SlowGuess/NeurIPS_2023_OCR huggingface dataset:
As a first step, Algorithm  $\square$  consists of a learning phase where the exact valuations are computed with confidence  $1 - \delta$ . Then, it starts the computation phase building the graph representation  $G$  based on the learned preferences. In order to compute an  $\varepsilon$ -FC partition, the algorithm has to pay attention to the density of  $G$ . In particular, the main intuition is to distinguish between instances based on the number of nodes having high out-degrees, treating the two arising cases separately. To this aim, observe first that the biggest fraction of (sampled) coalitions has a size close to  $\frac{n}{2}$ . As a consequence, if many nodes have low out-degree, a matching-like partition would be difficult to core-block, since a sampled coalition  $C$  will often contain very few neighbors of such nodes, compared to its size. Thus, such nodes would have a lower utility in  $C$ . On the other hand, if a sufficient amount of nodes have a high out-degree, we can form a clique large enough to make any partition containing it hard to core-block. In fact, with high probability, the sampled coalition  $C$  would contain at least one of such clique nodes, not connected to all the other agents in  $C$ . Therefore, it would have a lower utility moving to  $C$ .
First, we focus on the learning aspect and show that it is possible to learn the exact valuations in simple FHGs, upon sampling a number of sets polynomial in  $n$  and  $\log \frac{1}{\delta}$ .
Lemma 5.2. By sampling  $m \geq 16\log \frac{n}{\delta} + 4n$  sets from  $U(2^{N})$  it is possible to learn exactly the valuation functions  $v_{1}, \ldots, v_{n}$  with confidence  $1 - \delta$ .
The following definition is crucial for proving the main result.
Definition 5.3. Given a partition  $\pi$ , an agent  $i \in N$  is green with respect to  $\pi$  if
$$
\Pr_ {C \sim U \left(2 ^ {N}\right)} \left[ v _ {i} (C) > v _ {i} (\pi) \mid i \in C \right] \leq 2 ^ {- \Omega \left(n ^ {1 / 3}\right)}. \tag {2}
$$
Since a coalition containing a green agent w.r.t.  $\pi$  is unlikely to core-block  $\pi$ , if we manage to show that there are enough green agents in the partition returned by Algorithm  $\square$ , then we directly prove also its  $\varepsilon$ -fractional core-stability, as in this case any randomly drawn coalition contains at least one green agent with high probability. From now on, let  $\phi = |\{i \text{ s.t. } d_i \leq n - 31n^{2/3}\}|$  be as defined in Algorithm  $\square$ . As already informally explained above, the algorithm follows two different procedures based on  $\phi$  being greater equal or lower than  $\frac{n^{1/3}}{62}$ . The set Gr defined in Algorithm  $\square$  in both cases is meant to contain exclusively green agents w.r.t. the returned partition. We will start by proving that this is indeed true if  $\phi \leq \frac{n^{1/3}}{62}$ . Since we will never use in-degrees, from now on we will simply use the term degree in place of out-degree.
We first prove the correctness of the easier case of the algorithm in the else part of the if statement (line 19).
Lemma 5.4. Let  $\phi < \frac{n^{1/3}}{62}$  and  $\pi$  be the partition returned by Algorithm [1]. Then Gr contains only green agents w.r.t to  $\pi$ .
Proof. Let us say that an agent  $i \in N$  has high degree if  $d_i \geq n - 31n^{2/3}$ . Observe that, if such an agent is picked at any iteration of the algorithm, then at most  $31n^{2/3}$  agents are deleted from  $F$  at that iteration. By hypothesis, there are at least  $n - \frac{n^{1/3}}{62}$  agents with high degree. Since they are picked in non-increasing degree order, after  $t$  iterations the agents left in  $F$  with high degree are at least  $n - \frac{n^{1/3}}{62} - t \cdot 31n^{2/3}$ , which is strictly positive for  $t \leq \frac{n^{1/3}}{124}$ . This means that all agents in Gr
<sup>1</sup> Note that  $\frac{n}{2}$  is the average size of coalitions drawn from the uniform distribution.
