Coding theory revolves around the incorporation of redundancy into transmitted symbols, computation tasks, and stored data to guard against adversarial manipulation. However, error correction in coding theory is contingent upon a strict trust assumption. In the context of computation and storage, it is required that honest nodes outnumber adversarial ones by a certain margin. However, in several emerging real-world cases, particularly, in decentralized blockchain-oriented applications, such assumptions are often unrealistic. Consequently, despite the important role of coding in addressing significant challenges within decentralized systems, its applications become constrained. Still, in decentralized platforms, a distinctive characteristic emerges, offering new avenues for secure coding beyond the constraints of conventional methods. In these scenarios, the adversary benefits when the legitimate decoder recovers the data, and preferably with a high estimation error. This incentive motivates them to act rationally, trying to maximize their gains. In this paper, we propose a game theoretic formulation for coding, called the game of coding, that captures this unique dynamic where each of the adversaries and the data collector (decoder) have respective utility functions to optimize. The utility functions reflect the fact that both the data collector and the adversary are interested in increasing the chance of data being recoverable by the data collector. Moreover, the utility functions express the interest of the data collector to estimate the input with lower estimation error, but the opposite interest of the adversary. As a first, still highly non-trivial step, we characterize the equilibrium of the game for the repetition code with a repetition factor of 2 for a wide class of utility functions with minimal assumptions.

翻译：编码理论的核心在于通过向传输符号、计算任务和存储数据中引入冗余，以防范对抗性篡改。然而，编码理论中的纠错能力依赖于严格的信任假设。在计算与存储场景中，要求诚实节点数量以特定比例超过对抗节点。但在若干新兴的现实应用场景中，尤其是面向去中心化的区块链应用，此类假设往往难以成立。因此，尽管编码在应对去中心化系统中的重大挑战方面具有重要作用，其应用范围仍受到限制。值得注意的是，去中心化平台呈现出一种独特特性，为突破传统方法局限的安全编码提供了新路径。在这些场景中，当合法解码器成功恢复数据（尤其是以高估计误差恢复）时，对抗方将获得收益。这种激励机制促使对抗方采取理性行为以最大化自身收益。本文提出一种博弈论框架下的编码形式化模型——编码博弈，该模型刻画了对抗方与数据收集器（解码器）各自具有待优化效用函数的独特动态关系。效用函数反映了数据收集器与对抗方均关注数据可被收集器成功恢复的概率；同时，效用函数还表明数据收集器期望降低输入估计误差，而对抗方则具有相反的意图。作为首个高度非平凡的研究步骤，我们在最小假设条件下，针对广泛类别的效用函数，刻画了重复因子为2的重复编码博弈的均衡状态。