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：编码理论的核心是在传输符号、计算任务和存储数据中引入冗余，以抵御对抗性操纵。然而，编码理论中的纠错依赖于严格的信任假设。在计算和存储的背景下，要求诚实节点的数量超过对抗节点一定比例。但在若干新兴的实际场景中，特别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的重大挑战中扮演重要角色，其应用却受到限制。然而，在去中心化平台中，出现了一种独特特征，为超越传统方法约束的安全编码提供了新途径。在这些场景中，当合法解码器恢复数据且最好具有高估计误差时，对手将从中获益。这种激励促使对手理性行动，试图最大化其收益。本文提出了一种编码的博弈论框架，称为“游戏编码”，它捕捉了这种独特动态：对手和数据收集者（解码器）各自具有需要优化的效用函数。效用函数反映了数据收集者和对手都希望提高数据可恢复性的概率这一事实。此外，效用函数还表达了数据收集者希望以较低估计误差估计输入的偏好，而对手则持相反偏好。作为第一步，尽管仍具高度非平凡性，我们刻画了重复因子为2的重复码在广泛的一类效用函数（仅需最小假设）下的博弈均衡。