Algorithm 1: Stabilizing Simple FHGs
Input:  $N, S = \langle (S_j, \vec{v}(S_j)) \rangle_{j=1}^m$   
Output:  $\pi - a 2^{-\Omega(n^{1/3})}$ -fractional core-stable partition of  $N$   
1 Compute  $\vec{v}$  and build the corresponding graph  $G$   
2  $\pi \gets \{\{i\} | i \in N\}$   
3 Let  $d_i$  be the out-degree of each  $i$  in  $G$   
4  $\phi \gets |\{i \text{ s.t. } d_i \leq n - 31n^{2/3}\}|$   
5  $\mathrm{Gr} \gets \emptyset$   
6 if  $\phi \geq \frac{n^{1/3}}{62}$  then  
7 Sort agents in non-decreasing order of out-degree  
8  $H \gets \left[1, \ldots, \frac{n^{1/3}}{62}\right]$   
9 count  $\gets 1$   
10 while count  $\leq \frac{n^{1/3}}{124}$  do  
11 Select first remaining agent  $i \in H$  in the order  
12 Gr  $\gets$  Gr  $\cup \{i\}$   
13 Let  $F_i$  be a set of  $\lceil \frac{2d_i}{n-d_i} \rceil$  neighbors of  $i$ , giving priority to singleton agents in  $N \setminus H$   
14 Let  $C_i = \{i\} \cup \bigcup_{j \in F_i} \pi(j)$   
15  $\pi \gets \pi \cup \{C_i\} \setminus \{\pi(j) | j \in C_i\}$   
16  $H \gets H \setminus (F_i \cup \{i\})$   
17 count  $\gets$  count + 1  
18 All remaining singletons  $\pi(i) = \{i\}$  in  $\pi$  are grouped together  
19 else  
20  $F \gets N$   
21 for count = 1, ..., n $^{1/3}$  do  
22 Pick  $i \in F \setminus \mathrm{Gr}$  of maximal out-degree  
23 Delete from  $F$  all agents not in the neighborhood  $N_i$  of  $i$   
24 Gr  $\gets \mathrm{Gr} \cup \{i\}$   
25  $\pi \gets \{F, N \setminus F\}$   
26 return  $\pi$
have high degree, so that  $|F| \geq n - 31n^{2/3} \cdot \frac{n^{1/3}}{124} = \frac{3n}{4}$ . Moreover, since by construction at the end of the for loop each  $i \in \mathrm{Gr}$  is connected to all the other agents in  $F$ ,  $v_i(F) \geq 1 - \frac{4}{3n}$ , which is at least the utility  $i$  would have in a clique of  $\geq \frac{3n}{4}$  agents. Therefore, any coalition of size at most  $\frac{3n}{4}$  containing  $i$  cannot core block  $\pi$ . As a consequence, for each  $i \in \mathrm{Gr}$  it holds that:
$$
\operatorname * {P r} _ {C \sim U (2 ^ {N})} [ v _ {i} (C) > v _ {i} (F) | i \in C ] \leq \operatorname * {P r} _ {C \sim U (2 ^ {N})} \left[ | C | > \frac {3 n}{4} | i \in C \right] \leq e ^ {- \frac {n}{2 4}} <   2 ^ {- \Omega (n ^ {1 / 3})},
$$
where we used Equation (1) and the fact that  $n > 1$ .
The procedure returning a partition in the complementary case  $\phi \geq \frac{n^{1/3}}{62}$  is a bit more involved, and we divide the proof in different cases according to the degree of the agents added to Gr.
Lemma 5.5. Let  $\phi \geq \frac{n^{1/3}}{62}$  and let  $\pi$  be the partition returned by Algorithm  $\boxed{l}$  in this case. Then the following statements hold:
(i)  $H$  is never empty when executing line 11 of the while loop of Algorithm  $\square$ .
(ii) each agent  $i\in \mathrm{Gr}$  is green.
Proof of Theorem 5.7 We are now able to assemble all the above claims together to prove the main theorem. According to the above lemmas, in both the cases of Algorithm 1, i.e.  $\phi < \frac{n^{1/3}}{62}$  and  $\phi \geq \frac{n^{1/3}}{62}$ , the nodes put in Gr are green and  $\gamma := |\mathrm{Gr}| = n^{1/3} / 124$ .
