Probably approximately correct stability of allocations in uncertain coalitional games with private sampling

Multi-agent systems are pervasive across various fields, including engineering (Raja and Grammatico, 2021; Karaca and Kamgarpour, 2020; Han et al., 2019; Fele et al., 2017, 2018), economics, and social sciences (McCain, 2008). Even though agents often behave as self-interested entities, their limited ability to improve their utility can, in certain scenarios, motivate them to form coalitions to achieve a higher individual payoff. This situation can be modelled through coalitional games (Chalkiadakis et al., 2011). As each agent’s participation in a coalition is subject to their own payoff being maximised, a question emerges about how to allocate the total value of the coalition such that no agent deviates from it. This problem is known as stability of agents’ allocations.

In real-world scenarios, the value of a coalition is typically affected by uncertainty. The consideration of uncertainty in the coalitional values of a game finds its roots in seminal works such as Charnes and Granot (1976, 1977, 1973), which provided solution concepts to yield stable allocations against uncertainty. Suijs et al. (1999) discussed non-emptiness of the core for a particular class of stochastic games. Chalkiadakis and Boutilier (2004) and Ieong and Shoham (2008) addressed uncertainty through Bayesian learning methods, while Li and Conitzer (2015) investigated different solution concepts that maximize the probability of obtaining stable allocations. Chalkiadakis and Boutilier (2008) and Guéneron and Bonnet (2021) explored the dynamics of repeated stochastic coalitional games.

The connection between probably approximately correct (PAC) learning and uncertain coalitional games has been explored in Balcan et al. (2015). The spirit of Procaccia and Rosenschein (2006) is similar, using Vapnik-Chervonenkis (VC) theory to learn the winning coalitions for the class of the so-called simple games. Balcan et al. (2015) focused on a complementary problem where only a randomized subset of coalitions is considered. Both these works evaluate the sample complexity from the VC theory perspective, and hence their results suffer from the associated conservativeness. Pantazis et al. (2022a) leverage the scenario approach (Campi and Garatti, 2021, 2018a, 2018b; Garatti and Campi, 2022) to provide distribution-free guarantees on the stability of allocations in a PAC manner, based on samples from the exogenous uncertainty affecting the value functions. On a parallel line of research, Pantazis et al. (2023) propose a data-driven Wasserstein-based distributionally robust approach for allocations’ stability.

Unlike the aforementioned works, we consider a more general setting where the uncertainty data is privately drawn by each agent. As such, information is heterogeneous across agents, and the samples are regarded as a private resource. When data samples are commonly shared among agents, the work of Pantazis et al. (2022a) provide guarantees on stability of allocations. In particular, the developments in Pantazis et al. (2022a) rely on the notion of scenario core – a data-driven approximation of the robust core based on the hypothesis that all uncertainty samples are shared among the agents (Raja and Grammatico, 2021). Given this common set of samples, the perception of a stable allocation based on the available data is identical across agents. This is no longer the case for private sampling, as different samples of the exogenous uncertainty can result in a different perception of stability by each agent. We adopt a PAC learning approach and leverage the concept of compression (Margellos et al., 2015), i.e., the set of samples essential for the reconstruction of the scenario core and whose cardinality affects the generalization properties of the provided guarantees. We give a priori and a posteriori certificates for the entire scenario core obtained by private sampling, and propose a distributed algorithm to calculate a compression set.

We then focus on the specific allocation returned by some algorithm (akin, e.g., to the distributed payoff algorithm by Raja and Grammatico (2021)). We then provide a priori coalition stability guarantees for this allocation (Theorem 3). We show how the dimension of the problem – in our case the number of agents – plays a key role. Less conservative probabilistic bounds can often be obtained by taking an a posteriori approach (Campi et al., 2018), i.e., based on observation of the realized uncertainty. In this case, the probabilistic bounds can be significantly improved and even a tighter a priori bound can be obtained (see Theorem 4). Finally, we consider a relaxed $\zeta$-core to include nearby allocations and also address the case of empty core. We leverage recents results by Campi and Garatti (2021) and study the probabilistic stability of allocations in the $\zeta$-core.

