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Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods Aurlie Beynier, Nicolas Maudet, Simon Rey, Parham Shams LIP6, Sorbonne Universit, Paris, France Sylvain Bouveret LIG, Univ. GrenobleAlpes, Grenoble,

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Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

Aurélie Beynier, Nicolas Maudet, Simon Rey, Parham Shams

LIP6, Sorbonne Université, Paris, France

Sylvain Bouveret

LIG, Univ. Grenoble–Alpes, Grenoble, France

Michel Lemaître

Formerly Onera, Toulouse, France

18th Int. Conf. on Autonomous Agents and Multiagent Systems Montreal, Canada, 15th – 17th May, 2019

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Introduction

Fair division of indivisible goods...

2 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 3

Introduction

Fair division of indivisible goods...

2 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 4

Introduction

Fair division of indivisible goods...

2 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 5

Introduction

Fair division of indivisible goods...

2 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 6

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 7

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 8

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure.

> the rest > > the rest > the rest > > > the rest 3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure.

, 3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 10

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 11

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 12

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 13

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 14

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 15

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 16

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 19

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 20

Introduction

How to solve this problem

1 Ask the agents to give their preferences and use a (centralized) collective decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences.

In this work, we try to reconcile these approaches.

3 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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The setting

Fair division of indivisible goods

More formally, we have:

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 22

The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m}

  • 1
  • 2
  • 3

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m} a finite set of agents A = {1, . . . , n}

  • 1
  • 2
  • 3

agent 1 agent 2

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m} a finite set of agents A = {1, . . . , n} Additive preferences: → wi(j) (agent i, object j) .

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m} a finite set of agents A = {1, . . . , n} Additive preferences: → wi(j) (agent i, object j) → ui(X) =

j∈X wi(j).

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 u2({2, 3}) = 1 + 6 = 7

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 26

The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m} a finite set of agents A = {1, . . . , n} Additive preferences: → wi(j) (agent i, object j) → ui(X) =

j∈X wi(j).

We want:

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 27

The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m} a finite set of agents A = {1, . . . , n} Additive preferences: → wi(j) (agent i, object j) → ui(X) =

j∈X wi(j).

We want:

a complete allocation − → π : A → 2O...

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 − → π = {1}, {2, 3}

u1(− → π ) = 5 u2(− → π ) = 7

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 28

The setting

Fair division of indivisible goods

More formally, we have:

a finite set of objects O = {1, . . . , m} a finite set of agents A = {1, . . . , n} Additive preferences: → wi(j) (agent i, object j) → ui(X) =

j∈X wi(j).

We want:

a complete allocation − → π : A → 2O... ...which takes into account the agents’ preferences.

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 − → π = {1}, {2, 3}

u1(− → π ) = 5 u2(− → π ) = 7

4 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Part I

Sequences of sincere choices (aka picking sequences)

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A Sequential Protocol

Sequences of sincere choices

A simple protocol:

1 fix a sequence of agents σ 2 ask the agents to pick in turn their preferred object

Studied a lot. See among others:

Bouveret, S. and Lang, J. (2011).

A general elicitation-free protocol for allocating indivisible goods. In Walsh, T., editor, Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI-11), pages 73–78, Barcelona, Spain. IJCAI/AAAI.

Brams, S. J. and Taylor, A. D. (2000).

The Win-win Solution. Guaranteeing Fair Shares to Everybody.

  • W. W. Norton & Company.

Kohler, D. A. and Chandrasekaran, R. (1971).

A class of sequential games. Operations Research, 19(2):270–277. 6 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Allocations and sequences

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agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

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A Sequential Protocol

Allocations and sequences

  • 1
  • 2
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agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

Step 1 : agent 1 chooses o1

7 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Allocations and sequences

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

Step 1 : agent 1 chooses o1

7 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Allocations and sequences

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

Step 1 : agent 1 chooses o1 Step 2 : agent 2 chooses o3

7 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 35

A Sequential Protocol

Allocations and sequences

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

Step 1 : agent 1 chooses o1 Step 2 : agent 2 chooses o3

7 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 36

A Sequential Protocol

Allocations and sequences

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

Step 1 : agent 1 chooses o1 Step 2 : agent 2 chooses o3 Step 3 : agent 2 chooses o2

7 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Allocations and sequences

  • 1
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agent 1 5 4 2 agent 2 4 1 6 Sequence σ = 1, 2, 2

Step 1 : agent 1 chooses o1 Step 2 : agent 2 chooses o3 Step 3 : agent 2 chooses o2

Final allocation: − → π = {1}, {2, 3}

7 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
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agent 1 5 4 2 agent 2 4 1 6

8 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 First easy observation: {3}, {1, 2} is not sequenceable.

8 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 First easy observation: {3}, {1, 2} is not sequenceable. Why? No agent gets her preferred object!

8 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 First easy observation: {3}, {1, 2} is not sequenceable. Why? No agent gets her preferred object! Question: How can we decide if an allocation is sequenceable or not?

