Traditional coding theory guarantees valid decoding only if a minority of symbols are adversarially manipulated. In contrast, the game of coding framework ensures reliable decoding, even in the presence of an adversarial majority. This formulation is motivated by emerging permissionless applications, particularly decentralized machine learning (DeML), where computation tasks are outsourced to external volunteer nodes that are predominantly rational and reward-seeking. Prior investigations have analyzed the game of coding in the scalar setting. Since the results of most major computations in machine learning are vectors (e.g., computing the gradient of the loss for a machine learning model), we extend the framework in this paper to the general multi-dimensional Euclidean space. As a first, yet fundamental step, in this paper, we study a two-repetition code in which at least one node is controlled by a rational adversary, and we fully characterize the equilibrium and the optimal strategies of the players. Similar to the scalar case, this result serves as a cornerstone for addressing more general scenarios.

翻译：传统编码理论仅当少数符号受到对抗性操纵时才能保证有效解码。相比之下，“码局”框架即便在对抗方占据多数的情况下仍能确保可靠解码。这一框架的提出源于新兴的无许可应用场景，特别是去中心化机器学习（DeML），其中计算任务被外包给外部志愿节点，而这些节点主要具有理性和追求奖励的特性。已有研究针对标量场景分析了“码局”框架。由于机器学习中主要计算的结果多为向量（例如，计算机器学习模型损失函数的梯度），本文将该框架扩展至通用的多维欧氏空间。作为第一步基础性工作，本文研究了至少有一个节点受理性对抗方控制的二重复码，并完整刻画了该博弈中的纳什均衡及参与者的最优策略。与标量情形类似，该结果可作为解决更一般化场景的基石。