Asymptotic Existence of Fair Divisions for Groups
aa r X i v : . [ c s . G T ] J u l Asymptotic Existence of Fair Divisions forGroups
Pasin ManurangsiUC Berkeley Warut SuksompongStanford University
The problem of dividing resources fairly occurs in many practical situationsand is therefore an important topic of study in economics. In this paper, weinvestigate envy-free divisions in the setting where there are multiple playersin each interested party. While all players in a party share the same setof resources, each player has her own preferences. Under additive valuationsdrawn randomly from probability distributions, we show that when all groupscontain an equal number of players, a welfare-maximizing allocation is likelyto be envy-free if the number of items exceeds the total number of players bya logarithmic factor. On the other hand, an envy-free allocation is unlikelyto exist if the number of items is less than the total number of players. Inaddition, we show that a simple truthful mechanism, namely the randomassignment mechanism, yields an allocation that satisfies the weaker notionof approximate envy-freeness with high probability.
1. Introduction
Dividing resources among interested parties in a fair manner is a problem that commonlyoccurs in real-world situations and is consequently of fundamental importance. Countriesnegotiate over international issues, as Egypt and Israel did in 1978 over interests inthe Sinai Peninsula [9] and the U.S. and Panama in 1994 over those in the PanamaCanal. Likewise, divorced couples negotiate over their marital property, airlines overflight routes, and Internet clients over bandwidth and storage space. On a smallerscale, typical everyday tasks involving fair division include distributing household tasks,splitting a taxi fare, and sharing apartment rent [15]. Given its far-reaching and oftencritical applications, it should not come as a surprise that fair division has long been apopular topic of study in economics [12, 14, 22, 23, 28].To reason about fair division, we must carefully define what we mean for a division tobe “fair”. Several notions of fairness have been proposed in the literature. For example,a division is said to be proportional if every player values her allocation at least 1 /n timesher value for the whole set of items, where n denotes the number of players. Another1ommonly-used notion of fairness, and one that we will work with throughout the paper,is that of envy-freeness . A division of a set of items is said to be envy-free if each playervalues the set of items that she receives at least as much as the set of items that any otherplayer receives. When utilities are additive, an envy-free division is proportional andmoreover satisfies another important fairness notion called the maximin share criterion . While procedures for finding envy-free divisions have been proposed [7, 8, 16], an envy-free division does not necessarily exist in arbitrary settings. This can most easily be seenin the case where there are two players and one item that both players value positively,or more generally when the number of players exceeds the number of items and playershave positive values for items.The fair division literature has for the most part assumed that each interested partyconsists of a single player (or equivalently, several players represented by a single commonpreference). However, this assumption is too restrictive for many practical situations.Indeed, an outcome of a negotiation between countries may have to be approved bymembers of the cabinets of each country. Since valuations of outcomes are usuallysubjective, one can easily imagine a situation in which one member of a cabinet of acountry thinks that the division is fair while another member disagrees. Similarly, in adivorce case, different members of the family on the husband side and the wife side mayhave varying opinions on a proposed settlement. Another example is a large companyor university that needs to divide its resources among competing groups of agents (e.g.,departments in a university), since the agents in each group can have misaligned interests.Indeed, the professors who perform theoretical research may prefer more whiteboardsand open space in the department building, while those who engage in experimentalwork are more likely to prefer laboratories.In this paper, we study envy-free divisions when there are multiple players in eachgroup. Every player has her own preferences, and players in the same group can havevery different preferences. In this generalized setting, we consider a division to be envy-free if every player values the set of items assigned to her group at least as much as thatassigned to any other group.
