SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Randomized and Deterministic Maxim-share Approximations for Fractionally Subadditive Valuations
Hannaneh Akrami
Max Planck Institute for Informatics
Graduiertenschule Informatik
Universität des Saarlandes
hakrami@mpi-inf.mpg.de
Masoud Seddighin
Tehran Institute for Advanced Studies
m.seddighin@teias.institute
Kurt Mehlhorn
mehlhorn@mpi-inf.mpg.de
Golnoosh Shahkarami
gshahkar@mpi-inf.mpg.de
We consider the problem of guaranteeing maximin-share (MMS) when allocating a set of indivisible items to a set of agents with fractionally subadditive (XOS) valuations. For XOS valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than  $1/2$  of maximin-share to all the agents. Also, a deterministic allocation exists that guarantees 0.219225 of the maximin-share of each agent. Our results involve both deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to  $3/13 = 0.230769$ . We develop new ideas on allocating large items in our allocation algorithm which might be of independent interest. Furthermore, we investigate randomized algorithms and the Best-of-both-worlds fairness guarantees. We propose a randomized allocation that is  $1/4$ -MMS ex-ante and  $1/8$ -MMS ex-post for XOS valuations. Moreover, we prove an upper bound of  $3/4$  on the ex-ante guarantee for this class of valuations.
Fair allocation has been a central problem in economics for decades. This problem arises naturally in real-world applications such as advertising, negotiation, rent sharing, inheritance, etc (Caragiannis et al. (2019); Dehghani et al. (2018); Dickerson et al. (2014); Foley (1967); Nash Jr (1950); Varian (1973). In discrete fair division, the basic scenario is that we want to distribute a set  $\mathcal{M}$  of  $m$  indivisible items among  $n$  agents, such that the allocation is deemed fair by the agents. Each agent  $i$  has a valuation function  $v_{i}: 2^{\mathcal{M}} \to \mathbb{R}^{+}$  that represents her happiness for receiving a subset of items.
How do we evaluate fairness? This question has been the subject of intense debates in various contexts, including philosophy, economics, distributive justice, and mathematics. Since the introduction of the cake-cutting<sup>1</sup> problem Steinhaus (1948), scientists have suggested several different notions to evaluate fairness. The challenge is that a fairness notion must be both reasonable in terms of justice and implementable in practice. Conceived by Steinhaus, proportionality is one of the most natural
Cake-cutting is the continuous version of the fair allocation problem where the resource is a divisible cake.
and prominent notions. An allocation is proportional if the share allocated to every agent is worth at least  $1 / n$  of her value for the entire resource.
Unfortunately, despite many positive results on proportionality for cake-cutting, simple examples indicate that proportionality is not a proper fairness criterion for the case of indivisible items. For example, when there is one item and two agents, one agent receives nothing, though her proportional share is non-zero assuming that the item has positive value for both of the agents. An alternative form of proportionality adopted to deal with indivisibilities is Maximin-share (MMS). For every agent  $i$ , the maximin-share of agent  $i$ , denoted by  $\mathrm{MMS}_i$  is defined as follows:
$$
\mathsf{MMS}_{i} = \max_{\langle \Pi_{1},\Pi_{2},\dots ,\Pi_{n}\rangle \in \Omega}\min_{1\leq j\leq n}v_{i}(\Pi_{j}),
$$
where  $\Omega$  is the set of all partitions of  $\mathcal{M}$  into  $n$  parts. It is known that even when all the valuations are additive, MMS allocations, in which each agent  $i$  gets at least  $\mathrm{MMS}_i$ , need not exist Feige et al. (2022); Kurokawa et al. (2018); Procaccia and Wang (2014). Nevertheless, several studies in recent years show that it is possible to guarantee a constant factor of maximin-share to all agents for various classes of valuation functions, including additive, submodular, and fractionally subadditive. See Table 1 for the state-of-the-art guarantees on the maximin-share. For Submodular valuation class, a 10/27-MMS allocation can be computed in polynomial time. For all other classes of valuation functions mentioned in Table 1, the states-of-the-art are existential results.
Table 1: A summary of the results for MMS in different valuation classes.
