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

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

Jun 28, 2023 344 likes •753 views

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

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SLIDE 1

Rent Devision among Groups

Mohammad Ghodsi, Mohamad Latifian, Arman Mohammadi, Sadra Moradian, Masoud Seddighin

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SLIDE 2

Rent Division Problem

  • A house with n rooms,
  • n agents,
  • is the value of agent i for room j
  • Utility of agent i if she goes to room j

vi,j

ui,j = vi,j − rj

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SLIDE 3

Rent Division Problem

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SLIDE 4

Rent Division Problem

  • 10$
  • 15$
  • 13$
  • 7$
  • 11$
  • 2$
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SLIDE 5
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SLIDE 6

What if two or more people want to live together?

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SLIDE 7

Rent Division Among Groups (Model)

  • n groups,
  • Group i contains mi members,
  • n houses,
  • is the value of for ( )

Group 1 m1 ﹦2 Group 2 m2 ﹦4 Group 3 m3 ﹦3

G = {g1, g2, …, gn} H = {h1, h2, …, hn} vi,j,k 𝒪i = {ai,1, ai,2, …, ai,mi} ai,j hk

mi

∑

j=0

vi,j,k = 1 mi

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SLIDE 8

Rent Division Among Groups (Goal)

  • We seek to find a triple S = (A, R, D) where:
  • (Allocate one house to each group)
  • ( Rent of each house)
  • (Cost of each house for each agent)
  • Total payment of each group for each room sums up to that room’s

rent

  • Fair and individually rational

A : G → H R : H → ℝ≥0 D : (𝒪, H) → ℝ≥0

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SLIDE 9

Example

a1,1 a1,2 a1,3 h1 h2

v1,1,1 = 1/9

d1,1,1 = 1/6 u1,1,1 = − 1/18

* is the utility of if is assigned to

ui,j,k ai,j hk gi .

v1,2,1 = 1/6

d1,2,1 = 1/6 u1,2,1 = 0

v1,3,1 = 2/9

d1,3,1 = 1/6 u1,3,1 = 1/18

v1,1,2 = 2/9

d1,1,2 = 1/9 u1,1,2 = 1/9

v1,2,2 = 1/6

d1,2,2 = 1/9 u1,2,2 = 1/18

v1,3,2 = 1/9

d1,3,2 = 1/9 u1,3,2 = 0

R(h1) = 1/2 R(h2) = 1/3

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SLIDE 10

Example

a1,1 a1,2 a1,3 h1 h2

v1,1,1 = 1/9

d1,1,1 = 1/6 u1,1,1 = − 1/18

* is the utility of if is assigned to

ui,j,k ai,j hk gi .

v1,2,1 = 1/6

d1,2,1 = 1/6 u1,2,1 = 0

v1,3,1 = 2/9

d1,3,1 = 1/6 u1,3,1 = 1/18

v1,1,2 = 2/9

d1,1,2 = 1/9 u1,1,2 = 1/9

v1,2,2 = 1/6

d1,2,2 = 1/9 u1,2,2 = 1/18

v1,3,2 = 1/9

d1,3,2 = 1/9 u1,3,2 = 0

R(h1) = 1/2 R(h2) = 1/3

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SLIDE 11

Example

a1,1 a1,2 a1,3 h1 h2

v1,1,1 = 1/9

d1,1,1 = 1/6 u1,1,1 = − 1/18

* is the utility of if is assigned to

ui,j,k ai,j hk gi .

v1,2,1 = 1/6

d1,2,1 = 1/6 u1,2,1 = 0

v1,3,1 = 2/9

d1,3,1 = 1/6 u1,3,1 = 1/18

v1,1,2 = 2/9

d1,1,2 = 1/9 u1,1,2 = 1/9

v1,2,2 = 1/6

d1,2,2 = 1/9 u1,2,2 = 1/18

v1,3,2 = 1/9

d1,3,2 = 1/9 u1,3,2 = 0

R(h1) = 1/2 R(h2) = 1/3

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SLIDE 12

Example

a1,1 a1,2 a1,3 h1 h2

v1,1,1 = 1/9

d1,1,1 = 1/6 u1,1,1 = − 1/18

* is the utility of if is assigned to

ui,j,k ai,j hk gi .

