OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The division problem with maximal capacity constraints
G. Bergantiños,1 J. Massó,2 A. Neme3
1Universidade de Vigo
2Universitat Autónoma de Barcelona
3Universidad de San Luis
July, 2010
G. Bergantiños, J. Massó, A. Neme Presentation 1/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Outline
1 Outline
2 The classical division problem
3 The division problem with maximal capacity constraints
4 The uniform rule
5 Properties
6 Axiomatic characterizations
G. Bergantiños, J. Massó, A. Neme Presentation 2/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The model
There is an amount t ∈ R+ of some infinitely divisible commodity that has to beallocated among a set N of agents, who have preferences on the amount they receive.
A division problem (Sprumont, 1991) is a triple (N,�, t) where
N = {1, ..., n} denotes the set of agents.
t is the amount to divide.
�= (�i )i∈N . For each i ∈ N, �i denotes the continuous preference relation ofagent i over [0, t].We assume that �i is single peak on the interval [0, t] . Namely, there existspi ∈ [0, t] satisfying two conditions. First, if 0 ≤ x ≤ y ≤ pi , then y � x. Second, ifpi ≤ y ≤ x ≤ t, then y � x.pi is called the peak of agent i .
The division problem has been studied in many papers. For instance,Barbera-Jackson-Neme (1997), Ching (1994), Chun (2000, 2003), Dagan (1996),Sonmmez (1994), Thomson (1994, 1997), ...
G. Bergantiños, J. Massó, A. Neme Presentation 3/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The model
The set of feasible allocations of (N,�, t) is defined as
FA (N,�, t) =
8
<
:
(xi )i∈N :X
i∈N
xi = t and for each i ∈ N, xi ∈ [0, t]
9
=
;
.
A rule is a map f assigning to each problem (N,�, t) a vector
f (N,�, t) ∈ FA (N,�, t) .
G. Bergantiños, J. Massó, A. Neme Presentation 4/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
A Spanish example
Every year the university decides the number of hours each department must to teach.
The department must divide the hours among the member of the department.
The number of hours assigned to each member is bounded below and above. Below by0. Above by a number (called the maximal capacity) depending on the characteristicsof the member: full or partial time, bureaucratic tasks, research awards, ....
The department can teach all the hours assigned because the sum of the maximalcapacities of the members of the department is larger than the hours assigned.
The preferences of the professors are single peak on the number of hours they teach.Typically, the sum of the peaks of all professors is smaller than the number of hoursassigned to the department.Some peaks are 0 but others are the capacity (for instance, people that want to bepromoted).
This situation can not be included in the classical division problem.
G. Bergantiños, J. Massó, A. Neme Presentation 5/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Other examples
To divide the number of hours of a task among a group of workers.
To divide a bond among a group of investors.
To decide the amount invested by each agent in a joint project.
G. Bergantiños, J. Massó, A. Neme Presentation 6/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The model
There is an amount t ∈ R+ of some infinitely divisible commodity that has to beallocated among a set N of agents, who have preferences on the amount they receive.
A division problem with maximal capacity constraints is a 4-tuple (N, u,�, t) where
N = {1, ..., n} denotes the set of agents.
u = (ui )i∈N where for each i ∈ N, ui is the maximal capacity constraint of agent i .Agent i must receive at most ui .
�= (�i )i∈N . For each i ∈ N, �i is continuous and single peak on the interval[0, ui ] .As usually, pi denotes the peak.
t is the amount to divide.We assume that t ≤
X
i∈N
ui .
Notice that if ui = t for all i , then we are in the classical division problem.
G. Bergantiños, J. Massó, A. Neme Presentation 7/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The model
The set of feasible allocations of (N, u,�, t) is defined as
FA (N, u,�, t) =
8
<
:
(xi)i∈N :X
i∈N
xi = t and for each i ∈ N, xi ∈ [0, ui ]
9
=
;
.
A rule is a map f assigning to each problem (N, u,�, t) a vector
f (N, u,�, t) ∈ FA (N, u,�, t) .
G. Bergantiños, J. Massó, A. Neme Presentation 8/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Related models
Kibris (2003) SCW.The model is the same but the definition of a rule is different.He assumes free disposal. Thus, it is possible to assign less than t among the agents.Formally,
P
i∈N xi < t is possible.
Bergantiños, Massó and Neme (2009).They model a situation of voluntary participation instead of capacities.Formally,
The preferences are defined over a interval [li , ui ] instead of [0, ui ].Agents are interested only in shares inside the interval. Otherwise they can not beforced to participate.
It is possible to divide t among a subset of agents. Namely it is possible to excludeagents.
We must divide t among the agents. Otherwise no agent participate.
G. Bergantiños, J. Massó, A. Neme Presentation 9/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The uniform rule in classical problems
For classical problems, the uniform rule F U is defined as follows.
F Ui (N,�, t) =
8
<
:
min {β, pi} ifP
i∈Npi ≥ t
max {β, pi} ifP
i∈Npi < t.
where β is the unique number satisfyingX
i∈N
F Ui (N,�, t) = t.
Some examples. Let N = {1, 2} and t = 10.
1 Assume that (p1, p2) = (4, 7). Then β = 6 and F U (N,�, t) = (4, 6).
2 Assume that (p1, p2) = (1, 7). Then β = 3 and F U (N,�, t) = (3, 7).
The uniform rule F U is not a rule in our model because it could not be feasible.For instance in Case 2 when u1 = 2.
G. Bergantiños, J. Massó, A. Neme Presentation 10/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
The uniform rule in our problems
We extend the definition of the uniform rule to the division problem with maximalcapacity constraints.
F Ui (N, u,�, t) =
min {β, pi} ifP
i∈N pi ≥ tmin {max {β, pi} , ui} if
P
i∈N pi < t
where β is the unique number satisfyingX
i∈N
Fi (N, u,�, t) = t.
Some examples. N = {1, 2} , u = (2, 10) , and t = 10.
1 Assume that (p1, p2) = (4, 7). Then β = 6 and F U (N,�, t) = (4, 6).
2 Assume that (p1, p2) = (1, 7). Then β = 8 and F U (N,�, t) = (2, 8).
G. Bergantiños, J. Massó, A. Neme Presentation 11/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Generalizing classical properties 1
Efficiency (ef ). f (N, u,�, t) is not Pareto dominated in FA (N, u,�, t)
Strategy-proofness (sp). Agents do not improve by lying about their preferences.Namely, for each (N, u,�, t) , i ∈ N, and �′
i we have that
fi (N, u,�, t) �i fi`
N, u,`
�′i ,�N\{i}
´
, t´
.
Equal treatment of equals (ete). Agents with the same preferences must receive thesame.Namely, for each (N, u,�, t) , i , j ∈ N such that �i=�jwe have thatfi (N, u,�, t) = fj (N, u,�, t) .
G. Bergantiños, J. Massó, A. Neme Presentation 12/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Generalizing classical properties 2
Consistency (co) . Let (N, I,�, t) and S ⊂ N. Then, for all i ∈ S,
fi (N, I,�, t) = fi
0
@S, (I)S , (�)S , t −X
j∈N\S
fj (N, I,�, t)
1
A .
One side resource monotonicity (rm) .
1 Let (N, u,�, t) and (N, u,�, t ′) be such that t ≤ t ′ ≤P
i∈Npi . Then, for all i ∈ N,
fi (N, u,�, t ′) �i fi (N, u,�, t) .
2 Let (N, u,�, t) and (N, u,�, t ′) be such thatP
i∈Npi ≤ t ′ ≤ t. Then, for all i ∈ N,
fi (N, u,�, t ′) �i fi (N, u,�, t) .
G. Bergantiños, J. Massó, A. Neme Presentation 13/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
More classical properties: Envy-freeness
Envy freeness (envy) . Let (N, u,�, t), j ∈ N, and k ∈ N\ {j}
fj (N, u,�, t) �j fk (N, u,�, t) .
N = {1, 2} , u = (2, 10) , and t = 10. Thus,
FA (N, u,�, t) = {(x, 10 − x) : 0 ≤ x ≤ 2} .
No rule satisfies envy . Let �2 satisfying x ≻2 y when x ∈ [0, 2] and y ∈ [8, 10].