In Section 3, we investigate the asymptotic existence and non-existence of envy-freedivisions using a probabilistic model, previously used in the setting with one player pergroup [11]. We show that under additive valuations and other mild technical conditions,when all groups contain an equal number of players, an envy-free division is likely to existif the number of goods exceeds the total number of players by a logarithmic factor, nomatter whether the players are distributed into several groups of small size or few groupsof large size (Theorem 1). In particular, any allocation that maximizes social welfareis likely to be envy-free. In addition, when there are two groups with possibly unequalnumbers of players and the distribution on the valuation of each item is symmetric, anenvy-free division is likely to exist if the number of goods exceeds the total number of See, e.g., [6, 10] for the definition of the maximin share criterion. α -approximate envy-free with high probabilityfor any constant α ∈ [0 ,
1) (Theorem 4). Approximate envy-freeness means that eventhough a player may envy another player in the resulting division, the values of theplayer for her own allocation and for the other player’s allocation differ by no morethan a multiplicative factor of α . In other words, the player’s envy is relatively smallcompared to her value for her own allocation. The number of items required to obtainapproximate envy-freeness with high probability increases as we increase α . Our resultshows that it is possible to achieve truthfulness and approximate envy-freeness simulta-neously in a wide range of random instances, and improves upon the previous result forthe setting with one player per group [2] in several ways. Our results in Section 3 can be viewed as generalizations of previous results by Dickersonet al. [11], who showed asymptotic existence and non-existence under a similar modelbut in a more limited setting where each group has only one player. In particular, theseauthors proved that under certain technical conditions on the probability distributions,an allocation that maximizes social welfare is envy-free with high probability if thenumber of items is larger than the number of players by a logarithmic factor. In fact,their result also holds when the number of players stay constant, as long as the numberof items goes to infinity. Similarly, we show that a welfare-maximizing allocation is likelyto be envy-free if the number of items exceeds the number of players by a logarithmicfactor. While we require that the number of player per group goes to infinity, the numberof groups can stay small, even constant. On the non-existence front, Dickerson et al.showed that if the utility for each item is independent and identically distributed across3layers, then envy-free allocations are unlikely to exist when the number of items is largerthan the number of players by a linear fraction. On the other hand, our non-existenceresults apply to the regime where the number of items is smaller than the number ofplayers. Note that while this regime is uninteresting in Dickerson et al.’s setting sinceenvy-free allocations cannot exist, in our generalized setting an envy-free allocation canalready exist when the number of items is at least the number of groups .Besides the asymptotic results on envy-free divisions, results of this type have alsobeen shown for other fairness notions, including proportionality and the maximin sharecriterion. These two notions are weaker than envy-freeness when utilities are additive.Suksompong [25] showed that proportional allocations exist with high probability if thenumber of goods is a multiple of the number of players or if the number of goods growsasymptotically faster than the number of players. Kurokawa et al. [17] showed thatif either the number of players or the number of items goes to infinity, then an alloca-tion satisfying the maximin share criterion is likely to exist as long as each probabilitydistribution has at least constant variance. Amanatidis et al. [3] analyzed the rate ofconvergence for the existence of allocations satisfying the maximin share criterion whenthe utilities are drawn from the uniform distribution over the unit interval. Anothercommon approach for circumventing the potential nonexistence of divisions satisfyingcertain fairness concepts, which we do not discuss in this paper, is by showing approxima-tion guarantees for worst-case instances [3, 4, 18, 21, 26, 27]. In particular, Suksompong[26] investigated approximation guarantees for groups of agents using the maximin sharecriterion.Finally, a model was recently introduced that incorporates the element of resource allo-cation for groups [19, 24]. The model concerns the problem of finding a small “agreeable”subset of items, i.e., a small subset of items that a group of players simultaneously pre-fer to its complement. Nevertheless, in that model the preferences of only one groupof players are taken into account, whereas in our work we consider the preferences ofmultiple groups at the same time.
2. Preliminaries
Let a set N of n := gn ′ players be divided into g ≥ G , . . . , G g of n ′ playerseach, and let M := { , , . . . , m } denote the set of items. Let g ( i ) be the index of thegroup containing player i (i.e., player i belongs to the group G g ( i ) ). Each item willbe assigned to exactly one group, to be shared among the members of the group. Weassume that each player i ∈ N has a cardinal utility u i ( j ) for each item j ∈ M . Wemay suppose without loss of generality that u i ( j ) ∈ [0 , additive , i.e., u i ( M ′ ) = P j ∈ M ′ u i ( j ) for any player i ∈ N and any subset ofitems M ′ ⊆ M . The social welfare of an assignment is the sum of the utilities of all n players from the assignment.We are now ready to define the notion of envy-freeness. Denote the subsets of itemsthat are assigned to the g groups by M , . . . , M g , respectively.4 efinition 1. Player i in group G g ( i ) regards her allocation M g ( i ) as envy-free if u i ( M g ( i ) ) ≥ u i ( M j ) for every group G j = G g ( i ) . The assignment of the subsets M , . . . , M g to the g groups is called envy-free if every player regards her allocation as envy-free. Next, we list two probabilistic results that will be used in our proofs. We first statethe Chernoff bound, which gives us an upper bound on the probability that a sum ofindependent random variables is far away from its expected value.
Lemma 1 (Chernoff bound) . Let X , . . . , X r be independent random variables that arebounded in an interval [0 , , and let S := X + · · · + X r . We have Pr[ S ≥ (1 + δ ) E [ S ]] ≤ exp (cid:18) − δ E [ S ]3 (cid:19) , and, Pr[ S ≤ (1 − δ ) E [ S ]] ≤ exp (cid:18) − δ E [ S ]2 (cid:19) for every δ ≥ . Another lemma that we will use is the Berry-Esseen theorem. In short, it states thata sum of a sufficiently large number of independent random variables behaves similarlyto a normal distribution. On the surface, this sounds like the central limit theorem.However, the Berry-Esseen theorem relies on a slightly stronger assumption and deliversa more concrete bound, which is required for our purposes.
