June 2013
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Introduction
�
Legal procedures for disputed properties implement specific
quantitative treatments of claims in their final allocation stage
�
For instance:
�
The American Bankruptcy Law
�
Court settlements for car accidents
�
The U.S. Admiralty rule
�
Postcolonial reallocation of land properties
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Introduction
�
Normative foundations of those allocation schemes has been a
key subject in the literature of fair allocation
�
Nevertheless, most studies in this literature focus on:
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Introduction
�
Normative foundations of those allocation schemes has been a
key subject in the literature of fair allocation
�
Nevertheless, most studies in this literature focus on:
�
The problem of allocating a (disputed) single good (money),
without allowing for subsequent interactions among claimants
from such fair initial allocation (e.g., O’Neill, 1982; Moulin,
2000; Thomson, 2003)
�
The investigation of end-state fairness for a fixed initial
distribution of property rights without dispute (e.g., Pazner
and Schmeidler, 1974, 1978; Roemer, 1986; Thomson, 2011)
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Goal
�
Three categories of fairness in our environments:
�
fairness in the initial allocation of rights on disputed properties
�
fairness in the transaction of allocated rights
�
fairness of the end-state allocation
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Goal
�
�
Three categories of fairness in our environments:
�
fairness in the initial allocation of rights on disputed properties
�
fairness in the transaction of allocated rights
�
fairness of the end-state allocation
Our aim in this paper is to construct a comprehensive
framework for those environments in which one can investigate
the three categories of fairness, as well as their implications.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
How to achieve the goal
�
In a regime with some freedom of exchange:
�
�
�
An initial allocation regarding disputed properties is determined
through a legal procedure: initial allocation of rights
Agents may interact with their initial allocations: transaction
of allocated rights
And reach a final allocation: an end-state allocation
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
How to achieve the goal
�
In a regime with some freedom of exchange:
�
�
�
�
An initial allocation regarding disputed properties is determined
through a legal procedure: initial allocation of rights
Agents may interact with their initial allocations: transaction
of allocated rights
And reach a final allocation: an end-state allocation
Thus an end state is indeed determined through the
combination of two procedures:
�
�
The first (legal) procedure aims to be impartial discarding
information beyond legitimate claims and resource constraints
transaction of allocated rights
The second (exchange) procedure is based on individual
decision makings within the boundary of a socio-economic
institution the agents belong to
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
How to achieve the goal
�
In a regime with some freedom of exchange:
�
�
�
�
An initial allocation regarding disputed properties is determined
through a legal procedure: initial allocation of rights
Agents may interact with their initial allocations: transaction
of allocated rights
And reach a final allocation: an end-state allocation
Thus an end state is indeed determined through the
combination of two procedures:
�
�
The first (legal) procedure aims to be impartial discarding
information beyond legitimate claims and resource constraints
transaction of allocated rights
The second (exchange) procedure is based on individual
decision makings within the boundary of a socio-economic
institution the agents belong to
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Side goals
�
To extend the theory of fair allocation to the case in which
agents have claims on each good and the total endowment is
not sufficient to fully honor all individual claims
�
To extend the rationing model considering multiple goods and
allowing for individual preferences to be considered in an
exchange procedure that takes place after the application of a
rationing mechanism
�
Some of our results will also echo the discussion that arose
after Nozick’s entitlement theory of justice (e.g., Nozick,
1973)
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An economy e ≡ (N, Ω, c, R) is characterized by
�
�
�
�
�
A set of agents N ∈ N ,
A social endowment Ω ∈ RL
++ ,
A profile of individual claims c ≡ (ci )i∈N , and
A profile of (continuous, monotonic, and convex) preference
relations R ≡ (Ri )i∈N
�
Such that Ω � i∈N ci .
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An economy e ≡ (N, Ω, c, R) is characterized by
�
�
�
�
�
�
A set of agents N ∈ N ,
A social endowment Ω ∈ RL
++ ,
A profile of individual claims c ≡ (ci )i∈N , and
A profile of (continuous, monotonic, and convex) preference
relations R ≡ (Ri )i∈N
�
Such that Ω � i∈N ci .
