The game of coding is a new framework at the intersection of game theory and coding theory; designed to transcend the fundamental limitations of classical coding theory. While traditional coding theoretic schemes rely on a strict trust assumption, that honest nodes must outnumber adversarial ones to guarantee valid decoding, the game of coding leverages the economic rationality of actors to guarantee correctness and reliable decodability, even in the presence of an adversarial majority. This capability is paramount for emerging permissionless applications, particularly decentralized machine learning (DeML). However, prior investigations into the game of coding have been strictly confined to scalar computations, limiting their applicability to real world tasks where high dimensional data is the norm. In this paper, we bridge this gap by extending the framework to the general $N$-dimensional Euclidean space. We provide a rigorous problem formulation for vector valued computations and fully characterize the equilibrium strategies of the resulting high dimensional game. Our analysis demonstrates that the resilience properties established in the scalar setting are preserved in the vector regime, establishing a theoretical foundation for secure, large scale decentralized computing without honest majority assumptions.

翻译：编码博弈是博弈论与编码理论交叉领域的一个新框架，旨在超越经典编码理论的基本局限。传统编码理论方案依赖于严格的信任假设，即诚实节点必须多于对抗节点才能保证有效解码；而编码博弈则利用参与者的经济理性来保证正确性与可靠解码能力，即使在对抗性节点占多数的情况下依然成立。这种能力对于新兴的无许可应用至关重要，特别是去中心化机器学习（DeML）。然而，先前对编码博弈的研究严格局限于标量计算，限制了其在高维数据成为常态的现实任务中的适用性。本文通过将该框架扩展至一般的$N$维欧几里得空间来弥合这一差距。我们为向量值计算提供了严格的问题形式化描述，并完整刻画了由此产生的高维博弈的均衡策略。我们的分析表明，在标量场景中建立的抗干扰特性在向量体系中得以保持，从而为无需诚实多数假设的安全、大规模去中心化计算奠定了理论基础。