Lemma 2 (Berry-Esseen theorem [5, 13]) . Let X , . . . , X r be r independent and iden-tically distributed random variables, each of which has mean µ , variance σ , and thirdmoment ρ . Let S := X + · · · + X r . There exists an absolute constant C BE such that (cid:12)(cid:12)(cid:12)(cid:12) Pr[ S ≤ x ] − Pr y ∼N ( µr,σ r ) [ y ≤ x ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρC BE σ √ r for every x ∈ R . Note that N ( µr, σ r ) is the normal distribution with mean µr andvariance σ r , i.e., its probability density function is f ( x ) = 1 σ √ πr e − ( x − µr )22 σ r . Let us now state two assumptions on distributions of utilities; in Section 3 we willwork with the first and in Section 4 with the second. [A1]
For each item j ∈ M , the utilities u i ( j ) ∈ [0 ,
1] for i ∈ N are drawn indepen-dently at random from a distribution D j . Each distribution D j is non-atomic , i.e.,Pr[ u i ( j ) = x ] = 0 for every x ∈ [0 , σ min > D , . . . , D m are at least σ min . The third moment of a random variable X is defined as E [ | X − E [ X ] | ]. A2]
For each i ∈ N and j ∈ M , the utility u i ( j ) ∈ [0 ,
1] is drawn independentlyat random from a probability distribution D i,j . The mean of each distribution isbounded away from zero, i.e., there exists a constant µ min > E [ u i ( j )] ≥ µ min for every i ∈ N, j ∈ M .Note that assumption [A2] is weaker than [A1]. Indeed, in [A2] we do not require D i,j to be the same for every i . In addition, since u i ( j ) ∈ [0 ,
1] for all i ∈ N and j ∈ M , wehave E [ u i ( j )] ≥ E [ u i ( j ) ] ≥ E [ u i ( j ) ] − E [ u i ( j )] = Var( u i ( j )) . Hence, the condition thatthe means of the distributions are bounded away from zero follows from the analogouscondition on the variances.In Section 4, we consider the notion of approximate envy-freeness , which means thatfor each player, there is no allocation of another group for which the player’s utility isa certain (multiplicative) factor larger than the utility of the player for the allocation ofher own group. The notion is defined formally below.
Definition 2.
We write M p % αi M q for α ∈ [0 , if and only if u i ( M p ) ≥ αu i ( M q ) .Player i considers an assignment M , . . . , M g of items to the g groups α -approximateenvy-free if M g ( i ) % αi M p for every group p ∈ { , . . . , g } . We say that an assignment is α -approximate envy-free if it is α -approximate envy-free for every player i . Finally, we give the definition of a truthful mechanism, which we will use in Section 4.
Definition 3. A mechanism is a function that takes as input the utility of player i foritem j for all i ∈ N and j ∈ M , and outputs a (possibly random) assignment of items tothe groups. A mechanism is said to be truthful if every player always obtains the highestpossible (expected) utility by submitting her true utilities to the mechanism, regardless ofthe utilities that the remaining players submit.
3. Asymptotic Existence and Non-Existence of Fair Divisions
In this section, we study the existence and non-existence of fair divisions. First, weshow that, when m is Ω( n log n ), where Ω( · ) hides a sufficiently large constant, thereexists an envy-free division with high probability (Theorem 1). In particular, we provethat a welfare-maximizing allocation is likely to be envy-free. This gives rise to a simplealgorithm that finds such a fair division with high probability. We also extend ourexistence result to the case where there are two groups but the groups need not havethe same number of players; we show a similar result in this case, provided that eachdistribution D j satisfies an additional symmetry condition (Theorem 2).Moreover, on the non-existence front, we prove that when m is smaller than n , theprobability that a fair division exists is at most 1 /g n − m (Theorem 3). This has asconsequences that if the number of items is less than the total number of players by asuperconstant factor, or if the number of items is less than the total number of playersand the number of groups is large, then the probability that an envy-free division existsis low (Corollaries 1 and 2).We begin with our main existence result.6 heorem 1. Assume that [A1] holds. For any fixed σ min > , there exists a constant C > such that, for any sufficiently large n ′ , if m > Cn log n , then there exists anenvy-free assignment with high probability. In fact, we not only prove that an envy-free assignment exists but also give a simplegreedy algorithm that finds one such assignment with high probability. The algorithmidea is simple; we greedily assign each item to the group that maximizes the total utilityof the item with respect to the players in that group. This yields an allocation thatmaximizes social welfare. The allocation is therefore Pareto optimal, i.e., there existsno other allocation in which every player is weakly better off and at least one player isstrictly better off. The pseudocode of the algorithm is shown below.
Algorithm 1
Greedy Assignment Algorithm for Multiple Groups procedure Greedy–Assignment–Multiple let M = · · · = M g = ∅ . for each item j ∈ M do choose k ∗ from arg max k =1 ,...,g P p ∈ G k u p ( j ) let M k ∗ ← M k ∗ ∪ { j } end for end procedure The analysis of the algorithm contains similarities to that of the corresponding resultin the setting with one player per group [11]. However, significantly more technical carewill be required to handle our setting in which each group contains multiple players.This is reflected by our use of the Berry-Esseen theorem (Lemma 2). Here we providea proof sketch that contains all the high-level ideas but leaves out some tedious details,especially calculations; the full proof can be found in the appendix.
Proof sketch of Theorem 1.
