SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$\varepsilon$ -fractional core stability in Hedonic Games
Simone Fioravanti $^{1}$  Michele Flammini $^{1,2}$  Bojana Kodric $^{3}$  Giovanna Varricchio $^{2,4}$
<sup>1</sup> Gran Sasso Science Institute (GSSI), L'Aquila, Italy
<sup>2</sup> University of Calabria, Rende, Italy
$^{3}$  Ca' Foscari University of Venice, Venice, Italy
<sup>4</sup> Goethe-Universität, Frankfurt am Main, Germany
{simone.fioravanti, michele.flammini}@gssi.it
bojana.kodric@unive.it giovanna.varricchio@unical.it
Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences. According to these preferences, it is desirable that a coalition structure (i.e. a partition of agents into coalitions) satisfies some form of stability. The most well-known and natural of such notions is arguably core-stability. Informally, a partition is core-stable if no subset of agents would like to deviate by regrouping in a so-called core-blocking coalition. Unfortunately, core-stable partitions seldom exist and even when they do, it is often computationally intractable to find one. To circumvent these problems, we propose the notion of  $\varepsilon$ -fractional core-stability, where at most an  $\varepsilon$ -fraction of all possible coalitions is allowed to core-block. It turns out that such a relaxation may guarantee both existence and polynomial-time computation. Specifically, we design efficient algorithms returning an  $\varepsilon$ -fractional core-stable partition, with  $\varepsilon$  exponentially decreasing in the number of agents, for two fundamental classes of HGs: Simple Fractional and Anonymous. From a probabilistic point of view, being the definition of  $\varepsilon$ -fractional core equivalent to requiring that uniformly sampled coalitions core-block with probability lower than  $\varepsilon$ , we further extend the definition to handle more complex sampling distributions. Along this line, when valuations have to be learned from samples in a PAC-learning fashion, we give positive and negative results on which distributions allow the efficient computation of outcomes that are  $\varepsilon$ -fractional core-stable with arbitrarily high confidence.
Game-theoretic models of coalition formation have drawn significant interest in the last years, because of their ability to capture meaningful properties of multi-agent interactions. In Hedonic Games (HGs) [2, 16] agents gather together without any form of externality, that is only minding the internal composition of their groups. A solution is then a partition of the agents (or coalition structure) having some desirable properties, which typically stand for stability against some kinds of deviations. Among the many notions existing in the literature, one of the most fundamental is core stability [10, 31]. A partition is said to be core-stable or in the core if no subset of agents would benefit by regrouping and forming a so-called core-blocking coalition. Unfortunately, while being a quite natural requirement, it is notably very difficult to achieve [5, 25], even under the usual unrealistic assumption of full knowledge of agents' preferences. Furthermore, for the few classes of HGs in which the non-emptiness of the core has been established, a stable partition is usually hard to compute. Nonetheless, due to its significance, it is still desirable to find an approximation of the core.
The core was first considered in the setting of cooperative game theory, where the value of a coalition has to be allocated fairly between its members. In this scenario, the most well-known approximation to the core is the so-called strong  $\varepsilon$ -core [26], in which a blocking coalition increases the total value allocated to its members by at least  $\varepsilon$ . A derived notion is the one of least-core [22], i.e. the (strong)  $\varepsilon$ -core associated to the smallest possible value of  $\varepsilon$  guaranteeing its existence. An adaptation of these concepts to the context of HGs has been proposed in [17], where the authors define  $k$ -improvement core stability, requiring each member of a blocking coalition to increase her utility by a multiplicative factor strictly greater than  $k \geq 1$ . The same authors also propose to bound by a value  $q \geq 2$  the number of agents allowed to form a blocking coalition, obtaining what they call  $q$ -size core stability.