It remains to show that the output  $\pi$  of Algorithm 1 is a  $2^{-\Omega(n^{1/3})}$ -fractional core-stable partition:
$$
\begin{array}{l} \Pr_ {C \sim U (2 ^ {N})} [ C \text {b l o c k s} \pi ] \leq \Pr_ {C \sim U (2 ^ {N})} [ C \cap \mathrm {G r} = \emptyset ] + \Pr_ {C \sim U (2 ^ {N})} [ C \cap \mathrm {G r} \neq \emptyset \wedge C \text {b l o c k s} \pi ] \\ \leq \frac {2 ^ {n - \gamma}}{2 ^ {n}} + \left(1 - \frac {2 ^ {n - \gamma}}{2 ^ {n}}\right) \frac {1}{2 ^ {n ^ {1 / 3}}} \\ \leq \frac {1}{2 ^ {\gamma}} + \frac {1}{2 ^ {n ^ {1 / 3}}} \leq \frac {1}{2 ^ {\frac {n ^ {1 / 3}}{1 2 4} - 1}}. \\ \end{array}
$$
and that concludes the proof.
6 Anonymous Hedonic Games
In this section, we discuss the efficient computation of an  $\varepsilon$ -FC partition in the case of anonymous HGs, and focus on the more general class of  $\lambda$ -bounded distributions.
The main result of this section is given by the following:
Theorem 6.1. Given a  $\lambda$ -bounded distribution  $\mathcal{D}$  and a parameter  $\delta \in (0,1)$ , for any anonymous HG instance and with confidence  $1 - \delta$ , we can efficiently compute an  $\varepsilon$ -fractional core-stable partition for every  $\varepsilon \geq \frac{4\lambda}{2^{c(\lambda)}\sqrt[3]{n}}$ , where  $c(\lambda) = \frac{1}{\sqrt{13(\lambda + 1)}}$ .
Moreover, when agents' preferences are single-peaked, we can refine the bound on  $\varepsilon$  as follows.
Theorem 6.2. Given a  $\lambda$ -bounded distribution  $\mathcal{D}$  and a parameter  $\delta \in (0,1)$ , for any single-peaked anonymous HG instance and with confidence  $1 - \delta$ , we can efficiently compute an  $\varepsilon$ -fractional core-stable partition for every  $\varepsilon \geq 4 \cdot \frac{\lambda}{2^{n/4}}$ .
6.1 Distribution over coalitions sizes
Being anonymous preferences uniquely determined by coalition sizes, it is important to establish how the  $\lambda$ -bounded distribution  $\mathcal{D}$  impacts the probability of sampling coalitions of a given size.
Let  $X: 2^{N} \to [n] \cup \{\emptyset\}$  be the random variable corresponding to the size of  $C \sim \mathcal{D}$ , i.e.  $X(C) := |C|$ , and  $\mu := E[X]$  be its average. The following lemma shows some useful properties of  $X$ .
Lemma 6.3. Let  $X$  be the random variable representing the size of  $C \sim \mathcal{D}$ , with  $\mathcal{D}$ $\lambda$ -bounded, and let  $\mu \coloneqq E[X]$ . Then,
$$
\frac {n}{\lambda + 1} \leq \mu \leq \frac {\lambda n}{\lambda + 1}
$$
$$
\operatorname * {P r} _ {\mathcal {D}} \left[ | X - \mu | \geq \Delta \cdot \mu \right] \leq \frac {\varepsilon}{2},
$$
where  $\varepsilon$  is such that  $0 < \varepsilon < 1$  and  $\Delta$  is the quantity  $\sqrt{\frac{3(\lambda + 1)\log\frac{4}{\varepsilon}}{n}}$ .
Let us denote by  $I_{\mathcal{D}}(\varepsilon) \subseteq [n]$  the open interval  $((1 - \Delta)\mu, (1 + \Delta)\mu)$ , where  $\mu$  is the expected value for the size of a coalition sampled from  $\mathcal{D}$ . Lemma 6.3 implies that with probability at least  $1 - \varepsilon / 2$  we draw from  $\mathcal{D}$  a coalition whose size is in  $I_{\mathcal{D}}(\varepsilon)$ .