8 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 5 4 2 agent 2 4 1 6 First easy observation: {3}, {1, 2} is not sequenceable. Why? No agent gets her preferred object! Question: How can we decide if an allocation is sequenceable or not? First result: a precise characterization of sequenceable allocations. We can decide in time O(N × M2) if an allocation is sequenceable.

8 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 4 2 5 agent 2 2 1 8

9 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 4 2 5 agent 2 2 1 8 Second easy observation: {3}, {1, 2} is sequenceable but not Pareto-efficient.

9 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 4 2 5 agent 2 2 1 8 Second easy observation: {3}, {1, 2} is sequenceable but not Pareto-efficient. Why? It is dominated by {1}, {2, 3}

9 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

Properties of sequential allocations

  • 1
  • 2
  • 3

agent 1 4 2 5 agent 2 2 1 8 Second easy observation: {3}, {1, 2} is sequenceable but not Pareto-efficient. Why? It is dominated by {1}, {2, 3} However... Second result: Every Pareto-efficient allocation is sequenceable. But Pareto-efficiency ⇔ sequenceability.

9 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

A scale of efficiency

Second result: Every Pareto-efficient allocation is sequenceable. But Pareto-efficiency ⇔ sequenceability.

10 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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A Sequential Protocol

A scale of efficiency

Second result: Every Pareto-efficient allocation is sequenceable. But Pareto-efficiency ⇔ sequenceability. A scale of efficiency...

weaker stronger

Pareto-efficient Sequenceable Non Pareto-efficient Non sequenceable

10 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Part II

Negotiation...

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Cycle deals

Trading as an allocation procedure

Another allocation procedure.

Start from an initial allocation Let the agents trade objects

A particular kind of trading scheme: trading cycles

Sandholm, T. W. (1998).

Contract types for satisficing task allocation: I. theoretical results. In Sen, S., editor, Proceedings of the AAAI Spring Symposium: Satisficing Models, pages 68–75, Menlo Park, California. AAAI Press.

Shapley, L. and Scarf, H. (1974).

On cores and indivisibility. Journal of mathematical economics, 1(1):23–37. 12 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Trading cycles

13 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Trading cycles

13 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 53

Cycle deals

Trading cycles

13 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Trading cycles

(N, M)-cycle deal:

N: cycle length M: max number of objects involved in each trade

(in the example above, N = 4 and M = 1)

13 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Trading cycles

(N, M)-cycle deal:

N: cycle length M: max number of objects involved in each trade

(in the example above, N = 4 and M = 1) Interesting deals: improving deals.

13 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Trading cycles

(N, M)-cycle deal:

N: cycle length M: max number of objects involved in each trade

(in the example above, N = 4 and M = 1) Interesting deals: improving deals. Notion of efficiency: cycle-deal optimality.

13 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Deals and efficiency

OK, so, where is cycle-deal optimality in the scale of efficiency?

weaker stronger

Pareto-efficient Sequenceable Non Pareto-efficient Non sequenceable

14 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 58

Cycle deals

Deals and efficiency

OK, so, where is cycle-deal optimality in the scale of efficiency?

weaker stronger

Pareto-efficient Sequenceable Non Pareto-efficient Non sequenceable

Observations:

− → π Pareto-efficient ⇒ − → π (N, M)-cycle optimal (obvious)

14 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 59

Cycle deals

Deals and efficiency

OK, so, where is cycle-deal optimality in the scale of efficiency?

weaker stronger

Pareto-efficient Sequenceable Non Pareto-efficient Non sequenceable

Observations:

− → π Pareto-efficient ⇒ − → π (N, M)-cycle optimal (obvious) − → π (N, M)-cycle optimal ⇒ − → π (N′, M′)-cycle optimal for any N′ ≤ N and M′ ≤ M (obvious)

14 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 60

Cycle deals

Deals and efficiency

OK, so, where is cycle-deal optimality in the scale of efficiency?

weaker stronger

Pareto-efficient Sequenceable Non Pareto-efficient Non sequenceable

Observations:

− → π Pareto-efficient ⇒ − → π (N, M)-cycle optimal (obvious) − → π (N, M)-cycle optimal ⇒ − → π (N′, M′)-cycle optimal for any N′ ≤ N and M′ ≤ M (obvious)

Third result: − → π (n, 1)-cycle optimal ⇔ − → π sequenceable.

14 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Cycle deals

Deals and efficiency

OK, so, where is cycle-deal optimality in the scale of efficiency?

weaker stronger

Pareto-eff. Sequenceable (n, 1)-cyc-opt (n − 1, 1)-cyc-opt (2, 1)-cyc-opt Ineff.

Observations:

− → π Pareto-efficient ⇒ − → π (N, M)-cycle optimal (obvious) − → π (N, M)-cycle optimal ⇒ − → π (N′, M′)-cycle optimal for any N′ ≤ N and M′ ≤ M (obvious)

Third result: − → π (n, 1)-cycle optimal ⇔ − → π sequenceable.