However, the above stability concepts might still be fragile, especially when many coalitions could anyway benefit by deviating, even if not consistently according to the approximation/size factors. In fact, agents might be still inclined to subtle improvements. Interestingly, works like [11] experimentally show that, sampling random HGs, the fraction of instances with an empty core is small and decreases significantly as the number of agents increases. This could be related to the fact that, even in the instances with empty core, many or almost all the coalitions do not core-block. Consequently, the existence of outcomes having a relatively small number of blocking coalitions appears plausible.
In this work, we investigate the concept of  $\varepsilon$ -fractional core-stable partitions, i.e. partitions that can be core-blocked by only an  $\varepsilon$ -fraction of all possible coalitions. This notion bears some similarities with PAC-stabilizability as defined in [27]. In this work, the authors lifted for the first time the assumption of complete information in HGs and considered learning preferences and core-stable partitions from samples, employing the probably approximately correct (PAC) learning framework [30]. Specifically, an HGs class is PAC stabilizable if, after seeing a certain number of samples, it is possible to either determine that the core is empty or to return a partition that has probability at most  $\varepsilon$  to be core-blocked by a coalition sampled from the same distribution.
Our Contribution. In this paper, we introduce and study the  $\varepsilon$ -fractional core stability notion in Hedonic Games. We specifically investigate this concept on two of the most fundamental classes of HGs: Simple fractional and anonymous HGs.
Roughly speaking a partition of agents is  $\varepsilon$ -fractional core-stable,  $\varepsilon$ -FC in short, if at most an  $\varepsilon$  fraction of coalitions may core-block it. Such a definition has a natural probabilistic interpretation: Indeed, for an  $\varepsilon$ -FC partition,  $\varepsilon$  represents an upper bound to the probability of drawing a core-blocking coalition uniformly at random. Along this line, we broaden the definition to any distribution over coalitions by requiring that the probability of sampling a blocking coalition is at most  $\varepsilon$ . Worth noticing, if  $\varepsilon = 0$ , we are essentially requiring our solution to be core stable. Hence it is not possible in general to prove the existence and efficiently compute  $\varepsilon$ -FC outcomes, for values of  $\varepsilon$  that are sufficiently close to 0. In contrast, our aim is to efficiently compute  $\varepsilon$ -FC solutions for as small as possible values of  $\varepsilon$ , proving in turn also their existence.
Unfortunately, as a first result, we prove that by allowing arbitrary sampling distributions an  $\varepsilon$ -FC may fail to exist for constant values of  $\varepsilon$ . On the positive side, for the aforementioned classes of HGs, we show that it is possible to efficiently compute an  $\varepsilon$ -FC solution under the uniform distribution, with  $\varepsilon$  sub-exponentially small in the number of agents. In particular, in the case of anonymous HGs, we present an algorithm computing an  $\varepsilon$ -FC solution under the class of  $\lambda$ -bounded distributions, where the ratio of the probabilities of extracting any two coalitions is bounded by the parameter  $\lambda$ . Notably, this class includes the uniform distribution as the special case  $\lambda = 1$ , and can be considered a suitable extension of it. Our algorithms, besides guaranteeing sub-exponentially small values of  $\varepsilon$ , are designed to handle the possible incomplete knowledge of agents' preferences. In fact, in case preferences are unknown, the algorithms can use the sampling distribution to learn them in a PAC-learning fashion while maintaining the very same guarantees on  $\varepsilon$  with high confidence.
Core stability and Hedonic Games. Hedonic Games have captured considerable research attention from the scientific community over the years. One of the main goals in their study is understanding how to gather agents in a way they won't desire to modify the outcome. For this reason, several stability notions have been considered and studied such as core and Nash stability, or individual rationality. We refer to [2] for a comprehensive overview of the subject. Core stability is a fundamental
concept in multi-agent systems, first considered in the context of cooperative games [13, 19]. Its properties and the related complexity have been largely investigated in HGs [1, 10, 25, 28] and beyond, such as in house allocation [24], markets [8, 9] and many other settings. Recently Donahue and Kleinberg [14, 15] have modeled federated learning as an HG, where agents evaluate federating coalitions according to the expected mean squared error of the model they obtain by sharing data with the coalition's members. Works like [4, 12] have used core stability to study payoff allocation among team members in collaborative multi-agent settings.
