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Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods Sylvain Bouveret Ulle Endriss Jrme Lang Onera Toulouse University of Amsterdam Universit Paris Dauphine Third International Workshop on

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Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

Sylvain Bouveret

Onera Toulouse

Ulle Endriss

University of Amsterdam

Jérôme Lang

Université Paris Dauphine

Third International Workshop on Computational Social Choice Düsseldorf, September 13–16, 2010

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Introduction

The fair division problem

Given

a set of indivisible objects ❖ = {♦✶, . . . , ♦♠} and a set of agents ❆ = {✶, . . . , ♥} such that each agent has some preferences on the subsets of objects she may receive

Find

an allocation π : ❆ → ✷❖ such that π(✐) ∩ π(❥) for every ✐ = ❥ satisfying some fairness and efficiency criteria

2 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Separable ordinal preferences

We assume that the preferences are ordinal. Restriction: each agent specifies a linear order ⊲ on ❖ (single objects) N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞

❛❜❝ ❛❜ ❛❜ ❛❝

❛❜❝ ❛❜ ❳ ❨ ❩ ❳ ❨ ❳ ❩ ❨ ❩ ❛❜ ❛❝

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Ordinal preferences

Separable ordinal preferences

We assume that the preferences are ordinal. Restriction: each agent specifies a linear order ⊲ on ❖ (single objects) N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞

Problem: How to compare subsets of objects ? ❀ e.g ❛❜❝

?

≺≻ ❛❜; ❛❜

?

≺≻ ❛❝ ?

❛❜❝ ❛❜ ❳ ❨ ❩ ❳ ❨ ❳ ❩ ❨ ❩ ❛❜ ❛❝

3 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Separable ordinal preferences

We assume that the preferences are ordinal. Restriction: each agent specifies a linear order ⊲ on ❖ (single objects) N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞

Problem: How to compare subsets of objects ? ❀ e.g ❛❜❝

?

≺≻ ❛❜; ❛❜

?

≺≻ ❛❝ ?

1

Assume monotonicity ❀ e.g ❛❜❝ ≻ ❛❜. ❳ ❨ ❩ ❳ ❨ ❳ ❩ ❨ ❩ ❛❜ ❛❝

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Ordinal preferences

Separable ordinal preferences

We assume that the preferences are ordinal. Restriction: each agent specifies a linear order ⊲ on ❖ (single objects) N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞

Problem: How to compare subsets of objects ? ❀ e.g ❛❜❝

?

≺≻ ❛❜; ❛❜

?

≺≻ ❛❝ ?

1

Assume monotonicity ❀ e.g ❛❜❝ ≻ ❛❜.

2

Assume separability: if (❳ ∪ ❨ ) ∩ ❩ = ∅ then ❳ ≻ ❨ iff ❳ ∪ ❩ ≻ ❨ ∪ ❩. ❀ e.g ❛❜ ≻ ❛❝.

3 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Example

N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ Separability Monotonicity

❛❜❝❞ ❛❜❝ ❛❜❞ ❛❝❞ ❜❝❞ ❛❜ ❛❝ ❛❞ ❜❝ ❜❞ ❝❞ ❛ ❜ ❝ ❞ ∅

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Ordinal preferences

Example

N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ Separability Monotonicity

❛❜❝❞ ❛❜❝ ❛❜❞ ❛❝❞ ❜❝❞ ❛❜ ❛❝ ❛❞ ❜❝ ❜❞ ❝❞ ❛ ❜ ❝ ❞ ∅

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Ordinal preferences

Example

N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ Separability Monotonicity

❛❜❝❞ ❛❜❝ ❛❜❞ ❛❝❞ ❜❝❞ ❛❜ ❛❝ ❛❞ ❜❝ ❜❞ ❝❞ ❛ ❜ ❝ ❞ ∅

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Ordinal preferences

Example

N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ Separability Monotonicity

❛❜❝❞ ❛❜❝ ❛❜❞ ❛❝❞ ❜❝❞ ❛❜ ❛❝ ❛❞ ❜❝ ❜❞ ❝❞ ❛ ❜ ❝ ❞ ∅

4 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Example

N : ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ Separability Monotonicity

❛❜❝❞ ❛❜❝ ❛❜❞ ❛❝❞ ❜❝❞ ❛❜ ❛❝ ❛❞ ❜❝ ❜❞ ❝❞ ❛ ❜ ❝ ❞ ∅

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Ordinal preferences

Dominance

Proposition ❳ ≻N ❨ ⇔ ∃ an injective mapping of improvements ❨ → ❳. ❛ ❜ ❝ ❞ ❡ ❢

❛ ❝ ❞ ❜ ❝ ❡ ❛ ❞ ❡ ❜ ❝ ❢ ❛ ❝ ❞ ❜ ❝ ❡ ❢

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Ordinal preferences

Dominance

Proposition ❳ ≻N ❨ ⇔ ∃ an injective mapping of improvements ❨ → ❳. Example: N = ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ ⊲ ❡ ⊲ ❢

{ ❛ , ❝ , ❞ } ≻N { ❜ , ❝ , ❡ } { ❛ , ❞ , ❡ } and { ❜ , ❝ , ❢ } are incomparable. {❛, ❝, ❞} and {❜, ❝, ❡, ❢ } are incomparable.