�
When Ω = i∈N ci , e is referred to as an exchange economy
and c is called as a profile of individual endowments.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An economy e ≡ (N, Ω, c, R) is characterized by
�
�
�
�
�
�
�
A set of agents N ∈ N ,
A social endowment Ω ∈ RL
++ ,
A profile of individual claims c ≡ (ci )i∈N , and
A profile of (continuous, monotonic, and convex) preference
relations R ≡ (Ri )i∈N
�
Such that Ω � i∈N ci .
�
When Ω = i∈N ci , e is referred to as an exchange economy
and c is called as a profile of individual endowments.
�
When L = 1 and Ω < i∈N ci , e is referred to as a
bankruptcy problem.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
A social choice rule S : E → Z associates with each economy
a non-empty set of feasible allocations.
�
We are interested in social choice rules that are characterized
by the following two consecutive procedures:
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
A social choice rule S : E → Z associates with each economy
a non-empty set of feasible allocations.
�
We are interested in social choice rules that are characterized
by the following two consecutive procedures:
�
�
First, a claims adjudication procedure mapping non-preference
information (N, Ω, c) of each economy e ≡ (N, Ω, c, R) into a
profile of individual endowments ω ≡ (ωi )i∈N .
Thus, the claims adjudication procedure transforms the
economy e into an exchange economy e� ≡ (N, Ω, ω, R).
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
A social choice rule S : E → Z associates with each economy
a non-empty set of feasible allocations.
�
We are interested in social choice rules that are characterized
by the following two consecutive procedures:
�
�
�
First, a claims adjudication procedure mapping non-preference
information (N, Ω, c) of each economy e ≡ (N, Ω, c, R) into a
profile of individual endowments ω ≡ (ωi )i∈N .
Thus, the claims adjudication procedure transforms the
economy e into an exchange economy e� ≡ (N, Ω, ω, R).
Second, an exchange procedure determining final allocations
for the exchange economy obtained in the first procedure.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An assignment function ϕ assigns individual property rights
over the social endowment for each claims adjudication
problem and, thus, turns each economy e ≡ (N, Ω, c, R) into
an exchange economy e� ≡ (N, Ω, ϕ(N, Ω, c), R).
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An assignment function ϕ assigns individual property rights
over the social endowment for each claims adjudication
problem and, thus, turns each economy e ≡ (N, Ω, c, R) into
an exchange economy e� ≡ (N, Ω, ϕ(N, Ω, c), R).
�
Constrained equal award function:
ϕcea
il (N, Ω, c) = min{cil , µl },
�
where µl > 0 is chosen so that i∈N min{cil , µl } = Ωl .
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An assignment function ϕ assigns individual property rights
over the social endowment for each claims adjudication
problem and, thus, turns each economy e ≡ (N, Ω, c, R) into
an exchange economy e� ≡ (N, Ω, ϕ(N, Ω, c), R).
�
�
Constrained equal award function:
ϕcea
il (N, Ω, c) = min{cil , µl },
�
where µl > 0 is chosen so that i∈N min{cil , µl } = Ωl .
Constrained equal sacrifice function:
ϕcel
il (N, Ω, c) = max{0, cil − λl },
�
where λl > 0 is chosen so that i∈N max{0, cil − λl } = Ωl .
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An assignment function ϕ assigns individual property rights
over the social endowment for each claims adjudication
problem and, thus, turns each economy e ≡ (N, Ω, c, R) into
an exchange economy e� ≡ (N, Ω, ϕ(N, Ω, c), R).
�
�
�
Constrained equal award function:
ϕcea
il (N, Ω, c) = min{cil , µl },
�
where µl > 0 is chosen so that i∈N min{cil , µl } = Ωl .