PAC-stabilizability. Our definition of  $\varepsilon$ -fractional core stability is also strictly related to PAC stabilizability as defined in [27]. This notion was further investigated in several papers. Igarashi et al. [20] studied the case of HGs with underlying interaction networks. Jha and Zick [21] defined a general framework for learning game-theoretic solution concepts from samples. Trivedi and Hemachandra [29] considered learning and stabilizing HGs with noisy preferences. Recently, Fioravanti et al. [18] proposed to relax the requirements of PAC-stabilizability by considering only restricted distributions and showed stabilizability of  $\mathcal{W}$ -games under  $\lambda$ -bounded distributions.
In this section, we present the fundamental definitions for our work. Given a positive integer  $k$  we use  $[k]$  to denote the set  $\{1, \ldots, k\}$ .
3.1 Hedonic Games
Let  $N$  be a set of  $n$  agents. We call any non-empty subset  $C \subseteq N$  a coalition and any coalition of size one a singleton. We denote by  $\succ_{i}$  any binary preference relation of agent  $i$  over all coalitions containing  $i$ , which is reflexive, transitive, and complete. A Hedonic Game is then a pair  $H = (N, \succ)$ , where  $\succ = (\succ_{i}, \ldots, \succ_{n})$  is a preference profile, i.e., the collection of all agents' preferences.
Throughout this work, we will assume that preferences are expressed as real numbers by means of valuation functions  $v_{i}: 2^{N} \to \mathbb{R}$  for each  $i \in N$ . In other words, given two coalitions  $C, C'$  containing agent  $i$ ,  $v_{i}(C) \geq v_{i}(C')$  if and only if  $C \gtrsim_{i} C'$ . We will denote by  $\vec{v} = (v_{1}, \ldots, v_{n})$  the collections of agents' valuations and assume that  $v_{i}$  is not defined for  $C$  not containing  $i$  and write  $v_{i}(C) = \emptyset$ . A coalition structure  $\pi$  is a partition of agents into coalitions and  $\pi(i)$  denotes the coalition  $i$  is assigned to. We write  $v_{i}(\pi) = v_{i}(\pi(i))$  to denote the utility  $i$  gets in her assigned coalition  $\pi(i)$  inside  $\pi$ .
With this paper, we aim to relax the concept of core stability defined as follows.
Definition 3.1. Given a coalition structure  $\pi$ , a coalition  $C \subseteq N$  is said to core-block  $\pi$  if, for each  $i \in C$ ,  $v_{i}(C) > v_{i}(\pi)$ . A coalition structure  $\pi$  is said to be core-stable if no coalition  $C \subseteq N$  core-blocks it.
Simple fractional Hedonic Games. In fractional HGs (FHGs), first introduced in [3], every agent  $i \in N$  assigns a value  $v_{i}(j)$  to any other  $j \neq i$ , and then her evaluation for any coalition  $C \ni i$  is the average value ascribed to the members of  $C \setminus \{i\}$ . Formally:  $v_{i}(C) = \frac{\sum_{j \in C \setminus \{i\}} v_{i}(j)}{|C|}$ . A FHG is said to be simple if  $v_{i}(j) \in \{0,1\}$ , for each  $i, j \in N, i \neq j$ . A natural representation of these games is a directed and unweighted graph  $G = (V,E)$ , where  $V = N$  and  $(i,j) \in E$  if and only if  $v_{i}(j) = 1$ . Despite their name, Aziz et al. [3] show that simple FHGs capture the complexity of the entire class w.r.t. core-stability: In fact, deciding if a core-stable partition exists is  $\Sigma_2^p$ -complete.
