Gaming and Cooperation in Federated Learning: What Can Happen and How to Monitor It

Dongseok Kim, Hyoungsun Choi, Mohamed Jismy Aashik Rasool, Gisung Oh

Abstract:

The success of federated learning (FL) ultimately depends on how strategic participants behave
under partial observability, yet most formulations still treat FL as a static optimization
problem. We instead view FL deployments as governed strategic systems and develop an analytical
framework that separates welfare-improving behavior from metric gaming. Within
this framework, we introduce indices that quantify manipulability, the price of gaming, and
the price of cooperation, and we use them to study how rules, information disclosure, evaluation
metrics, and aggregator-switching policies reshape incentives and cooperation patterns.
We derive threshold conditions for deterring harmful gaming while preserving benign cooperation,
and for triggering auto-switch rules when early-warning indicators become critical.
Building on these results, we construct a design toolkit including a governance checklist and
a simple audit-budget allocation algorithm with a provable performance guarantee. Simulations
across diverse stylized environments and a federated learning case study consistently
match the qualitative and quantitative patterns predicted by our framework. Taken together,
our results provide design principles and operational guidelines for reducing metric
gaming while sustaining stable, high-welfare cooperation in FL platforms.

Submission Type: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

### Dear Action Editor

We thank you for the careful reading and for pointing out an important ambiguity in our participation model. In particular, your comment highlighted that the way the aggregate participation rate and individual participation decisions were written could be interpreted as a same-round circular definition, making it unclear how the next-round participation rate $x_{t+1}$ is generated from round $t$.

### Revisions in Section 5.1 (Participation dynamics)

To address this, we revised Section 5.1 to make the time indexing and update order explicit. The platform now observes the aggregate participation rate at the end of round $t$ ($x_t$), and each client decides whether to participate in the *next* round ($p_{i,t+1}$) based on the observed state $(x_t,\pi)$. This enforces the one-step update structure $x_t \rightarrow p_{i,t+1} \rightarrow x_{t+1}$ and ensures that the map $x_{t+1}=F(x_t;\pi)$ is a proper transition rule (dynamics), rather than an implicit same-round fixed-point definition. We also clarified the definition of the one-step net gain $\Delta U_{i,t+1}(x_t;\pi)$ that drives the threshold participation rule, and we separated the dynamics map from the subsequent fixed-point and local stability discussion.

### Revisions in Appendix A.2 (Proof consistency)

In addition, we updated Appendix A.2 to ensure full consistency with the revised definitions in the main text. In the proof of Proposition 5.4, the participation probability is now computed consistently with $F_\Theta$ as the CDF:
$$
\Pr(p_{i,t+1}=1\mid x_t,\pi)=\Pr(\theta_i \le \Delta U(x_t;\pi))=F_\Theta(\Delta U(x_t;\pi)).
$$
Accordingly, the expected aggregate participation update satisfies
$$
x_{t+1}
=\mathbb{E}\Big[\tfrac{1}{n}\sum_i \mathbb{I}\{p_{i,t+1}=1\}\,\Big|\,x_t,\pi\Big]
=F_\Theta(\Delta U(x_t;\pi))
=:F(x_t;\pi),
$$
which matches the participation map in Section 5.1 exactly and removes the previous complement and sign inconsistency.

Code: https://github.com/AndrewKim1997/gcfl

Assigned Action Editor: ~Ian_A._Kash1

Submission Number: 6735