We will first bound Pr[ M g ′ ≻ i M g ( i ) ] for each player i andeach group G g ′ = G g ( i ) ; we then use the union bound at the end to conclude Theorem 1.To bound Pr[ M g ′ ≻ i M g ( i ) ], we define a random variable A i,j to be u i ( j ) if item j isassigned to group G g ( i ) and zero otherwise. Similarly, define B g ′ i,j to be u i ( j ) if the itemis assigned to group G g ′ and zero otherwise.Intuitively, with respect to player i , A i,j is the utility contribution of item j to thegroup G g ( i ) . On the other hand, B g ′ i,j is the utility that is “lost” to group G g ′ . Inother words, M g ′ ≻ i M g ( i ) if and only if S A < S B , where S A = P j ∈ M A i,j and S B = P j ∈ M B g ′ i,j . We will use the Chernoff bound to estimate the probability of this event.To do so, we first need to bound E [ A i,j ] and E [ B g ′ i,j ].From symmetry between different groups, the probability that item j is assignedto each group is 1 /g . Thus, we have E [ A i,j ] = g E (cid:2) u i ( j ) | item j is assigned to G g ( i ) (cid:3) and E [ B g ′ i,j ] = g E (cid:2) u i ( j ) | item j is assigned to G g ′ (cid:3) . It is now fairly easy to see that E [ B g ′ i,j ] ≤ µ j /g , where µ j is the mean of D j ; the reason is that the expected value of7 i ( j ) when j is not assigned to G g ( i ) is clearly at most µ j . For convenience, we willassume in this proof sketch that E [ B g ′ i,j ] is roughly µ j /g .Now, we will bound the expected value of A i,j . For each p = 1 , . . . , g , let X p denotethe sum of the utilities of item j with respect to all players in G p . Due to symmetryamong players within the same group, we have E [ A i,j ] = 1 n ′ g E (cid:2) X g ( i ) | X g ( i ) = max { X , . . . , X g } (cid:3) = 1 n ′ g E [max { X , . . . , X g } ] . The latter equality comes from the symmetry between different groups.Now, we use the Berry-Esseen theorem (Lemma 2), which tells us that each of X , . . . , X g is close to N ( µ j n ′ , Ω( σ min n ′ )). With simple calculations, one can see that the expec-tation of the maximum of g identically independent random variables sampled from N ( µ j n ′ , Ω( σ min n ′ )) is µ j n ′ + Ω( σ min √ n ′ ). Roughly speaking, we also have E [ A i,j ] = µ j g + Ω (cid:18) σ min g √ n ′ (cid:19) . Having bounded the expectations of A i,j and B g ′ i,j , we are ready to apply the Chernoffbound. Let δ = Θ (cid:16) σ min µ j √ n ′ (cid:17) where Θ( · ) hides some sufficiently small constant. When n ′ is sufficiently large, we can see that (1 + δ ) E [ B g ′ i,j ] < (1 − δ ) E [ A i,j ], which implies that(1 + δ ) E [ S B ] < (1 − δ ) E [ S A ]. Using the Chernoff bound (Lemma 1) on S A and S B , wehave Pr[ S A ≤ (1 − δ ) E [ S A ]] ≤ exp (cid:18) − δ E [ S A ]2 (cid:19) , and, Pr[ S B ≥ (1 + δ ) E [ S B ]] ≤ exp (cid:18) − δ E [ S B ]3 (cid:19) . Thus, we have Pr[ S A < S B ] ≤ exp (cid:18) − δ E [ S A ]2 (cid:19) + exp (cid:18) − δ E [ S B ]3 (cid:19) ≤ (cid:18) − Ω (cid:18) σ min mnµ j (cid:19)(cid:19) (Since µ j ≤ ≤ (cid:18) − Ω (cid:18) σ min mn (cid:19)(cid:19) . Recall that Pr[ M g ′ ≻ i M g ( i ) ] = Pr[ S A < S B ] . Using the union bound for all i and all g ′ = g ( i ), the probability that the assignment output by the algorithm is not envy-freeis at most 2 n ( g −
1) exp (cid:18) − Ω (cid:18) σ min mn (cid:19)(cid:19) , /m when m ≥ Cn log n for some sufficiently large C . This completesthe proof sketch for the theorem.Unfortunately, the algorithm in Theorem 1 cannot be extended to give a proof for thecase where the groups do not have the same number of players. However, in a morerestricted setting where there are only two groups with potentially different numbers ofplayers and an additional symmetry condition on D , . . . , D m is enforced, a result similarto that in Theorem 1 can be shown, as stated in the theorem below. Theorem 2.
Assume that [A1] holds. Suppose that there are only two groups but notnecessarily with the same number of players; let n , n denote the numbers of players ofthe first and second group respectively (so n = n + n ). Assume also that D , . . . , D m are symmetric (around / ) , i.e., Pr X ∼D j (cid:20) X ≤ − x (cid:21) = Pr X ∼D j (cid:20) X ≥
12 + x (cid:21) for all x ∈ [0 , / . For any fixed σ min > , there exists a constant C > such that,for any sufficiently large n and n , if m > Cn log n , then there exists an envy-freeassignment with high probability. The algorithm is similar to that in Theorem 1; the only difference is that, insteadof assigning each item to the group with the highest total utility over its players, weassign the item to the group with the highest average utility, as seen in the pseudocodeof Algorithm 2.