Anonymous Hedonic Games. An HG is said to satisfy anonimity [6, 10] if agents evaluate coalitions on the sole basis of their size, i.e.,  $v_{i}(C) = v_{i}(C^{\prime})$  for any  $i \in N$  and any  $C, C^{\prime}$  containing  $i$  such that  $|C| = |C^{\prime}|$ . When considering anonymous HGs we will assume  $v_{i}: [n] \to \mathbb{R}$ . An anonymous HGs instance is said to be single-peaked if there exists a permutation  $(s_1, \ldots, s_n)$  of  $\{1, \ldots, n\}$  for which every agent  $i \in N$  admits a peak  $p(i) \in [n]$  such that  $h < k \leq p(i)$  or  $h > k \geq p(i)$  imply  $v_{i}(s_{k}) \geq v_{i}(s_{h})$ . Roughly speaking, the higher the distance from the peak in the given ordering, the lower the valuation for the coalition size. If the permutation is the identity function, i.e. the ordering is the usual one over  $\mathbb{N}$ , we say that the preference is single-peaked in the
natural ordering. For anonymous HGs, deciding if a core-stable partition exists has been shown to be NP-complete [5].
3.2 Epsilon-core, learning, and computation efficiency
Here we formalize our notion of  $\varepsilon$ -fractional core stability.
Definition 3.2. Given a parameter  $\varepsilon \in [0,1]$ , we say that a partition  $\pi$  is  $\varepsilon$ -fractional core-stable,  $\varepsilon$ -FC in short, if it holds that at most an  $\varepsilon$ -fraction of all possible coalitions are core-blocking for partition  $\pi$ , i.e.,
$$
\frac {\# o f c o r e - b l o c k i n g c o a l i t i o n s f o r \pi}{\# o f a l l p o s s i b l e c o a l i t i o n s} <   \varepsilon .
$$
This definition has a natural probabilistic interpretation: A partition is  $\varepsilon$ -FC if, by sampling u.a.r. a coalition, the probability of sampling a core-blocking one is at most  $\varepsilon$ . Such an interpretation inspired the following extension.
Definition 3.3. Given a parameter  $\varepsilon \in [0,1]$ , we say that a partition  $\pi$  is  $\varepsilon$ -fractional core-stable with respect to a distribution  $\mathcal{D}$  over  $2^{N}$  if
$$
\operatorname * {P r} _ {C \sim \mathcal {D}} \left[ C \text {c o r e b l o c k i n g f o r} \pi \right] <   \varepsilon .
$$
We may assume that agents' preferences are unknown and we need to learn them by sampling coalitions from a distribution  $\mathcal{D}$ . Consequently, our algorithms will have a learning phase, where preferences are learned by observing  $m$  samples, and a computation phase, where a stable outcome is computed upon the learned preferences. We say that an algorithm exactly learns a family  $\mathcal{T} \subseteq 2^{N}$  if after seeing a sample  $S$  it is able to determine the real valuation of any agent for every coalition in  $\mathcal{T}$ . Note that this does not necessarily mean that  $\mathcal{T} \subseteq S$ ; instead, by knowing the properties of the considered HG class, it must be possible to derive complete information of  $\mathcal{T}$  from  $\mathcal{S}$ . As an example, consider the anonymous HGs class: If  $\mathcal{T}$  is the family of coalitions of size  $s$ , in order to exactly learn it, for each agent  $i$  there must exist  $S \in S$  such that  $i \in S$  and  $|S| = s$ .
We aim to design polynomial-time algorithms for computing  $\varepsilon$ -FC solutions. However, to retrieve enough information on the preferences we may require a large number of samples. Hence, we say that an algorithm is efficient if its computation phase requires polynomial time in the instance size while the learning phase is polynomial in  $n$ ,  $1 / \varepsilon$ ,  $\log 1 / \delta$ , where  $\delta$  is a confidence parameter. Clearly, if the valuations are known in advance, an efficient algorithm requires polynomial time. Moreover, our algorithms will compute an  $\varepsilon$ -FC partition with confidence  $1 - \delta$  and the solution will turn out to be exact as soon as the true agents' preferences are given as input.