Valuation Class	Approximation Guarantee	Upper bound
Additive	3/4 + 1/3836 Akrami and Garg (2024)	39/40 Feige et al. (2022)
Submodular	10/27 Uziahu and Feige (2023)	3/4 Ghodsi et al. (2018)
Fractionally Subadditive	0.219225 Seddighin and Seddighin (2022)	1/2 Ghodsi et al. (2018)
Subadditive	1/log n log log n Seddighin and Seddighin (2022)	1/2 Ghodsi et al. (2018)
Let us revisit the instance with one item and two agents. Suppose that the item has value 6 for both agents. By definition, the proportional share of each agent is  $6/2 = 3$ , and since one agent receives no item, satisfying proportionality or any approximation of it is impossible. On the other hand, we have  $\mathrm{MMS}_1 = \mathrm{MMS}_2 = 0$ . Thus, allocating the item to any agent satisfies maximin-share. Indeed, we can circumvent the non-guaranteed existence of fair allocation by reducing our expectation of fairness to maximin-share. However, regardless of how we allocate the item, one agent receives one item, and the other receives nothing. Therefore, having one agent with zero utility is inevitable for any deterministic allocation in this example. The question then arises: can we do better? One way to improve the allocation is to use randomization and obtain a better guarantee in expectation (ex-ante). For example, we can allocate the item to each agent with probability  $1/2$ . This way, the expected utility of each agent is equal to 3. In economic terms, this allocation satisfies proportionality ex-ante. Note that one agent receives no item ex-post (that is, after fixing the outcome); however, it guarantees proportionality ex-ante to both agents.
Considering random allocations and ex-ante fairness makes the problem much handier. For instance, assuming there are  $n$  items and  $n$  agents, allocating each item to each agent with probability  $1 / n$  satisfies proportionality ex-ante. However, this randomized allocation has no ex-post fairness guarantee: with a non-zero probability, the outcome allocates all the items to one agent, and the rest of the agents receive no item. It is tempting to find allocations that simultaneously admit ex-ante and ex-post guarantees. The support of such an allocation is limited to outcomes with some desirable fairness guarantee. For example, consider the following random allocation: we choose a random permutation of these  $n$  items and allocate the  $i^{\text{th}}$  item in the permutation to agent  $i$ . This allocation satisfies proportionality ex-ante and maximin-share ex-post.
Recently, several studies have investigated randomized allocations with both ex-ante and ex-post guarantees. Some notable results with the focus on additive valuations are (i) an ex-ante envy-free and ex-post EF1 allocation algorithm Freeman et al. (2020), (ii) an ex-ante proportional and ex-post 1/2-MMS allocation algorithm Babaioff et al. (2022), and (iii) an ex-post 3/4-MMS and ex-ante 0.785-MMS allocation algorithm Akrami et al. (2023b). Very recently, Feldman et al. (2023) studied best-of-both-worlds for subadditive valuations and gave an allocation algorithm with ex-ante guarantee of 1/2-envy-freeness and ex-post guarantee of 1/2-EFX and EF1.
In this paper, our goal is to explore fair deterministic and randomized allocations for fractionally subadditive valuation functions. A valuation function  $v_{i}(\cdot)$  is fractionally subadditive (XOS), if there exists a set of several additive valuation functions  $u_{i,1}, u_{i,2}, \ldots, u_{i,\ell}: 2^{\mathcal{M}} \to \mathbb{R}_{\geq 0}$  such that for every set  $S$  we have  $v_{i}(S) = \max_{1 \leq k \leq \ell} u_{i,k}(S)$ . Fractionally subadditive is a super-class of different set functions such as additive, gross substitute, and submodular. In addition, fractionally subadditive is a subclass of subadditive set functions.
The fairness notion we consider is maximin-share. Indeed, we are looking for allocations that satisfy an approximation of maximin-share both ex-ante and ex-post for fractionally subadditive valuations. Of course, we expect our ex-ante guarantee to be stronger than the ex-post one. Moreover, we improve the ex-post guarantee (deterministically) for this class of valuations. We refer to the next subsection for an overview on our results and techniques.
In this paper, we provide improved guarantees for maximin-share in the fractionally subadditive setting. We investigate both randomized and deterministic allocation algorithms.
