arXiv subjects

Michel Grabisch

Publications and source records attributed to Michel Grabisch.

At least 19 recordsLinked to original sources

arXiv2026

Cooperative games with uncertainty: a survey

The main stream of the literature in cooperative game theory considers games as being deterministic. In this survey, we review the main models incorporating uncertainty in cooperative games, starting from the seminal paper of Charnes and Granot (1976) and finishing with our own contribution, called Bel coalitional games. The different models are explained and compared, making a focus on the notion of allocation and core, since all the described models propose a notion of core.

cs.GT↗

arXiv2026

Properties of the core and other solution concepts of Bel coalitional games in the ex-ante scenario

We study the properties of the core and other solution concepts of Bel coalitional games, that generalize classical coalitional games by introducing uncertainty in the framework. In this uncertain environment, we work with contracts, that specify how agents divide the values of the coalitions in the different states of the world. Every agent can have different a priori knowledge on the true state of the world, which is modeled through the Dempster-Shafer theory, while agents' preferences between contracts are modeled by the Choquet integral. We focus on the "ex-ante" scenario, when the contract is evaluated before uncertainty is resolved. We investigate the geometrical structure of the ex-ante core when agents have the same a priori knowledge which is a probability distribution. Finally, we define the (pre)nucleolus, the kernel and the bargaining set (a la Mas-Colell) in the ex-ante situation and we study their properties. It is found that the inclusion relations among these solution concepts are the same as in the classical case. Coincidence of the ex-ante core and the ex-ante bargaining set holds for convex Bel coalitional games, at the price of strengthening the definition of bargaining sets.

cs.GT↗

arXiv2026

On the closest balanced game

Cooperative games with nonempty core are called balanced, and the set of balanced games is a polyhedron. Given a game with empty core, we look for the closest balanced game, in the sense of the (weighted) Euclidean distance, i.e., the orthogonal projection of the game on the set of balanced games. Besides an analytical approach which becomes rapidly intractable, we propose a fast algorithm to find the closest balanced game, avoiding exponential complexity for the optimization problem, and being able to run up to 20 players. We show experimentally that the probability that the closest game has a core reduced to a singleton tends to 1 when the number of players grow. We provide a mathematical proof that the proportion of facets whose games have a non-singleton core tends to 0 when the number of players grow, by finding an expression of the aymptotic growth of the number of minimal balanced collections. This permits to prove mathematically the experimental result. Consequently, taking the core of the projected game defines a new solution concept, which we call least square core due to its analogy with the least core, and our result shows that the probability that this is a point solution tends to 1 when the number of players grow.

cs.GT↗

arXiv2025

Minimal balanced collections and their applications to core stability and other topics of game theory

Minimal balanced collections are a generalization of partitions of a finite set of n elements and have important applications in cooperative game theory and discrete mathematics. However, their number is not known beyond n = 4. In this paper we investigate the problem of generating minimal balanced collections and implement the Peleg algorithm, permitting to generate all minimal balanced collections till n = 7. Secondly, we provide practical algorithms to check many properties of coalitions and games, based on minimal balanced collections, in a way which is faster than linear programming-based methods. In particular, we construct an algorithm to check if the core of a cooperative game is a stable set in the sense of von Neumann and Morgenstern. The algorithm implements a theorem according to which the core is a stable set if and only if a certain nested balancedness condition is valid. The second level of this condition requires generalizing the notion of balanced collection to balanced sets.

cs.GT↗

arXiv2025

On the Set of Balanced Games

We study the geometric structure of the set of cooperative transferable utility games having a nonempty core, characterized by Bondareva and Shapley as balanced games. We show that this set is a nonpointed polyhedral cone, and we find the set of its extremal rays and facets. This study is also done for the set of balanced games whose value for the grand coalition is fixed, which yields an affine nonpointed polyhedral cone. Finally, the case of nonnegative balanced games with fixed value for the grand coalition is tackled. This set is a convex polytope, with remarkable properties. We characterize its vertices and facets, study the adjacency structure of vertices, develop an algorithm for generating vertices in a random uniform way, and show that this polytope is combinatorial and its adjacency graph is Hamiltonian. Last, we give a characterization of the set of games having a core reduced to a singleton. Funding: This work was supported by the Spanish Government [Grant PID2021-124933NB-I00].

