Fair allocation of indivisible goods
|
|
- Theresa Hopkins
- 5 years ago
- Views:
Transcription
1 Fair allocation of indivisible goods Gabriel Sebastian von Conta Technische Universitt Mnchen November 17, 2015 Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
2 Overview 1 Introduction 2 Preferences 3 Fairness x Efficiency 4 Computing allocations 5 Protocols Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
3 Introduction Objects denoted by : O = {o 1,..., o P } Objects can t be broken or divided in pieces Objects can t be shared N = {1,..., n} will be a set of n agents. An allocation is a function π : N 2 O Π mapping each agent to the bundle she receives, such that π(i) π(j) = whenever i j. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
4 Introduction The subset of objects π(i) will be called agent i s bundle (or share). When i N π(i) = O we say that the allocation is complete. a MultiAgent Resource Allocation setting (MARA setting for short) denotes a triple (N, O, R), where N is a finite set of agents, O is a finite set of indivisible and non-shareable objects and R is a sequence of n preference relations on the bundles of O. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
5 Introduction Fairness concepts are sometimes unreachable or really complex to find Can be relaxed permitting fractions of an object using a compensation (money) relaxing the assumption that every good must be alocated relaxing fairness criteria Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
6 Preferences What an agent prefer to get from the allocation Preference representation languages do not really transpose to the divisible case An agent can simply rank the objects that she wishes from the most to the less, showing his Individual preferences. This is a order i that can be either: A linear order i, in ordinal fashion, meaning that we can t know how much an agent i prefers an object o k i o l. A property that is often taken for granted in preference representation is monotonicity: A preference relation on 2 O is monotonic if and only if S S S S. Or a utility function ω i : O F, mapping each object to a score taken from a numerical set ( N, Q or R for sake of simplicity). Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
7 Preferences Unlike voting theory ranking itens generally is not enough. This can either be solved by: Lifting preferences to bundles of objects using natural assumptions Asking each agent to rank not only objects but also the bundles Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
8 Additive preferences Our first approach to represent is based on the property of Modularity A utility function is modular if and only if for each pair of bundles (S, S ) we have u(s S ) = u(s) + u(s ) u(s S ) An equivalent definition : For each bundle S: u(s) = u( ) + o S u({o}) Very strong property that forbids synergy between objects Lifts preferences over single objects to preference relations between bundles of objects of same cardinality. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
9 Additive preferences This can be circumvent by asking the agent to rank each bundle individually. This brings a problem! Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
10 Compact preference representation Using an intermediate language to represent preferences as closely as possible while maintaining compactness. They re defined as a pair L, I (L) that associates to each set of objects: a language L(O). an interpretation I L (O) that maps any well-formed formula ϕ of L(O) to a preorder ϕ of 2 O Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
11 Compact preference representation An trivial example is the bundle form, which is like a explicit representation Made of pairs S, u S, where S is a bundle of objects and u a non-zero numerical weight. The utility of a bundle S is just u S if S, u S belongs to the set, and 0 otherwise. Still yelds higher complexity, scaling to the numble of bundles Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
12 Compact preference representation A matter of representing the interesting preference relations Do the agent really express the comparison between each subset of a 42-objects bundle? Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
13 Compact preference representation k-additive representation language A k additive preference representation language is a set B of pairs S, w S where S O is a bundle of size at most k, and w S is a non-zero numerical weight. The utility of each formula is defined as: u(s) = S,w S B S S, S k Example : O = {o 1, o 2, o 3 }, weights: o 1, 2, o 2, 2, o 1 o 2, 2, o 2 o 3, 5 w S Succinctess of the language ensured by parameter k. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
14 Compact preference representation Graphical Model Two components a graphical component describing directed or undirected dependencies between variables a collection of local statements on single variables or small subsets of variables, compatible with the dependence structure The k-additive representation language can be seen as a generalized additive independence (GAI) representation with no graphical component associated where the size of the local relations (synergies) is explicitly bounded by k. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
15 Compact preference representation CI-NET Another Graphical representation is CP-nets Here, statements describes the agent s ordinal preferences on the values of the variables domain, given all the possible combinations of values of its parents ( CP standing here as Conditional preferences) An extension of this language is CI-nets, especially dedicated to represent ordinal preferences on sets of objects. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
16 Compact preference representation CI-nets Formally,a CI-net N is a set of CI statements (where CI stands for Conditional Importance) of the form S +, S : S1 S2 Informal reading : if I have all the items in S + and none of those in S, I prefer obtaining all items in S1 to obtaining all those in S2, all other things being equal (ceteris paribus). Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
17 Compact preference representation CI-nets Example : Let O = {o 1, o 2, o 3, o 4 } be a set of objects, and let N be the CI-net defined by the two following CI-statements: S1 = (o 1, : o 4 o 2 o 3 ); S2 = (, o 1 : o 2 o 3 o 4 ) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
