arXiv subjects

Domenico Mergoni Cecchelli

Publications and source records attributed to Domenico Mergoni Cecchelli.

9 recordsLinked to original sources

arXiv2026

Refundable Deposits: How to Restore Cooperation in Finitely Repeated Games

While infinitely repeated games admit a rich set of Nash equilibria, finitely repeated games typically have a much smaller and often inefficient one. We show how to enlarge this set using deposits: in each period a player may place a refundable sum with a neutral intermediary, returned when the game ends and forfeited following a deviation. Paying these deposits is voluntary and incentive compatible at every stage, so no commitment by the players is assumed, the only commitment required being that of the intermediary to a refund rule fixed before play begins. The mechanism sustains payoff profiles more efficient than those of the standard equilibria, without altering the underlying game and without transfers between players. We demonstrate it on the prisoner's dilemma, a congestion game, and a public goods game, all settings where cooperation cannot emerge in the standard finitely repeated version. We also apply it to a dynamic common-pool resource, suggesting that the construction extends beyond repeated stage-games.

cs.GT↗

arXiv2026

A generalised transference principle

The last two decades have witnessed a growing trend towards proving sparse random analogues of combinatorial theorems. One unified approach to proving such theorems, formalised by Conlon and Gowers [Ann. of Math. 2016], involves establishing a 'transference principle' which allows one to translate between robust properties in the dense setting and the sparse $p$-random setting, provided $p$ is not too small. Our results provide a more general transference theorem, extending the results of Conlon and Gowers and also those of Schacht [Ann. of Math. 2016]. Among a variety of other applications, we use this to obtain a sparse counting lemma for graphs and hypergraphs which are not necessarily strictly balanced. Our method achieves asymptotically optimal bounds on the probability $p$, and the probability of success.

math.CO↗

arXiv2026

The Bounds of Algorithmic Collusion; $Q$-learning, Gradient Learning, and the Folk Theorem

We explore the behaviour emerging from learning agents repeatedly interacting strategically for a wide range of learning dynamics, including $Q$-learning, projected gradient, replicator and log-barrier dynamics. Going beyond the better understood classes of potential games and zero-sum games, we consider the setting of a general repeated game with finite recall under different forms of monitoring. We obtain a Folk Theorem-style result and characterise the set of payoff vectors that can be obtained by these dynamics, discovering a wide range of possibilities for the emergence of algorithmic collusion. Achieving this requires a novel technical approach, which, to the best of our knowledge, yields the first convergence result for multi-agent $Q$-learning algorithms in repeated games.

cs.GT↗

arXiv2025

Simultaneous Best-Response Dynamics in Random Potential Games

This paper examines the convergence behaviour of simultaneous best-response dynamics in random potential games. We provide a theoretical result showing that, for two-player games with sufficiently many actions, the dynamics converge quickly to a cycle of length two. This cycle lies within the intersection of the neighbourhoods of two distinct Nash equilibria. For three players or more, simulations show that the dynamics converge quickly to a Nash equilibrium with high probability. Furthermore, we show that all these results are robust, in the sense that they hold in non-potential games, provided the players' payoffs are sufficiently correlated. We also compare these dynamics to gradient-based learning methods in near-potential games with three players or more, and observe that simultaneous best-response dynamics converge to a Nash equilibrium of comparable payoff substantially faster.

cs.GT↗

arXiv2025

On product Schur triples in the integers

Schur's theorem states that in any $k$-colouring of the set of integers $[n]$ there is a monochromatic solution to $a+b=c$, provided $n$ is sufficiently large. Abbott and Wang studied the size of the largest subset of $[n]$ such that there is a $k$-colouring avoiding a monochromatic $a+b=c$. In other directions, the minimum number of $a+b=c$ in $k$-colourings of $[n]$ and the probability threshold in random subsets of $[n]$ for the property of having a monochromatic $a+b=c$ in any $k$-colouring were investigated. In this paper, we study natural generalisations of these streams to products $ab=c$, in a deterministic, random, and randomly perturbed environments.

math.CO↗

arXiv2024

Dirac's theorem for graphs of bounded bandwidth

We provide an optimal sufficient condition, relating minimum degree and bandwidth, for a graph to contain a spanning subdivision of the complete bipartite graph $K_{2,\ell}$. This includes the containment of Hamilton paths and cycles, and has applications in the random geometric graph model. Our proof provides a greedy algorithm for constructing such structures.

math.CO↗

arXiv2023

Graphs with large minimum degree and no small odd cycles are $3$-colourable

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a stronger result that works with a slightly smaller minimum degree.

math.CO↗

arXiv2022

The Ramsey numbers of squares of paths and cycles

The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the edges of the complete graph $K_N$ are coloured with red and blue, there exists either a red copy of $G$ or a blue copy of $H$. We prove that for all sufficiently large $n$ we have \[R(P_{3n}^2,P_{3n}^2)=R(P_{3n+1}^2,P_{3n+1}^2)=R(C_{3n}^2,C_{3n}^2)=9n-3\mbox{ and } R(P_{3n+2}^2,P_{3n+2}^2)=9n+1.\] We also show that for any $γ>0$ and $Δ$ there exists $β>0$ such that the following holds. If $G$ can be coloured with three colours such that all colour classes have size at most $n$, the maximum degree $Δ(G)$ of $G$ is at most $Δ$, and $G$ has bandwidth at most $βn$, then $R(G,G)\le (3+γ)n$.

math.CO↗

arXiv2022

Density of small diameter subgraphs in $K_r$-free graphs

We denote by $\text{ex}(n, H, F)$ the maximum number of copies of $H$ in an $n$-vertex graph that does not contain $F$ as a subgraph. Recently, Grzesik, Győri, Salia, Tompkins considered conditions on $H$ under which $\text{ex}(n, H, K_r)$ is asymptotically attained at a blow-up of $K_{r-1}$, and proposed a conjecture. In this note we disprove their conjecture.

math.CO↗