Laplace principle for large population games with control interaction

Shuffling large decks of cards and the Bernoulli-Laplace urn model

In card games, in casino games with multiple decks of cards and in cryptography, one is sometimes faced with the following problem: how can a human (as opposed to a computer) shuffle a large deck of cards? The procedure we study is to break the deck into several reasonably sized piles, shuffle each thoroughly, recombine the piles, do some simple deterministic operation, for instance a cut, and repeat. This process can also be seen as a generalised Bernoulli-Laplace urn model. We use coupling arguments and spherical function theory to derive upper and bounds on the mixing times of these Markov …

Convergence of large population games to mean field games with interaction through the controls

This work considers stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. We develop a new framework to prove convergence of finite-player games to the asymptotic mean field game. Our approach is based on the concept of propagation of chaos for forward and backward weakly interacting particles which we investigate by stochastic analysis methods, and which appear to be of independent interest. These propagation of chaos arguments allow to derive moment and concentration bounds f…

Mean Field Games with Branching

Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time which may lead to different Nash equilibria. For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the mean field game problem. As a first step, we study a simple model using …

Regularized Bayesian best response learning in finite games

We introduce the notion of regularized Bayesian best response (RBBR) learning dynamic in heterogeneous population games. We obtain such a dynamic via perturbation by an arbitrary lower semicontinuous, strongly convex regularizer in Bayesian population games with finitely many strategies. We provide a sufficient condition for the existence of rest points of the RBBR learning dynamic, and hence the existence of regularized Bayesian equilibrium in Bayesian population games. These equilibria are shown to approximate the original Bayesian equilibria for vanishingly small perturbations. We also expl…

Infinite horizon for symetric strategy population game

To predict the behavior of a population game when time becomes very long, the process that characterizes the evolution of our game dynamics must be reversible. Known games satisfying this are 2 strategy games as well as potential games with an exponential protocol. We will try to extend the study of infinite horizons for what are called symetric strategy games.

平均场正倒向随机微分方程的强解及其在多群体平均场博弈中的应用

Strong solutions of mean-field FBSDEs and their applications to multi-population mean-field games

本文研究具有可测系数且允许不连续和非利普希茨条件的平均场FBSDE强解存在性，并借助随机最大值原理和鞅方法将其应用于考虑群体规模变化的多群体平均场博弈纳什均衡分析。

随机微分博弈中的种群规模

On the population size in stochastic differential games

本文综述随机微分博弈可解性的最新进展，重点介绍纯概率方法，并探讨种群规模对分析的影响及有限玩家博弈向平均场博弈的收敛。

Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well as the coefficient functions of the state dynamics are shared among the agents within each population, they can be different population by population. We study the mean field limit for the three different situations; (i) every agent is non-cooperative; (ii) the agents within eac…

On Gradient like Properties of Population games, Learning models and Self Reinforced Processes

We consider ordinary differential equations on the unit simplex of $\RR^n$ that naturally occur in population games, models of learning and self reinforced random processes. Generalizing and relying on an idea introduced in \cite{DF11}, we provide conditions ensuring that these dynamics are gradient like and satisfy a suitable "angle condition". This is used to prove that omega limit sets and chain transitive sets (under certain smoothness assumptions) consist of equilibria; and that, in the real analytic case, every trajectory converges toward an equilibrium. In the reversible case, the dynam…

The limits of weak selection and large population size in evolutionary game theory

Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the wN limit, in which weak selection is applied before the large population limit, and the Nw limit, in which the order is reversed.…

基于基础概率论的群体博弈微分方程近似方法

Differential Equation Approximations for Population Games using Elementary Probability

针对群体博弈平均场近似方法现有证明依赖高等数学、门槛较高的问题，本文提出仅用本科概率论知识的更简洁推导方式，降低了该分析工具的理解难度。