[PPT] - Fair Division of Indivisible Goods on a Graph II Sylvain Bouveret PowerPoint Presentation

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Fair Division of Indivisible Goods on a Graph II

Sylvain Bouveret

LIG – Grenoble INP, Univ. Grenoble-Alpes, France

Katarína Cechlárová

P.J. Šafárik University, Slovakia

Edith Elkind, Ayumi Igarashi, Dominik Peters

University of Oxford, UK

Advances in Fair Division, Высшая Школа Экономики, Санкт-Петербург, August 10, 2017

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Introduction

Fair division of indivisible items

A traditional fair division problem...

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Introduction

Fair division of indivisible items

A traditional fair division problem... Given

a set of indivisible objects O = {o1, . . . , om} a set of agents A = {1, . . . , n} each agent has additive preferences on the objects

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SLIDE 4

Introduction

Fair division of indivisible items

A traditional fair division problem... Given

a set of indivisible objects O = {o1, . . . , om} a set of agents A = {1, . . . , n} each agent has additive preferences on the objects

Find

an allocation π : A → 2O such that π(i) ∩ π(j) = ∅ for every i = j satisfying some fairness and efficiency criteria

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Introduction

A typical example

A common facility to be time-shared...

a common summer house a scientific experimental device an Earth observing satellite ...

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Introduction

A typical example

A common facility to be time-shared...

a common summer house a scientific experimental device an Earth observing satellite ...

Time-sharing with predefined timeslots

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Introduction

A typical example

Predefined timeslots → indivisible items

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Introduction

A typical example

Predefined timeslots → indivisible items

ts1 ts2 ts3 ts4 ts5 ts6 ts7 time

Agent 1 Agent 2 Agent 3

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Introduction

A typical example

Predefined timeslots → indivisible items

ts1 ts2 ts3 ts4 ts5 ts6 ts7 time

Agent 1 Agent 2 Agent 3

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Introduction

A typical example

Predefined timeslots → indivisible items

ts1 ts2 ts3 ts4 ts5 ts6 ts7 time

Agent 1 Agent 2 Agent 3 Fair? Maybe...

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Introduction

A typical example

Predefined timeslots → indivisible items

ts1 ts2 ts3 ts4 ts5 ts6 ts7 time

Agent 1 Agent 2 Agent 3 Fair? Maybe... Admissible? Probably not...

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Introduction

Time slots vs cake shares

NB: Can also represent a cake with predefined cut points...

ts1 ts2 ts3 ts4 ts5 ts6 ts7 time ts1 ts2 ts3 ts4 ts5 ts6 ts7

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Introduction

Another typical example

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Introduction

Another typical example

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Introduction

Another typical example

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Introduction

Fair division of a graph

Given

a set of indivisible objects O = {o1, . . . , om} a set of agents A = {1, . . . , n} each agent has additive preferences on the objects

Find

an allocation π : A → 2O such that π(i) ∩ π(j) = ∅ for every i = j satisfying some fairness and efficiency criteria

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Introduction

Fair division of a graph

Given

a set of indivisible objects O = {o1, . . . , om} a set of agents A = {1, . . . , n} each agent has additive preferences on the objects a neighbourhood relation R ⊆ O × O defining a graph of objects G

Find

an allocation π : A → 2O such that π(i) ∩ π(j) = ∅ for every i = j satisfying some fairness and efficiency criteria such that π(i) is connected in G for every i

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Introduction

Fairness

The fairness concepts we study:

Proportionality:1 ui(π(i)) ≥ 1

n for every i

Envy-freeness:2 ui(π(i)) ≥ ui(π(j)) for every (i, j) Max-min share: ui(π(i)) ≥ uMMS(i) for every i, where uMMS

i

= max−

→ π minj∈N ui(πj) 1Equal-division-lower-bound 2No-envy

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Proportionality

Proportionality: the bad news...

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Proportionality

Proportionality: the bad news... Proposition Prop-CFD is NP-complete even if G is a path. Idea: Reduction from Exact-3-Cover.

v 1

T1

v 2

T1

v 3

T1

v 1

T2

v 2

T2

v 3

T2

. . . v 1

Tr

v 2

Tr

v 3

Tr

b1 b2 . . . bs w 11 / 21 Fair Division of Indivisible Goods on a Graph II

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Proportionality

Proportionality: the bad news... Proposition Prop-CFD is NP-complete even if G is a path. Idea: Reduction from Exact-3-Cover.

v 1

T1

v 2

T1

v 3

T1

v 1

T2

v 2

T2

v 3

T2

. . . v 1

Tr

v 2

Tr

v 3

Tr

b1 b2 . . . bs w

Some good news: Proposition Prop-CFD can be solved in polynomial time if G is a star.

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Proportionality

Proportionality: good news

Proportionality: the good news...