6.2 Properties of  $I_{\mathcal{D}}(\varepsilon)$
The interval  $I_{\mathcal{D}}(\varepsilon)$  will play a central role in the computation of an  $\varepsilon$ -FC partition. However, under the uncertainty of a distribution  $\mathcal{D}$ , our algorithms need to i) estimate it (in fact,  $I_{\mathcal{D}}(\varepsilon)$  is uniquely determined by the value  $\mu$  which is unknown) and ii) learn exactly the agents' valuations for coalitions whose size is in  $I_{\mathcal{D}}(\varepsilon)$ . The technical proofs of this subsection are deferred to the Appendix.
Being  $\mu$  unknown, we will work on a superset of  $I_{\mathcal{D}}(\varepsilon)$  which can be estimated by simply knowing that  $\mathcal{D}$  is  $\lambda$ -bounded. Let  $S = \{S_j\}_{j=1}^m$  be a sample of size  $m$  drawn from  $\mathcal{D}$ , and let  $\bar{\mu} = \frac{1}{m}\sum_{j}|S_j|$  be the frequency estimator. By the Hoeffding bound (see Theorem 4.13 in [23]), it is possible to show that we can estimate  $\mu$  with high confidence, as stated in the following.
Lemma 6.4. Given any two constants  $\alpha >0,\delta < 1$ , if  $m > \frac{n^2\log 2 / \delta}{2\varepsilon^2}$ , then:
$$
\Pr_ {\mathcal {S} \sim \mathcal {D} ^ {m}} [ | \bar {\mu} - \mu | <   \alpha ] \geq 1 - \delta . \tag {3}
$$
As a consequence, we can determine a good superset of  $I_{\mathcal{D}}(\varepsilon)$  as the interval with extreme points  $(1 \pm \Delta) (\bar{\mu} \pm \alpha)$ .
We now turn our attention to the exact learning of  $I_{\mathcal{D}}(\varepsilon)$ .
Lemma 6.5. By sampling  $m = \frac{2\lambda(1 + \lambda)n^2\log n^2 / \delta}{\varepsilon}$  sets from  $\mathcal{D}$  it is possible to learn exactly the valuations in  $I_{\mathcal{D}}(\varepsilon)$ , with confidence  $1 - \delta$ .
6.3 Computing an  $\varepsilon$ -fractional core-stable partition for bounded distributions
Our algorithm will consist of a learning phase and a computation phase.
During the learning phase, the algorithm passes through a sample of size  $m = \frac{2\lambda(1 + \lambda)n^2\log n^2 / \delta}{\varepsilon}$  and for each coalition  $C$  in the sample of size  $c$  and  $i\in C$  it stores the value  $v_{i}(c)$ . Let us denote by  $\mathcal{X}$  the coalition sizes that have been learned exactly during this learning phase, that is, the coalition sizes for which the algorithm learned the valuations of each agent. By the previous lemma, with confidence  $1 - \delta$ ,  $I_{\mathcal{D}}(\varepsilon)\subseteq \mathcal{X}$ .
During the computation phase, our algorithm will make sure that a sufficiently large amount of agents have a very small probability of being involved in a core blocking coalition. Such agents will be called green agents and are defined as follows. An agent  $i$  is said to be green in a partition  $\pi$  w.r.t.  $I \subseteq [n]$ , if  $|\pi(i)|$  maximizes  $v_i(s)$  for  $s \in I$ . We will show that if the probability of sampling coalitions of sizes  $s \in I$  is high enough, then green agents do not want to deviate from their actual partition with high probability as well. As a consequence, a partition containing many green agents is difficult to core-block, as shown in the following:
Lemma 6.6. For any partition  $\pi$  of agents and  $I \subseteq [n]$  such that  $I_{\mathcal{D}}(\varepsilon) \subseteq I$ . If  $\pi$  contains at least  $\log_2 \frac{2\lambda}{\varepsilon}$  green agents w.r.t.  $I$ , then,  $\pi$  is  $\varepsilon$ -fractional core-stable.
