Determine the optimal values of general load-balancing games

Determine the exact classical and quantum values of the load-balancing nonlocal game with two parties, odd input cardinality $n_{\mathrm{in}}=2k+1$, channel-output cardinality $n_{\mathrm{out}}=2k+1$, and capacity condition $s+t\leq0$, for general $k$ and the specified utility function.

Background

The load-balancing game models two transmitters that must select channels without communication, matching channels when their combined data rate is within capacity and choosing distinct channels when it exceeds capacity. The paper studies the game numerically as an application of BGD.

A special two-input version is solved exactly in the paper, but the general odd-input family used in the numerical experiments remains unresolved. Exact values would quantify the optimal classical and quantum coordination performance of the broader load-balancing model.

References

The classical and quantum values are not known in general for load balancing problems.

— Efficiently Optimizing the Quantum Value of Bell Inequalities using Batched Gradient Descent  (2610.01699 - Xu et al., 1 Oct 2026) in Section 7, subsection “Load balancing $(2,n_{\mathrm{in}},n_{\mathrm{out}})$”