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Proportionality

Proportionality: good news

Proportionality: the good news... Proposition Prop-CFD is XP

what is it?

with respect to the number of agent types if G is a path. Idea: dynamic programming algorithm (parameters: number of remaining vertices and number of agents of each type to satisfy)

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Proportionality

Proportionality: good news

Proportionality: the good news... Proposition Prop-CFD is XP

what is it?

with respect to the number of agent types if G is a path. Idea: dynamic programming algorithm (parameters: number of remaining vertices and number of agents of each type to satisfy) Proposition Prop-CFD is FPT

what is it?

with respect to the number of agents if G is a tree. Idea: run through all the possible ways of partioning a tree.

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Envy-freeness

Envy-freeness: bad news

Proposition EF-CFD is NP-complete even if:

G is a path G is a star

Idea:

Path: (Similar) reduction from Exact-3-Cover Star: Reduction from Independent set.

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Envy-freeness

Envy-freeness: good news

Proposition EF-CFD is XP with respect to the number of agent types if G is a path. Idea: “Guess” the utility received by each type, and use the previous dynamic programming algorithm (used for proportionality).

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Max-min share

Formal definition: ui(π(i)) ≥ uMMS(i) for every i, where uMMS

i

= max−

→ π minj∈N ui(πj)

More about MMS? 15 / 21 Fair Division of Indivisible Goods on a Graph II

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Max-min share

Formal definition: ui(π(i)) ≥ uMMS(i) for every i, where uMMS

i

= max−

→ π minj∈N ui(πj)

More about MMS?

Known facts for classical fair division:

An MMS allocation almost always exists Counter-examples are rare and intricate [Procaccia and Wang, 2014, Kurokawa et al., 2016]

Kurokawa, D., Procaccia, A. D., and Wang, J. (2016).

When can the maximin share guarantee be guaranteed? In AAAI’16, pages 523–529.

Procaccia, A. D. and Wang, J. (2014).

Fair enough: Guaranteeing approximate maximin shares. In ACM EC’14, pages 675–692. 15 / 21 Fair Division of Indivisible Goods on a Graph II

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Max-min share

Max-min share and graphs

Interestingly, as soon as there are connectivity constraints, it is easy to find an instance with no MMS allocation.

Show me the instance 16 / 21 Fair Division of Indivisible Goods on a Graph II

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Max-min share

Max-min share and graphs

Proposition If G is a tree, every agent can compute her MMS share uMMS

i

in polynomial time. Idea: “guess” the value by binary search and “move a knife” along the tree

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Max-min share

Max-min share and graphs

Proposition If G is a tree, every agent can compute her MMS share uMMS

i

in polynomial time. Idea: “guess” the value by binary search and “move a knife” along the tree Proposition If G is a tree, an MMS allocation always exists and can be found in polynomial time. Idea:

Every agent computes uMMS

i

We apply a discrete analogue of the last diminisher procedure

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

A

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

A

B does nothing

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

C

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

A

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

B

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Max-min share

Finding an MMS allocation

Intuition of the procedure on a path...

1
2
3
4
5
6
7

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Max-min share

Finding an MMS allocation

Last diminisher on a tree (intuition)... r The first player proposes a bundle.

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Max-min share

Finding an MMS allocation

Last diminisher on a tree (intuition)... r Other players may diminish the bundle.

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Max-min share

Finding an MMS allocation

Last diminisher on a tree (intuition)... r The last-diminisher receives the bundle.

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Conclusion

Take-away message

Fair division of indivisible items with connectivity constraints Negative (NP-completeness) general results But, also positive ones for simple yet interesting cases

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Conclusion

Take-away message

Fair division of indivisible items with connectivity constraints Negative (NP-completeness) general results But, also positive ones for simple yet interesting cases Path:

Proportionality: NP-complete, but XP with respect to the number of agent types and FPT with respect to the number of agents Envy-freeness: NP-complete, but XP with respect to the number of agent types Max-min share: polynomial (and guaranteed to exist)

Tree:

Proportionality: NP-complete, but FPT with respect to the number of agents Envy-freeness: NP-complete Max-min share: polynomial (and guaranteed to exist)

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Conclusion

Future work

Other fairness concepts? Other preference representations? Other topological constraints (nicely shaped shares)?

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Conclusion

Future work

Other fairness concepts? Other preference representations? Other topological constraints (nicely shaped shares)?

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Thank you for your attention

Questions?

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Conclusion

Max-min share

Proportionality is nice, but sometimes too demanding for indivisible goods → e.g. 2 agents, 1 object

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Conclusion

Max-min share

Proportionality is nice, but sometimes too demanding for indivisible goods → e.g. 2 agents, 1 object Max-min share (MMS):

Introduced recently [Budish, 2011]; not so much studied so far. Idea: in the cake-cutting case, proportionality = the best share an agent can hopefully get for sure in a “I cut, you choose (I choose last)” game. Same game for indivisible goods → MMS.

Budish, E. (2011).

The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6). 23 / 28 Fair Division of Indivisible Goods on a Graph II

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Conclusion

Max-min share

Idea: in the cake-cutting case, proportionality = the best share an agent can hopefully get for sure in a “I cut, you choose (I choose last)” game. Max-min share The max-min share of an agent i is equal to: uMMS

i

= max

− → π

min

j∈N ui(πj)

An allocation − → π satisfies max-min share (MMS) if every agent gets at least her max-min share.