math.CO↗

arXiv2024

Polynomial representation of TU-games

We propose in this paper a polynomial representation of TU-games, fuzzy measures, capacities, and more generally set functions. Our representation needs a countably infinite set of players and the natural ordering of finite sets of $\mathbb{N}$, defined recursively. For a given basis of the vector space of games, we associate to each game $v$ a formal polynomial of degree at most $2^n-1$ whose coefficients are the coordinates of $v$ in the given basis. By the fundamental theorem of algebra, $v$ can be represented by the roots of the polynomial. We present some new families of games stemming from this polynomial context, like the irreducible games, the multiplicative games and the cyclotomic games.

math.CO↗

arXiv2023

An improvement of Random Node Generator for the uniform generation of capacities

Capacity is an important tool in decision-making under risk and uncertainty and multi-criteria decision-making. When learning a capacity-based model, it is important to be able to generate uniformly a capacity. Due to the monotonicity constraints of a capacity, this task reveals to be very difficult. The classical Random Node Generator (RNG) algorithm is a fast-running speed capacity generator, however with poor performance. In this paper, we firstly present an exact algorithm for generating a $n$ elements' general capacity, usable when $n < 5$. Then, we present an improvement of the classical RNG by studying the distribution of the value of each element of a capacity. Furthermore, we divide it into two cases, the first one is the case without any conditions, and the second one is the case when some elements have been generated. Experimental results show that the performance of this improved algorithm is much better than the classical RNG while keeping a very reasonable computation time.

cs.DM↗

arXiv2023

Player-centered incomplete cooperative games

The computation of a solution concept of a cooperative game usually employs values of all coalitions. However, in some applications, the values of some of the coalitions might be unknown due to high costs associated with their determination or simply because it is not possible to determine them exactly. We introduce a method to approximate standard solution concepts based only on partial characteristic function of the cooperative game. In this paper, we build on our previous results and generalise the results of our methods to a significantly larger class of structures of incomplete information.

cs.GT↗

arXiv2022

A $k$-additive Choquet integral-based approach to approximate the SHAP values for local interpretability in machine learning

Besides accuracy, recent studies on machine learning models have been addressing the question on how the obtained results can be interpreted. Indeed, while complex machine learning models are able to provide very good results in terms of accuracy even in challenging applications, it is difficult to interpret them. Aiming at providing some interpretability for such models, one of the most famous methods, called SHAP, borrows the Shapley value concept from game theory in order to locally explain the predicted outcome of an instance of interest. As the SHAP values calculation needs previous computations on all possible coalitions of attributes, its computational cost can be very high. Therefore, a SHAP-based method called Kernel SHAP adopts an efficient strategy that approximate such values with less computational effort. In this paper, we also address local interpretability in machine learning based on Shapley values. Firstly, we provide a straightforward formulation of a SHAP-based method for local interpretability by using the Choquet integral, which leads to both Shapley values and Shapley interaction indices. Moreover, we also adopt the concept of $k$-additive games from game theory, which contributes to reduce the computational effort when estimating the SHAP values. The obtained results attest that our proposal needs less computations on coalitions of attributes to approximate the SHAP values.

cs.LG↗

arXiv2022

An approximation algorithm for random generation of capacities

Capacities on a finite set are sets functions vanishing on the empty set and being monotonic w.r.t. inclusion. Since the set of capacities is an order polytope, the problem of randomly generating capacities amounts to generating all linear extensions of the Boolean lattice. This problem is known to be intractable even as soon as $n>5$, therefore approximate methods have been proposed, most notably one based on Markov chains. Although quite accurate, this method is time consuming. In this paper, we propose the 2-layer approximation method, which generates a subset of linear extensions, eliminating those with very low probability. We show that our method has similar performance compared to the Markov chain but is much less time consuming.

cs.DM↗

arXiv2022

The threshold model with anticonformity under random sequential updating

We study an asymmetric version of the threshold model with anticonformity under asynchronous update mode that mimics continuous time. We study this model on a complete graph using three different approaches: mean-field approximation, Monte Carlo simulation, and the Markov chain approach. The latter approach yields analytical results for arbitrarily small systems, in contrast to the mean-field approach, which is strictly correct only for an infinite system. We show that for sufficiently large systems, all three approaches produce the same results, as expected. We consider two cases: (1) homogeneous, in which all agents have the same tolerance threshold, and (2) heterogeneous, in which the thresholds are given by a beta distribution parametrized by two positive shape parameters $α$ and $β$. The heterogeneous case can be treated as a generalized model that reduces to a homogeneous model in special cases. We show that particularly interesting behaviors, including social hysteresis and critical mass, arise only for values of $α$ and $β$ that yield the shape of the distribution observed in real social systems.

physics.soc-ph↗

arXiv2021

Diffusion in large networks

We investigate the phenomenon of diffusion in a countably infinite society of individuals interacting with their neighbors in a network. At a given time, each individual is either active or inactive. The diffusion is driven by two characteristics: the network structure and the diffusion mechanism represented by an aggregation function. We distinguish between two diffusion mechanisms (probabilistic, deterministic) and focus on two types of aggregation functions (strict, Boolean). Under strict aggregation functions, polarization of the society cannot happen, and its state evolves towards a mixture of infinitely many active and infinitely many inactive agents, or towards a homogeneous society. Under Boolean aggregation functions, the diffusion process becomes deterministic and the contagion model of Morris (2000) becomes a particular case of our framework. Polarization can then happen. Our dynamics also allows for cycles in both cases. The network structure is not relevant for these questions, but is important for establishing irreducibility, at the price of a richness assumption: the network should contain infinitely many complex stars and have enough space for storing local configurations. Our model can be given a game-theoretic interpretation via a local coordination game, where each player would apply a best-response strategy in a random neighborhood.

cs.SI↗

arXiv2020

An unsupervised capacity identification approach based on Sobol' indices

In many ranking problems, some particular aspects of the addressed situation should be taken into account in the aggregation process. An example is the presence of correlations between criteria, which may introduce bias in the derived ranking. In these cases, aggregation functions based on a capacity may be used to overcome this inconvenience, such as the Choquet integral or the multilinear model. The adoption of such strategies requires a stage to estimate the parameters of these aggregation operators. This task may be difficult in situations in which we do not have either further information about these parameters or preferences given by the decision maker. Therefore, the aim of this paper is to deal with such situations through an unsupervised approach for capacity identification based on the multilinear model. Our goal is to estimate a capacity that can mitigate the bias introduced by correlations in the decision data and, therefore, to provide a fairer result. The viability of our proposal is attested by numerical experiments with synthetic data

eess.SP↗

arXiv2019

A model of anonymous influence with anti-conformist agents

We study a stochastic model of anonymous influence with conformist and anti-conformist individuals. Each agent with a `yes' or `no' initial opinion on a certain issue can change his opinion due to social influence. We consider anonymous influence, which depends on the number of agents having a certain opinion, but not on their identity. An individual is conformist/anti-conformist if his probability of saying `yes' increases/decreases with the number of `yes'-agents. We focus on three classes of aggregation rules (pure conformism, pure anti-conformism, and mixed aggregation rules) and examine two types of society (without, and with mixed agents). For both types we provide a complete qualitative analysis of convergence, i.e., identify all absorbing classes and conditions for their occurrence. Also the pure case with infinitely many individuals is studied. We show that, as expected, the presence of anti-conformists in a society brings polarization and instability: polarization in two groups, fuzzy polarization (i.e., with blurred frontiers), cycles, periodic classes, as well as more or less chaotic situations where at any time step the set of `yes'-agents can be any subset of the society. Surprisingly, the presence of anti-conformists may also lead to opinion reversal: a majority group of conformists with a stable opinion can evolve by a cascade phenomenon towards the opposite opinion, and remains in this state.

cs.SI↗

arXiv2018

The cone of supermodular games on finite distributive lattices

In this article we study supermodular functions on finite distributive lattices. Relaxing the assumption that the domain is a powerset of a finite set, we focus on geometrical properties of the polyhedral cone of such functions. Specifically, we generalize the criterion for extremality and study the face lattice of the supermodular cone. An explicit description of facets by the corresponding tight linear inequalities is provided.

math.CO↗

arXiv2018

An interaction index for multichoice games

Models in Multicriteria Decision Analysis (MCDA) can be analyzed by means of an importance index and an interaction index for every group of criteria. We consider first discrete models in MCDA, without further restriction, which amounts to considering multichoice games, that is, cooperative games with several levels of participation. We propose and axiomatize an interaction index for multichoice games. In a second part, we consider the continuous case, supposing that the continuous model is obtained from a discrete one by means of the Choquet integral.

cs.GT↗

arXiv2017

Game Theoretic Interaction and Decision: A Quantum Analysis

An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, Fourier transformation of cooperative games becomes meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.

quant-ph↗

arXiv2017

Axiomatization of an importance index for $k$-ary games

We consider MultiCriteria Decision Analysis models which are defined over discrete attributes, taking a finite number of values. We do not assume that the model is monotonically increasing with respect to the attributes values. Our aim is to define an importance index for such general models, considering that they are equivalent to $k$-ary games (multichoice games). We show that classical solutions like the Shapley value are not suitable for such models, essentially because of the efficiency axiom which does not make sense in this context. We propose an importance index which is a kind of average variation of the model along the attributes. We give an axiomatic characterization of it.

cs.GT↗