18 Compact preference representation Logic-based languages Another family of compact representation languages, which is not based on synergies. Based on propositional formulas. Given a set of objects O, we will denote by L O the propositional language built on the propositional operators, and and one propositional variable for each object in O. Each formula ϕ of L O represent a goal that a agent wants to achieve. From a bundle S there is a logical interpretation I (S) that set all propositional variables of an object in S to if it is in S and to otherwise. A bundle S satisfies a goal ϕ(s = ϕ) if and only if I (S) = ϕ Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
19 Compact preference representation Logic-based languages A agent is only happy if a goal is satisfied. Not very subtle, agent can t express between two different objects. This can be extended Adding a goal base, so that the agent can have multiple goals, and we can know which bundle is better by counting how many goals are satisfied. Adding weight to the goals, so that the agent can prioritize it. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
20 Compact preference representation Logic-based languages A formula Weighted logic-based preference representation language is a set of pairs ϕ, w ϕ, where ϕ is a well-formed formula of the propositional language L O, and w ϕ is a non-zero numerical weight. Given a formula in this language, the utility of each bundle S is: u(s) = w ϕ (ϕ,w ϕ) S =ϕ Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
21 Compact preference representation Logic-based languages Example: Let O = {o 1, o 2, o 3 } be a set of objects. The goal = { o 1 o 2, 1, o 2 o 3, 2 } is a compact representation of an utility function Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
22 Compact preference representation Multiagent Resource Allocation Setting With preferences now defined, we can update the definition of MARA setting proposed before. An ordinal MARA setting is a triple N, O, R, where N is a finite set of agents, O is a finite set of objects, and R is a set { 1,..., n } of preorders on 2 O An cardinal MARA setting is defined replacing the set R by a set U = {u 1,..., u n } of utility functions on 2 O Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
23 Fairness x Efficiency First two notions of fairness: maximin allocations and envy-free allocations. Maximin is only defined on cardinal-mara settings, where we need to compare the well-being of agents. { } Defining Maximin : max = min u i(π(i)) π Π i N Defining envy-freeness: π(i) i π(j) for all agents i, j N Efficiency Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
24 Fairness x Efficiency Maxmin allocations Not all maxmin allocations are Pareto-optimal, but at least of them need must be Price of fairness is defined as the ratio between the total utility of the optimal utilitarian allocation over the total utility of the best maxmin optimal allocation. Price of fairness for maxmin allocations is unbounded(caragiannis et al., 2012) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
25 Fairness x Efficiency envy-freeness Trivial case : Empty bundle Is also not necessarily Pareto-efficient Example : O = {o 1, o 2, o 3, o 4 }, u 1 (S) = 1 {o1 o 2 } and u 2 (S) = 1 {o3 o 4 } Don t always exist Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
26 Fairness x Efficiency Proportionality Each agent should get at least the n th of the total utility she would have received if she were alone For normalized utility, becomes maxmin Can t be always found Maxmin share (Budish,2011) Example: 2 agents, Objects {o 1, o 2, o 3, o 4 }, u 1 (o 1 ) = 7, u 1 (o 2 ) = 2, u 3 (o 3 ) = 6, u 4 (o 4 ) = 10. Agent s 1 maxmin share is Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
27 Fairness x Efficiency In the case of additive preferences, Envy-freeness implies proportionality and proportionality implies maxmin share guarantee. (Bouveret and Lematre, 2014) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
28 Computing Fair Allocations Is quite challenging Input include preference profiles encoded in a given representation language If preference profile is represented with a formula that is superpolynomial in p and n, even for easy decision problems, finding a fair allocation remains prohibitive. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
29 Computing Fair Allocations Maxmin allocations Without assumptions, computing an optimal maxmin allocation, and even an approximation, is hard (Golovin, 2005) Argument based on the partition problem, a well known NP-Complete problem Even for basic settings, it is still challenging Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
30 Computing Fair Allocations Santa Claus problem As an example we have the Santa Claus Problem (Bansal and Sviridenko, 2006), based on acardinal-mara setting, with modular utility functions Santa Claus has p gifts to allocate to n children, having modular preferences, try to allocate the gifts so as to maximize the utility of the unhappiest child - Same as maximin allocation - Remains NP-hard (Bezkov and Dani, 2005) Can be solved by a linear program, but then solving the relaxation of it, assuming divisible goods results in a infinite integrality gap(ratio between fractional and integral optimum) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
31 Computing Fair Allocations Computing envy-free or low envy allocations Easy algorithm : Throw everything away! Not efficient It is computationally hard to decide whether an envy-free complete allocation exists (Lipton et al., 2004) Combined with pareto-optimality, problem lies above NP (Bouveret and Lang, 2008) Also in additive domains Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
32 Computing Fair Allocations Computing envy-free or low envy allocations More realistic to minimize the degree of envy of the agents Defined here as by Lipton et al. (2004) e i,j (π) = max { 0, u i (π(j)) u i (π(i)) } is the envy of each agent for other agents e(π) = max{e i,j (π) i, j N} is the envy of the allocation Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
33 Computing Fair Allocations Computing envy-free or low envy allocations Allocations with bounded maximal envy can be obtained by taking the maximal marginal utility, α The marginal utility of a good o j, given an agent i and a bundle S is the amount of additional utility given by this object when added to the bundle The maximal marginal utility is just the maximal value among all objects, agents and bundles In an additive setting, it s just the highest utility an agent gives to an object Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