3.3 Chernoff Bound and  $\lambda$ -bounded distributions
The analysis of our algorithms is mainly probabilistic and will strongly rely on the classical Chernoff bound (see, e.g., Chapter 4 in [23]) that we summarize hereafter. Let  $X = \sum_{i=1}^{n} X_i$  be a sum of independent Poisson trials of mean  $\mu$ , then for any constant  $b \in (0,1)$  it holds:
$$
\Pr \left[ X \geq (1 + b) \mu \right] \leq e ^ {- \mu b ^ {2} / 3} \quad \text {a n d} \quad \Pr \left[ X \geq (1 - b) \mu \right] \leq e ^ {- \mu b ^ {2} / 2}. \tag {1}
$$
As we already mentioned, we will study the  $\varepsilon$ -fractional core stability subject to distributions  $\mathcal{D}$  over  $2^{N}$ . We will see that, while for general distributions it is not possible to guarantee good enough values of  $\varepsilon$ , it is indeed possible for  $\lambda$ -bounded distributions. This class of distributions has been first introduced in a learning context by [7], where they consider general continuous distributions over bounded subsets of  $\mathbb{R}^d$ . The idea is that a distribution in this class has constraints (parameterized by the value  $\lambda$ ) on how much the probability density function can vary.
Definition 3.4. A distribution  $\mathcal{D}$  over  $2^{N}$  is said to be  $\lambda$ -bounded, if there exists  $\lambda \geq 1$  such that, for every two coalitions  $C_1, C_2$ , it holds that
$$
\operatorname * {P r} _ {C \sim \mathcal {D}} \left[ C = C _ {1} \right] \leq \lambda \operatorname * {P r} _ {C \sim \mathcal {D}} \left[ C = C _ {2} \right].
$$
A straightforward consequence of this definition is that no coalition has null probability of being sampled. Moreover, it can be noted that, while setting  $\lambda = 1$  we obtain the uniform distribution
over  $2^{N}$ , as  $\lambda \to +\infty$  every distribution can be considered  $\lambda$ -bounded up to an approximation factor. Thus, in order to keep the original intended purpose of this definition, in the rest of this work we will consider  $\lambda$  to be constant with respect to  $n$ .
The following lemma, an adaptation of Lemma 4 in [7], will be useful in our computations.
Lemma 3.5. Let distribution  $\mathcal{D}$  be  $\lambda$ -bounded. Let  $\mathcal{F} \subseteq \mathcal{P}(2^N)$  be a family of subsets and let  $a = |\mathcal{F}| / 2^n$ . Then, the following inequalities hold:
$$
\frac {a}{a + \lambda (1 - a)} \leq \Pr_ {C \sim \mathcal {D}} [ C \in \mathcal {F} ] \leq \frac {\lambda a}{\lambda a + 1 - a}.
$$
4 Impossibility results
In this section, we explore the boundaries of feasible  $\varepsilon$  for simple fractional and anonymous HGs according to general,  $\lambda$ -bounded, and uniform distributions.
Simple fractional Hedonic Games. We start by showing that when dealing with arbitrary distributions, an  $\varepsilon$ -FC may not exist for  $\varepsilon$  exponentially small w.r.t. the number of agents. Specifically, this impossibility result holds for constant values of  $\varepsilon$ . The informal intuition is that, being the distribution arbitrary, one may choose it in an adversarial way with respect to a partition having an empty core.
Proposition 4.1. There exists a distribution  $\mathcal{D}$  and a simple fractional HG instance such that no  $\varepsilon$ -fractional core-stable solution w.r.t.  $\mathcal{D}$  exists for  $\varepsilon \leq 1/2^{40}$ .
