Sharing an increase of the rent fairly Rodrigo A. Velez Texas A&M University

June 2016

Rodrigo A. Velez (Texas A&M)

1 / 13

Summary



A group of roommates rent a house in an envy-free way.



Rent increases.



This paper characterizes the ways in which the roommates can redistribute the rooms and rent, so no-envy is preserved and they all contribute to the higher rent.



Axiomatic/Normative study.

Rodrigo A. Velez (Texas A&M)

2 / 13

Model ◮

N: a group of n agents



A: a set of n objects



consumption bundle (xi , α) ∈ R × A



Ri : preference on R × A (money is desirable and no object is infinitely better than another, in terms of money).



Preferences need not to be quasi-linear.



m: budget (only variable) P z ≡ (x, µ), x ≡ (xi )i∈N , xi = m, and µ : N → A a bijection.

◮ ◮ ◮

Zm set of allocations with budget m m ∈ R 7→ f (m) ∈ Zm

Rodrigo A. Velez (Texas A&M)

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Axioms

z ∈ Zm ◮

Efficiency: there is no z ′ ∈ Zm that Pareto dominates z.



No-envy (non-contestability): ∀ {i, j} zi Ri zj .



if f only selects envy-free allocations we write f ∈ F

Rodrigo A. Velez (Texas A&M)

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Axioms

f is budget-monotone at m if ◮

∀l < m, ∀i ∈ N, fi (m) Pi fi (l)



∀h > m, ∀i ∈ N, fi (h) Pi fi (m)

f is budget-monotone if it is budget-monotone at each m ∈ R (Moulin and Thomson, 1988)

Rodrigo A. Velez (Texas A&M)

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There are envy-free budget-monotone rules

Alkan, Demange, and Gale (1991, Econometrica) introduce four alternative constructions.

Rodrigo A. Velez (Texas A&M)

5 / 13

There are envy-free budget-monotone rules

VALUE MAXMIN:

fix u ≡ (ui )i∈N representation of R   f (m) ∈ arg max min ui (zi ) z∈Fm

Rodrigo A. Velez (Texas A&M)

i∈N

5 / 13

There are envy-free budget-monotone rules

VALUE MINMAX:

fix u ≡ (ui )i∈N representation of R   f (m) ∈ arg min max ui (zi ) z∈Fm

Rodrigo A. Velez (Texas A&M)

i∈N

5 / 13

There are envy-free budget-monotone rules

MONEY MAXMIN:

fix h ≡ (hα )α∈A strictly increasing functions   f (m) ∈ arg max min hα (xα ) z≡(x,µ)∈Fm

Rodrigo A. Velez (Texas A&M)

α∈A

5 / 13

There are envy-free budget-monotone rules

MONEY MINMAX:

fix h ≡ (hα )α∈A strictly increasing functions   f (m) ∈ arg min max hα (xα ) z≡(x,µ)∈Fm

Rodrigo A. Velez (Texas A&M)

α∈A

5 / 13

Theorem f ∈ F . The following are equivalent: 1. f is budget-monotone 2. f is value maxmin 3. f is value minmax 4. f is money maxmin 5. f is money minmax

Rodrigo A. Velez (Texas A&M)

6 / 13

Proof



1 ⇒ (2, 3, 4, 5) calibration exercise.



(2, 3, 4, 5) ⇒ 1 alternative direct proof. “The main advantage to our proof [...] is that because of its constructive nature we are able to derive qualitative properties of fair allocations which do not seem to be obtainable by the usual fixed points methods.” (ADG, 1991)

Rodrigo A. Velez (Texas A&M)

7 / 13

Intuitive solutions may violate budget-monotonicity, e.g., Utilitarian and equal-compensation (Tadenuma and Thomson 1995 T&D) rules

Rodrigo A. Velez (Texas A&M)

8 / 13

Theorem f ∈ F and M ⊆ R the set of budgets at which f is budget-monotone. There is g that coincides with f on M and is weakly budget monotone, i.e., ∀l < m < h, i ∈ N, gi (h) Ri gi (m) Ri gi (l).

Rodrigo A. Velez (Texas A&M)

9 / 13

Theorem f ∈ F , M ⊆ R the set of budgets at which f is budget-monotone, and δ > 0. There is g and ∆ ⊂ R covered by a set of measure δ, s. t. g coincides with f on M \ ∆ and is budget monotone.

Rodrigo A. Velez (Texas A&M)

10 / 13

Conditions on preferences guaranteeing that, for each negative budget, there is an envy-free allocation at which each agent pays rent ◮

Su (1999): all agents are indifferent between any two objects with no money.

Rodrigo A. Velez (Texas A&M)

11 / 13

Conditions on preferences guaranteeing that, for each negative budget, there is an envy-free allocation at which each agent pays rent ◮

Su (1999): all agents are indifferent between any two objects with no money.

Proposition Fix R. The following are equivalent 1. ∀m ∈ R, m < 0, there is (x, µ) ∈ Fm s. t. x << 0. 2. ∀m ∈ R, m > 0, there is (x, µ) ∈ Fm s. t. x >> 0. 3. there is (0, µ) ∈ F0 4. ∀S ⊆ N, |∪i∈S {α ∈ A : ∀β ∈ A, (0, α) Ri (0, β)}| ≥ |S|

Rodrigo A. Velez (Texas A&M)

11 / 13

Continuity

Proposition f ∈ F budget-monotone 1. Fix (ui )i∈N continuous representation of R. Then, utility of each agent is a continuous function of m. 2. Consumption of money of agent who gets object α is a continuous function of m. This result allows one to extend existence of budget-monotone, envy-free, efficient rules when |A| > n.

Rodrigo A. Velez (Texas A&M)

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Thanks.

Rodrigo A. Velez (Texas A&M)

13 / 13

Sharing an increase of the rent fairly

redistribute the rooms and rent, so no-envy is preserved and they all contribute to the higher rent. ▷ Axiomatic/Normative study. Rodrigo A. Velez (Texas A&M).

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