34 Computing Fair Allocations Computing envy-free or low envy allocations Theorem (Lipton et al., 2004) It is always possible to find an allocation whose envy is bounded by α, the maximal marginal utility of the problem. Create an envy graph associated with an allocation π where nodes are agents and an edge from i to j when i envies j A cycle in the envy graph can be rotated, breaking the envy cycle at some point, since the utility of each agent on the cycle increases at each rotation Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
35 Computing Fair allocations Computing envy-free or low envy allocations Considering the following procedure Allocate goods one by one First one allocated arbitrarily After round k, with k + 1 objects and envy bounded by α, we build the envy graph Rotate bundles as previously described Find agent i whom no-one envies Allocate object o k+1 to agent i Envy is at most α Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
36 Computing Fair allocations Computing envy-free or low envy allocations Example : Let O = {o 1, o 2, o 3, o 4, o 5 } be a set of objects and {1, 2, 3} three agents and following additive preferences S o 1 o 2 o 3 o 4 o 5 u 1 (S) u 2 (S) u 3 (S) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
37 Computing Fair allocations Computing envy-freeness It s not possible to compute a minimal bound of envy. Can be circumvet with the use of minimum envy-ratio max{1, u i (π j ) u i (π i ) } Ordinal notion Possible π(i) i π(j) and Necessary π(j) i π(i) envy (Bouveret et al., 2010) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
38 Protocols for Fair allocation Until now we only saw centralized approaches Two major drawbacks (i) Elicitation process can be really expensive or agents my not want to reveal fully their preferences (ii) Agents may not accept a solution computed as a black-box About one, as we already saw, if the preference is not modular, the communication load to compute fairness criteria is a barrier Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
39 Protocols for Fair allocation The adjusted winner procedure Winning phase Each good goes to the agent who values it the most Now, either u 1 = u 2 and we re done There is an agent r, the richest and an agent p the poorest Adjusting phase transfers goods from richest to poorest, in increasing order of the ratio u r (o) u p(o) Stops when u r = u p or the richest agent becomes the poorest This happened due to an object g Split g to attain the same utility for both agents, giving the richest u p(g)+u p((π(p)\{g}) u r (π(r)\{g}) u r (g+u p(g)) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
40 Protocols for fair allocation The adjusted Winner procedure Theorem (Brams and Taylor, 2000) The adjusted winner procedure returns an equitable, envy-free, and Pareto-optimal allocation. Example : O = {o 1, o 2, o 3, o 4, o 5 } be a set of objects. S o 1 o 2 o 3 o 4 o 5 u 1 (S) u 2 (S) Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
41 Protocols for fair allocation The undercut procedure Guarantees to find an envy-free allocation among two agents, whenever one exists Take as input an ordinal information(ranking of items) That allow us to rank only some bundles, for example if o 1 o 2 o 3, we don t know if o 1 o 2 o 3, but know that o 1 o 2 o 1 o 3 Composed of two phases Generation phase Each agent name their preferred item, if the items are different, they are allocated, if not, the item goes to a contested pile. Iterates until there is no other item. By now each agent values their bundle more than the bundle of the other agent We now need to find a split of the contested pile that results in a envy-free allocation Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
42 Protocols for fair allocation The undercut procedure A minimal bundle for an agent i is a bundle of items that is worth at least 50% of the full set of items(we say that the bundle is envy-free(ef) to agent i and it is not possible to find another bundle less preferred to it which would also be EF The protocol then chooses a minimal bundle as a proposal (lets assume S 1 for agent 1) Agent 2 can either accept the proposal, complement it, or take undercut the proposal, modifying the proposed split Theorem ((Brams et al., 2012)) If agents differ on at least one minimal bundle, then an envy-free allocation exists and the undercut protocol returns it. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
43 Protocols for fair allocation The undercut procedure Example(Brams et al.,2012): Agents with same preference o 1 o 2 o 3 o 4 o 5 All items go to contested pile Agent 1 minimal bundle: o 1 o 2 Agent 2 minimal bundle: o 3 o 4 o 5 minimal bundles differ, there must be a envy-free location. Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
44 What else is out there? Other representations : bidding languages Other fairness measures Protocols for more than two agents Avoiding agents manipulation Different agent priority Agents entering allocation sequentially Further restrictions Gabriel Sebastian von Conta (TUM) Chapter 12 November 17, / 44
The efficiency of fair division
The efficiency of fair division Ioannis Caragiannis Christos Kaklamanis Panagiotis Kanellopoulos Maria Kyropoulou Research Academic Computer Technology Institute and Department of Computer Engineering
More informationLecture 12. Applications to Algorithmic Game Theory & Computational Social Choice. CSC Allan Borodin & Nisarg Shah 1
Lecture 12 Applications to Algorithmic Game Theory & Computational Social Choice CSC2420 - Allan Borodin & Nisarg Shah 1 A little bit of game theory CSC2420 - Allan Borodin & Nisarg Shah 2 Recap: Yao s
More informationCardinal and Ordinal Preferences. Introduction to Logic in Computer Science: Autumn Dinner Plans. Preference Modelling
Cardinal and Ordinal Preferences Introduction to Logic in Computer Science: Autumn 2007 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam A preference structure represents
More information8. Cake cutting. proportionality and envy-freeness. price of fairness. approximate envy-freeness. Cake cutting 8-1
Cake cutting 8-1 8. Cake cutting proportionality and envy-freeness price of fairness approximate envy-freeness Cake cutting model a cake is the interval [0; 1] a peice of cake is X [0; 1] each of n agents
More informationFairness Criteria for Fair Division of Indivisible Goods
Sylvain Bouveret LIG (STeamer) Ensimag / Grenoble INP Michel Lemaître Formerly Onera Toulouse Séminaire de l équipe ROSP du laboratoire G-SCOP Grenoble, March 24, 2016 Introduction A fair division problem...
More informationTwo Desirable Fairness Concepts for Allocation of Indivisible Objects under Ordinal Preferences
Two Desirable Fairness Concepts for Allocation of Indivisible Objects under Ordinal Preferences HARIS AZIZ and SERGE GASPERS and SIMON MACKENZIE and TOBY WALSH Data61 and UNSW Fair allocation of indivisible
More informationIntroduction to Formal Epistemology Lecture 5
Introduction to Formal Epistemology Lecture 5 Eric Pacuit and Rohit Parikh August 17, 2007 Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 5 1 Plan for the Course Introduction,
More informationFair Division of a Graph
Sylvain Bouveret LIG - Grenoble INP, France sylvain.bouveret@imag.fr Fair Division of a Graph Katarína Cechlárová P.J. Šafárik University, Slovakia katarina.cechlarova@upjs.sk Edith Elkind University of
More informationDistributed Optimization. Song Chong EE, KAIST
Distributed Optimization Song Chong EE, KAIST songchong@kaist.edu Dynamic Programming for Path Planning A path-planning problem consists of a weighted directed graph with a set of n nodes N, directed links
More informationFair Allocation of Indivisible Goods: Improvements and Generalizations
Fair Allocation of Indivisible Goods: Improvements and Generalizations MOHAMMAD GHODSI, Sharif University of Technology and Institute for Research in Fundamental Sciences (IPM) School of Computer Science
More informationConditional Importance Networks: A Graphical Language for Representing Ordinal, Monotonic Preferences over Sets of Goods
Conditional Importance Networks: A Graphical Language for Representing Ordinal, Monotonic Preferences over Sets of Goods Sylvain Bouveret ONERA-DTIM Toulouse, France sylvain.bouveret@onera.fr Ulle Endriss
More informationExact and Approximate Equilibria for Optimal Group Network Formation
Exact and Approximate Equilibria for Optimal Group Network Formation Elliot Anshelevich and Bugra Caskurlu Computer Science Department, RPI, 110 8th Street, Troy, NY 12180 {eanshel,caskub}@cs.rpi.edu Abstract.
More informationFair 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
More informationCS 598RM: Algorithmic Game Theory, Spring Practice Exam Solutions
CS 598RM: Algorithmic Game Theory, Spring 2017 1. Answer the following. Practice Exam Solutions Agents 1 and 2 are bargaining over how to split a dollar. Each agent simultaneously demands share he would
More informationFairly Allocating Many Goods with Few Queries
Fairly Allocating Many Goods with Few Queries Hoon Oh Computer Science Department Carnegie Mellon University Ariel D. Procaccia Computer Science Department Carnegie Mellon University Warut Suksompong Department
More informationThe Price to Pay for Forgoing Normalization in Fair Division of Indivisible Goods
The Price to Pay for Forgoing Normalization in Fair Division of Indivisible Goods Pascal Lange, Nhan-Tam Nguyen, and Jörg Rothe Institut für Informatik Heinrich-Heine-Universität Düsseldorf 40225 Düsseldorf,
More informationFair Divsion in Theory and Practice
Fair Divsion in Theory and Practice Ron Cytron (Computer Science) Maggie Penn (Political Science) Lecture 1: Introduction 1 Some things to watch for on these slides Definition 1 definition: A concept that
More informationManipulating picking sequences
ECAI 2014 T. Schaub et al. (Eds.) 2014 The Authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial
More informationPareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints Ayumi Igarashi Kyushu University Fukuoka, Japan igarashi@agent.inf.kyushu-u.ac.jp Dominik Peters University of Oxford Oxford,
More informationOn low-envy truthful allocations
On low-envy truthful allocations Ioannis Caragiannis, Christos Kaklamanis, Panagiotis Kanellopoulos, and Maria Kyropoulou Research Academic Computer Technology Institute and Department of Computer Engineering
More informationComputing Desirable Partitions in Additively Separable Hedonic Games
Computing Desirable Partitions in Additively Separable Hedonic Games Haris Aziz, Felix Brandt, Hans Georg Seedig Institut für Informatik, Technische Universität München, 80538 München, Germany Abstract
More informationControl of Fair Division
Haris Aziz Data61 and UNSW Australia haris.aziz@data61.csiro.au Control of Fair Division Ildikó Schlotter Budapest University of Technology and Economics ildi@cs.bme.hu Toby Walsh Data61 and UNSW Australia
More informationarxiv: v2 [cs.gt] 6 Jun 2017
Sylvain Bouveret LIG - Grenoble INP, France sylvain.bouveret@imag.fr Fair Division of a Graph Katarína Cechlárová P.J. Šafárik University, Slovakia katarina.cechlarova@upjs.sk Edith Elkind University of
More informationMaximin Share Allocations on Cycles
Maximin Share Allocations on Cycles Zbigniew Lonc 1, Miroslaw Truszczynski 2 1 Warsaw University of Technology, Poland 2 University of Kentucky, USA zblonc@mini.pw.edu.pl, mirek@cs.uky.edu Abstract The
More informationWhen Can the Maximin Share Guarantee Be Guaranteed?
