The present study provides a closed-form characterisation of Nash equilibria in N-player binary games with unidirectional dependencies. While general network games are PPAD-complete, prior work has established that trees or paths admit polynomial-time solutions via dynamic programming. We provide a deterministic characterisation for the subclass of directed cycle graphical games, demonstrating that non-zero boundary incentives linearize the topology into a feed-forward propagation. Under this Robust Incentive Structure, resolution is achieved in O(N) time: strict dominance guarantees a unique equilibrium; in its absence, pure strategy equilibria are governed by the Parity Condition, while a unique fully mixed equilibrium is guaranteed via induced payoff indifference. For non-robust regimes, we deliver branching rules. The transition-matrix formulation evaluates the search tree size beforehand. This transparency enables the inverse design of target equilibria in circular networks, making explicit the mechanics that remain opaque in numerical solvers.