14 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Conclusion

A taste of the other results

A parallel scale for weakly improving cycles “Complexity” of deals necessary to reach a Pareto-optimal allocation

15 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Conclusion

A taste of the other results

A parallel scale for weakly improving cycles “Complexity” of deals necessary to reach a Pareto-optimal allocation Restricted domains (interesting things happen)

15 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Conclusion

A taste of the other results

A parallel scale for weakly improving cycles “Complexity” of deals necessary to reach a Pareto-optimal allocation Restricted domains (interesting things happen) Link between efficiency and fairness properties:

envy-freeness CEEI in particular: CEEI ⇒ sequenceable, but CEEI ⇒ Pareto-efficient!

15 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 65

Conclusion

A taste of the other results

A parallel scale for weakly improving cycles “Complexity” of deals necessary to reach a Pareto-optimal allocation Restricted domains (interesting things happen) Link between efficiency and fairness properties:

envy-freeness CEEI in particular: CEEI ⇒ sequenceable, but CEEI ⇒ Pareto-efficient!

Experiments

0.1 1 10 100 1000 10000

  • MMS

PFS mMS EF CEEI 0.2 0.4 0.6 0.8 1 NS Swap Seq PO NS (proportion) Swap (proportion) Seq (proportion) PO (proportion) 0.1 1 10 100 1000 10000

  • MMS

PFS mMS EF CEEI 0.2 0.4 0.6 0.8 1 NS Swap Seq PO NS (proportion) Swap (proportion) Seq (proportion) PO (proportion)

15 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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Conclusion

Take-away message

A scale of efficiency that (kind of) reconciles central allocation, distributed allocation, and picking sequences.

16 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 67

Conclusion

Take-away message

A scale of efficiency that (kind of) reconciles central allocation, distributed allocation, and picking sequences.

Inefficient allocation This is really bad: simple trades can improve it

16 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 68

Conclusion

Take-away message

A scale of efficiency that (kind of) reconciles central allocation, distributed allocation, and picking sequences.

Inefficient allocation This is really bad: simple trades can improve it Swap [(2,1)-cycle] optimal The simplest trades cannot improve the allocation

16 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 69

Conclusion

Take-away message

A scale of efficiency that (kind of) reconciles central allocation, distributed allocation, and picking sequences.

Inefficient allocation This is really bad: simple trades can improve it Swap [(2,1)-cycle] optimal The simplest trades cannot improve the allocation . . .

16 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 70

Conclusion

Take-away message

A scale of efficiency that (kind of) reconciles central allocation, distributed allocation, and picking sequences.

Inefficient allocation This is really bad: simple trades can improve it Swap [(2,1)-cycle] optimal The simplest trades cannot improve the allocation . . . Sequenceable / (n, 1)-cycle optimal Almost Pareto-efficient...

16 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 71

Conclusion

Take-away message

A scale of efficiency that (kind of) reconciles central allocation, distributed allocation, and picking sequences.

Inefficient allocation This is really bad: simple trades can improve it Swap [(2,1)-cycle] optimal The simplest trades cannot improve the allocation . . . Sequenceable / (n, 1)-cycle optimal Almost Pareto-efficient... Pareto-efficient The best we can do

16 / 17 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 72

Thank you

Want to see more?

http://recherche.noiraudes.net/en/cycle-deals.php Pictures (shamefully) borrowed without permission from ADN (https://drawthesimpsons.tumblr.com/)

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SLIDE 73

Appendix

CEEI and efficiency

What we already know... Bouveret and Lemaître, 2015 Every CEEI allocation is Pareto-optimal if preferences are strict on shares.

Sylvain Bouveret and Michel Lemaître.

Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259–290, 2016. 18 / 18 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 74

Appendix

CEEI and efficiency

What we already know... Bouveret and Lemaître, 2015 Every CEEI allocation is Pareto-optimal if preferences are strict on shares.

Sylvain Bouveret and Michel Lemaître.

Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259–290, 2016.

No longer true if preferences are not strict on shares.

18 / 18 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 75

Appendix

CEEI and efficiency

What we already know... Bouveret and Lemaître, 2015 Every CEEI allocation is Pareto-optimal if preferences are strict on shares.

Sylvain Bouveret and Michel Lemaître.

Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259–290, 2016.

No longer true if preferences are not strict on shares.    2 3 3 2 2 3 4 1 4 2 4   

18 / 18 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 76

Appendix

CEEI and efficiency

What we already know... Bouveret and Lemaître, 2015 Every CEEI allocation is Pareto-optimal if preferences are strict on shares.

Sylvain Bouveret and Michel Lemaître.

Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259–290, 2016.

No longer true if preferences are not strict on shares.    2 3 3 2 2 3 4 1 4 2 4    Price vector: 0.5, 1, 1, 0.5.

18 / 18 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

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SLIDE 77

Appendix

CEEI and efficiency

What we already know... Bouveret and Lemaître, 2015 Every CEEI allocation is Pareto-optimal if preferences are strict on shares.

Sylvain Bouveret and Michel Lemaître.

Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30(2):259–290, 2016.

No longer true if preferences are not strict on shares.    2 3 3 2 2 3 4 1 4 2 4   

18 / 18 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods