Fair allocation of indivisible goods

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