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, called 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 to increase the chance of data being recoverable at 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 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、计算任务及存储数据中引入冗余，以抵御恶意篡改。然而，编码理论中的纠错机制依赖于严格的信任假设——在计算与存储场景中，要求诚实节点数量以特定比例超过恶意节点。但在若干新兴现实案例中，特别是在去中心化区块链应用中，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核心挑战中具有重要价值，其应用仍受到制约。值得注意的是，去中心化平台展现出独特特性，为突破传统方法的限制实现安全编码提供了新路径。在此类场景中，当合法解码器成功恢复数据时（尤其当估计误差较高时），攻击者反而能从中获益。这种激励机制促使攻击者采取理性行为，试图最大化自身收益。本文提出一种名为"编码博弈"的博弈论框架，用以刻画这种独特动态：攻击者与数据收集者（解码器）均具有待优化的效用函数。效用函数揭示了双方对提升数据可恢复性的共同诉求，同时反映了数据收集者希望降低输入估计误差，而攻击者期望增大估计误差的相反利益。作为初始但极具挑战性的探索，本文在重复因子为2的重复编码场景下，针对满足最小假设的宽泛效用函数族，刻画了该博弈的均衡特性。