Most General Winning Secure Equilibria Synthesis in Graph Games
Satya Prakash Nayak, Anne-Kathrin Schmuck
TL;DR
The paper addresses co-synthesis in multi-player graph games with ω-regular specifications by generalizing secure equilibria to Most General Winning Secure Equilibria (GWSE), enabling distributed assume-guarantee realizations. It introduces a formal GWSE framework, an iterative semi-algorithm using adequately permissive assumptions (APA) to synthesize GWSE, and an exponential-time parity-based algorithm; then presents a terminating, optimized approach using UCAs to ensure practicality on parity games. The results show that locally satisfiable specifications can guarantee a global winning secure equilibrium, supporting robust, decentralized implementations for reactive systems. This work advances distributed synthesis by replacing punitive equilibria with contract-like, assumption-driven guarantees that are implementable and resilient to variation in local strategies.
Abstract
This paper considers the problem of co-synthesis in $k$-player games over a finite graph where each player has an individual $ω$-regular specification $φ_i$. In this context, a secure equilibrium (SE) is a Nash equilibrium w.r.t. the lexicographically ordered objectives of each player to first satisfy their own specification, and second, to falsify other players' specifications. A winning secure equilibrium (WSE) is an SE strategy profile $(π_i)_{i\in[1;k]}$ that ensures the specification $φ:=\bigwedge_{i\in[1;k]}φ_i$ if no player deviates from their strategy $π_i$. Distributed implementations generated from a WSE make components act rationally by ensuring that a deviation from the WSE strategy profile is immediately punished by a retaliating strategy that makes the involved players lose. In this paper, we move from deviation punishment in WSE-based implementations to a distributed, assume-guarantee based realization of WSE. This shift is obtained by generalizing WSE from strategy profiles to specification profiles $(\varphi_i)_{i\in[1;k]}$ with $\bigwedge_{i\in[1;k]}\varphi_i = φ$, which we call most general winning secure equilibria (GWSE). Such GWSE have the property that each player can individually pick a strategy $π_i$ winning for $\varphi_i$ (against all other players) and all resulting strategy profiles $(π_i)_{i\in[1;k]}$ are guaranteed to be a WSE. The obtained flexibility in players' strategy choices can be utilized for robustness and adaptability of local implementations. Concretely, our contribution is three-fold: (1) we formalize GWSE for $k$-player games over finite graphs, where each player has an $ω$-regular specification $φ_i$; (2) we devise an iterative semi-algorithm for GWSE synthesis in such games, and (3) obtain an exponential-time algorithm for GWSE synthesis with parity specifications $φ_i$.