Secure Equilibrium in Game Theory

• Secure Equilibrium (SE) is a refinement of Nash equilibrium that evaluates deviations lexicographically—first maximizing a player's own outcome then minimizing opponents’ gains.
• SE is applied in various settings including turn-based, weighted, and ω-regular games, supporting synthesis methods like assume-guarantee synthesis and ensuring robustness against malevolent deviations.
• Algorithmic synthesis of SE utilizes punishment strategies and compactness principles, with complexity and memory requirements varying significantly across different game models.

Secure equilibrium (SE) is a refinement of Nash equilibrium for non-zero-sum games in which unilateral deviations are evaluated lexicographically: a player first seeks to improve their own outcome, and, when their own outcome is unchanged, they prefer deviations that worsen the outcomes of other players. In the literature, SE is studied in turn-based graph games, weighted games, games with finitely many outcomes, and games with $\omega$-regular specifications, and it is specifically developed for assume-guarantee synthesis and has already been applied in this context (Pril et al., 2014). The concept sits between ordinary strategic stability and adversarial robustness: it excludes not only profitable selfish deviations, but also “malevolent” tie-preserving deviations that merely harm others (Roux et al., 2017).

1. Definition and equilibrium-theoretic position

SE strengthens Nash equilibrium (NE) by refining each player’s preference relation. In the two-player finite-outcome setting, an SE is an NE in the malevolent game obtained by lexicographically refining each player’s preference: a player prefers an outcome not just by their own payoff, but, in case of ties, by minimizing the opponent’s payoff (Roux et al., 2017). In weighted two-player games, this is expressed as the requirement that each player tries to maximize their own objective and, among optimal choices, select the one most disadvantageous to the other (Bruyère et al., 2014).

In quantitative reachability games, where each player $i$ has a goal set $\mathsf{F}_i \subseteq V$ and aims to reach it as soon as possible, the cost of a play $\rho$ is the first hitting time of $\mathsf{F}_i$, or $+\infty$ if the goal is never reached: $\mathit{cost}_i(\rho) = \begin{cases} l & \text{if %%%%6%%%% is the least index such that %%%%7%%%%}\ +\infty & \text{otherwise.} \end{cases}$ For cost profiles $x=(x_i)_{i\in\Pi}$ and $y=(y_i)_{i\in\Pi}$, player $j$ uses the preference

$i$0

and a strategy profile is secure when no unilateral deviation yields a strictly preferred profile under $i$1 (Brihaye et al., 2012).

In games with Boolean or $i$2-regular objectives, the same idea is encoded on payoff vectors in $i$3: player $i$4 first prefers switching from loss to win, and, if their own bit is unchanged, prefers decreasing the payoff bits of other players. Formally,

$i$5

which makes SE a Nash-like stability notion for rational but potentially hostile agents (Nayak et al., 2024).

This common structure suggests a unifying view: SE is not defined by a single payoff domain, but by a lexicographic refinement of unilateral rationality whose primary component is self-interest and whose secondary component is opponent harm.

2. Game models and semantic settings

The most studied setting in the supplied literature is the turn-based multiplayer quantitative reachability game on a finite directed graph, where each play is an infinite path and each player minimizes the delay before first reaching their goal set (Brihaye et al., 2012). This model is finite-state, graph-based, and explicitly quantitative.

A second major setting is that of two-player non zero-sum infinite duration games played on weighted graphs. There each edge carries a pair of rational rewards, and play values are aggregated using measures such as $i$6, $i$7, $i$8, $i$9, mean-payoff, and discounted sum (Bruyère et al., 2014). The secure-equilibrium notion extends from the Boolean setting to this quantitative setting without changing the lexicographic tie-breaking principle.

A broader existence theory is developed for multi-player perfect information turn-based games. Two classes are singled out: games with possibly probabilistic transitions, countable state and finite action spaces, and bounded and continuous payoff functions; and games with only deterministic transitions, arbitrary state and action spaces, and Borel payoff functions with a finite range, in particular qualitative Borel payoff functions (Pril et al., 2014). These results place SE in a much wider descriptive-set-theoretic and stochastic framework than finite graph games alone.

SE also appears in two-player games in normal form with finitely many outcomes, where the decisive structural assumption is determinacy of all derived win/lose games (Roux et al., 2017). In that setting, Muller games, parity games, and more general priority games are treated as instances. More recent work studies $\mathsf{F}_i \subseteq V$0-player games over finite graphs where each player has an individual $\mathsf{F}_i \subseteq V$1-regular specification $\mathsf{F}_i \subseteq V$2, and SE is defined with respect to lexicographically ordered satisfaction patterns of these specifications (Nayak et al., 2024). Another line focuses on constrained SE existence in multi-player games with objectives drawn from Büchi, co-Büchi, Parity, Streett, Rabin, and Muller classes, where the constraint specifies exactly which players must win (Mizuno et al., 2 Sep 2025).