Randomized Allocations For randomized allocations, we take one step toward extending the best-of-both-worlds idea for fairness concepts to valuations more general than additive. For the additive setting, Babaioff et al. (2022) proved the existence of randomized allocations that are proportional ex-ante and  $1/2$ -MMS ex-post. However, as we show in Section 3, though guaranteeing proportionality ex-ante is easy for XOS valuations, this notion is not always a proper choice as a fairness criterion. In fact, for some instances, proportionality can be as small as  $O(1/n)$  of the MMS value, which is highly undesirable. Therefore, here we focus on guaranteeing MMS both ex-ante and ex-post. In contrast to the additive setting, for XOS valuations guaranteeing MMS ex-ante is not easy. Recall that in the additive setting, a fractional allocation that allocates a fraction  $1/n$  of each item to each agent is proportional and consequently MMS. However, for some XOS instances this allocation is  $O(1/n)$ -MMS (Observation 3.2). Indeed, we show that there are instances for which guaranteeing MMS ex-ante is not possible. More precisely, we show that there are instances that no randomized allocation can guarantee a factor better than  $3/4$  of maximin-share to all agents (Lemma 3.3). On the positive side, we propose an algorithm that finds a randomized allocation which is  $1/4$ -MMS ex-ante and  $1/8$ -MMS ex-post (Theorem 1.1). The idea to prove the ex-ante approximation guarantee of our allocation is inspired by Ghodsi et al. (2018). In the beginning of the algorithm, as long as there exists a remaining agent  $i$  who likes a remaining item  $b_j$  at least  $1/2$  of her MMS value, we allocate  $b_j$  to  $i$  and remove them from  $\mathcal{M}$  and  $\mathcal{N}$ . By the end of this phase, we get an upper bound on the value of the remaining items for the remaining agents. Let  $\bar{v}_i(\cdot) = \min \left\{\frac{1}{2}, v_i(\cdot)\right\}$  for all agents  $i$ . In the second phase, we find a half-integral allocation that maximizes social welfare with respect to these new valuations  $\bar{v}_i$ . For every  $i$ ,  $\bar{v}_i(\cdot)$  is the same as  $v_i(\cdot)$ , except that for every bundle  $X$  with  $v_i(X) > 1/2$  we have  $\bar{v}_i(X) = 1/2$ . Then, we convert this fractional allocation into a randomized one. To prove the ex-post guarantee, the fact that the allocation is half-integral plays a key role.
Theorem 1.1. There exists a randomized allocation that is  $1/4$ -MMS ex-ante and  $1/8$ -MMS ex-post.
Deterministic Allocations Since the impossibility result on MMS by Procaccia and Wang (2014), there has been a considerable effort in the line of improving the approximate MMS factor for various classes of the valuation functions Akrami and Garg (2024); Akrami et al. (2023a); Barman and Krishnamurthy (2020); Garg and Taki (2021); Ghodsi et al. (2018); Kurokawa et al. (2018). However, all these results and their techniques apply to less general classes of valuation functions than XOS. In this work, we give a deterministic algorithm which outputs an allocation with 3/13-MMS guarantee when valuation functions are XOS. This way, we improve the state-of-the-art on approximate MMS for XOS valuations from 0.219225 Seddighin and Seddighin (2022) to  $3 / 13 = 0.230769$ . In order to do so, like in our randomized algorithm, we first allocate large items. However, in addition to allocating single large items, here we also allocate pairs and triples of items if they satisfy some agent up to 3/13 factor of their MMS value. This way we get a stronger upper bound on the value of most of the remaining items. In the last step, we output an allocation which maximizes the social welfare with respect to valuations  $\bar{v}_i(\cdot) = \min \left\{\frac{6}{13}, v_i(\cdot)\right\}$  for the remaining agents and items. Note that this last step is also in correspondence with the last step of the randomized algorithm. Here we cap the value of the bundles with 6/13 (instead of 1/2) and we output an integral allocation with maximum
social welfare (instead of a half-integral allocation with maximum social welfare). Moreover, here we need to do a more careful analysis to show that the output is indeed a 3/13-MMS allocation.
Theorem 1.2. Given an instance with XOS valuations, there exists a 3/13-MMS allocation.