Constrained equal sacrifice function:
ϕcel
il (N, Ω, c) = max{0, cil − λl },
�
where λl > 0 is chosen so that i∈N max{0, cil − λl } = Ωl .
Proportional function:
c
� il
ϕpro
.
il (N, Ω, c) = Ωl ×
j∈N cjl
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An exchange rule F : Ē → Z associates with each exchange
economy e ≡ (N, Ω, ω, R) ∈ Ē a non-empty set of feasible
allocations.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An exchange rule F : Ē → Z associates with each exchange
economy e ≡ (N, Ω, ω, R) ∈ Ē a non-empty set of feasible
allocations.
�
The best known one is Walrasian exchange rule associating
with each exchange economy (N, ω, R) the set of Walrasian
equilibrium allocations.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The model
�
An exchange rule F : Ē → Z associates with each exchange
economy e ≡ (N, Ω, ω, R) ∈ Ē a non-empty set of feasible
allocations.
�
The best known one is Walrasian exchange rule associating
with each exchange economy (N, ω, R) the set of Walrasian
equilibrium allocations.
�
The composition of an assignment function ϕ and an
exchange rule F gives rise to a social choice rule S such that
for each e ≡ (N, Ω, c, R) ∈ E,
S(e) ≡ F (N, Ω, ϕ(N, Ω, c), R).
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Assignment Functions
�
�
Resource Monotonicity: ϕ(N, c, Ω� ) ≥ ϕ(N, c, Ω), for Ω� ≥ Ω.
Consistency: For each (N, Ω, c) and N � � N ,
�
ϕ(N � , cN � ,
ϕi (N, c, Ω)) = ϕN � (N, c, Ω).
i∈N �
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Assignment Functions
�
�
Resource Monotonicity: ϕ(N, c, Ω� ) ≥ ϕ(N, c, Ω), for Ω� ≥ Ω.
Consistency: For each (N, Ω, c) and N � � N ,
�
ϕ(N � , cN � ,
ϕi (N, c, Ω)) = ϕN � (N, c, Ω).
i∈N �
�
Zero-Award-Out-Consistency: For each (N, Ω, c), i ∈ N , and
l ∈ L, if ϕil (N, c, Ω) = 0 then, for each j ∈ N \{i},
�
ϕjl (N \{i}, cN \{i} ,
ϕj (N, c, Ω)) = ϕjl (N, c, Ω).
j∈N \{i}
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Assignment Functions
�
�
Resource Monotonicity: ϕ(N, c, Ω� ) ≥ ϕ(N, c, Ω), for Ω� ≥ Ω.
Consistency: For each (N, Ω, c) and N � � N ,
�
ϕ(N � , cN � ,
ϕi (N, c, Ω)) = ϕN � (N, c, Ω).
i∈N �
�
Zero-Award-Out-Consistency: For each (N, Ω, c), i ∈ N , and
l ∈ L, if ϕil (N, c, Ω) = 0 then, for each j ∈ N \{i},
�
ϕjl (N \{i}, cN \{i} ,
ϕj (N, c, Ω)) = ϕjl (N, c, Ω).
j∈N \{i}
�
Full-Award-Out-Consistency: For each (N, Ω, c), i ∈ N , and
l ∈ L, if ϕil (N, c, Ω) = cil then, for each j ∈ N \{i}
�
ϕjl (N \{i}, cN \{i} ,
ϕj (N, c, Ω)) = ϕjl (N, c, Ω).
j∈N \{i}
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Exchange Rules
�
Individual Rationality: For each (N, ω, R), i ∈ N , and
z ∈ F (N, ω, R), zi Ri ωi .
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Social Choice Rules
�
No-Envy: For each e ≡ (N, Ω, c, R), and z ∈ S (e), there is no
pair i, j ∈ N such that
zj P i z i .
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Social Choice Rules
�
No-Envy: For each e ≡ (N, Ω, c, R), and z ∈ S (e), there is no
pair i, j ∈ N such that
zj P i z i .