5 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Dominance

Proposition ❳ ≻N ❨ ⇔ ∃ an injective mapping of improvements ❨ → ❳. Example: N = ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ ⊲ ❡ ⊲ ❢

{ ❛ , ❝ , ❞ } ≻N { ❜ , ❝ , ❡ } { ❛ , ❞ , ❡ } and { ❜ , ❝ , ❢ } are incomparable. {❛, ❝, ❞} and {❜, ❝, ❡, ❢ } are incomparable.

5 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Dominance

Proposition ❳ ≻N ❨ ⇔ ∃ an injective mapping of improvements ❨ → ❳. Example: N = ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ ⊲ ❡ ⊲ ❢

{ ❛ , ❝ , ❞ } ≻N { ❜ , ❝ , ❡ } { ❛ , ❞ , ❡ } and { ❜ , ❝ , ❢ } are incomparable. {❛, ❝, ❞} and {❜, ❝, ❡, ❢ } are incomparable.

?

5 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Ordinal preferences

Dominance

Proposition ❳ ≻N ❨ ⇔ ∃ an injective mapping of improvements ❨ → ❳. Example: N = ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ ⊲ ❡ ⊲ ❢

{ ❛ , ❝ , ❞ } ≻N { ❜ , ❝ , ❡ } { ❛ , ❞ , ❡ } and { ❜ , ❝ , ❢ } are incomparable. {❛, ❝, ❞} and {❜, ❝, ❡, ❢ } are incomparable.

Brams, S. J., Edelman, P. H., and Fishburn, P. C. (2004).

Fair division of indivisible items. Theory and Decision, 5(2):147–180.

Brams, S. J. and King, D. (2005).

Efficient fair division—help the worst off or avoid envy? Rationality and Society, 17(4):387–421. 5 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Fairness and efficiency

Envy-freeness

  • Fairness. . .

✶ ♥

✐ ❥ ✐

✐

❥

✶ ♥

✐ ❥ ❥

✐

✐ ✐ ❥ ✐

✐

❥

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Fairness and efficiency

Envy-freeness

  • Fairness. . .

Envy-freeness: ≻✶, . . . , ≻♥ total strict orders, allocation π. π envy-free ⇔ ∀✐, ❥, π(✐) ≻✐ π(❥)

✶ ♥

✐ ❥ ❥

✐

✐ ✐ ❥ ✐

✐

❥

6 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Fairness and efficiency

Envy-freeness

  • Fairness. . .

Envy-freeness: ≻✶, . . . , ≻♥ total strict orders, allocation π. π envy-free ⇔ ∀✐, ❥, π(✐) ≻✐ π(❥) When ≻✶, . . . , ≻♥ are partial orders ?

✐ ❥ ❥

✐

✐ ✐ ❥ ✐

✐

❥

6 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Fairness and efficiency

Envy-freeness

  • Fairness. . .

Envy-freeness: ≻✶, . . . , ≻♥ total strict orders, allocation π. π envy-free ⇔ ∀✐, ❥, π(✐) ≻✐ π(❥) When ≻✶, . . . , ≻♥ are partial orders ? ❀ Envy-freeness becomes a modal notion Possible and necessary Envy-freeness

π is Possibly Envy-Free iff for all ✐, ❥, we have π(❥) ≻✐ π(✐); π is Necessary Envy-Free iff for all ✐, ❥, we have π(✐) ≻✐ π(❥).

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Fairness and efficiency

Pareto-efficiency

  • Efficiency. . .

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Fairness and efficiency

Pareto-efficiency

  • Efficiency. . .

Complete allocation. Pareto-efficiency

7 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Fairness and efficiency

Pareto-efficiency

  • Efficiency. . .

Classical Pareto dominance π′ dominates π if for all ✐, π′(✐) ✐ π(✐) and for some ❥, π′(❥) ≻❥ π(❥) Extended to possible and necessary Pareto dominance.

π is possibly Pareto-efficient (PPE) if there exists no allocation π′ such that π′ necessarily dominates π. π′ is necessarily Pareto-efficient (NPE) if there exists no allocation π′ such that π′ possibly dominates π.

