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Andrea Collevecchio

Publications and source records attributed to Andrea Collevecchio.

At least 19 recordsLinked to original sources
arXiv2025

Finding a Nash equilibrium of a random win-lose game in expected polynomial time

A long-standing open problem in algorithmic game theory asks whether or not there is a polynomial time algorithm to compute a Nash equilibrium in a random bimatrix game. We study random win-lose games, where the entries of the $n\times n$ payoff matrices are independent and identically distributed (i.i.d.) Bernoulli random variables with parameter $p=p(n)$. We prove that, for nearly all values of the parameter $p=p(n)$, there is an expected polynomial-time algorithm to find a Nash equilibrium in a random win-lose game. More precisely, if $p\sim cn^{-a}$ for some parameters $a,c\ge 0$, then there is an expected polynomial-time algorithm whenever $a\not\in \{1/2, 1\}$. In addition, if $a = 1/2$ there is an efficient algorithm if either $c \le e^{-52} 2^{-8} $ or $c\ge 0.977$. If $a=1$, then there is an expected polynomial-time algorithm if either $c\le 0.3849$ or $c\ge \log^9 n$.

cs.GT↗
arXiv2025

Once-Reinforced and Self-Interacting Random Walks beyond exchangeability

We present the first rigorous quantitative analysis of once-reinforced random walks (ORRW) on general graphs, based on a novel change of measure formula.~This enables us to prove large deviations estimates for the range of the walk to have cardinality of the order $N^{d/(d+2)}$ in dimension larger than or equal than two. We also prove that ORRW is transient on all non-amenable graphs for small reinforcement.~Moreover, we study the shape of oriented ORRW on euclidean lattices. We also provide a new approach to the study of general self-interacting random walk, which we apply to random walk in random environment, reinforced processes on oriented graphs, including the directed ORRW.

math.PR↗
arXiv2024

Basins of Attraction in Two-Player Random Ordinal Potential Games

We consider the class of two-person ordinal potential games where each player has the same number of actions $K$. Each game in this class admits at least one pure Nash equilibrium and the best-response dynamics converges to one of these pure Nash equilibria; which one depends on the starting point. So, each pure Nash equilibrium has a basin of attraction. We pick uniformly at random one game from this class and we study the joint distribution of the sizes of the basins of attraction. We provide an asymptotic exact value for the expected basin of attraction of each pure Nash equilibrium, when the number of actions $K$ goes to infinity.

cs.GT↗
arXiv2024

Finding pure Nash equilibria in large random games

Best Response Dynamics (BRD) is a class of strategy updating rules to find Pure Nash Equilibria (PNE) in a game. At each step, a player is randomly picked, and the player switches to a "best response" strategy based on the strategies chosen by others, so that the new strategy profile maximises their payoff. If no such strategy exists, a different player will be chosen randomly. When no player wants to change their strategy anymore, the process reaches a PNE and will not deviate from it. On the other hand, either PNE may not exist, or BRD could be "trapped" within a subgame that has no PNE. We consider a random game with $N$ players, each with two actions available, and i.i.d. payoffs, in which the payoff distribution may have an atom, i.e. ties are allowed. We study a class of random walks in a random medium on the $N$-dimensional hypercube induced by the random game. The medium contains two types of obstacles corresponding to PNE and traps. The class of processes we analyze includes BRD, simple random walks on the hypercube, and many other nearest neighbour processes. We prove that, with high probability, these processes reach a PNE before hitting any trap.

math.PR↗
arXiv2024

Strongly vertex-reinforced jump process on graphs with bounded degree

We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function $w$ is reciprocally integrable and non-decreasing, then the process visits only a finite number of vertices. In the case where $w$ is approximately equal to a super-linear polynomial, we show that the process eventually gets stuck on a star-shaped subgraph and there is exactly one vertex with unbounded local time.

math.PR↗
arXiv2020

Long range one-cookie random walk with positive speed

We study one-dimensional excited random walks with non-nearest neighbor jumps. When the process is at a vertex that has not been visited before, its next transition has a positive drift to the right, possibly with long jumps. Whenever the process visits a vertex that has already been visited in the past, its next transition is the one of a simple symmetric random walk. We give a sufficient condition for the process to have positive speed.

math.PR↗
arXiv2020

When "Better" is better than "Best"

We consider two-player normal form games where each player has the same finite strategy set. The payoffs of each player are assumed to be i.i.d. random variables with a continuous distribution. We show that, with high probability, the better-response dynamics converge to pure Nash equilibrium whenever there is one, whereas best-response dynamics fails to converge, as it is trapped.

econ.TH↗
arXiv2020

Vertex-reinforced jump process on the integers with nonlinear reinforcement

We consider a non-linear vertex-reinforced jump process (VRJP($w$)) on $\mathbb{Z}$ with an increasing measurable weight function $w:[1,\infty)\to [1,\infty)$ and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP($w$) depending on the integrability of the reciprocal of $w$. In particular, we prove that if $\int_1^{\infty} \frac{\text{d}u}{w(u)} =\infty$ then the process is recurrent, i.e. it visits each vertex infinitely often and all local times are unbounded. On the other hand, if $\int_1^{\infty} \frac{\text{d} u}{w(u)} <\infty$ and there exists a $\rho>0$ such that $t \mapsto w(t)^{\rho}\int_t^{\infty}\frac{\text{d}u}{w(u)}$ is non-increasing then the process will eventually get stuck on exactly three vertices, and there is only one vertex with unbounded local time. We also show that if the initial weights are all the same, VRJP on $\mathbb{Z}$ cannot be transient, i.e. there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov [Probab. Theory Relat. Fields (2002)] who showed that VRJP with linear reinforcement on $\mathbb{Z}$ is recurrent.