v1,2,1 = 1/6

d1,2,1 = 1/6 u1,2,1 = 0

v1,3,1 = 2/9

d1,3,1 = 1/6 u1,3,1 = 1/18

v1,1,2 = 2/9

d1,1,2 = 1/9 u1,1,2 = 1/9

v1,2,2 = 1/6

d1,2,2 = 1/9 u1,2,2 = 1/18

v1,3,2 = 1/9

d1,3,2 = 1/9 u1,3,2 = 0

R(h1) = 1/2 R(h2) = 1/3

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SLIDE 13

Example

a1,1 a1,2 a1,3 h1 h2

v1,1,1 = 1/9

d1,1,1 = 1/6 u1,1,1 = − 1/18

* is the utility of if is assigned to

ui,j,k ai,j hk gi .

v1,2,1 = 1/6

d1,2,1 = 1/6 u1,2,1 = 0

v1,3,1 = 2/9

d1,3,1 = 1/6 u1,3,1 = 1/18

v1,1,2 = 2/9

d1,1,2 = 1/9 u1,1,2 = 1/9

v1,2,2 = 1/6

d1,2,2 = 1/9 u1,2,2 = 1/18

v1,3,2 = 1/9

d1,3,2 = 1/9 u1,3,2 = 0

R(h1) = 1/2 R(h2) = 1/3

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SLIDE 14

Fairness Criteria

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SLIDE 15

Fairness Criteria

  • Weak envy-free

  • At least one agent of each group does not envy
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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
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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
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SLIDE 18

How to Divide the Rent?

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SLIDE 19

How to Divide the Rent?

  • Equal cost-sharing policy

  • Share the rent equally among group members
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SLIDE 20

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
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SLIDE 21

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
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SLIDE 22

Example

a3,1 a3,2 a3,3 h1 h2

v1,1,1 = 1/9

d1,1,1 = 1/6 u1,1,1 = − 1/18

* is the utility of if is assigned to

ui,j,k ai,j hk gi .

v1,1,1 = 1/6

d1,1,1 = 1/6 u1,1,1 = 0

v1,1,1 = 2/9

d1,1,1 = 1/6 u1,1,1 = 1/18

v1,1,1 = 2/9

d1,1,1 = 1/9 u1,1,1 = 1/9

v1,1,1 = 1/6

d1,1,1 = 1/9 u1,1,1 = 1/18

v1,1,1 = 1/9

d1,1,1 = 1/9 u1,1,1 = 0

R(h1) = 1/2 R(h2) = 1/3

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SLIDE 23

Results So Far

Equal Proportional Free Weak Aggregate Strong

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SLIDE 24

Results So Far

Equal Proportional Free Weak Aggregate Strong

Inconsistent

The equal cost-sharing policy and strong envy-freeness are inconsistent

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SLIDE 25

Results So Far

Equal Proportional Free Weak Aggregate Strong

Inconsistent Inconsistent

Proportional cost-sharing policy and strong envy-freeness are inconsistent

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SLIDE 26

Results So Far

Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent

Equal cost-sharing policy and aggregate-envy-freeness are consistent Proportional cost-sharing policy and aggregate-envy-freeness are consistent

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SLIDE 27

Results So Far

Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent

Strong envy-freeness implies weak envy-freeness. Strong envy-freeness implies aggregate envy-freeness.

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SLIDE 28

Results So Far

Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent Consistent Consistent

Strong envy-freeness implies weak envy-freeness. Strong envy-freeness implies aggregate envy-freeness.

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SLIDE 29

Results So Far

Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent Consistent Consistent Consistent Consistent

Free cost-sharing policy is a generalization of proportional cost-sharing policy

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SLIDE 30

Results So Far

Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent

?

Consistent Consistent Consistent Consistent

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SLIDE 31

Main Theorem

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SLIDE 32

Main Theorem

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

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SLIDE 33

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 .

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SLIDE 34

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 .

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SLIDE 35

Results

Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent Consistent Consistent Consistent Consistent Consistent

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SLIDE 36

Not Pre-determined Groups

  • n houses with capacity m
  • Set of agents
  • is the value of house j for agent ai

𝒪 = {a1, a2, …, al}

vi,j

Strong envy-freeness is consistent with equal cost- sharing when groups are not pre-determined.

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SLIDE 37

Conclusion

Pre-determined Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent Consistent Consistent Consistent Consistent Consistent

Not Pre-determined Equal Proportional Free Weak Aggregate Strong

Consistent Consistent Consistent Consistent Consistent Consistent Consistent Consistent

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SLIDE 38

Conclusion

Pre-determined Equal Proportional Free Weak Aggregate Strong

Consistent Inconsistent Inconsistent Consistent Consistent Consistent Consistent Consistent Consistent

Not Pre-determined Equal Proportional Free Weak Aggregate Strong

Consistent Consistent Consistent Consistent Consistent Consistent Consistent Consistent

?

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SLIDE 39

Thanks for your time