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Conclusion

Max-min share: examples

Example: 3 objects {1, 2, 3}, 2 agents {1, 2}. Preferences: 1 2 3 agent 1 5 4 2 agent 2 4 1 6

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Conclusion

Max-min share: examples

Example: 3 objects {1, 2, 3}, 2 agents {1, 2}. Preferences: 1 2 3 agent 1 5 4 2 → uMMS

1

= 5 (with cut {1}, {2, 3}) agent 2 4 1 6 → uMMS

2

= 5 (with cut {1, 2}, {3})

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Conclusion

Max-min share: examples

Example: 3 objects {1, 2, 3}, 2 agents {1, 2}. Preferences: 1 2 3 agent 1 5 4 2 → uMMS

1

= 5 (with cut {1}, {2, 3}) agent 2 4 1 6 → uMMS

2

= 5 (with cut {1, 2}, {3}) MMS evaluation: − → π = {1}, {2, 3} → u1(π1) = 5 ≥ 5; u2(π2) = 7 ≥ 5 ⇒ MMS satisfied

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Conclusion

Max-min share: examples

Example: 3 objects {1, 2, 3}, 2 agents {1, 2}. Preferences: 1 2 3 agent 1 5 4 2 → uMMS

1

= 5 (with cut {1}, {2, 3}) agent 2 4 1 6 → uMMS

2

= 5 (with cut {1, 2}, {3}) MMS evaluation: − → π = {1}, {2, 3} → u1(π1) = 5 ≥ 5; u2(π2) = 7 ≥ 5 ⇒ MMS satisfied − → π ′′ = {2, 3}, {1} → u1(π′′

1 ) = 6 ≥ 5; u2(π′′ 2 ) = 4 < 5 ⇒ MMS not

satisfied

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Conclusion

Max-min share: examples

Example: 3 objects {1, 2, 3}, 2 agents {1, 2}. Preferences: 1 2 3 agent 1 5 4 2 → uMMS

1

= 5 (with cut {1}, {2, 3}) agent 2 4 1 6 → uMMS

2

= 5 (with cut {1, 2}, {3}) MMS evaluation: − → π = {1}, {2, 3} → u1(π1) = 5 ≥ 5; u2(π2) = 7 ≥ 5 ⇒ MMS satisfied − → π ′′ = {2, 3}, {1} → u1(π′′

1 ) = 6 ≥ 5; u2(π′′ 2 ) = 4 < 5 ⇒ MMS not

satisfied Example: 2 agents, 1 object.

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Conclusion

Max-min share: examples

Example: 3 objects {1, 2, 3}, 2 agents {1, 2}. Preferences: 1 2 3 agent 1 5 4 2 → uMMS

1

= 5 (with cut {1}, {2, 3}) agent 2 4 1 6 → uMMS

2

= 5 (with cut {1, 2}, {3}) MMS evaluation: − → π = {1}, {2, 3} → u1(π1) = 5 ≥ 5; u2(π2) = 7 ≥ 5 ⇒ MMS satisfied − → π ′′ = {2, 3}, {1} → u1(π′′

1 ) = 6 ≥ 5; u2(π′′ 2 ) = 4 < 5 ⇒ MMS not

satisfied Example: 2 agents, 1 object. uMMS

1

= uMMS

2

= 0 → every allocation satisfies MMS! Not very satisfactory, but can we do much better?

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Conclusion

MMS counterexample

v1 v2 v3 v4 v5 v6 v7 v8 Players 1 & 2 1 4 4 1 3 2 2 3 Players 3 & 4 4 4 1 3 2 2 3 1 P1:

v3 v2 v1 v8 v7 v6 v5 v4

P2:

v3 v2 v1 v8 v7 v6 v5 v4

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Conclusion

MMS counterexample

v1 v2 v3 v4 v5 v6 v7 v8 Players 1 & 2 1 4 4 1 3 2 2 3 Players 3 & 4 4 4 1 3 2 2 3 1 P1:

v3 v2 v1 v8 v7 v6 v5 v4

P2:

v3 v2 v1 v8 v7 v6 v5 v4

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Conclusion

Slice-wise polynomiality

Definition A problem is slice-wise polynomial (XP) with respect to a parameter k if ∃f , computable function, s.t. each instance I of this problem can be solved in time |I|f (k). Intuition: once k is fixed, f (k) can be large, but is fixed. Hence, I can be solved in polynomial time (but the degree of the polynome can be large).

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Conclusion

Fixed-parameter tractability

Definition A problem is fixed-parameter tractable (FPT) with respect to a pa- rameter k if ∃f , computable function, s.t. each instance I of this problem can be solved in time f (k) × poly(|I|). Intuition: once k is fixed, f (k) can be large, but is fixed. I can be solved in polynomial time and the degree of the polynome remains the same for every k. NB: FPT is strictly contained in XP.

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