The division problem with maximal capacity constraints · The division problem with maximal capacity constraints The uniform rule Properties Axiomatic characterizations The model

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




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





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), ...





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) .





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.





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.





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.





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) .





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.





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).


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.





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.







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) .





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) .





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.





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.





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.





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.





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.





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.


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The division problem with maximal capacity constraints · The division problem with maximal capacity constraints The uniform rule Properties Axiomatic characterizations The model