When Can the Maximin Share Guarantee Be Guaranteed? David Kurokawa Computer Science Department Carnegie Mellon University Ariel D. Procaccia Computer Science Department Carnegie Mellon University Junxing
More informationManipulating picking sequences
Manipulating picking sequences Sylvain Bouveret and Jérôme Lang Abstract Picking sequences are a natural way of allocating indivisible items to agents in a decentralized manner: at each stage, a designated
More informationFair Public Decision Making
1 Fair Public Decision Making VINCENT CONITZER, Duke University RUPERT FREEMAN, Duke University NISARG SHAH, Harvard University We generalize the classic problem of fairly allocating indivisible goods
More informationFair Assignment of Indivisible Objects under Ordinal Preferences
Fair Assignment of Indivisible Objects under Ordinal Preferences Haris Aziz, Serge Gaspers, Simon Mackenzie, Toby Walsh NICTA and University of New South Wales, Sydney 2052, Australia arxiv:1312.6546v4
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
12. LOCAL SEARCH gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley h ttp://www.cs.princeton.edu/~wayne/kleinberg-tardos
More informationWhen Can the Maximin Share Guarantee Be Guaranteed?
When Can the Maximin Share Guarantee Be Guaranteed? David Kurokawa Ariel D. Procaccia Junxing Wang Abstract The fairness notion of maximin share (MMS) guarantee underlies a deployed algorithm for allocating
More informationLecture 6: Communication Complexity of Auctions
Algorithmic Game Theory October 13, 2008 Lecture 6: Communication Complexity of Auctions Lecturer: Sébastien Lahaie Scribe: Rajat Dixit, Sébastien Lahaie In this lecture we examine the amount of communication
More information1 Submodular functions
CS 369P: Polyhedral techniques in combinatorial optimization Instructor: Jan Vondrák Lecture date: November 16, 2010 1 Submodular functions We have already encountered submodular functions. Let s recall
More informationAlgorithmic Game Theory Introduction to Mechanism Design
Algorithmic Game Theory Introduction to Mechanism Design Makis Arsenis National Technical University of Athens April 216 Makis Arsenis (NTUA) AGT April 216 1 / 41 Outline 1 Social Choice Social Choice
More informationThe Adjusted Winner Procedure : Characterizations and Equilibria
The Adjusted Winner Procedure : Characterizations and Equilibria Simina Brânzei Aarhus University, Denmark Joint with Haris Aziz, Aris Filos-Ratsikas, and Søren Frederiksen Background Adjusted Winner:
More informationSanta Claus Schedules Jobs on Unrelated Machines
Santa Claus Schedules Jobs on Unrelated Machines Ola Svensson (osven@kth.se) Royal Institute of Technology - KTH Stockholm, Sweden March 22, 2011 arxiv:1011.1168v2 [cs.ds] 21 Mar 2011 Abstract One of the
More informationPareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints Ayumi Igarashi Kyushu University Fukuoka, Japan igarashi@agent.inf.kyushu-u.ac.jp Dominik Peters University of Oxford Oxford,
More informationCO759: Algorithmic Game Theory Spring 2015
CO759: Algorithmic Game Theory Spring 2015 Instructor: Chaitanya Swamy Assignment 1 Due: By Jun 25, 2015 You may use anything proved in class directly. I will maintain a FAQ about the assignment on the
More informationExact and Approximate Equilibria for Optimal Group Network Formation
Noname manuscript No. will be inserted by the editor) Exact and Approximate Equilibria for Optimal Group Network Formation Elliot Anshelevich Bugra Caskurlu Received: December 2009 / Accepted: Abstract
More informationEx-Ante Stable Lotteries
Ex-Ante Stable Lotteries Jan Christoph Schlegel Faculty of Business and Economics, University of Lausanne, Switzerland jschlege@unil.ch Abstract We study the allocation of indivisible objects (e.g. school
More informationAn Equivalence result in School Choice
An Equivalence result in School Choice Jay Sethuraman May 2009 Abstract The main result of the paper is a proof of the equivalence of single and multiple lottery mechanisms for the problem of allocating
More informationApproximation Algorithms and Mechanism Design for Minimax Approval Voting
Approximation Algorithms and Mechanism Design for Minimax Approval Voting Ioannis Caragiannis RACTI & Department of Computer Engineering and Informatics University of Patras, Greece caragian@ceid.upatras.gr
More informationLecture 2: Connecting the Three Models
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 2: Connecting the Three Models David Mix Barrington and Alexis Maciel July 18, 2000
More informationApproximation Algorithms and Mechanism Design for Minimax Approval Voting 1
Approximation Algorithms and Mechanism Design for Minimax Approval Voting 1 Ioannis Caragiannis, Dimitris Kalaitzis, and Evangelos Markakis Abstract We consider approval voting elections in which each
More informationMulticlass Classification-1
CS 446 Machine Learning Fall 2016 Oct 27, 2016 Multiclass Classification Professor: Dan Roth Scribe: C. Cheng Overview Binary to multiclass Multiclass SVM Constraint classification 1 Introduction Multiclass
More informationA New Ex-Ante Efficiency Criterion and Implications for the Probabilistic Serial Mechanism
A New Ex-Ante Efficiency Criterion and Implications for the Probabilistic Serial Mechanism Battal Doğan Serhat Doğan Kemal Yıldız First Draft: September 2014 Current Draft: May 10, 2016 Abstract For probabilistic
More informationCMPSCI 611 Advanced Algorithms Midterm Exam Fall 2015