We now introduce a weak version of envy .Weak envy freeness (wenvy) . Let (N, u,�, t), j ∈ N, k ∈ N\ {j} such thatfk (N, u,�, t) ≻j fj (N, u,�, t) , and (xi )i∈N where xj = fk (N, u,�, t) ,xk = fj (N, u,�, t) , and xi = fi (N, u,�, t) otherwise. Then,
(xi )i∈N /∈ FA (N, u,�, t) .
In the classical model envy and wenvy are equivalents but in our model they are not.
G. Bergantiños, J. Massó, A. Neme Presentation 14/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
More classical properties: Individually rational from equal division
Individually rational from equal division (ir) . Let (N, u,�, t) and i ∈ N. Then,
fi (N, u,�, t) �it
|N|.
Assume that N = {1, 2} , u = (2, 10) , p2 = 5, and t = 10. Thus,
FA (N, u,�, t) = {(x, 10 − x) : 0 ≤ x ≤ 2} .
If f satisfies ir , then f2 (N, u,�, t) = 5, which means that f (N, u,�, t) /∈ FA (N, u,�, t).
We now introduce a weak version of ir .Weak individually rational from equal division (wir) . Let (N, u,�, t) such that„
t
|N|
«
i∈N∈ FA (N, u,�, t) and i ∈ N. Then, fi (N, u,�, t) �i
t
|N|.
In the classical model, ir and wir are the same property but in our model they are not.
G. Bergantiños, J. Massó, A. Neme Presentation 15/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
New property: upper bound monotonicity 1
We introduce a property about u, the new element of our model (when comparing withthe classical division problem).
Upper bound monotonicity (ubm) . Let (N, u,�, t) and (N, u′,�′, t) be such that foreach i ∈ N\ {k} , ui = u′
i , �i=�′i , uk < u′
k < t, and �k coincides with �′k on [0, uk ] .
Then,fi (N, u,�, t) ≥ min
˘
fi`
N, u′,�′, t´
, ui¯
for each i ∈ N.
G. Bergantiños, J. Massó, A. Neme Presentation 16/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
New property: upper bound monotonicity 2
The previous condition can be rewritten as
fi (N, u,�, t) ≥ fi`
N, u′,�′, t´
when i 6= k and
fk (N, u,�, t) ≥ min˘
fk`
N, u′,�′, t´
, uk¯
.
For agent k , two cases are possible:
1 uk ≥ fk (N, u′,�′, t). Then, fk (N, u,�, t) ≥ fk (N, u′,�′, t) and hence
f (N, u,�, t) = f`
N, u′,�′, t´
.
2 uk < fk (N, u′,�′, t). Then,
fk (N, u,�, t) = uk .
If a rule f satisfies ubm, then f also satisfies the classical property of independence ofirrelevant alternatives.
G. Bergantiños, J. Massó, A. Neme Presentation 17/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Characterizations with strategy-proofness
Classical problems
1 Sprumont (1991) Econometrica.F U is the unique rule satisfying ef , sp, and ete,.
2 Sprumont (1991) Econometrica.F U is the unique rule satisfying ef , sp, and envy ,.
Our problemsTheorem 1 .
1 F U is the unique rule satisfying ef , sp, ete, and ubm.
2 F U is the unique rule satisfying ef , sp, wenvy , and ubm.
The properties used in Theorem 1 are independent.
G. Bergantiños, J. Massó, A. Neme Presentation 18/19
OutlineThe classical division problem
The division problem with maximal capacity constraintsThe uniform rule
PropertiesAxiomatic characterizations
Characterizations with consistency
Classical problems
1 Thomson (1994) JET, Dagan (1996) JET.F U is the unique rule satisfying ef , ir , and co.
2 Sonmez (1994) EL.F U is the unique rule satisfying rm, ir , and co,.
Our problemsTheorem 2 .
1 F U is the unique rule satisfying ef , wir , co, and ubm.
2 F U is the unique rule satisfying rm, wir , co, and ubm.
The properties used in Theorem 2 are independent.
G. Bergantiños, J. Massó, A. Neme Presentation 19/19