# Enabling Long-Term Cooperation in Cross-Silo Federated Learning: A Repeated Game Perspective

arXiv: https://arxiv.org/abs/2106.11814 · HTML: https://mlsanity.com/paper/2106.11814/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 68 references that are in the map.
- First posted on arXiv 22 Jun 2021.

## Builds on (most cited first, 25 of 25)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [On the Convergence of FedAvg on Non-IID Data](https://mlsanity.com/paper/1907.02189/) (2019)
- [Agnostic Federated Learning](https://mlsanity.com/paper/1902.00146/) (2019)
- [Adaptive Federated Learning in Resource Constrained Edge Computing Systems](https://mlsanity.com/paper/1804.05271/) (2018)
- [Federated Learning in Mobile Edge Networks: A Comprehensive Survey](https://mlsanity.com/paper/1909.11875/) (2019)
- [Personalized Cross-Silo Federated Learning on Non-IID Data](https://mlsanity.com/paper/2007.03797/) (2020)
- [Broadband Analog Aggregation for Low-Latency Federated Edge Learning (Extended Version)](https://mlsanity.com/paper/1812.11494/) (2018)
- [Federated Learning for Internet of Things: Recent Advances, Taxonomy, and Open Challenges](https://mlsanity.com/paper/2009.13012/) (2020)
- [Cost-Effective Federated Learning Design](https://mlsanity.com/paper/2012.08336/) (2020)
- [Free-rider Attacks on Model Aggregation in Federated Learning](https://mlsanity.com/paper/2006.11901/) (2020)
- [Free-riders in Federated Learning: Attacks and Defenses](https://mlsanity.com/paper/1911.12560/) (2019)
- [FMore: An Incentive Scheme of Multi-dimensional Auction for Federated Learning in MEC](https://mlsanity.com/paper/2002.09699/) (2020)
- [Device Sampling for Heterogeneous Federated Learning: Theory, Algorithms, and Implementation](https://mlsanity.com/paper/2101.00787/) (2021)
- [Throughput-Optimal Topology Design for Cross-Silo Federated Learning](https://mlsanity.com/paper/2010.12229/) (2020)
- [FOCUS: Dealing with Label Quality Disparity in Federated Learning](https://mlsanity.com/paper/2001.11359/) (2020)
- [Energy-Efficient Federated Edge Learning with Joint Communication and Computation Design](https://mlsanity.com/paper/2003.00199/) (2020)
- [Adaptive Federated Learning With Gradient Compression in Uplink NOMA](https://mlsanity.com/paper/2003.01344/) (2020)
- [An Exploratory Analysis on Users' Contributions in Federated Learning](https://mlsanity.com/paper/2011.06830/) (2020)
- [Towards Causal Federated Learning For Enhanced Robustness and Privacy](https://mlsanity.com/paper/2104.06557/) (2021)
- [Communication Efficient Federated Learning over Multiple Access Channels](https://mlsanity.com/paper/2001.08737/) (2020)
- [Differentially private cross-silo federated learning](https://mlsanity.com/paper/2007.05553/) (2020)
- [Network Utility Maximization based on Incentive Mechanism for Truthful Reporting of Local Information](https://mlsanity.com/paper/1909.03938/) (2019)
- [Demand Side Management in Smart Grids using a Repeated Game Framework](https://mlsanity.com/paper/1311.1887/) (2013)
- [Reputation and Pricing Dynamics in Online Markets](https://mlsanity.com/paper/2103.16100/) (2021)

## Built on by (most cited first, 14 of 14)

- [Mechanisms that Incentivize Data Sharing in Federated Learning](https://mlsanity.com/paper/2207.04557/) (2022)
- [Evaluating and Incentivizing Diverse Data Contributions in Collaborative Learning](https://mlsanity.com/paper/2306.05592/) (2023)
- [Fair yet Asymptotically Equal Collaborative Learning](https://mlsanity.com/paper/2306.05764/) (2023)
- [On the Effect of Defections in Federated Learning and How to Prevent Them](https://mlsanity.com/paper/2311.16459/) (2023)
- [Research in Collaborative Learning Does Not Serve Cross-Silo Federated Learning in Practice](https://mlsanity.com/paper/2510.12595/) (2025)
- [Welfare and Fairness Dynamics in Federated Learning: A Client Selection Perspective](https://mlsanity.com/paper/2302.08976/) (2023)
- [Learner Referral for Cost-Effective Federated Learning Over Hierarchical IoT Networks](https://mlsanity.com/paper/2307.09977/) (2023)
- [GTFLAT: Game Theory Based Add-On For Empowering Federated Learning Aggregation Techniques](https://mlsanity.com/paper/2212.04103/) (2022)
- [Adaptive incentive for cross-silo federated learning: A multi-agent reinforcement learning approach](https://mlsanity.com/paper/2302.07493/) (2023)
- [Price of Stability in Quality-Aware Federated Learning](https://mlsanity.com/paper/2310.08790/) (2023)
- [Incentive Mechanism Design for Distributed Ensemble Learning](https://mlsanity.com/paper/2310.08792/) (2023)
- [TACO: Tackling Over-correction in Federated Learning with Tailored Adaptive Correction](https://mlsanity.com/paper/2504.17528/) (2025)
- [Degree of Staleness-Aware Data Updating in Federated Learning](https://mlsanity.com/paper/2508.16931/) (2025)
- [Gaming and Cooperation in Federated Learning: What Can Happen and How to Monitor It](https://mlsanity.com/paper/2509.02391/) (2025)

## Abstract (arXiv)

Cross-silo federated learning (FL) is a distributed learning approach where clients of the same interest train a global model cooperatively while keeping their local data private. The success of a cross-silo FL process requires active participation of many clients. Clients in cross-silo FL aim to optimize their long-term benefits by selfishly choosing their participation levels. While there has been some work on incentivizing clients to join FL, the analysis of clients' long-term selfish participation behaviors in cross-silo FL remains largely unexplored. In this paper, we analyze the selfish participation behaviors of heterogeneous clients in cross-silo FL. Specifically, we model clients' long-term selfish participation behaviors as an infinitely repeated game. For the stage game SPFL, we derive the unique Nash equilibrium (NE), and propose a distributed algorithm for each client to calculate its equilibrium participation strategy. We show that at the NE, clients fall into at most three categories: (i) free riders, (ii) a unique partial contributor (if exists), and (iii) contributors. For the long-term interactions among clients, we derive a cooperative strategy for clients which minimizes the number of free riders while increasing the amount of local data for model training. We show that enforced by a punishment strategy, such a cooperative strategy is a subgame perfect Nash equilibrium (SPNE) of the infinitely repeated game, under which some clients who are free riders at the NE of the stage game choose to be (partial) contributors. We further propose an algorithm to calculate the optimal SPNE which minimizes the number of free riders while maximizing the amount of local data for model training. Simulation results show that our derived optimal SPNE can effectively reduce the number of free riders by up to 99.3% and increase the amount of local data for model training by up to 82.3%.