NAME: CMPSCI 611 Advanced Algorithms Midterm Exam Fall 015 A. McGregor 1 October 015 DIRECTIONS: Do not turn over the page until you are told to do so. This is a closed book exam. No communicating with
More information34.1 Polynomial time. Abstract problems
< Day Day Up > 34.1 Polynomial time We begin our study of NP-completeness by formalizing our notion of polynomial-time solvable problems. These problems are generally regarded as tractable, but for philosophical,
More informationCS 781 Lecture 9 March 10, 2011 Topics: Local Search and Optimization Metropolis Algorithm Greedy Optimization Hopfield Networks Max Cut Problem Nash
CS 781 Lecture 9 March 10, 2011 Topics: Local Search and Optimization Metropolis Algorithm Greedy Optimization Hopfield Networks Max Cut Problem Nash Equilibrium Price of Stability Coping With NP-Hardness
More informationAhlswede Khachatrian Theorems: Weighted, Infinite, and Hamming
Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Yuval Filmus April 4, 2017 Abstract The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of
More informationCORE AND NO-TREAT EQUILIBRIUM IN TOURNAMENT GAMES WITH EXTERNALITIES
CORE AND NO-TREAT EQUILIBRIUM IN TOURNAMENT GAMES WITH EXTERNALITIES RYO MIZUNO Abstract. We consider a situation where coalitions are formed to divide a resource. As in real life, the value of a payoff
More informationFundamentals of Operations Research. Prof. G. Srinivasan. Indian Institute of Technology Madras. Lecture No. # 15
Fundamentals of Operations Research Prof. G. Srinivasan Indian Institute of Technology Madras Lecture No. # 15 Transportation Problem - Other Issues Assignment Problem - Introduction In the last lecture
More informationAlgorithmic Game Theory and Applications
Algorithmic Game Theory and Applications Lecture 18: Auctions and Mechanism Design II: a little social choice theory, the VCG Mechanism, and Market Equilibria Kousha Etessami Reminder: Food for Thought:
More informationGame Theory: Spring 2017
Game Theory: Spring 2017 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today In this second lecture on mechanism design we are going to generalise
More informationA Note on Object Allocation under Lexicographic Preferences
A Note on Object Allocation under Lexicographic Preferences Daniela Saban and Jay Sethuraman March 7, 2014 Abstract We consider the problem of allocating m objects to n agents. Each agent has unit demand,
More informationOptimal Proportional Cake Cutting with Connected Pieces
Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence Optimal Proportional Cake Cutting with Connected Pieces Xiaohui Bei Institute for Interdisciplinary Information Sciences Tsinghua
More informationarxiv: v1 [cs.gt] 10 Apr 2018
Individual and Group Stability in Neutral Restrictions of Hedonic Games Warut Suksompong Department of Computer Science, Stanford University 353 Serra Mall, Stanford, CA 94305, USA warut@cs.stanford.edu
More informationGame Theory: Spring 2017
Game Theory: Spring 2017 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today The coming four lectures are about cooperative game theory, where
More informationP, NP, NP-Complete, and NPhard
P, NP, NP-Complete, and NPhard Problems Zhenjiang Li 21/09/2011 Outline Algorithm time complicity P and NP problems NP-Complete and NP-Hard problems Algorithm time complicity Outline What is this course
More informationChapter 4: Computation tree logic
INFOF412 Formal verification of computer systems Chapter 4: Computation tree logic Mickael Randour Formal Methods and Verification group Computer Science Department, ULB March 2017 1 CTL: a specification
More informationarxiv: v1 [cs.gt] 6 Apr 2016
Efficiency and Sequenceability in Fair Division of Indivisible Goods with Additive Preferences Sylvain Bouveret and Michel Lemaître arxiv:1604.01734v1 [cs.gt] 6 Apr 2016 Abstract In fair division of indivisible
More informationPossible and Necessary Allocations via Sequential Mechanisms
Possible and Necessary Allocations via Sequential Mechanisms Haris Aziz, Toby Walsh, and Lirong Xia 1 NICTA and UNSW, Sydney, NSW 2033, Australia haris.aziz@nicta.com.au 2 NICTA and UNSW, Sydney, NSW 2033,
More informationStable and Pareto optimal group activity selection from ordinal preferences
Int J Game Theory (2018) 47:1183 1209 https://doiorg/101007/s00182-018-0612-3 ORIGINAL PAPER Stable and Pareto optimal group activity selection from ordinal preferences Andreas Darmann 1 Accepted: 21 January
More informationPreliminaries. Introduction to EF-games. Inexpressivity results for first-order logic. Normal forms for first-order logic
Introduction to EF-games Inexpressivity results for first-order logic Normal forms for first-order logic Algorithms and complexity for specific classes of structures General complexity bounds Preliminaries
More informationExistence of Nash Networks in One-Way Flow Models
Existence of Nash Networks in One-Way Flow Models pascal billand a, christophe bravard a, sudipta sarangi b a CREUSET, Jean Monnet University, Saint-Etienne, France. email: pascal.billand@univ-st-etienne.fr
More informationProof Techniques (Review of Math 271)
Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
Coping With NP-hardness Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you re unlikely to find poly-time algorithm. Must sacrifice one of three desired features. Solve
More informationQuestion 1. (p p) (x(p, w ) x(p, w)) 0. with strict inequality if x(p, w) x(p, w ).
