Two Algorithms for Additive and Fair Division of Mixed Manna Martin - - PowerPoint PPT Presentation

Two Algorithms for Additive and Fair Division of Mixed Manna

Martin Aleksandrov and Toby Walsh The 43rd KI Conference, 2020

• M. Aleksandrov and T. Walsh

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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Additive and fair division of mixed manna

Fair division refers to the task of allocating resources to agents1. a set of agents N = {1, 2, . . . , n} a set of indivisible items O = {o1, o2, . . . , om} (real-valued) ∀i ∈ N, o ∈ O, ui(o) ∈ R (additive) ∀i ∈ N, B ⊆ O, ui(B) =

• ∈B ui(o)
• 1H. Steinhaus. The problem of fair division. Econometrica 16, 101–104 (1948).

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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EF1 & EFX (Budish et al. 2011, Caragiannis et al. 2016)

Allocation: A = (A1, . . . , An) gives different Aj to each j ∈ [n]

Envy-freeness up to some item (EF1)

A is EF1 if, for a, b ∈ [n] with ua(Aa) < ua(Ab), ∃o ∈ Aa ∪ Ab: ua(Aa \ {o}) ≥ ua(Ab \ {o}).

Envy-freeness up to any item (EFX)

A is EFX if, for a, b ∈ [n], (1) ∀o ∈ Aa, ua(o) < 0: ua(Aa \ {o}) ≥ ua(Ab) and (2) ∀o ∈ Ab, ua(o) > 0: ua(Aa) ≥ ua(Ab \ {o}). EFX ⇒ EF1

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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EF13 & EFX3

Good allocation: A+ = (A+

1 , . . . , A+ n ) s.t. A+ j = {o ∈ A|uj(o) > 0}

Bad allocation: A− = (A−

1 , . . . , A− n ) s.t. A− j = {o ∈ A|uj(o) < 0}

Goodsj = Agent j O = Badsj = EFX3 requires that A is EFX, A+ is EFX and A− is EFX EF13 requires that A is EF1, A+ is EF1 and A− is EF1 EFX3 ⇒ EFX EF13 ⇒ EF1 EFX3 ⇒ EF13

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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Motivation

Alice 1 1

• 1
• 1

Bob 1 1 1

• 1
• 1

Mary 1 1

• 1
• 1

For EF1 (EFX) and PO, A gives , , to Bob, to Alice and to Mary. A also maximizes the egalitarian welfare. Problem: But, Bob has to do both chores! EF13 (EFX3) avoids this. For EF13 (EFX3) and PO, B gives , to Bob, , to Alice and to Mary.

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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The Modified Double Round-Robin (MDRR) Algorithm

The DRR algorithm2.

• 1. DRR is not PO with −1/0/1!
• 2. MDRR is PO with −α/0/β!
• 2ACIW. Fair allocation of indivisible goods and chores. IJCAI-19, 53–59.

(TU Berlin) Additive and Fair Division of Mixed Manna

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General additive utilities (i.e. ua(o) ∈ R)

Theorem 1

MDDR returns an EF13 allocation in O(max{m2, mn}) time.

Proposition 1

There are problems where no allocation is EFX3. a b c agent 1 −1 −1 2 agent 2 −1 −1 2 By contradiction: To achieve EFX3, we must share a, b among 1,2. Say, 1 gets a, c and 2 gets b. This is not EFX: u2(∅) = 0 < 1 = u2({a, c}).

(TU Berlin) Additive and Fair Division of Mixed Manna

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Additive utilities with identical magnitudes

utilities with identical magnitudes a b c agent 1 10 1 −4 agent 2 −10 −1 4 identical utilities (e.g. money) a b c agent 1 10 1 4 agent 2 10 1 4

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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The Minimax Algorithm

(TU Berlin) Additive and Fair Division of Mixed Manna

• M. Aleksandrov and T. Walsh

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Additive utilities with identical magnitudes

Theorem 2

MDRR returns an EF13 and PO allocation in O(max{m2, mn}) time.

Theorem 3

Minimax returns an EFX and PO allocation in O(max{m log m, mn}) time.

Corollary 1

With identical utilities, Minimax returns an EFX and PO allocation. Note: With identical utilities, the “egal-sequential” algorithm3.

3Aziz, H., Rey, S.: Almost group envy-free allocation of indivisible goods and chores.

IJCAI-20. 3–45.

(TU Berlin) Additive and Fair Division of Mixed Manna M. Aleksandrov and T. Walsh 10 / 12

Ternary additive utilities (i.e. ua(o) ∈ {−α, 0, β})

Theorem 4

MDRR returns an EF13 and PO allocation in O(max{m2, mn}) time.

Theorem 5

Minimax returns an EFX and PO allocation in O(max{m log m, mn}) time.

Theorem 6

With −α/0/α utilities, MDRR returns an EFX3 and PO allocation. Note: With −α/0/α utilities, the “ternary-flow” algorithm4.

4Aziz, H., Rey, S.: Almost group envy-free allocation of indivisible goods and chores.

IJCAI-20. 3–45.

(TU Berlin) Additive and Fair Division of Mixed Manna M. Aleksandrov and T. Walsh 11 / 12

Summary

property general ident.mag. −α/0/β −α/0/α utilities utilities utilities utilities EF13 , A1, P (T1) EF13 & PO

• pen

, A1, P (T2) , A1, P (T4) EFX & PO

• pen

, A2, P (T3) , A2, P (T5) EFX3 × (P1) EFX3 & PO , A1, P (T6) Table: Key: -possible, ×-not possible, P-polynomial time, α, β ∈ R>0 : α = β, A1-The Modified Double Round-Robin Algorithm, A2-The Minimax Algorithm.

Thank you!

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