We consider a coalitional game with $N$ agents identified by the set $\pazocal{N}=\{1,\dots,N\}$. We denote the number of possible subcoalitions, excluding the grand coalition, by $M$, i.e., $M=2^{N}-1$. The worth of a coalition $S\subseteq\pazocal{N}$ is given by the so called value function; here we consider transferable utility problems, i.e., the value of coalitions is expressed by a real value which can be split among participating agents. We posit that the value of each coalition is subject to uncertainty. Then, the value of a coalition $S\subset\pazocal{N}$ is a function $u_{S}:\Xi\to\mathbb{R}$ that given an uncertainty realization $\xi\in\Xi$ returns the total payoff for the agents in $S$. The value of the grand coalition is the deterministic quantity $u{N}\in\mathbb{R}$.

The uncertain coalitional game is then defined as the tuple $G_{\Xi}=\langle\pazocal{N},\{u_{S}\}_{S\subseteq\pazocal{N}},\Xi,\mathbb{P}\rangle$, where $\mathbb{P}$ is some unknown probability measure over $\Xi$. A vector $\bm{x}\coloneqq(x_{i})_{i\in\pazocal{N}}\in\mathbb{R}^{N}$, where $x_{i}$ is the payoff received by agent $i$, is called an allocation. For a given uncertainty realization $\xi\in\Xi$, an allocation is strictly rational for the members of $S$ if $\sum_{i\in S}x_{i}>u_{S}(\xi)$; the latter implies that agents have an incentive to form this coalition for this particular uncertainty realization. An allocation is efficient if $\sum_{i\in\pazocal{N}}x_{i}=u{N}$. Efficient allocations such that there are no incentives for agents to deviate the grand coalition are called stable. The set of all such allocations is the core of the game.

The notion of robust core is proposed to account for the presence of uncertainty; see, e.g., Pantazis et al. (2022a); Raja and Grammatico (2021); Nedić and Bauso (2013). We formally define it as $C(G_{\Xi})\coloneqq\{\bm{x}\in\mathbb{R}^{N}\colon\textstyle\sum_{i\in\pazocal{N}}x_{i}=u{N},\,\textstyle\sum_{i\in S}x_{i}\geq\max_{\xi\in\Xi}u_{S}(\xi)\text{ for all }S\subset\pazocal{N}\}$. For any possible realization of the uncertainty $\xi\in\Xi$, any allocation $\bm{x}\in C(G_{\Xi})$ gives the agents no incentive to defect from the grand coalition and form sub-coalitions. Unfortunately, computing explicitly the robust core is hard, as we assume no knowledge on the uncertainty support $\Xi$ (nor on the underlying probability distribution $\mathbb{P}$). To circumvent this challenge, we adopt a data-driven methodology and approximate the robust core by drawing a finite number $K$ of independent and identically distributed (i.i.d.) samples $\bm{\xi}\coloneqq(\xi^{(1)},\ldots,\xi^{(K)})\in\Xi^{K}$, where $\Xi^{K}$ denotes the $K$-fold cartesian product of $\Xi$; we refer to vectors $\bm{\xi}$ as multi-samples. This constitutes the scenario game $G_{K}=\langle\pazocal{N},\{u_{S}\}_{S\subseteq\pazocal{N}},\bm{\xi}\rangle$, whose core is the set $\{\bm{x}\in\mathbb{R}^{N}:\sum_{i\in\pazocal{N}}x_{i}=u{N}\text{ and }\sum_{i\in S}x_{i}\geq\max_{k=1,\dots K}u_{S}(\xi^{(k)}),\,\forall S\subset\pazocal{N}\}$, referred to as the scenario core.

We now take the notion of allocation stability a step further, considering the more general setting where every agent $i$ has only access to a private set of samples $\bm{\xi}_{i}$ from $\Xi$. Let $\bm{\xi}_{i}\in\Xi^{K_{i}}$ be the multi-sample privately drawn by agent $i$. We say that an allocation $\bm{x}$ of $G_{K}$ is stable with respect to $\bm{\xi}_{i}$ if $\sum_{i\in\pazocal{N}}x_{i}=u{N}$ and $\sum_{i\in S}x_{i}\geq\max_{k=1,\dots K_{i}}u_{S}(\xi_{i}^{(k)})$, $\forall S\subset\pazocal{N}$ s.t. $S\supseteq\{i\}$.

This immediately leads to the following extension of the scenario core:

Unless differently specified, we assume the following conditions hold throughout the paper:

In the following, we let $K=\sum_{i\in\pazocal{N}}K_{i}$, and $\bm{\xi}=(\bm{\xi}_{i})_{i=1}^{N}$. Also, for brevity, we will use $S\supseteq\{i\}$ to denote all $S\subset\pazocal{N}$ which allow agent $i$ as a member.