Proof. Let  $C \sim \mathcal{D}$  and  $c = |C|$  be its size. We denote by  $\mathcal{E}_1$  the event that  $C$  core blocks  $\pi$ , and by  $\mathcal{E}_2$  then the event that  $C$  does not contain green agents. By definition of green agents, if  $c \in I$  and  $C$  contains at least one green agent w.r.t.  $I$ , then it cannot core block  $\pi$ . Formally speaking,  $\overline{\mathcal{E}_2} \wedge c \in I \Rightarrow \overline{\mathcal{E}_1}$  and therefore  $\mathcal{E}_1 \Rightarrow \mathcal{E}_2 \lor c \notin I$ . As a consequence,
$$
\operatorname * {P r} _ {\mathcal {D}} \left[ \mathcal {E} _ {1} \right] \leq \operatorname * {P r} _ {\mathcal {D}} \left[ c \notin I \vee \mathcal {E} _ {2} \right] \leq \operatorname * {P r} _ {\mathcal {D}} \left[ c \notin I \right] + \operatorname * {P r} _ {\mathcal {D}} \left[ \mathcal {E} _ {2} \right].
$$
Being  $I_{\mathcal{D}}(\varepsilon)\subseteq I$  , by Lemma 6.3 and Lemma 3.5 we finally get  $\operatorname *{Pr}_{\mathcal{D}}[\mathcal{E}_1]\leq \frac{\varepsilon}{2} +\lambda \frac{2^{n - \log_2\frac{2\lambda}{\varepsilon}}}{2^n} = \varepsilon .$
We finally sketch the proof of Theorem 6.1, for a rigorous proof see the Appendix. The key idea is to create a partition  $\pi$  by grouping as many agents as possible in a coalition of preferred size among a set of sizes, say  $I$ , that may occur with particularly high probability. Hence, by Lemma 6.6 we only need to find appropriate  $I$  and  $\pi$  such that the number of green agents for  $\pi$  w.r.t.  $I$  is at least  $\log_2\frac{2\lambda}{\varepsilon}$ .
Proof of Theorem 6.1 Let us start by defining the set  $I$ . By Lemma 6.4, we can provide an estimation  $\bar{\mu}$  of  $\mu$  such that  $\mu \in (\bar{\mu} - \alpha, \bar{\mu} + \alpha)$  for a certain  $\alpha$  with confidence  $1 - \delta$ . As a consequence, if we can consider  $I$  as the intersection of the interval having extreme points  $(1 \pm \Delta)$ $(\bar{\mu} \pm \alpha)$  and the set  $\mathcal{X}$  of sizes exactly learned during the learning phase. By Lemma 6.5, with confidence  $1 - \delta$ , we can say that  $I_{\mathcal{D}}(\varepsilon) \subseteq \mathcal{X}$  since with such confidence  $I_{\mathcal{D}}(\varepsilon)$  has been learned. By union bound, with confidence  $1 - 2\delta$ ,  $I_{\mathcal{D}}(\varepsilon) \subseteq I$ . W.l.o.g. we can assume our confidence to be  $1 - \delta$  by replacing  $\delta$
with  $\delta /2$ . Moreover, being  $|I|\leq 2\left(\Delta \bar{\mu} +\alpha\right)$  and  $\bar{\mu}\geq 1$  and choosing  $\alpha = \min \left\{\frac{1}{2\sqrt{n}},\frac{n}{\lambda + 1}\right\}$  we can derive the following bound:
$$
| I | \leq n \sqrt {\frac {1 3 (\lambda + 1) \log \frac {4}{\varepsilon}}{n}}.
$$
By the pigeonhole principle, there is one  $s \in I$  such that at least  $\frac{n}{|I|}$  agents prefer  $s$  over the other coalition sizes in  $I$ . Using the hypothesis on  $\varepsilon$ , we can conclude  $\frac{n}{|I|} \geq \log_2\frac{2\lambda}{\varepsilon}$ .
Let us now create the desired  $\pi$ . Let  $q, r$  be two positive integers s.t.  $n = qs + r$  and  $r < s$ . Giving priorities to the agents having  $s$  as the most preferred size in  $I$ , we create  $q$  coalitions of size  $s$  and a coalition of size  $r$ . In this way, at least  $\log_2\frac{2\lambda}{\varepsilon}$  agents are in a coalition having the most preferred size within  $I$  and, therefore, they are green agents for  $\pi$  w.r.t.  $I$ . By Lemma 6.6 the thesis follows.
