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 adversary and the data collector (decoder) have a utility function 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向传输符号、计算任务和存储数据中引入冗余，以抵御敌对篡改。然而，编码理论中的纠错机制依赖于严格的信任假设。在计算和存储场景中，要求诚实节点在数量上以一定比例超过敌对节点。但在若干新兴的现实应用场景中，尤其是面向去中心化区块链的应用中，此类假设往往不切实际。因此，尽管编码在解决去中心化系统重大挑战中扮演重要角色，其应用仍受到限制。不过，去中心化平台呈现出独特特征，为超越传统方法约束的安全编码开辟了新途径。在这些场景中，当合法解码器成功恢复数据时（且最好存在较大估计误差），攻击者反而能获益。这种激励机制促使攻击者理性行动，试图最大化自身收益。本文提出一种编码的博弈论形式化框架——称为"编码博弈"，该框架捕捉了这种独特动态：攻击者和数据收集者（解码器）各自拥有待优化的效用函数。效用函数反映了数据收集者与攻击者都希望提升数据可恢复概率的共同目标，同时体现数据收集者期望以较低估计误差评估输入而攻击者持相反立场的特性。作为初步但极具挑战性的步骤，我们针对重复因子为2的重复码，在最小化假设条件下刻画了广泛效用函数类对应的博弈均衡。