Both of our results are constructive; however, their running times are not polynomially bounded. One step that cannot be done in polynomial time is computing the MMS values of the agents and another one is computing an allocation with maximum social welfare.
1.2 Further Related Work
Maximin-share fairness notion. Budish (2011) introduced the MMS notion. As we already mentioned, there are instances with additive valuations for which no MMS allocation exists Feige et al. (2022); Kurokawa et al. (2018); Procaccia and Wang (2014). Therefore, several studies considered the approximation guarantees. For additive valuations, the first approximation guarantee, 2/3-MMS, is given by Kurokawa et al. (2018). The approximation factor improved over time by Ghodsi et al. (2018) to  $3/4$ , and then to  $3/4 + o(1)$  by Garg and Taki (2021) and Akrami et al. (2023a). The best-known result for this valuation classed is  $3/4 + 3/3836$  which was very recently proposed by Akrami and Garg (2024). Moreover, there are several works on MMS approximations for submodular valuations Barman and Krishnamurthy (2020); Ghodsi et al. (2018), and subadditive valuations Ghodsi et al. (2018); Seddighin and Seddighin (2022). Also, Farhadi et al. (2019) proved the tight bound of  $1/n$ -MMS for the setting where agents can have different entitlements.
Fractionally subadditive valuations. A 1/5-MMS approximation algorithm for fractionally subadditive valuations proposed by Ghodsi et al. (2018), and improved later to 0.2173913-MMS approximation by Seddighin and Seddighin (2022). Fractionally subadditive valuations for hereditary set systems have been studied by Li and Vetta (2018). Different other notions of fairness have been studied for this class of valuation. Feige (2009) studied maximizing social welfare. Also, Hoefer et al. (2023) proved the existence of randomized allocations which are proportional ex-ante and proportional up to one item (PROP1) ex-post.
Best-of-both-worlds fairness. Aleksandrov et al. (2015) first explored this line of research in the context of the food bank problem for a special case of additive valuations. Freeman et al. (2020) proposed a randomized polynomial-time algorithm for additive valuations that is envy-free ex-ante and EF1 ex-post. Afterward, Aziz (2020) modified this algorithm to also get the Probabilistic Serial fractional outcome Bogomolnaia and Moulin (2001) as well as a weak notion of efficiency with a simpler proof. Babaioff et al. (2022) provided an allocation algorithm in which the expected value of each agent's bundle is at least a  $1/n$ -fraction of her value for the set of all items (ex-ante proportional) and ex-post proportional up to one item and 1/2-MMS allocations for the additive valuations. Feldman et al. (2023) gave an allocation algorithm for subadditive valuations with guarantees of  $1/2$ -envy-freeness ex-ante and  $1/2$ -EFX and EF1 ex-post. Moreover, studies have been conducted for additive valuations with binary margins Aziz (2020); Halpern et al. (2020), and for matroid rank valuations Aziz et al. (2023); Babaioff et al. (2021).
A fair division instance is denoted by  $\mathcal{I} = (\mathcal{N},\mathcal{M},\mathcal{V})$  where  $\mathcal{N} = [n]$  is a set of  $n$  agents,  $\mathcal{M}$  is a set of  $m$  indivisible items and  $\mathcal{V} = (v_{1},\ldots ,v_{n})$  is a vector of valuation functions. Each agent  $i$  has a valuation function  $v_{i}:2^{\mathcal{M}}\to \mathbb{R}_{\geq 0}$  that represents her value for every bundle of items. Thus, for every set  $S$  of items,  $v_{i}(S)$  shows the value of  $S$  to agent  $i$ . We denote the  $j^{\mathrm{th}}$  item by  $b_{j}$ . An allocation of items is a partition of  $\mathcal{M}$  into  $n$  parts (i.e., bundles) where the  $i^{\mathrm{th}}$  bundle is the share allocated to agent  $i$ . For an allocation  $\mathcal{A}$ , we denote the bundle of agent  $i$  by  $\mathcal{A}_i$ . Here we assume that the valuations are monotone, i.e., for every two sets  $S$  and  $T$  such that  $S\subseteq T$ , we have  $v_{i}(S)\leq v_{i}(T)$  and normalized, i.e.,  $v_{i}(\emptyset) = 0$  for all  $i\in \mathcal{N}$ . Our discussion in this paper involves two classes of valuation functions: additive and fractionally subadditive valuation functions. A valuation function  $v_{i}(\cdot)$  is additive, if for every set  $S$  of items, we have  $v_{i}(S) = \sum_{b_{j}\in S}v_{i}(\{b_{j}\})$ .