Proof. Simple fractional HGs have been shown to have an empty core [3]. In particular, the authors provide an instance  $\mathcal{I}$  with 40 agents having an empty core. We can extend this instance to an instance  $\mathcal{I}'$ , with  $N' = [n]$  being the set of agents, still having an empty core. Let  $N = \{1, \ldots, 40\}$  be the set of agents in  $\mathcal{I}$ . Their mutual preferences remain the same, while they value 0 all the other agents in  $N' \setminus N$ . In turn, the agents in  $N' \setminus N$  have mutual preferences equal to 1, and they value 0 all the agents in  $N$ .  $\mathcal{I}'$  has an empty core and, in particular, for any coalition structure  $\pi$  there exists a core blocking coalition in  $2^{N} \cup \{\{41, \ldots, n\}\} \setminus \{\emptyset\}$ . In fact, on the one hand, the agents in  $\{41, \ldots, n\}$  will form a blocking coalition whenever we return a partition where they are not in the same coalition. On the other hand, if  $\{41, \ldots, n\}$  is a coalition of the considered partition, no matter how the agents in  $N$  will be partitioned, there exists a blocking coalition in  $2^{N} \setminus \{\emptyset\}$  because the instance  $\mathcal{I}$  has empty core. In conclusion, by choosing  $\mathcal{D}$  as the uniform distribution over  $2^{N'} \cup \{\{41, \ldots, n\}\} \setminus \{\emptyset\}$ , for any coalition structure  $\pi$  we have  $\operatorname*{Pr}_{C \sim \mathcal{D}}[C$  blocking for  $\pi] > 1/2^{40}$  and the thesis follows.
Similarly, by applying Lemma 3.5 we can easily derive the following generalization.
Corollary 4.2. Given a parameter  $\lambda$  there exists a bounded distribution with parameter  $\lambda$  such that in simple fractional HGs no  $\varepsilon$ -core exists for  $\varepsilon < \frac{\lambda}{2^{40}(\lambda - 1) + 2^n}$ .
Anonymous Hedonic Games. In the following, we consider single-peaked anonymous HGs. Clearly, all the provided results hold for the more general class of anonymous HGs. Following the same approach of the previous section, knowing that for anonymous HGs there exists an instance with seven agents and single-peaked preferences having empty core [6], we can show the following.
Proposition 4.3. There exists a distribution  $\mathcal{D}$  and an anonymous (single-peaked) HG instance such that for every  $\varepsilon \leq 1/2^7$  there is no  $\varepsilon$ -fractional core w.r.t.  $\mathcal{D}$ .
Corollary 4.4. Given a parameter  $\lambda$ , there exists a  $\lambda$ -bounded distribution such that for  $\varepsilon < \frac{\lambda}{2^7(\lambda - 1) + 2^n}$  no  $\varepsilon$ -fractional core-stable solution exists in anonymous single-peaked HGs.
As a consequence, both for simple fractional and anonymous HGs, under uniform distributions, where  $\lambda = 1$ , and constant values of  $\lambda$ , the provided bound becomes exponentially small. Hence, in the rest of this paper, we will be focusing on these specific classes of distributions.
5 Simple Fractional Hedonic Games
In this section, we present an algorithm returning an  $\varepsilon$ -FC for simple FHGs with  $\varepsilon$  exponentially small in the number of agents, and prove its correctness, as summarized in the following claim.
Theorem 5.1. Given a parameter  $\delta \in (0,1)$ , for any simple FHG instance and with confidence  $1 - \delta$ , we can efficiently compute an  $\varepsilon$ -fractional core-stable partition for every  $\varepsilon \geq 2^{-\Omega(n^{1/3})}$ .
5.1 Computing an  $\varepsilon$ -fractional core-stable partition
We will be working only with the uniform distribution over all possible coalitions denoted by  $U(2^{N})$ .
The proof of the theorem is quite involved and needs several steps. To facilitate the reader, the section is organized as follows: First, we provide a high-level explanation of Algorithm  $\mathbb{I}$  for computing an  $\varepsilon$ -FC partition, then we show various parts of the proof as separate lemmas, and finally, we assemble all together to prove Theorem 5.1.