On the basis of privately available data, we wish to provide guarantees on the probability that allocations $\bm{x}\in C(G_{K})$ will remain stable (i.e., within $C(G_{K})$) for any future, yet unseen, uncertainty realization. Capitalizing on Pantazis et al. (2022b); Fabiani et al. (2022); Pantazis et al. (2022a), we define two probabilistic notions of instability. In particular, the first refers to a particular allocation in the core, whereas the second involves the entire set $C(G_{K})$.

$\mathbb{V}(C(G_{K}))$ thus denotes the probability with respect to the realizations of $\xi$ that, for some $S$ with value function $u_{S}(\xi)$, at least one of the allocations in the scenario core will become unstable, i.e., will be dominated by the option of defecting from the grand coalition to form $S$.

To bound the probability of core instability in a PAC fashion, we introduce two key concepts from statistical learning theory, namely the algorithm and the compression set (Margellos et al., 2015). The latter refers to the fact that only a subset of data from $\bm{\xi}$ may be sufficient to produce the same scenario core. As we will see later this underpins the quality of the provided probabilistic stability guarantees.

Any compression set of least cardinality is called minimal compression set. Another important notion used in our derivations is the support rank (Schildbach et al., 2012).

In words, the support rank of an uncertain constraint is equal to the dimension of the problem at hand minus the dimension of the maximal unconstrained space of the constraint.

Let $\bm{x}^{\ast}$ be the unique allocation in $C(G_{K})$ which maximises some convex utility function $f(\cdot)$.¹¹1Uniqueness of the maximiser is without loss of generality. If multiple maximisers are possible, a convex tie-break rule can be applied to single out a unique solution. Recalling the notion of support rank as per Definition 2.4, the following result holds for $\bm{x}^{*}$.

Proof: Similarly to the proofline of Theorem 1, we have that

To obtain the last inequality consider the following optimization program

We then group the constraints depending on which agent they correspond to. Let $X_{i}(\xi)\coloneqq\{\bm{x}\in\mathbb{R}^{N}\colon\sum_{i\in S}x_{i}\geq u_{S}(\xi),\,\forall S\supseteq\{i\}\}$. With $\beta_{i}$ defined as above, from Schildbach et al. (2012) we have

which concludes the proof. $\blacksquare$

Note that the confidence parameter $\beta_{i}$ in Theorem 3 depends on the number of samples $K_{i}$ of agent $i\in\pazocal{N}$ and the violation level $\epsilon_{i}$ set individually by the agent to determine the probability of satisfaction of the rationality constraints. Most importantly, $\beta_{i}$ depends on the so called support rank $\rho_{i}$ of the coalitional constraints corresponding to $i\in\pazocal{N}$, which is in any instance upper bounded by the dimension of the decision variable in (8). Theorem 3 is an a priori result and in certain cases can be conservative. The following result provides an improved probabilistic bound, based on the observation of the uncertainty realization. Let

Note that (9) is compatible with (3), and allows to explicitly compute $\epsilon_{i}$ as a function of $\beta_{i}$.

Recall that, as in Theorem 4, $s_{i,K}$ is the cardinality of the (sub)compression set associated with the rationality constraints relative to all $S\supseteq\{i\}$, quantified a posteriori, e.g., by means of the procedure illustrated in Campi et al. (2018, §II).

Finally, an a priori bound can be derived from Theorem 4 by considering the worst case among all a posteriori observable compressions $\{s_{i,K}\}$; this bound is nontrivial by noticing that $\sum_{i\in\pazocal{N}}s_{i,K}\leq N$, which follows from Calafiore and Campi (2006, Th. 3).

In this work we considered uncertain coalitional games and proposed a data-driven methodology to study the stability of allocations for the general setting where uncertainty is privately sampled by the agents. Future work will investigate stability of allocations through a distributionally robust framework, as well as analyse the case where the value of the grand coalition can also be affected by uncertainty.

F. Fele gratefully acknowledges support from grant RYC2021-033960-I funded by MCIN/AEI/ 10.13039/501100011033 and European Union NextGenerationEU/PRTR, as well as from grant PID2022-142946NA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by ERDF A way of making Europe.