arXiv subjects

Dylan Laplace Mermoud

Publications and source records attributed to Dylan Laplace Mermoud.

12 recordsLinked to original sources

arXiv2026

The algebraic structures of social organizations: the operad of cooperative games

The main goal of this paper is to settle a conceptual framework for cooperative game theory in which the notion of composition/aggregation of games is the defining structure. This is done via the mathematical theory of algebraic operads: we start by endowing the collection of all cooperative games with any number of players with an operad structure, and we show that it generalises all the previous notions of sums, products and compositions of games considered by Owen, Shapley, von Neumann and Morgenstern, and many others. Furthermore, we explicitly compute this operad in terms of generators and relations, showing that the Möbius transform map induces a canonical isomorphism between the operad of cooperative games and the operad that encodes commutative triassociative algebras. In other words, we prove that any cooperative game is a linear combination of iterated compositions of the 2-player bargaining game and the 2-player dictator games. We show that many interesting classes of games (simple, balanced, capacities a.k.a fuzzy measures and convex functions, totally monotone, etc) are stable under compositions, and thus form suboperads. In the convex case, this gives by the submodularity theorem a new operad structure on the family of all generalized permutahedra. Finally, we focus on how solution concepts in cooperative game theory behave under composition: we study the core of a composite and describe it in terms of the core of its components, and we give explicit formulas for the Shapley value and the Banzhaf index of a compound game.

math.CO↗

arXiv2026

Scaling QAOA: transferring optimal adiabatic schedules from small-scale to large-scale variational circuits

The Quantum Approximate Optimization Algorithm (QAOA) is a leading approach for combinatorial optimization on near-term quantum devices, yet its scalability is limited by the difficulty of optimizing $2p$ variational parameters for a large number $p$ of layers. Recent empirical studies indicate that optimal QAOA angles exhibit concentration and transferability across problem sizes. Leveraging this observation, we propose a schedule-learning framework that transfers spectral-gap-informed adiabatic control strategies from small-scale instances to larger systems. Our method extracts the spectral gap profile of small problems and constructs a continuous schedule governed by $\partial_t s = κg^q(s)$, where $g(s)$ is the instantaneous gap and $(κ, q)$ are global hyperparameters. Discretizing this schedule yields closed-form expressions for all QAOA angles, reducing the classical optimization task from $2p$ parameters to only $2$, independent of circuit depth. This drastic parameter compression mitigates classical optimization overhead and reduces sensitivity to barren plateau phenomena. Numerical simulations on random QUBO and 3-regular MaxCut instances demonstrate that the learnt schedules transfer effectively to larger systems while achieving competitive approximation ratios. Our results suggest that gap-informed schedule transfers provide a scalable and parameter-efficient strategy for QAOA.

quant-ph↗

arXiv2026

Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond

We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the $\mathtt{cx}$ gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, \textsc{QuPer}, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with $256$ variables, requiring $20$ qubits.

quant-ph↗

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

Diffusion in dynamic networks with time-varying inputs to allocate responsibility

Responsibility in complex networks extends beyond direct actions: players should also bear responsibility for the indirect effects within their supply chains or network. We introduce a novel framework to allocate responsibility for indirect environmental, social, and economic impacts across a dynamic network. Unlike static approaches, our framework accounts for the evolving structure of supply chains, financial systems, and other interconnected systems, where relationships change over time. We use the time-dependent Laplacian matrix to capture how responsibility propagates through the network, revealing a diffusion process that aligns with key axioms of fairness: linearity, efficiency, symmetry, and the independent player property. We show that approximating the responsibility measure preserves these properties, supporting the use of our framework as a rigorous method to allocate responsibility in real-world networks.

econ.TH↗

arXiv2025

Sampled-Based Guided Quantum Walk: Non-variational quantum algorithm for combinatorial optimization

We introduce SamBa-GQW, a novel quantum algorithm for solving binary combinatorial optimization problems of arbitrary degree with no use of any classical optimizer. The algorithm is based on a continuous-time quantum walk on the solution space represented as a graph. The walker explores the solution space to find its way to vertices that minimize the cost function of the optimization problem. The key novelty of our algorithm is an offline classical sampling protocol that gives information about the spectrum of the problem Hamiltonian. Then, the extracted information is used to guide the walker to high quality solutions via a quantum walk with a time-dependent hopping rate. We investigate the performance of SamBa-GQW on several quadratic problems, namely MaxCut, maximum independent set, portfolio optimization, and higher-order polynomial problems such as LABS, MAX-$k$-SAT and a quartic reformulation of the travelling salesperson problem. We empirically demonstrate that SamBa-GQW finds high quality approximate solutions on problems up to a size of $n=20$ qubits by only sampling $n^2$ states among $2^n$ possible decisions. SamBa-GQW compares very well also to other guided quantum walks and QAOA.