3. Existence results and separations

The existence theory for SE is strongly model-dependent. The following table summarizes the main existence results stated in the cited works.

Setting	Main SE-related result	Citation
Multiplayer quantitative reachability games	Existence of SE remains an open problem; every multiplayer game admits an SPE	(Brihaye et al., 2012)
Two-player quantitative reachability games	SE always exists; every game admits an SPSE	(Brihaye et al., 2012)
Two-player finite-outcome determined games	Determinacy of all derived win/lose games implies existence of NE, and hence SE under lexicographic preferences	(Roux et al., 2017)
Two-player weighted games	SE always exists for sup, inf, lim sup, lim inf, mean-payoff, and discounted sum measures	(Bruyère et al., 2014)
Multi-player perfect information games	SE exists in the two broad classes specified by bounded continuous or finite-range Borel payoffs	(Pril et al., 2014)

Two distinctions are central. First, in quantitative reachability games there is a sharp separation between ordinary subgame perfection and secure subgame perfection: every multiplayer game admits a subgame perfect equilibrium (SPE), but the existence of SE in the multiplayer case remained and is still an open problem, and for more than two players the existence of subgame perfect secure equilibrium (SPSE) is not settled (Brihaye et al., 2012). Second, the difficulty of lifting two-player arguments to the multiplayer setting is tied to the fact that the preference relation used in the relevant proof methods is total only in two-player games (Brihaye et al., 2012).

These statements do not amount to a general nonexistence result for multiplayer SE. On the contrary, existence is proved in broad multi-player perfect-information classes: if the state space is countable, all action spaces are finite, and all payoffs are continuous and bounded, then there exists a secure equilibrium; if transitions are deterministic, state and action spaces are arbitrary, and each payoff is Borel-measurable with a finite range, then there exists a secure equilibrium (Pril et al., 2014). The open problems are therefore specific to particular game models and refinements, rather than to SE in general.

4. Refinements and related solution concepts

SE has generated several refinements designed to control behavior after histories or to encode stronger synthesis guarantees. In quantitative reachability games, a subgame perfect equilibrium is a profile that is a Nash equilibrium in every subgame, and a subgame perfect secure equilibrium is a profile that is a secure equilibrium in every subgame. The hierarchy explicitly stated in the literature is

$\mathsf{F}_i \subseteq V$3

with existence of SPSE established for two-player quantitative reachability games (Brihaye et al., 2012).

In specification-based graph games, a winning secure equilibrium (WSE) is an SE strategy profile that ensures the conjunction $\mathsf{F}_i \subseteq V$4 if no player deviates from the prescribed profile. The security clause is formulated so that if a unilateral deviation would prevent some other player from winning, it also causes the deviator to lose (Nayak et al., 2024). This turns SE from a purely stability-oriented notion into a co-synthesis notion that preserves the fully winning outcome under rational deviations.

The same work introduces most general winning secure equilibria (GWSE), which generalize WSE from strategy profiles to specification profiles $\mathsf{F}_i \subseteq V$5 with $\mathsf{F}_i \subseteq V$6. The defining conditions are: preservation of the cooperative solution set, individual realizability of each local specification in a zero-sum sense, and the guarantee that every profile of locally realizing strategies forms a WSE (Nayak et al., 2024). This shifts the emphasis from explicit retaliatory strategies to distributed assume-guarantee realizability.

A distinct refinement vocabulary appears in multi-player perfect information games, where the paper introduces sum-secure equilibrium and strongly secure equilibrium for technical purposes. A profile is sum-secure when any best-response tie by player $\mathsf{F}_i \subseteq V$7 does not decrease the sum of the other players’ payoffs, and strongly secure when no opponent is worse off under such a tie-preserving deviation (Pril et al., 2014). These notions clarify the spectrum between ordinary NE and the full security requirement.

5. Algorithmic synthesis and complexity

The algorithmics of SE divide into existence checking, strategy synthesis, and constrained variants. For multiplayer quantitative reachability games, the existence of an SE in the infinite game is equivalent to the existence of an SE with certain “good” properties in a finite truncation of the game tree $\mathsf{F}_i \subseteq V$8 up to depth

$\mathsf{F}_i \subseteq V$9

The enforced properties are goal-optimality and deviation-optimality, and the resulting guess-and-check procedure yields that deciding the existence of a secure equilibrium is in ExpSpace. Whenever an SE exists, a finite-memory SE exists as well, and the same approach extends to threshold constraints on costs (Brihaye et al., 2012).

In weighted games, the main synthesis theorem states that in every two-player non zero-sum weighted game with $\rho$0, $\rho$1, $\rho$2, $\rho$3, or $\rho$4 payoff function, there exists a secure equilibrium with memory at most $\rho$5, and for $\rho$6 and $\rho$7 payoff function there exists a secure equilibrium with memory at most $\rho$8. Secure equilibria can be synthesized in pseudo-polynomial time for $\rho$9, $\mathsf{F}_i$0, and $\mathsf{F}_i$1, and in polynomial time for $\mathsf{F}_i$2, $\mathsf{F}_i$3, $\mathsf{F}_i$4, and $\mathsf{F}_i$5. The constrained existence problem is in NP $\mathsf{F}_i$6 co-NP for $\mathsf{F}_i$7 and $\mathsf{F}_i$8, in P for $\mathsf{F}_i$9, $+\infty$0, $+\infty$1, and $+\infty$2, while the discounted-sum case remains open (Bruyère et al., 2014).