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Computing envy-free allocations

Envy-freeness and efficiency

complete PPE NPE Efficiency

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Computing envy-free allocations

Envy-freeness and efficiency

complete PPE NPE Efficiency PEF NEF Fairness

8 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations

Envy-freeness and efficiency

complete PPE NPE Efficiency PEF X X X NEF X X X Fairness

8 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations

Envy-freeness and efficiency

complete PPE NPE Efficiency PEF X X X NEF X X X Fairness Envy-freeness and efficiency cannot always be satisfied simultaneously

8 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations

Envy-freeness and efficiency

complete PPE NPE Efficiency PEF X X X NEF X X X Fairness Envy-freeness and efficiency cannot always be satisfied simultaneously Questions:

under which conditions is it guaranteed that there exists a allocation that satisfies Fairness and Efficiency ? how hard it is to determine whether such an allocation exists?

8 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Possible envy-freeness

Complete possibly envy-free allocations

complete PPE NPE PEF X X X NEF X X X ♥ ♠ ❦ ♠ ✷♥ ❦

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Computing envy-free allocations – Possible envy-freeness

Complete possibly envy-free allocations

complete PPE NPE PEF X X X NEF X X X Result ♥ agents, ♠ objects, ❦ distinct goods are top-ranked by some agent. ∃ complete PEF allocation ⇔ ♠ ≥ ✷♥ − ❦. Constructive proof (algorithm/protocol)

9 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Possible envy-freeness

Example

N✶: ❛ ⊲ ❜ ⊲ ❝ ⊲ ❞ ⊲ ❡ ⊲ ❢ N✷: ❛ ⊲ ❞ ⊲ ❜ ⊲ ❝ ⊲ ❡ ⊲ ❢ N✸: ❜ ⊲ ❛ ⊲ ❝ ⊲ ❞ ⊲ ❢ ⊲ ❡ N✹: ❜ ⊲ ❛ ⊲ ❞ ⊲ ❡ ⊲ ❢ ⊲ ❝ (❦ = ✷; ♠ = ✻ ≥ ✷♥ − ❦) Consider the agents in order 1 > 2 > 3 > 4:

first step: give ❛ to 1; give ❜ to 3; 1 and 3 leave the room; second step: give ❞ to 2; give ❝ to 4; third step: give ❡ to 4; give ❢ to 2.

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Computing envy-free allocations – Possible envy-freeness

PPE-PEF allocations

complete PPE NPE PEF X X X NEF X X X

11 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Possible envy-freeness

PPE-PEF allocations

complete PPE NPE PEF X X X NEF X X X Result ∃ PPE-PEF allocation ⇔ ∃ complete, PEF allocation. Based on the characterization of efficient allocations in [Brams and King, 2005].

Brams, S. J. and King, D. (2005).

Efficient fair division—help the worst off or avoid envy? Rationality and Society, 17(4):387–421. 11 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Possible envy-freeness

NPE-PEF allocations

complete PPE NPE PEF X X X NEF X X X

12 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Possible envy-freeness

NPE-PEF allocations

complete PPE NPE PEF X X X NEF X X X Complexity of the existence of NPE-PEF allocations: open.

12 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Necessary envy-freeness

Complete NEF allocations

complete PPE NPE PEF X X X NEF X X X

Two easy necessary conditions:

distinct top ranked objects; ♠ is a multiple of ♥.

Complete allocation

deciding whether there exists a complete NEF allocation is ◆P-complete (even if ♠ = ✷♥). the problem falls down in P for two agents

(hardness by reduction from [X3C])

13 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Computing envy-free allocations – Necessary envy-freeness

Pareto-efficient-NEF allocations

complete PPE NPE PEF X X X NEF X X X Possible and necessary Pareto-efficiency

existence of a PPE-NEF allocation: ◆P-complete existence of a NPE-NEF allocation: ◆P-hard but probably not in ◆P (Σ♣

✷-completeness conjectured).

14 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Conclusion

Results and beyond

Fair division with incomplete ordinal preferences:

separable and monotone ordinal preferences; modal Pareto-efficiency and Envy-freeness.

complete PPE NPE PEF P

(algorithm)

P

(algorithm)

? NEF ◆P-complete ◆P-complete

(P for 2 agents)

◆P-hard

(Σ♣

✷-completeness

conjectured)

15 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods

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Conclusion

Results and beyond

Fair division with incomplete ordinal preferences:

separable and monotone ordinal preferences; modal Pareto-efficiency and Envy-freeness.

complete PPE NPE PEF P

(algorithm)

P

(algorithm)

? NEF ◆P-complete ◆P-complete

(P for 2 agents)

◆P-hard

(Σ♣

✷-completeness

conjectured)

Beyond separable preferences ? CI-nets [Bouveret et al., 2009]. ❀ Even dominance is PSPACE-complete.

Bouveret, S., Endriss, U., and Lang, J. (2009).

Conditional importance networks: A graphical language for representing ordinal, monotonic preferences over sets of goods. In Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI’09), pages 67–72, Pasadena, California. 15 / 15 Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods