Hypergraph Minority Game with Local Hyperedge Payoffs

We provide a theoretical derivation of the Hypergraph Minority Game with Local Hyperedge Payoffs (HMG-L), in which $N$ adaptive agents compete simultaneously in multiple overlapping groups modeled as hyperedges of a static hypergraph $\Hyper=(\Vset,\Eset)$. Each hyperedge constitutes an independent local minority game, and agents accumulate payoffs across all groups to which they belong. We derive the continuum-time limit of the score dynamics, from which we obtain a set of coupled nonlinear stochastic differential equations for the agents' strategy polarization variables. The deterministic drift is shown to derive from a global cost function that generalizes the standard Minority Game Hamiltonian to hypergraph-structured interactions. We perform a sparse-annealed replica analysis of the stationary state for the case of a $k$-uniform, $d$-regular random hypergraph, obtaining the saddle-point equations within the replica-symmetric ansatz, an explicit replicon stability criterion, and Bethe/cavity equations for sparse corrections. The leading sparse-regime transition occurs on a critical surface $\alphacrit(k,d)$, while the globally coupled MG value $\alphacrit\simeq0.3374$ is recovered only in the separate single-hyperedge limit. We derive expressions for the order parameters -- global volatility $\sigma^2$, predictability $\theta$, hyperedge frustration $F_e$, and frozen fraction $\phi$ -- and discuss their scaling behavior near criticality. The Fokker-Planck equation governing finite-$N$ fluctuations is presented, and the noise covariance matrix is computed from the hypergraph structure. Limiting cases ($k\to N$, $k\to2$, $d\to\infty$) are analyzed in detail, establishing connections to the standard MG, networked MG, and parallel MG models.