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This paper considers the problem of co-synthesis in $k$-player games over afinite graph where each player has an individual $\omega$-regular specification$\phi_i$. In this context, a secure equilibrium (SE) is a Nash equilibriumw.r.t. the lexicographically ordered objectives of each player to first satisfytheir own specification, and second, to falsify other players' specifications.A winning secure equilibrium (WSE) is an SE strategy profile$(\pi_i)_{i\in[1;k]}$ that ensures the specification$\phi:=\bigwedge_{i\in[1;k]}\phi_i$ if no player deviates from their strategy$\pi_i$. Distributed implementations generated from a WSE make components actrationally by ensuring that a deviation from the WSE strategy profile isimmediately punished by a retaliating strategy that makes the involved playerslose.  In this paper, we move from deviation punishment in WSE-based implementationsto a distributed, assume-guarantee based realization of WSE. This shift isobtained by generalizing WSE from strategy profiles to specification profiles$(\varphi_i)_{i\in[1;k]}$ with $\bigwedge_{i\in[1;k]}\varphi_i = \phi$, whichwe call most general winning secure equilibria (GWSE). Such GWSE have theproperty that each player can individually pick a strategy $\pi_i$ winning for$\varphi_i$ (against all other players) and all resulting strategy profiles$(\pi_i)_{i\in[1;k]}$ are guaranteed to be a WSE. The obtained flexibility inplayers' strategy choices can be utilized for robustness and adaptability oflocal implementations.  Concretely, our contribution is three-fold: (1) we formalize GWSE for$k$-player games over finite graphs, where each player has an $\omega$-regularspecification $\phi_i$; (2) we devise an iterative semi-algorithm for GWSEsynthesis in such games, and (3) obtain an exponential-time algorithm for GWSEsynthesis with parity specifications $\phi_i$.

Most General Winning Secure Equilibria Synthesis in Graph Games