University of California, Davis Date: August 24, 2017 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Please answer any three
More informationNo-envy in Queueing Problems
No-envy in Queueing Problems Youngsub Chun School of Economics Seoul National University Seoul 151-742, Korea and Department of Economics University of Rochester Rochester, NY 14627, USA E-mail: ychun@plaza.snu.ac.kr
More information3.7 Cutting plane methods
3.7 Cutting plane methods Generic ILP problem min{ c t x : x X = {x Z n + : Ax b} } with m n matrix A and n 1 vector b of rationals. According to Meyer s theorem: There exists an ideal formulation: conv(x
More informationarxiv: v1 [cs.gt] 30 May 2017
Truthful Allocation Mechanisms Without Payments: Characterization and Implications on Fairness * Georgios Amanatidis Georgios Birmpas George Christodoulou Evangelos Markakis arxiv:705.0706v [cs.gt] 30
More informationCIS 2033 Lecture 5, Fall
CIS 2033 Lecture 5, Fall 2016 1 Instructor: David Dobor September 13, 2016 1 Supplemental reading from Dekking s textbook: Chapter2, 3. We mentioned at the beginning of this class that calculus was a prerequisite
More information2 A 3-Person Discrete Envy-Free Protocol
Envy-Free Discrete Protocols Exposition by William Gasarch 1 Introduction Whenever we say something like Alice has a piece worth α we mean it s worth α TO HER. The term biggest piece means most valuable
More informationChapter VI. Relations. Assumptions are the termites of relationships. Henry Winkler
Chapter VI Relations Assumptions are the termites of relationships. Henry Winkler Studying relationships between objects can yield important information about the objects themselves. In the real numbers,
More informationMultiagent Systems Motivation. Multiagent Systems Terminology Basics Shapley value Representation. 10.
Multiagent Systems July 2, 2014 10. Coalition Formation Multiagent Systems 10. Coalition Formation B. Nebel, C. Becker-Asano, S. Wöl Albert-Ludwigs-Universität Freiburg July 2, 2014 10.1 Motivation 10.2
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationGraphical Hedonic Games of Bounded Treewidth
Graphical Hedonic Games of Bounded Treewidth Dominik Peters Department of Computer Science University of Oxford, UK dominik.peters@cs.ox.ac.uk Abstract Hedonic games are a well-studied model of coalition
More informationQuery and Computational Complexity of Combinatorial Auctions
Query and Computational Complexity of Combinatorial Auctions Jan Vondrák IBM Almaden Research Center San Jose, CA Algorithmic Frontiers, EPFL, June 2012 Jan Vondrák (IBM Almaden) Combinatorial auctions
More informationLecture 18: P & NP. Revised, May 1, CLRS, pp
Lecture 18: P & NP Revised, May 1, 2003 CLRS, pp.966-982 The course so far: techniques for designing efficient algorithms, e.g., divide-and-conquer, dynamic-programming, greedy-algorithms. What happens
More informationarxiv: v1 [cs.ds] 2 Jan 2009
On Allocating Goods to Maximize Fairness Deeparnab Chakrabarty Julia Chuzhoy Sanjeev Khanna February 18, 2013 arxiv:0901.0205v1 [cs.ds] 2 Jan 2009 Abstract Given a set A of m agents and a set I of n items,
More informationOptimal Auctions with Correlated Bidders are Easy
Optimal Auctions with Correlated Bidders are Easy Shahar Dobzinski Department of Computer Science Cornell Unversity shahar@cs.cornell.edu Robert Kleinberg Department of Computer Science Cornell Unversity
More informationOn Maximizing Welfare when Utility Functions are Subadditive
On Maximizing Welfare when Utility Functions are Subadditive Uriel Feige October 8, 2007 Abstract We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming
princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Matt Weinberg Scribe: Sanjeev Arora One of the running themes in this course is
More informationOptimal Reallocation under Additive and Ordinal Preferences
Optimal Reallocation under Additive and Ordinal Preferences Haris Aziz, Péter Biró, Jérôme Lang, Julien Lesca, and Jérôme Monnot Abstract Reallocating resources to get mutually beneficial outcomes is a
More informationKidney Exchange with Immunosuppressants
Kidney Exchange with Immunosuppressants Youngsub Chun Eun Jeong Heo Sunghoon Hong October 2, 2015 Abstract This paper investigates implications of introducing immunosuppressants (or suppressants) in kidney
