SIAM Review, Volume 66, Issue 2, Page 353-353, May 2024.
In this issue the Education section presents two contributions. The first paper, “The Poincaré Metric and the Bergman Theory,” by Steven G. Krantz, discusses the Poincaré metric on the unit disc in the complex space and the Bergman metric on an arbitrary domain in any dimensional complex space. To define the Bergman metric the notion of Bergman kernel is crucial. Some striking properties of the Bergman kernel are discussed briefly, and it is calculated when the domain is the open unit ball. The Bergman metric is invariant under biholomorphic maps. The paper ends by discussing several attractive applications. To incorporate invariance within models in applied science, in particular for machine learning applications, there is currently a considerable interest in non-Euclidean metrics, in invariant (under some actions) metrics, and in reproducing kernels, mostly in the real-valued framework. The Bergman theory (1921) is a special case of Aronszajn's theory of Hilbert spaces with reproducing kernels (1950). Invariant metrics are used, in particular, in the study of partial differential equations. Complex-valued kernels have some interesting connections to linear systems theory. This article sheds some new light on the Poincaré metric, the Bergman kernel, the Bergman metric, and their applications in a manner that helps the reader become accustomed to these notions and to enjoy their properties. The second paper, “Dynamics of Signaling Games,” is presented by Hannelore De Silva and Karl Sigmund and is devoted to much-studied types of interactions with incomplete information, analyzing them by means of evolutionary game dynamics. Game theory is often encountered in models describing economic, social, and biological behavior, where decisions can not only be shaped by rational arguments, but may also be influenced by other factors and players. However, it is often restricted to an analysis of equilibria. In signaling games some agents are less informed than others and try to deal with it by observing actions (signals) from better informed agents. Such signals may be even purposely wrong. This article offers a concise guided tour of outcomes of evolutionary dynamics in a number of small dimensional signaling games focusing on the replicator dynamics, the best-reply dynamics, and the adaptive dynamics (dynamics of behavioral strategies whose vector field follows the gradient of the payoff vector). Furthermore, for the model of evolution of populations of players, the authors compare these dynamics. Several interesting examples illustrate that even simple adaptation processes can lead to nonequilibrium outcomes and endless cycling. This tutorial is targeted at graduate/Ph.D. students and researchers who know the basics of game theory and want to learn examples of signaling games, together with evolutionary game theory.

中文翻译：

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 教育


《SIAM 评论》，第 66 卷，第 2 期，第 353-353 页，2024 年 5 月。

在本期中，教育部分提出了两项​​贡献。第一篇论文“庞加莱度量和伯格曼理论”由 Steven G. Krantz 撰写，讨论了复空间中单位圆盘上的庞加莱度量和任意维复空间中任意域上的伯格曼度量。为了定义伯格曼度量，伯格曼核的概念至关重要。简要讨论了伯格曼核的一些显着性质，并在域为开单位球时进行了计算。伯格曼度量在双全纯映射下是不变的。本文最后讨论了几个有吸引力的应用。为了将不变性纳入应用科学模型中，特别是机器学习应用，目前人们对非欧几里得度量、不变（在某些操作下）度量和再现内核（主要是在实值框架中）产生了相当大的兴趣。 Bergman 理论（1921）是 Aronszajn 的带有再生核的 Hilbert 空间理论（1950）的一个特例。不变度量尤其用于偏微分方程的研究。复值核与线性系统理论有一些有趣的联系。本文对庞加莱度量、伯格曼核、伯格曼度量及其应用提供了一些新的见解，帮助读者习惯这些概念并享受它们的特性。第二篇论文“信号博弈的动力学”由 Hannelore De Silva 和 Karl Sigmund 提出，致力于对不完全信息的交互类型进行大量研究，并通过进化博弈动力学的方式对其进行分析。 博弈论经常出现在描述经济、社会和生物行为的模型中，其中决策不仅可以由理性论证决定，还可能受到其他因素和参与者的影响。然而，它通常仅限于平衡分析。在信号博弈中，一些智能体比其他智能体了解得更少，并尝试通过观察信息更丰富的智能体的动作（信号）来处理它。这些信号甚至可能是故意错误的。本文提供了一些小维信号游戏中进化动力学结果的简明导览，重点关注复制动态、最佳回复动态和自适应动态（行为策略的动态，其向量场遵循收益梯度）向量）。此外，对于玩家群体的进化模型，作者比较了这些动态。几个有趣的例子表明，即使是简单的适应过程也可能导致不平衡的结果和无休止的循环。本教程针对研究生/博士。了解博弈论基础知识并希望学习信号博弈示例以及进化博弈论的学生和研究人员。