Definition 2.1 (XOS). A valuation function  $v_{i}(\cdot)$  is fractionally subadditive (XOS), if there exists a set of several additive valuation functions  $u_{i,1}, u_{i,2}, \ldots, u_{i,\ell}: 2^{\mathcal{M}} \to \mathbb{R}_{\geq 0}$  such that for every set  $S$  we have  $v_{i}(S) = \max_{1 \leq k \leq \ell} u_{i,k}(S)$ .
Given an allocation  $\mathcal{A}$ , we denote by  $u_{i,i'}$ , an additive function of  $v_i$  that defines  $v_i(\mathcal{A}_i)$ , i.e.,  $v_i(\mathcal{A}_i) = u_{i,i'}(\mathcal{A}_i)$ . Another term we frequently use in this paper is contribution; which is defined to evaluate the marginal value of one set to another.
Definition 2.2 (Contribution). For every sets  $S, T$  of items such that  $S \subseteq T$ , we define the marginal contribution of  $S$  to  $T$  with respect to valuation function  $v$ , denoted by  $C_v^T(S)$  as  $C_v^T(S) = v(T) - v(T \setminus S)$ , i.e., the contribution of  $S$  to  $T$  is the decrease in the value when  $S$  is removed from  $T$ .
With abuse of notation, for an allocation  $\mathcal{A}$  of items to agents with valuation vector  $\mathcal{V} = (v_{1},\dots ,v_{n})$  and every set  $S$  of items, we define the contribution of  $S$  to  $\mathcal{A}$  with respect to  $\mathcal{V}$ , denoted by  $C_{\mathcal{V}}^{\mathcal{A}}(S)$  as  $C_{\mathcal{V}}^{\mathcal{A}}(S) = \sum_{1\leq i\leq n}C_{v_i}^{\mathcal{A}_i}(\mathcal{A}_i\cap S)$ . However, since an XOS valuation function might include many additive functions, this equation is not always practical. Therefore, we bound  $C_{\mathcal{V}}^{\mathcal{A}}(S)$  as follows.
Observation 2.3. Given an allocation  $\mathcal{A}$  of items to agents with valuation vector  $\mathcal{V}$ , and every set  $S$  of items, we have  $C_{\mathcal{V}}^{\mathcal{A}}(S) \leq \sum_{1 \leq i \leq n} u_{i,i'} (\mathcal{A}_i \cap S)$ .
Our goal is to allocate the items to the agents in a fair manner. Here we discuss two share-based notions of fairness, namely proportionality, and maximin-share. For agent  $i$ , we define the proportional share of agent  $i$ , denoted by  $\pi_i$  as  $\pi_i = v_i(\mathcal{M}) / n$ . We also define the maximin-share of agent  $i$ , denoted by  $\mathrm{MMS}_i$  as
$$
\mathsf{MMS}_{i} = \max_{\langle \Pi_{1},\Pi_{2},\ldots ,\Pi_{n}\rangle \in \Omega}\min_{1\leq j\leq n}v_{i}(\Pi_{j}),
$$
where  $\Omega$  is the set of all partitions of  $\mathcal{M}$  into  $n$  bundles. Moreover, if for a partition  $\Pi = \langle \Pi_1,\Pi_2,\dots ,\Pi_n\rangle$  of  $\mathcal{M}$  into  $n$  bundles we have  $v_{i}(\Pi_{j})\geq \mathrm{MMS}_{i}$  for all  $j\in [n]$ , we say  $\Pi$  is an "MMS partition" of agent  $i$ . For brevity, in the rest of the paper, we assume that the valuations are scaled so that for each agent  $i$ , we have  $\mathrm{MMS}_i = 1$ . We also define approximate versions of these two notions as follows. For a constant  $\alpha >0$ , we say an allocation is  $\alpha$ -proportional, if it guarantees to each agent  $i$  a bundle with value at least  $\alpha \pi_{i}$ . Likewise, in an  $\alpha$ -MMS allocation, the value of the share allocated to each agent  $i$  is at least  $\alpha$ .
