Rent Devision among Groups Mohammad Ghodsi, Mohamad Latifian, Arman - PowerPoint PPT Presentation

Rent Devision among Groups Mohammad Ghodsi, Mohamad Latifian, Arman Mohammadi, Sadra Moradian, Masoud Seddighin

Rent Division Problem • A house with n rooms, • n agents, • is the value of agent i for room j v i , j • Utility of agent i if she goes to room j u i , j = v i , j − r j

Rent Division Problem

Rent Division Problem -13$ -10$ -15$ -2$ -7$ -11$

What if two or more people want to live together?

Rent Division Among Groups (Model) • n groups, G = { g 1 , g 2 , …, g n } • Group i contains m i members, 𝒪 i = { a i ,1 , a i ,2 , …, a i , m i } • n houses, H = { h 1 , h 2 , …, h n } • is the value of for ( ) m i v i , j , k = 1 v i , j , k a i , j ∑ h k m i j =0 Group 1 Group 2 Group 3 m 1 ﹦ 2 m 2 ﹦ 4 m 3 ﹦ 3

Rent Division Among Groups (Goal) • We seek to find a triple S = (A, R, D) where: • (Allocate one house to each group) A : G → H • ( Rent of each house) R : H → ℝ ≥ 0 • (Cost of each house for each agent) D : ( 𝒪 , H ) → ℝ ≥ 0 • Total payment of each group for each room sums up to that room’s rent • Fair and individually rational

Example a 1,1 a 1,2 a 1,3 v 1,1,1 = 1/9 v 1,2,1 = 1/6 v 1,3,1 = 2/9 h 1 d 1,1,1 = 1/6 d 1,2,1 = 1/6 d 1,3,1 = 1/6 u 1,1,1 = − 1/18 u 1,2,1 = 0 u 1,3,1 = 1/18 R ( h 1 ) = 1/2 v 1,1,2 = 2/9 v 1,2,2 = 1/6 v 1,3,2 = 1/9 h 2 d 1,1,2 = 1/9 d 1,2,2 = 1/9 d 1,3,2 = 1/9 u 1,1,2 = 1/9 u 1,2,2 = 1/18 u 1,3,2 = 0 R ( h 2 ) = 1/3 * is the utility of if is assigned to u i , j , k a i , j g i . h k

Example a 1,1 a 1,2 a 1,3 v 1,1,1 = 1/9 v 1,2,1 = 1/6 v 1,3,1 = 2/9 h 1 d 1,1,1 = 1/6 d 1,2,1 = 1/6 d 1,3,1 = 1/6 u 1,1,1 = − 1/18 u 1,2,1 = 0 u 1,3,1 = 1/18 R ( h 1 ) = 1/2 v 1,1,2 = 2/9 v 1,2,2 = 1/6 v 1,3,2 = 1/9 h 2 d 1,1,2 = 1/9 d 1,2,2 = 1/9 d 1,3,2 = 1/9 u 1,1,2 = 1/9 u 1,2,2 = 1/18 u 1,3,2 = 0 R ( h 2 ) = 1/3 * is the utility of if is assigned to u i , j , k a i , j g i . h k

Example a 1,1 a 1,2 a 1,3 v 1,1,1 = 1/9 v 1,2,1 = 1/6 v 1,3,1 = 2/9 h 1 d 1,1,1 = 1/6 d 1,2,1 = 1/6 d 1,3,1 = 1/6 u 1,1,1 = − 1/18 u 1,2,1 = 0 u 1,3,1 = 1/18 R ( h 1 ) = 1/2 v 1,1,2 = 2/9 v 1,2,2 = 1/6 v 1,3,2 = 1/9 h 2 d 1,1,2 = 1/9 d 1,2,2 = 1/9 d 1,3,2 = 1/9 u 1,1,2 = 1/9 u 1,2,2 = 1/18 u 1,3,2 = 0 R ( h 2 ) = 1/3 * is the utility of if is assigned to u i , j , k a i , j g i . h k

Example a 1,1 a 1,2 a 1,3 v 1,1,1 = 1/9 v 1,2,1 = 1/6 v 1,3,1 = 2/9 h 1 d 1,1,1 = 1/6 d 1,2,1 = 1/6 d 1,3,1 = 1/6 u 1,1,1 = − 1/18 u 1,2,1 = 0 u 1,3,1 = 1/18 R ( h 1 ) = 1/2 v 1,1,2 = 2/9 v 1,2,2 = 1/6 v 1,3,2 = 1/9 h 2 d 1,1,2 = 1/9 d 1,2,2 = 1/9 d 1,3,2 = 1/9 u 1,1,2 = 1/9 u 1,2,2 = 1/18 u 1,3,2 = 0 R ( h 2 ) = 1/3 * is the utility of if is assigned to u i , j , k a i , j g i . h k

