Natalia Jurga, Ian D. Morris
In the 1988 textbook "Fractals Everywhere" M. Barnsley introduced an algorithm for generating fractals through a random procedure which he called the "chaos game". Using ideas from the classical theory of covering times of Markov chains we prove an asymptotic formula for the expected time taken by this procedure to generate a $δ$-dense subset of a given self-similar fractal satisfying the open set condition.
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Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the determinis…
The chaos game representation (CGR) is an interesting method to visualize one-dimensional sequences. In this paper, we show how to construct a chaos game representation. The applications mentioned here are biological, in which CGR was able to uncover patterns in DNA or proteins that were previously unknown. We also show how CGR might be introduced in the classroom, either in a modelling course or in a dynamical systems course. Some sequences that are tested are taken from the Online Encyclopedia of Integer Sequences, and others are taken from sequences that arose mainly from a course in experi…
This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen c…
Fictitious Play in $3\times 3$ games: chaos and dithering behaviour
在60年代沙普利提供了一个例子，两个球员虚构的游戏与周期性的行为。在这个游戏中，玩家$A$的目标是复制$B$的行为，玩家$B$的目标是在玩家$A$之前玩一个。本文继续研究一类博弈，通过引入外部参数推广了Shapley的例子，证明了博弈中存在大量的周期性和混沌行为，参与者在不同的策略之间摇摆不定。所有这一切的原因，是存在一个周期性轨道（包括播放混合策略），这是{ \em“抖动类型”}：这样的轨道既不是吸引，排斥或鞍型，因为附近的轨道抖动接近和远离它的方式，这是令人想起一个随机游动运动。我们证明了这种行为的开集的游戏。
In this paper we provide extensions to hyperbolic numbers plane of the classical Chaos game algorithm and the Shannon entropy. Both notions connected with that of probability with values in hyperbolic number, introduced by D. Alpay et al \cite{eluna}. Within this context, particular attention has been paid to the interpretation of the hyperbolic valued probabilities and the hyperbolic extension of entropy as well.
The chaos game on a general iterated function system from a topological point of view
我们研究了与使用随机轨道近似迭代函数系统的吸引子有关的组合问题，目的是澄清轨道生成过程中的随机过程的作用。Baire类对应的几乎必然收敛，拓扑和概率方法之间的联系被观察到。
A non-logarithmic approach to the rate of convergence of the deterministic chaos game
本文从新角度探讨混沌游戏算法收敛到迭代函数系统的吸引子的速率，证明了典型驱动器的恢复速率与函数ψ的极限行为一致，并扩展了Leśniak等人2024年的主要定理。
Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos
研究乘法权重更新算法在博弈中呈现李 - 约克混沌时的长期行为，利用自然不变测度提供框架，以双策略拥塞博弈为例，能对动态全面统计表征，涵盖多种行为谱，实现统计可预测性。
We study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and any fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to become u…
We establish that algorithmic experiments in zero-sum games "fail miserably" to confirm the unique, sharp prediction of maxmin equilibration. Contradicting nearly a century of economic thought that treats zero-sum games nearly axiomatically as the exemplar symbol of economic stability, we prove that no meaningful prediction can be made about the day-to-day behavior of online learning dynamics in zero-sum games. Concretely, Multiplicative Weights Updates (MWU) with constant step-size is Lyapunov chaotic in the dual (payoff) space. Simply put, let's assume that an observer asks the agents playin…
We study the network replicator equation and characterize its fixed points on arbitrary graph structures for $2 \times 2$ symmetric games. We show a relationship between the asymptotic behavior of the network replicator and the existence of an independent vertex set in the graph and also show that complex behavior cannot emerge in $2 \times 2$ games. This links a property of the dynamical system with a combinatorial graph property. We contrast this by showing that ordinary rock-paper-scissors (RPS) exhibits chaos on the 3-cycle and that on general graphs with $\geq 3$ vertices the network repl…