More informationAssigning indivisible and categorized items
Assigning indivisible and categorized items Lirong Xia Computer Science Department Rensselaer Polytechnic Institute 110 8th Street, Troy, NY 12180, USA xial@cs.rpi.edu Abstract In this paper, we study
More informationarxiv: v3 [cs.gt] 11 Nov 2017
Almost Envy-Freeness with General Valuations Benjamin Plaut Tim Roughgarden arxiv:1707.04769v3 [cs.gt] 11 Nov 2017 {bplaut, tim}@cs.stanford.edu Stanford University Abstract The goal of fair division is
More informationA Polynomial-Time Algorithm for Pliable Index Coding
1 A Polynomial-Time Algorithm for Pliable Index Coding Linqi Song and Christina Fragouli arxiv:1610.06845v [cs.it] 9 Aug 017 Abstract In pliable index coding, we consider a server with m messages and n
More informationThe Consumer, the Firm, and an Economy
Andrew McLennan October 28, 2014 Economics 7250 Advanced Mathematical Techniques for Economics Second Semester 2014 Lecture 15 The Consumer, the Firm, and an Economy I. Introduction A. The material discussed
More informationVoting on Multiattribute Domains with Cyclic Preferential Dependencies
Voting on Multiattribute Domains with Cyclic Preferential Dependencies Lirong Xia Department of Computer Science Duke University Durham, NC 27708, USA lxia@cs.duke.edu Vincent Conitzer Department of Computer
More information1 Primals and Duals: Zero Sum Games
CS 124 Section #11 Zero Sum Games; NP Completeness 4/15/17 1 Primals and Duals: Zero Sum Games We can represent various situations of conflict in life in terms of matrix games. For example, the game shown
More informationOn Regular and Approximately Fair Allocations of Indivisible Goods
On Regular and Approximately Fair Allocations of Indivisible Goods Diodato Ferraioli DIAG Sapienza Università di Roma 00185 Rome, Italy ferraioli@dis.uniroma1.it Laurent Gourvès Jérôme Monnot CNRS, LAMSADE
More informationProbabilistic Models. Models describe how (a portion of) the world works
Probabilistic Models Models describe how (a portion of) the world works Models are always simplifications May not account for every variable May not account for all interactions between variables All models
More informationCooperative Games. M2 ISI Systèmes MultiAgents. Stéphane Airiau LAMSADE
Cooperative Games M2 ISI 2015 2016 Systèmes MultiAgents Stéphane Airiau LAMSADE M2 ISI 2015 2016 Systèmes MultiAgents (Stéphane Airiau) Cooperative Games 1 Why study coalitional games? Cooperative games
More informationOn the Complexity of Computing an Equilibrium in Combinatorial Auctions
On the Complexity of Computing an Equilibrium in Combinatorial Auctions Shahar Dobzinski Hu Fu Robert Kleinberg April 8, 2014 Abstract We study combinatorial auctions where each item is sold separately
More informationPareto Optimality in Coalition Formation
Pareto Optimality in Coalition Formation Haris Aziz Felix Brandt Paul Harrenstein Department of Informatics Technische Universität München 85748 Garching bei München, Germany {aziz,brandtf,harrenst}@in.tum.de
More informationLecture notes 2: Applications
Lecture notes 2: Applications Vincent Conitzer In this set of notes, we will discuss a number of problems of interest to computer scientists where linear/integer programming can be fruitfully applied.
More informationTHE REPRESENTATION THEORY, GEOMETRY, AND COMBINATORICS OF BRANCHED COVERS
THE REPRESENTATION THEORY, GEOMETRY, AND COMBINATORICS OF BRANCHED COVERS BRIAN OSSERMAN Abstract. The study of branched covers of the Riemann sphere has connections to many fields. We recall the classical
More informationUpper and Lower Bounds on the Number of Faults. a System Can Withstand Without Repairs. Cambridge, MA 02139
Upper and Lower Bounds on the Number of Faults a System Can Withstand Without Repairs Michel Goemans y Nancy Lynch z Isaac Saias x Laboratory for Computer Science Massachusetts Institute of Technology
More informationEssential facts about NP-completeness:
CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions
More informationA solution to the random assignment problem over the complete domain
A solution to the random assignment problem over the complete domain Akshay-Kumar Katta Jay Sethuraman January 2004 Abstract We consider the problem of allocating a set of indivisible objects to agents
More information