Example a 1,1 a 1,2 a 1,3 v 1,1,1 = 1/9 v 1,2,1 = 1/6 v 1,3,1 = 2/9 h 1 d 1,1,1 = 1/6 d 1,2,1 = 1/6 d 1,3,1 = 1/6 u 1,1,1 = − 1/18 u 1,2,1 = 0 u 1,3,1 = 1/18 R ( h 1 ) = 1/2 v 1,1,2 = 2/9 v 1,2,2 = 1/6 v 1,3,2 = 1/9 h 2 d 1,1,2 = 1/9 d 1,2,2 = 1/9 d 1,3,2 = 1/9 u 1,1,2 = 1/9 u 1,2,2 = 1/18 u 1,3,2 = 0 R ( h 2 ) = 1/3 * is the utility of if is assigned to u i , j , k a i , j g i . h k

Fairness Criteria

Fairness Criteria • Weak envy-free 
 - At least one agent of each group does not envy

Fairness Criteria • Weak envy-free 
 - At least one agent of each group does not envy • Aggregate envy-free 
 - The representative agent of each group does not envy

Fairness Criteria • Weak envy-free 
 - At least one agent of each group does not envy • Aggregate envy-free 
 - The representative agent of each group does not envy • Strong envy-free 
 - Nobody envies any other option

How to Divide the Rent?

How to Divide the Rent? • Equal cost-sharing policy 
 - Share the rent equally among group members

How to Divide the Rent? • Equal cost-sharing policy 
 - Share the rent equally among group members • Proportional cost-sharing policy 
 - Share the rent proportional to the values

How to Divide the Rent? • Equal cost-sharing policy 
 - Share the rent equally among group members • Proportional cost-sharing policy 
 - Share the rent proportional to the values • Free cost-sharing policy 
 - No rule on the rent sharing method among group members

Example a 3,1 a 3,2 a 3,3 v 1,1,1 = 1/9 v 1,1,1 = 1/6 v 1,1,1 = 2/9 h 1 d 1,1,1 = 1/6 d 1,1,1 = 1/6 d 1,1,1 = 1/6 u 1,1,1 = − 1/18 u 1,1,1 = 0 u 1,1,1 = 1/18 R ( h 1 ) = 1/2 v 1,1,1 = 2/9 v 1,1,1 = 1/6 v 1,1,1 = 1/9 h 2 d 1,1,1 = 1/9 d 1,1,1 = 1/9 d 1,1,1 = 1/9 u 1,1,1 = 1/9 u 1,1,1 = 1/18 u 1,1,1 = 0 R ( h 2 ) = 1/3 * is the utility of if is assigned to u i , j , k a i , j g i . h k

Results So Far Equal Proportional Free Weak Aggregate Strong

Results So Far Equal Proportional Free Weak Aggregate Inconsistent Strong The equal cost-sharing policy and strong envy-freeness are inconsistent

Results So Far Equal Proportional Free Weak Aggregate Inconsistent Inconsistent Strong Proportional cost-sharing policy and strong envy-freeness are inconsistent

Results So Far Equal Proportional Free Weak Consistent Consistent Aggregate Inconsistent Inconsistent Strong Equal cost-sharing policy and aggregate-envy-freeness are consistent Proportional cost-sharing policy and aggregate-envy-freeness are consistent

Results So Far Equal Proportional Free Weak Consistent Consistent Aggregate Inconsistent Inconsistent Strong Strong envy-freeness implies weak envy-freeness. Strong envy-freeness implies aggregate envy-freeness.

Results So Far Equal Proportional Free Consistent Consistent Weak Consistent Consistent Aggregate Inconsistent Inconsistent Strong Strong envy-freeness implies weak envy-freeness. Strong envy-freeness implies aggregate envy-freeness.

Results So Far Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Inconsistent Inconsistent Strong Free cost-sharing policy is a generalization of proportional cost-sharing policy

Results So Far Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Inconsistent Inconsistent ? Strong

Main Theorem

Main Theorem Strong envy-freeness is consistent with free cost sharing policy.

Main Theorem Strong envy-freeness is consistent with free cost sharing policy. A strong-envy-free allocation-triple with free cost sharing policy that maximizes total rent over all answers can be found using following LP .

Main Theorem Strong envy-freeness is consistent with free cost sharing policy. A strong-envy-free allocation-triple with free cost sharing policy that maximizes total rent over all answers can be found using following LP .

Results Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Inconsistent Inconsistent Consistent Strong

Not Pre-determined Groups • n houses with capacity m • Set of agents 𝒪 = { a 1 , a 2 , …, a l } • is the value of house j for agent a i v i , j Strong envy-freeness is consistent with equal cost- sharing when groups are not pre-determined.

Conclusion Pre-determined Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Inconsistent Inconsistent Consistent Strong Not Pre-determined Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Consistent Consistent Strong

Conclusion Pre-determined Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Inconsistent Inconsistent Consistent Strong Not Pre-determined Equal Proportional Free Consistent Consistent Consistent Weak Consistent Consistent Consistent Aggregate Consistent Consistent ? Strong

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