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arXiv · 2606.06625

N-Player Binary Games with Unidirectional Dependencies: Cycle Robustness and Induced Indifference

Jose Maria Sanchez-Saez·Nana Odishelidze·Francisco Criado-Aldeanueva
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Abstract

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.

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Jose Maria Sanchez-Saez, Nana Odishelidze, Francisco Criado-Aldeanueva. 2026-06-04. N-Player Binary Games with Unidirectional Dependencies: Cycle Robustness and Induced Indifference. https://doi.org/10.1016/j.cnsns.2026.110151

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