For finite-outcome two-player games, the constructive proof of the determinacy-to-equilibrium transfer theorem computes secure equilibria for non-zero-sum priority games in quasi-polynomial time. The explicit corollary stated is that positional secure equilibria can be computed in $+\infty$3 (Roux et al., 2017). This is notable because the proof is formalized in both Coq and Isabelle, and the computation emerges from the constructive content of the formal proof.

In the $+\infty$4-regular synthesis setting, GWSE synthesis is addressed by an iterative semi-algorithm and, for parity specifications, by an exponential-time algorithm. The semi-algorithm may not terminate for general $+\infty$5-regular objectives, while the parity-specialized method terminates because the underlying game structure does not change per iteration and runs in time $+\infty$6 (Nayak et al., 2024).

For constrained SE existence in multi-player games, the core characterization is given in terms of coalition winning regions. With a target payoff vector $+\infty$7, one defines sets $+\infty$8 and $+\infty$9, constructs a subarena $\mathit{cost}_i(\rho) = \begin{cases} l & \text{if %%%%6%%%% is the least index such that %%%%7%%%%}\ +\infty & \text{otherwise.} \end{cases}$0 from coalition-enforceable retaliation conditions, and obtains

$\mathit{cost}_i(\rho) = \begin{cases} l & \text{if %%%%6%%%% is the least index such that %%%%7%%%%}\ +\infty & \text{otherwise.} \end{cases}$1

This yields the stated upper bounds: co-NP for Büchi, NP for co-Büchi, and PSPACE for parity and Muller objectives; the listed lower bounds include P-hardness for Büchi/co-Büchi in the two-player case, and NP-hardness and co-NP-hardness for parity in the two-player case (Mizuno et al., 2 Sep 2025).

6. Proof methods, applications, and open problems

The proof techniques behind SE results are diverse but structurally related. In multiplayer quantitative reachability games, the existence of SPE is proved by compactness and Kuhn’s theorem, passing from SPEs on finite truncations to the infinite game; the SPSE proof in the two-player case uses similar ideas but relies on totality properties that do not extend to arbitrary numbers of players (Brihaye et al., 2012). In two-player finite-outcome games, the transfer theorem proceeds through determinacy of all derived win/lose games and a generalized lift of the preference order to subsets of outcomes; the proof is constructive and machine-verified in Coq and Isabelle (Roux et al., 2017).

Weighted-game proofs reduce SE synthesis to zero-sum lexicographic payoff games and rely on punishment strategy construction: if a player deviates from the main plan, the other switches to a strategy that minimizes the deviator’s lexicographic payoff. The construction exploits positional or uniform determinacy, depending on the payoff measure, and the prefix-linear or prefix-independent behavior of the relevant objective functions (Bruyère et al., 2014). In multi-player perfect-information games, one proof line uses an action elimination procedure and punishment strategies in a core game, while another uses a payoff transformation

$\mathit{cost}_i(\rho) = \begin{cases} l & \text{if %%%%6%%%% is the least index such that %%%%7%%%%}\ +\infty & \text{otherwise.} \end{cases}$2

to reduce secure-equilibrium existence to Nash-equilibrium existence in a perturbed game (Pril et al., 2014).

SE’s applications are concentrated in model checking, controller synthesis, automata-theoretic verification, and assume-guarantee synthesis. The literature emphasizes that SE is useful when one wants guarantees not only against self-interested deviations but also against hostile best-response deviations (Roux et al., 2017). WSE-based implementations realize this via deviation punishment, whereas GWSE shifts to a distributed assume-guarantee realization in which local specifications, rather than retaliatory strategies, define the implementation contract (Nayak et al., 2024).

Several open problems remain explicit. The existence of secure equilibria in multiplayer quantitative reachability games is still open, as is the existence of SPSE for more than two players in that model (Brihaye et al., 2012). In weighted games, the constrained existence problem for discounted sum remains open and is linked to a longstanding open problem about discounted sums over infinite sequences (Bruyère et al., 2014). In GWSE synthesis, termination is not guaranteed for general $\mathit{cost}_i(\rho) = \begin{cases} l & \text{if %%%%6%%%% is the least index such that %%%%7%%%%}\ +\infty & \text{otherwise.} \end{cases}$3-regular objectives (Nayak et al., 2024). These boundaries mark SE as a mature but still actively differentiated research area: existence is well understood in several two-player and perfect-information settings, while multiplayer graph-theoretic refinements continue to expose nontrivial gaps between equilibrium, security, and subgame perfection.