Finally, we turn our attention to a special but really popular subclass of anonymous HGs. In this case, thanks to the further assumption of single-peakedness of agents' preferences we can achieve better values of  $\varepsilon$ . Because of space constraints, the proof of Theorem 6.2 is deferred to the Appendix.
We introduced and studied the  $\varepsilon$ -fractional core stability concept, a natural relaxation of core stability where only a small ratio (or a small probability mass) of coalitions is allowed to core-block.
We investigated this concept on two fundamental classes of HGs: Simple FHGs and anonymous HGs. For both these classes the problem of deciding the existence of a core-stable partition is notably hard: NP-complete for anonymous HGs and even  $\Sigma_2^p$ -complete for simple FHGs. While in Section 4 we showed that very small value of  $\varepsilon$  and the choice of the distributions pose limits to the existence of  $\varepsilon$ -FC solutions, we have still been able to obtain positive results for both classes under different assumptions on the considered distributions. For simple FHGs we showed that, when sampling from a uniform distribution (that corresponds to  $\varepsilon$ -FC as in Definition 3.2), it is possible to always construct an  $\varepsilon$ -FC under the assumption that  $\varepsilon$  does not decrease faster than a sub-exponential value. For anonymous HGs instead, we were able to show the existence of  $\varepsilon$ -FC for a much broader class of distributions (i.e. the  $\lambda$ -bounded ones), under a similar condition over  $\varepsilon$ . These encouraging results show that, while having a natural probabilistic nature, the notion of  $\varepsilon$ -FC stability is flexible enough to allow positive results in very complex settings where core-stability is usually considered unattainable. Moreover, its very definition and connection with the concept of PAC stabilizability makes it resilient to possible uncertainty of agents' preferences, increasing its usability in applications.
Limitations of  $\varepsilon$ -fractional core stability. Beyond the core, many stability concepts have been introduced and studied in the literature of HGs. Among the others, we mention individual rationality (IR) which is considered the minimum requirement of stability: It postulates that no agent strictly prefers being alone, forming a singleton, rather than being in their current coalition. Clearly, core stability implies IR; in fact, if the outcome is not IR there must exist an agent able to form a blocking coalition on her own. Despite IR being implied by core-stability, this is not the case for  $\varepsilon$ -FC and the algorithms presented in this paper do not necessarily satisfy IR. Specifically, in the case of simple fractional HGs, any outcome is IR and therefore it is our proposed solution; on the other hand, in the case of anonymous HGs, our algorithm provides an IR solution if coalitions of size 1 are less preferred than any other coalition size. That said, we believe it is possible to modify our algorithm, under the reasonable assumption that the value of the singleton coalition is known for each agent, in a way that also IR is satisfied, at the cost of a possible worse lower bound on  $\varepsilon$ .
Future directions. This contribution has a high potential for future work. A first natural question is whether it is possible to extend our positive results for simple FHGs to the more general class of  $\lambda$ -bounded distributions. We believe that such an extension is attainable but nonetheless non-trivial. Moreover, although we showed that exponentially small values of  $\varepsilon$  are not possible, it is still worth investigating which is the best guarantee for general distributions. More broadly, there are several HG classes that have not been considered in our work, and understanding for which  $\varepsilon$  an  $\varepsilon$ -FC partition exists is certainly of interest. As discussed in the previous paragraph, it would be definitely of interest to come up with an algorithm for anonymous HGs guaranteeing the solutions to be IR. Generally speaking, it would be worth studying  $\varepsilon$ -FC in conjunction with IR in other HGs classes.
We acknowledge the support of the PNRR MIUR project FAIR - Future AI Research (PE00000013), Spoke 9 - Green-aware AI, the PNRR MIUR project VITALITY (ECS00000041), Spoke 2 ASTRA - Advanced Space Technologies and Research Alliance, the Italian MIUR PRIN 2017 project ALGADIMAR - Algorithms, Games, and Digital Markets (2017R9FHSR_002) and the DFG, German Research Foundation, grant (Ho 3831/5-1).
We thank the anonymous reviewers for their insightful comments and suggestions, which helped us to improve the quality of the manuscript.