�
Sacrifice-No-Envy: For each e ≡ (N, Ω, c, R), and z ∈ S (e),
there is no pair i, j ∈ N , such that ci + (zj − cj ) ∈ R+ and
[ci + (zj − cj )] Pi zi .
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Axioms of Social Choice Rules
�
No-Envy: For each e ≡ (N, Ω, c, R), and z ∈ S (e), there is no
pair i, j ∈ N such that
zj P i z i .
�
Sacrifice-No-Envy: For each e ≡ (N, Ω, c, R), and z ∈ S (e),
there is no pair i, j ∈ N , such that ci + (zj − cj ) ∈ R+ and
[ci + (zj − cj )] Pi zi .
�
Relative-Sacrifice-No-Envy: For each e ≡ (N, Ω, c, R), and
z ∈ S (e), there is no pair i, j ∈ N , such that
ci ×
Biung-Ghi Ju and Juan Moreno-Ternero
zj
Pi zi .
cj
Fair Allocation of Disputed Properties
The results
For each e ≡ (N, Ω, c, R), let ω ed (N, Ω) =
E0
≡ {(N, Ω, c, R) ∈ E : 0 ≤
ω ed (N, Ω)
Biung-Ghi Ju and Juan Moreno-Ternero
�
Ω
Ω
|N | , . . . , |N |
≤ c}.
�
Fair Allocation of Disputed Properties
and
The results
For each e ≡ (N, Ω, c, R), let ω ed (N, Ω) =
E0
≡ {(N, Ω, c, R) ∈ E : 0 ≤
ω ed (N, Ω)
Theorem
�
Ω
Ω
|N | , . . . , |N |
≤ c}.
�
and
An assignment mechanism satisfies resource monotonicity and
full-award-out-consistency, and, when combined with an
individually rational exchange rule, leads to a social choice rule
satisfying no-envy on E 0 if and only if it is the constrained equal
awards mechanism.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The results
For each e ≡ (N, Ω, c, R), let ω ed (N, Ω) =
E0
≡ {(N, Ω, c, R) ∈ E : 0 ≤
ω ed (N, Ω)
Theorem
�
Ω
Ω
|N | , . . . , |N |
≤ c}.
�
and
An assignment mechanism satisfies resource monotonicity and
full-award-out-consistency, and, when combined with an
individually rational exchange rule, leads to a social choice rule
satisfying no-envy on E 0 if and only if it is the constrained equal
awards mechanism.
�
The claims domain C 0 is the maximal domain on which an
assignment mechanism and an individually rational exchange
rule combined together can generate envy-free allocations.
�
Moreover, on this maximal claims domain, individual
rationality, no-envy, and efficiency are compatible.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The results
�
For each e ≡ (N, Ω, c, R), let ω es (N, Ω, c) = ci −
�
and E ∗ ≡ {e = (N, Ω, c, R) ∈ E : 0 ≤ ω es (N, Ω, c) ≤ c}.
Biung-Ghi Ju and Juan Moreno-Ternero
�
ci −Ω
|N |
i∈N
i∈N
Fair Allocation of Disputed Properties
The results
�
For each e ≡ (N, Ω, c, R), let ω es (N, Ω, c) = ci −
�
�
ci −Ω
|N |
i∈N
i∈N
and E ∗ ≡ {e = (N, Ω, c, R) ∈ E : 0 ≤ ω es (N, Ω, c) ≤ c}.
Theorem
An assignment mechanism satisfies resource-monotonicity,
zero-award-out consistency and, when combined with the
Walrasian exchange rule, leads to a social choice rule satisfying
strong sacrifice-no-envy on E ∗ if and only if it is the constrained
equal losses mechanism
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The results
�
For each e ≡ (N, Ω, c, R), let ω es (N, Ω, c) = ci −
�
�
ci −Ω
|N |
i∈N
i∈N
and E ∗ ≡ {e = (N, Ω, c, R) ∈ E : 0 ≤ ω es (N, Ω, c) ≤ c}.