math.PR↗
arXiv2020

Three steps mixing for general random walks on the hypercube at criticality

We introduce a general class of random walks on the $N$-hypercube, study cut-off for the mixing time, and provide several types of representation for the transition probabilities. We observe that for a sub-class of these processes with long range (i.e. non-local) there exists a critical value of the range that allows an "almost-perfect" mixing in at most three steps. In other words, the total variation distance between the three steps transition and the stationary distribution decreases geometrically in $N$, which is the dimension of the hypercube. In some cases, the walk mixes almost-perfectly in exactly two steps. Notice that a well-known result (Theorem 1 in Diaconis and Shahshahani (1986)) shows that there exist no random walk on Abelian groups (such as the hypercube) which mixes perfectly in exactly two steps.

math.PR↗
arXiv2019

Functional central limit theorem for random walks in random environment defined on regular trees

We study Random Walks in an i.i.d. Random Environment (RWRE) defined on $b$-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the process is defined on $b$-regular trees, with $ b \ge 4$, substantially improving the results of the first author (see Theorem 3 of Collevecchio (2006)).

math.PR↗
arXiv2019

Pure Nash Equilibria and Best-Response Dynamics in Random Games

In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

cs.GT↗
arXiv2019

Phase transitions for edge-reinforced random walks on the half-line

We study the behaviour of a class of edge-reinforced random walks {on $\mathbb{Z}_+$}, with heterogeneous initial weights, where each edge weight can be updated only when the edge is traversed from left to right. We provide a description for different behaviours of this process and describe phase transitions that arise as trade-offs between the strength of the reinforcement and that of the initial weights. Our result aims to complete the ones given by Davis~\cite{Davis89, Davis90}, Takeshima~\cite{Takeshima00, Takeshima01} and Vervoort~\cite{Vervoort00}.

math.PR↗
arXiv2019

A deterministic walk on the randomly oriented Manhattan lattice

Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic walk is then started at the origin and at each step moves diagonally to the nearest vertex in the direction of the horizontal and vertical lines of the present location. This definition can be generalized, in a natural way, to larger dimensions, but we mainly focus on the two dimensional case. In this context the process localizes on two vertices at all large times, almost surely. We also provide estimates for the tail of the length of paths, when the walk is defined on the two dimensional lattice. In particular, the probability of the path to be larger than $n$ decays sub-exponentially in $n$. It is easy to show that higher dimensional paths may not localize on two vertices but will still eventually become periodic, and are therefore bounded.

math.PR↗
arXiv2018

The branching-ruin number as critical parameter of random processes on trees

The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees. First, we study random walks on random conductances on trees, when the conductances have an heavy tail at $0$, parametrized by some $p>1$, where $1/p$ is the exponent of the tail. We prove a phase transition recurrence/transience with respect to $p$ and identify the critical parameter to be equal to the branching-ruin number of the tree. Second, we study a multi-excited random walk on trees where each vertex has $M$ cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).

math.PR↗
arXiv2018

A note on recurrence of the Vertex reinforced jump process and fractional moments localization

We give a simple proof for recurrence of vertex reinforced jump process on \(\mathbb{Z}^d\), under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on \ \(\mathbb{Z}^d\) is {recurrent} for strong reinforcement. Finally, we prove that the \(H^{(2|2)}\) model on \(\mathbb{Z}^d\) localizes at strong disorder. Even though these results are well-known, we propose a unified approach, {which also has the advantage to provide shorter proofs}, and relies on estimating fractional moments, introduced by Aizenman and Molchanov.

math.PR↗
arXiv2017

Invariance principle for biased Boostrap Random Walk

Our main goal is to study a class of processes whose increments are generated via a cellular automata rule. Given the increments of a simple biased random walk, a new sequence of (dependent) Bernoulli random variables is produced. It is built, from the original sequence, according to a cellular automata rule. Equipped with these two sequences, we construct two more according to the same cellular automata rule. The construction is repeated a fixed number of times yielding an infinite array ($\{-K,\ldots,K\}\times\N$) of (dependent) Bernoulli random variables. %In turn, using this new sequence as input, we reiterate a construction of a new sequence. %This process is repeated a finite number of times resulting in an infinite array, herein called the "downward process", of highly dependent Bernoulli random variables. The process of taking partial products can be reversed allowing to construct new sequences, herein called the "upward process", the characteristics of which are different from the downward process. Taking partial sums of these sequences, we obtain a $(2K+1)$-dimensional process whose increments belong to the state space $\{-1,1\}^{2K+1}$. The aim of the paper is to study the long term behaviour of this process. In particular, we establish transience/recurrence properties and prove an invariance principle. The limiting behaviour of these processes depends strongly on the direction of the iteration, and exhibits few surprising features. This work is motivated by an earlier investigation in which the starting sequence is symmetric.

math.PR↗
arXiv2017

The branching-ruin number and the critical parameter of once-reinforced random walk on trees

The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. computable) criterion characterizing the recurrence/transience of a larger class of self-interacting walks on trees, providing the complete picture for their phase transition.

math.PR↗