Randomized allocation. In this paper, we also consider randomized allocations. A randomized allocation is a distribution over a set of deterministic allocations. For a randomized allocation  $\mathcal{R}$ , we denote by  $D(\mathcal{R})$  the set of allocations in the support of  $\mathcal{R}$ . Given a randomized allocation  $\mathcal{R}$ , the expected welfare of agent  $i$  for  $\mathcal{R}$  is defined as  $v_{i}(\mathcal{R}) = \sum_{\mathcal{A} \in D(\mathcal{R})} v_{i}(\mathcal{A}_{i}) \cdot p_{\mathcal{A}}$ , where  $p_{\mathcal{A}}$  is the probability of allocation  $\mathcal{A}$  in  $\mathcal{R}$ .
Fractional allocation. En route to proving our results, we leverage another relaxed form of allocation called fractional allocation. In a fractional allocation, we ignore the indivisibility assumption and treat each item as a divisible one. Formally, a fractional allocation  $\mathcal{F}$  is a set of  $nm$  variables  $f_{i,j}$  indicating the fraction of item  $b_{j}$  allocated to agent  $i$ . Therefore, we expect allocation  $\mathcal{F}$  to satisfy the following constraints:
$$
\forall_ {j} \quad \sum_ {1 \leq i \leq n} f _ {i, j} \leq 1
$$
(we have one unit of each item)
$$
\forall_ {i, j} \quad 0 \leq f _ {i, j}
$$
(each agent receives a non-negative share of each item)
A fractional allocation is complete if  $\sum_{i}f_{i,j} = 1$  for all items  $b_{j}$ , i.e., all items are completely allocated. Given a fractional allocation  $\mathcal{F}$ , we define the utility of agent  $i$  for  $\mathcal{F}$  in the same way as we calculate it for integral allocations:  $v_{i}(\mathcal{F}) = \max_{k}\sum_{1\leq j\leq m}u_{i,k}(\{b_{j}\})f_{i,j}$ . Complete fractional allocations give rise to randomized allocations in the standard way, i.e., the probability  $p_{\mathcal{A}}$  of an allocation  $\mathcal{A}$  is defined as
$$
p _ {\mathcal {A}} = \prod_ {1 \leq i \leq n} \prod_ {b _ {j} \in \mathcal {A} _ {i}} f _ {i j}. \tag {1}
$$
Lemma 2.4. Let  $\mathcal{F}$  be a complete fractional allocation and let randomized allocation  $\mathcal{R}$  be defined by (1). Then given an XOS valuation function  $v_{i}, v_{i}(\mathcal{R}) \geq v_{i}(\mathcal{F})$  for all  $1 \leq i \leq n$ .
We also need to redefine contribution for fractional allocations and fractional bundles. Suppose that  $\mathcal{F}$  is a fractional allocation and  $S$  is a fractional set of items. Since items are fractionally allocated, the term contribution must be defined more precisely. For example, suppose that set  $S$  consists of a
fraction 0.4 of item  $b_{j}$ , and in allocation  $\mathcal{F}$ , 0.2 of  $b_{j}$  belongs to agent  $i_{1}$ , 0.5 of  $b_{j}$  belongs to agent  $i_{2}$ , and 0.3 of  $b_{j}$  belongs to agent  $i_{3}$ . Here we should exactly determine that this 0.4 fraction of item  $b_{j}$  in  $S$  corresponds to the share of which agent or agents. One reasonable strategy is to choose the share of each agent in a way that after removal of  $S$  from  $\mathcal{F}$  we have the smallest possible decrease in the social welfare. Based on this strategy, assuming that  $s_{j}$  is the fraction of item  $b_{j}$  in  $S$ , we define the contribution of  $S$  to  $\mathcal{F}$ , denoted by  $C_{\mathcal{V}}^{\mathcal{F}}(S)$  as the answer of the following optimization program:
$$
\text {m i n i m i z e} \sum_ {1 \leq i \leq n} v _ {i} (\mathcal {F}) - v _ {i} \left(\mathcal {F} ^ {\prime}\right)
$$