We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do for regular IFSs. We study sufficient conditions which guarantee that the closure of the target set is a local attractor for the IFS. As a corollary, we establish necessary and sufficient conditions for the IFS having a global attractor. We give an example of a non-regular IFS whose target set is nonempty, showing that our approach gives rise to a "new class" of sem…
The emergence of chaos in population game dynamics induced by comparisons
该研究证明在离散时间下，通过构造修正协议并选择时间步长，可使$2\ imes2$反协调博弈动力学呈现Li-Yorke混沌，导致系统不可预测，且该特性内嵌于任何模仿性修正协议中。
We present volume analyses of Multiplicative Weights Updates (MWU) and Optimistic Multiplicative Weights Updates (OMWU) in zero-sum as well as coordination games. Such analyses provide new insights into these game dynamical systems, which seem hard to achieve via the classical techniques within Computer Science and Machine Learning. The first step is to examine these dynamics not in their original space (simplex of actions) but in a dual space (aggregate payoff space of actions). The second step is to explore how the volume of a set of initial conditions evolves over time when it is pushed for…
We study the dynamics of simple congestion games with two resources where a continuum of agents behaves according to a version of Experience-Weighted Attraction (EWA) algorithm. The dynamics is characterized by two parameters: the (population) intensity of choice $a>0$ capturing the economic rationality of the total population of agents and a discount factor $σ\in [0,1]$ capturing a type of memory loss where past outcomes matter exponentially less than the recent ones. Finally, our system adds a third parameter $b \in (0,1)$, which captures the asymmetry of the cost functions of the two resour…
Routing games are amongst the most studied classes of games. Their two most well-known properties are that learning dynamics converge to equilibria and that all equilibria are approximately optimal. In this work, we perform a stress test for these classic results by studying the ubiquitous dynamics, Multiplicative Weights Update, in different classes of congestion games, uncovering intricate non-equilibrium phenomena. As the system demand increases, the learning dynamics go through period-doubling bifurcations, leading to instabilities, chaos and large inefficiencies even in the simplest case …
Stability of Multi-Agent Learning in Competitive Networks: Delaying the Onset of Chaos
该研究摒弃零和博弈假设，通过参数化网络博弈的统计方法分析竞争网络中Q-Learning动力学的收敛性，发现其稳定性仅依赖网络连通性而非智能体总数，且特定网络下增规模不会提升混沌行为概率。
On the discrete-time origins of the replicator dynamics: From convergence to instability and chaos
本文比较三种具有相同复制者动力学连续极限的离散时间博弈学习模型，揭示二阶效应导致收敛、不稳定乃至Li-Yorke混沌等截然不同的长期行为，从而界定了复制者动力学的预测适用范围。
Games are natural models for multi-agent machine learning settings, such as generative adversarial networks (GANs). The desirable outcomes from algorithmic interactions in these games are encoded as game theoretic equilibrium concepts, e.g. Nash and coarse correlated equilibria. As directly computing an equilibrium is typically impractical, one often aims to design learning algorithms that iteratively converge to equilibria. A growing body of negative results casts doubt on this goal, from non-convergence to chaotic and even arbitrary behaviour. In this paper we add a strong negative result to…
A periodic behavior is a well observed phenomena in biological and economical systems. We show that evolutionary games on graphs with imitation dynamics can display periodic behavior for an arbitrary choice of game theoretical parameters describing social-dilemma games. We construct graphs and corresponding initial conditions whose trajectories are periodic with an arbitrary minimal period length. We also examine a periodic behavior of evolutionary games on graphs with the underlying graph being an acyclic (tree) graph. Astonishingly, even this acyclic structure allows for arbitrary long perio…
Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems
本文研究高维动力系统的在线状态估计问题，确立了集合卡尔曼滤波器在混沌系统及机器学习替代动力学模型中的长时间精度，证明了其估计误差在长时间范围内保持较小，验证了机器学习预测模型在集合数据同化中的有效性。
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