Coding theory plays a crucial role in ensuring data integrity and reliability across various domains, from communication to computation and storage systems. However, its reliance on trust assumptions for data recovery poses significant challenges, particularly in emerging decentralized systems where trust is scarce. To address this, the game of coding framework was introduced, offering insights into strategies for data recovery within incentive-oriented environments. The focus of the earliest version of the game of coding was limited to scenarios involving only two nodes. This paper investigates the implications of increasing the number of nodes in the game of coding framework, particularly focusing on scenarios with one honest node and multiple adversarial nodes. We demonstrate that despite the increased flexibility for the adversary with an increasing number of adversarial nodes, having more power is not beneficial for the adversary and is not detrimental to the data collector, making this scheme sybil-resistant. Furthermore, we outline optimal strategies for the data collector in terms of accepting or rejecting the inputs, and characterize the optimal noise distribution for the adversary.

Differentially private distributed mean estimation (DP-DME) is a fundamental building block in privacy-preserving federated learning, where a central server estimates the mean of $d$-dimensional vectors held by $n$ users while ensuring $(\epsilon,\delta)$-DP. Local differential privacy (LDP) and distributed DP with secure aggregation (SecAgg) are the most common notions of DP used in DP-DME settings with an untrusted server. LDP provides strong resilience to dropouts, colluding users, and malicious server attacks, but suffers from poor utility. In contrast, SecAgg-based DP-DME achieves an $O(n)$ utility gain over LDP in DME, but requires increased communication and computation overheads and complex multi-round protocols to handle dropouts and malicious attacks. In this work, we propose CorDP-DME, a novel DP-DME mechanism that spans the gap between DME with LDP and distributed DP, offering a favorable balance between utility and resilience to dropout and collusion. CorDP-DME is based on correlated Gaussian noise, ensuring DP without the perfect conditional privacy guarantees of SecAgg-based approaches. We provide an information-theoretic analysis of CorDP-DME, and derive theoretical guarantees for utility under any given privacy parameters and dropout/colluding user thresholds. Our results demonstrate that (anti) correlated Gaussian DP mechanisms can significantly improve utility in mean estimation tasks compared to LDP -- even in adversarial settings -- while maintaining better resilience to dropouts and attacks compared to distributed DP.