A compositional approach to game-theoretic economic modelling

博弈论是对不确定条件下的战略互动和决策的数学研究。它可以说是微观经济学的核心工具，也被广泛应用于进化生物学、网络安全和军事战略等应用领域。组合性是软件科学最基本的思想之一，它是指系统的行为应该可以通过其组件的行为来理解的原则。组合性允许以模块化的方式设计、实现、分析和测试大型复杂系统，并允许模块在不同的上下文中重用。没有这一点，现代软件工程就不可能存在。然而，游戏模型并不是组合的，通常必须完整地制作，而不是将标准组件组合在一起。因此，博弈论建模是一个缓慢的过程，因为待建模领域的小变化可能导致模型所需的大变化。特别是，这意味着博弈论模型目前不太适合软件实现。这个项目涉及到一种新的博弈论方法，这种方法是组合的，因此有望在目前不可能的规模上为经济学家和博弈论的其他用户提供软件支持。具体来说，它应该能够以一种允许重用现有工作的方式来指定、模拟、解决和更普遍地推理游戏。例如，一个典型的经济系统具有层次结构，从代理人到家庭到市场再到经济，并且应该有可能以直接使用较低层次的现有模型的方式逐步构建每个层次的模型。这种方法背后的数学技术和概念主要来自证明论（数学逻辑的一部分）和编程语言理论（理论计算机科学的一部分）。幸运的是，用户不需要学习这种复杂的（对他们来说）不熟悉的理论，因为它也可以将数学隐藏在称为弦图的直观图形语言背后，弦图最近由于在量子信息理论，语言学和抽象代数中的应用而得到广泛研究。这意味着博弈论软件可以是图形化和直观的，但仍然有一个强大的理论基础。该项目的目的是开发这些经济应用所需的数学理论，以一种利用该领域理论与应用之间密切关系的方式，同时使用一个工作示例（基于智能电网建模）来持续测试组合性的实际好处。这个项目的很大一部分理论部分将涉及用博弈论中的各种已知概念扩展选择函数理论，例如重复博弈，不完全信息和不同解的概念，这些可以在任何博弈论的标准文本中找到。这将主要包括将现有理论推广到新的框架。

Game theory is the mathematical study of strategic interaction and decision-making under uncertainty. It is arguably the central tool of microeconomics, and is also widely used in evolutionary biology, cybersecurity and military strategy, among other application areas. Compositionality, one of the most fundamental ideas of software science, is the principle that the behaviour of a system should be understandable in terms of the behaviour of its components. Compositionality allows large, complex systems to be designed, implemented, analysed and tested in a modular way, and allows modules to be reused in different contexts. Without this, modern software engineering would be impossible.Game models, however, are not compositional, and generally must be produced in their entirety rather than by combining standard components. As a result, game-theoretic modelling is a slow process, as small variations in the domain to be modelled can lead to large changes needed in the model. In particular this means that game-theoretic models are currently not well-suited to software implementation.This project concerns a new approach to game theory which is compositional, and therefore promises the possibility of software support for economists and other users of game theory on a scale that is currently impossible. Specifically, it should be possible to specify, simulate, solve and more generally reason about games in a way that allows the reuse of existing work. For example a typical economic system has a hierarchical structure, from agents to households to markets to economies, and it should be possible to gradually build models of each level in a way that directly uses existing models of the lower levels.The mathematical techniques and concepts underlying this approach mostly come from proof theory (part of mathematical logic) and the theory of programming languages (part of theoretical computer science). Fortunately there is no need for users to learn this sophisticated and (to them) unfamiliar theory, because it is also possible to hide the mathematics behind an intuitive graphical language known as string diagrams, which have been widely studied recently due to applications in quantum information theory, linguistics and abstract algebra. This means that game theoretic software can be graphical and intuitive, but still have a strong theoretical underpinning.The purpose of this project is to develop the mathematical theory needed for these economic applications, in a way that exploits the close relationship between theory and applications in this area, while using a worked example (based on modelling of smart energy grids) to provide a continual test of the practical benefits of compositionality.A large part of the theoretical part of this project will involve extending the theory of selection functions with various known concepts in game theory, such as repeated games, imperfect information and different solution concepts, which can be found in any standard text on game theory. This will largely consist of generalising existing theory to the new framework.