Theorem
An assignment mechanism satisfies resource-monotonicity,
zero-award-out consistency and, when combined with the
Walrasian exchange rule, leads to a social choice rule satisfying
strong sacrifice-no-envy on E ∗ if and only if it is the constrained
equal losses mechanism
�
For each N ∈ N with |N | = 2, the claims domain C ∗ (N ) is
the maximal domain with population N on which an
assignment mechanism combined with Walrasian exchange
rule can generate sacrifice-envy-free allocations.
�
Moreover, on this maximal claims domain, individual
rationality, sacrifice-no-envy, and efficiency are compatible.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The results
Theorem
An assignment mechanism, when combined with an individually
rational exchange rule, leads to a social choice rule satisfying
relative-sacrifice-no-envy if and only if it is the proportional
mechanism.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
The results
Theorem
An assignment mechanism, when combined with an individually
rational exchange rule, leads to a social choice rule satisfying
relative-sacrifice-no-envy if and only if it is the proportional
mechanism.
�
On the universal claims domain, there exists an assignment
mechanism and an individually rational exchange rule such
that, when combined together, can generate relative-envy-free
allocations.
�
However, individual rationality, relative-no-envy and efficiency
are incompatible.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Basic summary
�
Fairness based on no-envy (and efficiency) can be attained
through an egalitarian resolution of property rights dispute
�
The egalitarian resolution is the ONLY way of satisfying
fairness based on no-envy (and efficiency)
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
�
Our framework is useful to review Nozick’s position and to
investigate how his alternative approach, after a concrete
formulation, can be related to theories of end-state justice
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
�
Our framework is useful to review Nozick’s position and to
investigate how his alternative approach, after a concrete
formulation, can be related to theories of end-state justice
�
Justice in acquisition: Not properly developed by Nozick
�
Justice in transfer: Interpreted by Nozick as voluntariness
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
�
Our framework is useful to review Nozick’s position and to
investigate how his alternative approach, after a concrete
formulation, can be related to theories of end-state justice
�
Justice in acquisition: Not properly developed by Nozick
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
�
Our framework is useful to review Nozick’s position and to
investigate how his alternative approach, after a concrete
formulation, can be related to theories of end-state justice
�
Justice in acquisition: Not properly developed by Nozick
�
Our assignment mechanisms can be interpreted as a way to
develop his proposal in our context
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
�
Our framework is useful to review Nozick’s position and to
investigate how his alternative approach, after a concrete
formulation, can be related to theories of end-state justice
�
Justice in transfer: Interpreted by Nozick as voluntariness
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Nozick criticizes theories of end-state justice (patterned
justice), such as fairness principles based on no-envy
�
His alternative proposal advocates justice determined not by
pattern but by respecting rights
�
Our framework is useful to review Nozick’s position and to
investigate how his alternative approach, after a concrete
formulation, can be related to theories of end-state justice
�
Justice in transfer: Interpreted by Nozick as voluntariness
�
Voluntariness can be interpreted as individual rationality
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Entitlement theory as a way of implementing end-state
justice:
�
Three formulations of egalitarianism in the initial acquisition,
together with the principle of just transfer (voluntary transfer),
allow us to achieve end-state justice (three notions of no-envy).
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties
Nozick’s Entitlement Theory and End-State Justice
�
Entitlement theory as a way of implementing end-state
justice:
�
�
Three formulations of egalitarianism in the initial acquisition,
together with the principle of just transfer (voluntary transfer),
allow us to achieve end-state justice (three notions of no-envy).
End-state justice as a selection criterion for a principle of just
acquisition:
�
Three formulations of end-state fairness, together with the
principle of just transfer (voluntary transfer), allow us to
pin-down three versions of egalitarianism in the initial
acquisition.
Biung-Ghi Ju and Juan Moreno-Ternero
Fair Allocation of Disputed Properties