Algorithm 2
Greedy Assignment Algorithm for Two Possibly Unequal-Sized Groups procedure Greedy–Assignment–Two let M = M = ∅ . for each item j ∈ M do choose k ∗ from arg max k =1 , P p ∈ Gk u p ( j ) n k let M k ∗ ← M k ∗ ∪ { j } end for end procedure The proof is essentially the same as that of Theorem 1 after the random variablesdefined are changed corresponding to the modification in the algorithm. For instance, A i,j is now defined as A i,j = u i ( j ) · " g ( i ) = arg max k =1 , P p ∈ G k u p ( j ) n k where [ E ] denotes an indicator variable for event E . There is nothing special about the number 1 /
2; a similar result holds if the distributions are supportedon a subset of an interval [ a, b ] and are symmetric around ( a + b ) /
2, for some 0 < a < b . Proposition 1.
Let X and X denote P p ∈ G u p ( j ) /n and P p ∈ G u p ( j ) /n respec-tively. Then, Pr[ X ≥ X ] = 12 . Proof.
To show this, observe first that, since D j is symmetric over 1 /
2, the distributionsof X and X are also symmetric over 1 /
2. Let f and f be the probability densityfunctions of X and X respectively, we havePr[ X ≥ X ] = Z Z x f ( x ) f ( y ) dydx = Z Z x f (1 − x ) f (1 − y ) dydx = Z Z x f ( x ) f ( y ) dydx = Pr[ X ≥ X ] . Hence, Pr[ X ≥ X ] = Pr[ X ≥ X ] = 1 /
2, as desired.Next, we state and prove an upper bound for the probability that an envy-free as-signment exists when the number of players exceeds the number of items. Such anassignment obviously does not exist under this condition if every group contains onlyone player. In fact, the theorem holds even without the assumption that the variancesof D , . . . , D m are at least σ min > Theorem 3.
Assume that [A1] holds. If m < n , then there exists an envy-free assign-ment with probability at most /g n − m .Proof. Suppose that m ≤ n −
1, and fix an assignment M , . . . , M g . We will bound theprobability that this assignment is envy-free. Consider any player i in group G g ( i ) . Theprobability that the assignment is envy-free for this particular player is the probabilitythat the total utility of the player for the bundle M g ( i ) is no less than that for otherbundles M j . This can be written as follows:Pr u i (1) ∈D ,...,u i ( m ) ∈D m X l ∈ M g ( i ) u i ( l ) = max k =1 ,...,g X l ∈ M k u i ( l ) . For each j = 1 , . . . , g , define p j as p j = Pr x ∈D ,...,x m ∈D m X l ∈ M j x l = max k =1 ,...,g X l ∈ M k x l . i is p g ( i ) .Since u i (1) , . . . , u i ( m ) is chosen independently of u i ′ (1) , . . . , u i ′ ( m ), for every i ′ = i ,the probability that this assignment is envy-free for every player is simply the productof the probability that the assignment is envy-free for each player, i.e., n Y i =1 p g ( i ) = g Y j =1 p n ′ j . Using the inequality of arithmetic and geometric means, we arrive at the followingbound: g Y j =1 p n ′ j ≤ g g X j =1 p j n ′ g . Recall our assumption that the distributions D j are non-atomic. Hence we may assumethat the events P l ∈ M j x l = max k =1 ,...,g P l ∈ M k x l are disjoint for different j . This impliesthat P gj =1 p j = 1. Thus, the probability that this fixed assignment is envy-free is atmost g g X j =1 p j n ′ g = (cid:18) g (cid:19) n ′ g = 1 g n . Finally, since each assignment is envy-free with probability at most 1 /g n and there are g m possible assignments, by union bound the probability that there exists an envy-freeassignment is at most 1 /g n − m . This completes the proof of the theorem.The following corollaries can be immediately derived from Theorem 3. They say thatan envy-free allocation is unlikely to exist when the number of items is less than thenumber of players by a superconstant factor, or when the number of items is less thanthe number of players and the number of groups is large. Corollary 1.
Assume that [A1] holds. When m = n − ω (1) , the probability that thereexists an envy-free assignment converges to zero as n → ∞ . Corollary 2.
Assume that [A1] holds. When m < n , the probability that there exists anenvy-free assignment converges to zero as g → ∞ .
