Efficiency, Sequenceability and Deal-Optimality in Fair Division of - PowerPoint PPT Presentation

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

Introduction Fair division of indivisible goods... Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 2 / 17 �

Introduction Fair division of indivisible goods... Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 2 / 17 �

Introduction Fair division of indivisible goods... Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 2 / 17 �

Introduction Fair division of indivisible goods... Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 2 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. > the rest > the rest > > > > the rest > the rest Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. , Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 decision making procedure. 2 Start from a random allocation and ask the agents to negotiate. 3 Use an interactive protocol like picking sequences . Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

Introduction How to solve this problem Ask the agents to give their preferences and use a (centralized) collective 1 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. Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 3 / 17 �

The setting Fair division of indivisible goods More formally, we have: Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

The setting Fair division of indivisible goods More formally, we have: a finite set of objects O = { 1 , . . . , m } o 1 o 2 o 3 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

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 } o 1 o 2 o 3 agent 1 agent 2 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

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: → w i ( j ) (agent i , object j ) . o 1 o 2 o 3 agent 1 5 4 2 agent 2 4 1 6 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

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: → w i ( j ) (agent i , object j ) → u i ( X ) = � j ∈X w i ( j ). o 1 o 2 o 3 u 2 ( { 2 , 3 } ) = 1 + 6 = 7 agent 1 5 4 2 agent 2 4 1 6 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

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: → w i ( j ) (agent i , object j ) → u i ( X ) = � j ∈X w i ( j ). We want: o 1 o 2 o 3 agent 1 5 4 2 agent 2 4 1 6 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

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: → w i ( j ) (agent i , object j ) → u i ( X ) = � j ∈X w i ( j ). We want: a complete allocation − → π : A → 2 O ... o 1 o 2 o 3 − → π = �{ 1 } , { 2 , 3 }� agent 1 5 4 2 u 1 ( − → π ) = 5 agent 2 4 1 6 u 2 ( − → π ) = 7 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

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: → w i ( j ) (agent i , object j ) → u i ( X ) = � j ∈X w i ( j ). We want: a complete allocation − → π : A → 2 O ... ...which takes into account the agents’ preferences. o 1 o 2 o 3 − → π = �{ 1 } , { 2 , 3 }� agent 1 5 4 2 u 1 ( − → π ) = 5 agent 2 4 1 6 u 2 ( − → π ) = 7 Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods 4 / 17 �

Part I Sequences of sincere choices (aka picking sequences)