quant-ph↗

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

Attraction of the core and the cohesion flow

We adopt a continuous-time dynamical system approach to study the evolution of the state of a game driven by the willingness to reduce the total dissatisfaction of the coalitions about their payment. Inspired by the work of Grabisch and Sudhölter about core stability, we define a vector field on the set of preimputations from which is defined, for any preimputation, a cohesion curve describing the evolution of the state. We prove that for each preimputation, there exists a unique cohesion curve. Subsequently, we show that, for the cohesion flow of a balanced game, the core is the unique minimal attractor of the flow, the realm of which is the whole preimputation set. These results improve our understanding of the ubiquity of the core in the study of cooperative games with transferable utility.

econ.TH↗

arXiv2024

On the Formation of Steady Coalitions

This paper studies the formation of the grand coalition of a cooperative game by investigating its possible internal dynamics. Each coalition is capable of forcing all players to reconsider the current state of the game when it does not provide sufficient payoff. Different coalitions may ask for contradictory evolutions, leading to the impossibility of the grand coalition forming. In this paper, we give a characterization of the impossibility, for a given state, of finding a new state dominating the previous one such that each aggrieved coalition has a satisfactory payoff. To do so, we develop new polyhedral tools related to a new family of polyhedra, appearing in numerous situations in cooperative game theory.

econ.TH↗

arXiv2024

Projection onto the core: An optimal reallocation to correct market failure

This paper provides formulae and algorithms to compute the projection onto the core of a preimputation outside it. The core of a game is described using an exponential number of linear constraints, and we cannot know beforehand which are redundant or defining the polytope. We apply these new results to market games, a class of games in which every game has a nonempty core. Given an initial state of the game represented by a preimputation, it is not guaranteed that the state of the game evolves toward the core following the dynamics induced by the domination relations. Our results identify and compute the most efficient side payment that acts on a given state of the game and yields its closest core allocation. Using this side payment, we propose a way to evaluate the failure of a market to reach a state of the economy belonging to the core, and we propose a new solution concept consisting of preimputations that minimizes this failure.

math.MG↗

arXiv2024

Combinatorics on Social Configurations

In cooperative game theory, the social configurations of players are modeled by balanced collections. The Bondareva-Shapley theorem, perhaps the most fundamental theorem in cooperative game theory, characterizes the existence of solutions to the game that benefit everyone using balanced collections. Roughly speaking, if the trivial set system of all players is one of the most efficient balanced collections for the game, then the set of solutions from which each coalition benefits, the so-called core, is non-empty. In this paper, we discuss some interactions between combinatorics and cooperative game theory that are still relatively unexplored. Indeed, the similarity between balanced collections and uniform hypergraphs seems to be a relevant point of view to obtain new properties on those collections through the theory of combinatorial species.

cs.GT↗

arXiv2023

Geometry of Set Functions in Game Theory: Combinatorial and Computational Aspects

The main ambition of this thesis is to contribute to the development of cooperative game theory towards combinatorics, algorithmics and discrete geometry. Therefore, the first chapter of this manuscript is devoted to highlighting the geometric nature of the coalition functions of transferable utility games and spotlights the existing connections with the theory of submodular set functions and polyhedral geometry. To deepen the links with polyhedral geometry, we define a new family of polyhedra, called the basic polyhedra, on which we can apply a generalized version of the Bondareva-Shapley Theorem to check their nonemptiness. To allow a practical use of these computational tools, we present an algorithmic procedure generating the minimal balanced collections, based on Peleg's method. Subsequently, we apply the generalization of the Bondareva-Shapley Theorem to design a collection of algorithmic procedures able to check properties or generate specific sets of coalitions. In the next chapter, the connections with combinatorics are investigated. First, we prove that the balanced collections form a combinatorial species, and we construct the one of k-uniform hypergraphs of size p, as an intermediary step to construct the species of balanced collections. Afterwards, a few results concerning resonance arrangements distorted by games are introduced, which gives new information about the space of preimputations and the facial configuration of the core. Finally, we address the question of core stability using the results from the previous chapters. Firstly, we present an algorithm based on Grabisch and Sudhölter's nested balancedness characterization of games with a stable core, which extensively uses the generalization of the Bondareva-Shapley Theorem introduced in the second chapter. Secondly, a new necessary condition for core stability is described, based on the application ...

cs.GT↗