4. Truthful Mechanism for Approximate Envy-Freeness
While the algorithms in Section 3 translate to mechanisms that yield with high proba-bility envy-free divisions that are compatible with social welfare assuming that playersare truth-telling, the resulting mechanisms suffer from the setback that they are easilymanipulable. Indeed, since they aim to maximize (total or average) welfare, strategicplayers will declare their values for the items to be high, regardless of what the actual11alues are. This presents a significant disadvantage: Implementing these mechanisms inmost practical situations, where we do not know the true valuations of the players andhave no reason to assume that they will reveal their valuations in a honest manner, canlead to potentially undesirable outcomes.In this section, we work with the weaker notion of approximate envy-freeness andshow that a simple truthful mechanism yields an approximately envy-free assignmentwith high probability. In particular, we prove that the random assignment mechanism,which assigns each item to a player chosen uniformly and independently at random, islikely to produce such an assignment. In the setting where each group consists of onlyone player, Amanatidis et al. [2] showed that when the distribution is as above and thenumber of items m is large enough compared to n , the random assignment mechanismyields an approximately envy-free assignment with high probability. Our statement isan analogous statement for the case where each group can have multiple players. Theorem 4.
Assume that [A2] holds. For every α ∈ [0 , , there exists a constant C depending only on α and µ min such that, if m > Cg log n , then the random assignment,where each item j ∈ M is assigned independently and uniformly at random to a group,is α -approximate envy-free with high probability. Before we prove Theorem 4, we note some ways in which our result is stronger thanthat of Amanatidis et al.’s apart from the fact that multiple players per group are allowedin our setting. First, Amanatidis et al. required D i,j to be the same for all j , whichwe do not assume here. Next, they only showed that the random assignment is likelyto be approximately proportional , a weaker notion that is implied by approximate envy-freeness. Moreover, in their result, m needs to be as high as Ω( n ), whereas in our case itsuffices for m to be in the range Ω( g log n ). Finally, we also derive a stronger probabilisticbound; they showed a “success probability” of the algorithm of 1 − O ( n /m ), while oursuccess probability is 1 − exp( − Ω( m/g )). Proof of Theorem 4.
For each player i ∈ N , each item j ∈ M and each p ∈ { , . . . , g } , let A pi,j be a random variable representing the contribution of item j ’s utility with respectto player i to group G p , i.e., A pi,j is u i ( j ) if item j is assigned to group G p and is zerootherwise.Define S pi := P j ∈ M A pi,j . Observe that each player i considers the assignment to be α -approximate envy-free if and only if S g ( i ) i ≥ αS pi for every p . Let δ = − α α ; from thischoice of δ and since E [ S pi ] is equal for every p , we can conclude that S g ( i ) i ≥ αS pi isimplied by S g ( i ) i ≥ (1 − δ ) E [ S g ( i ) i ] and S pi ≤ (1 + δ ) E [ S pi ]. In other words, we can boundthe probability that the random assignment is not α -approximate envy-free as follows.Pr[ ∃ i ∈ N, p ∈ { , . . . , g } : S g ( i ) i < αS pi ] ≤ X i ∈ N,p ∈{ ,...,g } Pr[ S g ( i ) i < αS pi ] ≤ X i ∈ N,p ∈{ ,...,g } Pr[ S g ( i ) i < (1 − δ ) E [ S g ( i ) i ] ∨ S pi > (1 + δ ) E [ S g ( i ) i ]]12 X i ∈ N,p ∈{ ,...,g } (Pr[ S g ( i ) i < (1 − δ ) E [ S g ( i ) i ]] + Pr[ S pi > (1 + δ ) E [ S pi ]]) . Since S pi = P j ∈ M A pi,j and A pi,j ’s are independent and lie in [0 , α -approximate envy-free is at most X i ∈ N,p ∈{ , ··· ,g } exp − δ E [ S g ( i ) i ]]2 ! + exp (cid:18) − δ E [ S pi ]]3 (cid:19) . Finally, observe that E [ S pi ] = X j ∈ M E [ A pi,j ] = X j ∈ M g E [ u i ( j )] ≥ mµ min g . This means that the desired probability is bounded above by X i ∈ N,p ∈{ ,...,g } exp (cid:18) − δ mµ min g (cid:19) + exp (cid:18) − δ mµ min g (cid:19) ≤ ng exp (cid:18) − δ mµ min g (cid:19) ≤ exp (cid:18) − δ mµ min g + 3 log n (cid:19) . When m > (cid:16) µ min δ (cid:17) g log n , the above expression is at most exp( − Ω( m/g )), conclud-ing our proof.
5. Concluding Remarks
In this paper, we study a generalized setting for fair division that allows interested partiesto contain multiple players, possibly with highly differing preferences. This settingallows us to model several real-world cases of fair division that cannot be done under thetraditional setting. We establish almost-tight bounds on the number of players and itemsunder which a fair division is likely or unlikely to exist. Furthermore, we consider theissue of truthfulness and show that a simple truthful mechanism produces an assignmentthat is approximately envy-free with high probability.While the assumptions of additivity and independence are somewhat restrictive andmight not apply fully to settings in the real world, our results give indications as to whatwe can expect if the assumptions are relaxed, such as if a certain degree of dependenceis introduced. An interesting future direction is to generalize the results to settings withmore general valuations. In particular, if the utility functions are low-degree polynomials,then one could try applying the invariance principle [20], which is a generalization of theBerry-Esseen theorem that we use. 13e end the paper with some questions that remain after this work. A natural questionis whether we can generalize our existence and non-existence results (Theorems 1 and 3)to the setting where the groups do not contain the same number of players. This non-symmetry between the groups seems to complicate the approaches that we use in thispaper. For example, it breaks the greedy algorithm used in Theorem 1. Nevertheless, itmight still be possible to prove existence of an envy-free division using other algorithmsor without relying on a specific algorithm.Another direction for future research is to invent procedures for computing envy-freedivisions, whenever such divisions exist, for the general setting where each group containsmultiple players and players have arbitrary (not necessarily additive) preferences. Evenprocedures that only depend on rankings of single items [8] do not appear to extendeasily to this setting. Indeed, if a group contains two players whose preferences areopposite of each other, it is not immediately clear what we should assign to the group.It would be useful to have a procedure that produces a desirable outcome, even for asmall number of players in each group.Lastly, one could explore the limitations that arise when we impose the condition oftruthfulness, an important property when we implement the mechanisms in practice.For instance, truthful allocation mechanisms have recently been characterized in thecase of two players [1], and it has been shown that there is a separation between truthfuland non-truthful mechanisms for approximating maximin shares [2]. In our setting, anegative result on the existence of a truthful mechanism that yields an envy-free divisionwith high probability would provide such a separation as well, while a positive result inthis direction would have even more consequences for practical applications.
Acknowledgments
The authors thank the anonymous reviewers for their helpful feedback. Warut Suksom-pong acknowledges support from a Stanford Graduate Fellowship.
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A. Appendix
A.1. Proof of Theorem 1
First we list the following well-known fact, which allows us to easily determine the meanof a random variable from its cumulative density function.16 roposition 2.
Let X be a non-negative random variable. Then E [ X ] = Z ∞ Pr[ X ≥ x ] dx. To analyze the algorithm, consider any player i and any group g ′ = g ( i ). We will firstbound the probability that M g ′ ≻ i M g ( i ) . To do this, for each item j ∈ M , define A i,j as A i,j = u i ( j ) · g ( i ) = arg max k =1 ,...,g X p ∈ G k u p ( j ) , where 1 h g ( i ) = arg max k =1 ,...,g P p ∈ G k u p ( j ) i is an indicator random variable that indi-cates whether g ( i ) = arg max k =1 ,...,g P p ∈ G k u p ( j ). Similarly, define B g ′ i,j as B g ′ i,j = u i ( j ) · g ′ = arg max k =1 ,...,g X p ∈ G k u p ( j ) . Moreover, suppose that D j has mean µ j and variance σ j .Notice that, with respect to player i , A i,j is the utility that item j contributes to g ( i ) whereas B g ′ i,j is the utility that item j contributes to g ′ . In other words, M g ′ ≻ i M g ( i ) if and only if P j ∈ M A i,j < P j ∈ M B g ′ i,j . To bound Pr (cid:2) M g ′ ≻ i M g ( i ) (cid:3) , we willfirst bound E [ A i,j ] and E h B g ′ i,j i . Then, we will use the Chernoff bound to boundPr hP j ∈ M A i,j < P j ∈ M B g ′ i,j i .Observe that, due to symmetry, we can conclude that E u i ( j ) · g ′ = arg max k =1 ,...,g X p ∈ G k u p ( j ) = E u i ( j ) · g ′′ = arg max k =1 ,...,g X p ∈ G k u p ( j ) for any g ′′ = g ( i ). Thus, we can now rearrange B g ′ i,j as follows: E h B g ′ i,j i = 1 g − X g ′′ = g ( i ) E u i ( j ) · g ′′ = arg max k =1 ,...,g X p ∈ G k u p ( j ) = 1 g − E u i ( j ) X g ′′ = g ( i ) g ′′ = arg max k =1 ,...,g X p ∈ G k u p ( j ) = 1 g − E u i ( j ) − g ( i ) = arg max k =1 ,...,g X p ∈ G k u p ( j ) . E h B g ′ i,j i = 1 g − µ j − E [ A i,j ]) . (1)Now, consider A i,j . Again, due to symmetry, we have E [ A i,j ] = 1 n ′ X i ′ ∈ G g ( i ) E u i ′ ( j ) · g ( i ) = arg max k =1 ,...,g X p ∈ G k u p ( j ) = 1 n ′ E X i ′ ∈ G g ( i ) u i ′ ( j ) · g ( i ) = arg max k =1 ,...,g X p ∈ G k u p ( j ) . Let S denote the distribution of the sum of n ′ independent random variables, eachdrawn from D j . It is obvious that P p ∈ G k u p ( j ) is drawn from S independently for each k . In other words, E [ A i,j ] can be written as E [ A i,j ] = 1 n ′ E [ X · [ X = max { X , . . . , X g } ]] . The expectation on the right is taken over X , . . . , X g sampled independently from S .From symmetry among X , . . . , X g , we can further derive the following: E [ A i,j ] = 1 n ′ Pr [ X = max { X , . . . , X g } ] E [ X | X = max { X , . . . , X g } ]= 1 n ′ g E [ X | X = max { X , . . . , X g } ]= 1 n ′ g E [max { X , . . . , X g } ] . Consider the distribution of max { X , . . . , X g } . Let us call this distribution Y . Noticethat E [max { X , . . . , X g } ] is just the mean of Y , i.e., E [ A i,j ] = 1 n ′ g E Y ∼Y [ Y ] . (2)To bound this, let F S and F Y be the cumulative density functions of S and Y respec-tively. Notice that F Y ( x ) = F S ( x ) g for all x . Applying Proposition 2 to S and Y yieldsthe following: E S ∼S [ S ] = Z ∞ (1 − F S ( x )) dx, and, E Y ∼Y [ Y ] = Z ∞ (1 − F S ( x ) g ) dx.
18y taking the difference of the two, we have E Y ∼Y [ Y ] = E S ∼S [ S ] + Z ∞ F S ( x ) (cid:0) − F S ( x ) g − (cid:1) dx. To bound the right hand side, recall that S is just the distribution of the sum of n ′ independent random variables sampled according to D j . Note that the third momentof D j is at most 1 because it is bounded in [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F S ( x ) − Pr y ∼N ( µ j n ′ ,σ j n ′ ) [ y ≤ x ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C BE σ j √ n ′ . for all x ∈ R . When n ′ is sufficiently large, the right hand side is at most 0.1.Moreover, it is easy to check that Pr y ∼N ( µ j n ′ ,σ j n ′ ) [ y ≤ x ] ∈ [0 . , .
85] for every x ∈ h µ j n ′ , µ j n ′ + σ j √ n ′ i . Hence, F S ( x ) ∈ [0 . , .
95] for every x ∈ h µ j n ′ , µ j n ′ + σ j √ n ′ i .Now, we can bound E Y ∼Y [ Y ] as follows: E Y ∼Y [ Y ] = E S ∼S [ S ] + Z ∞ F S ( x ) (cid:0) − F S ( x ) g − (cid:1) dx = µ j n ′ + Z ∞ F S ( x ) (cid:0) − F S ( x ) g − (cid:1) dx (Since F S ( x ) (cid:0) − F S ( x ) g − (cid:1) ≥ ≥ µ j n ′ + Z µ j n ′ + σ j √ n ′ µ j n ′ F S ( x ) (cid:0) − F S ( x ) g − (cid:1) dx ≥ µ j n ′ + Z µ j n ′ + σ j √ n ′ µ j n ′ (0 . . dx = µ j n ′ + σ j √ n ′ / σ j ≥ σ min ) ≥ µ j n ′ + σ min √ n ′ / . Plugging the above inequality into equation (2), we can conclude that E [ A i,j ] = 1 n ′ g E Y ∈Y [ Y ] ≥ µ j g + σ min g √ n ′ . From this and equation (1), we have E h B g ′ i,j i = 1 g − µ j − E [ A i,j ]) ≤ g − (cid:18) µ j − µ j g (cid:19) = µ j g . Now, define C g ′ i,j as C g ′ i,j = B g ′ i,j + (cid:16) µ j /g − E h B g ′ i,j i(cid:17) . Notice E h C g ′ i,j i = µ j /g .As stated earlier, M g ′ ≻ i M g ( i ) if and only if P j ∈ M A i,j < P j ∈ M B g ′ i,j . Let S A = P j ∈ M A i,j , S B = P j ∈ M B g ′ i,j , S C = P j ∈ M C g ′ i,j and let δ = σ min µ j √ n ′ . Notice that, since19e assume that the variance of D j is positive, µ j is also non-zero, which means that δ iswell-defined. Using Chernoff bound (Lemma 1) on S A and S C , we havePr[ S A ≤ (1 − δ ) E [ S A ]] ≤ exp (cid:18) − δ E [ S A ]2 (cid:19) , and, Pr[ S C ≥ (1 + δ ) E [ S C ]] ≤ exp (cid:18) − δ E [ S C ]3 (cid:19) . Moreover, when n ′ is large enough, we have (1 − δ ) E [ S A ] ≥ (1 + δ ) E [ S C ]. Thus, wehave Pr[ S A < S C ] ≤ exp (cid:18) − δ E [ S A ]2 (cid:19) + exp (cid:18) − δ E [ S C ]3 (cid:19) ≤ exp (cid:18) − δ mµ j g (cid:19) + exp (cid:18) − δ mµ j g (cid:19) ≤ (cid:18) − σ min m gn ′ µ j (cid:19) (Since µ j ≤ ≤ (cid:18) − σ min m n (cid:19) . Due to how C g ′ i,j is defined, we have Pr[ S A < S C ] ≥ Pr[ S A < S B ] = Pr[ M g ′ ≻ i M g ( i ) ] . Using the union bound for all i and all g ′ = g ( i ), the probability that the assignmentoutput by the algorithm is not envy-free is at most2 n ( g −