Strong Equilibrium in Game Theory

• Strong Equilibrium is a game theory concept defined by stability against any coalition deviation, ensuring that no group of players can jointly improve their outcomes.
• It plays a critical role in applications such as network formation, job scheduling, and distributed systems, offering insights into dynamic coordination and robust resource allocation.
• Computational challenges arise with mixed-strategy SEs, while pure-strategy SEs can be verified efficiently and maintain resilience under stochastic dynamics.

A strong equilibrium (SE) is a solution concept in game theory that generalizes Nash equilibrium by requiring resilience not only to unilateral deviations but also to joint, coordinated deviations by arbitrary coalitions of players. This concept lies at the intersection of strategic stability, computational complexity, stochastic evolutionary dynamics, and applications in distributed systems, control, and resource allocation. SE imposes strict conditions for coalition-proof strategy selection, yielding rich theoretical properties and nontrivial computational challenges.

1. Formal Definition and Foundation

Strong equilibrium was introduced by Aumann as a refinement of Nash equilibrium (NE) for $n$-player strategic games. Formally, given a game with player set $N = \{1, \ldots, n\}$, pure action spaces $A_i$, utility functions $u_i$, and joint action profiles $a^*\in A = \prod_{i} A_i$, an action profile is a strong Nash equilibrium if:

• For all coalitions $S \subseteq N$, there does not exist an alternate profile $a$ with $a_i = a_i^*$ for all $i \notin S$ and $u_i(a) > u_i(a^*)$ for all $i \in S$.

Hence, SE is stable against any strict Pareto-improving deviation by a coalition, with other players’ strategies fixed. The notion of strict strong Nash equilibrium (SSSE) strengthens this to include cases where all members are at least as well off and one is strictly better off.

Any SE is necessarily a Nash equilibrium, but the converse does not hold. SE may not exist in many games: the condition is generically "too strong" except in special game classes (e.g., potential games with convexity or resource selection games under partitions (Caskurlu et al., 2019)).

2. Coalitional Dynamics and Stochastic Stability

Coalitional better-response (CBR) dynamics provide a process for equilibrium selection under coordinated deviations (Avrachenkov et al., 2015). At each time step, a randomly formed coalition may deviate to a new action profile if all members strictly benefit. The process is modeled as a Markov chain over joint action profiles, where transitions correspond to coalitional joint deviations.

• Absorbing states of this chain are SE profiles: once reached, no beneficial coalition deviation exists.
• If SE does not exist, the process may converge to a closed cycle—a set of profiles reachable via cycling coalitional deviations, yet impossible to escape via such deviations.

To model mistakes (mutations) in coordination, the CBR dynamics is perturbed: with small probability, a coalition may make a suboptimal deviation. The resulting perturbed Markov chain $P^\varepsilon$ is irreducible and has a unique stationary distribution. Stochastic stability is defined as the limiting probability that the process selects particular recurrent sets (SE or closed cycles) as error probability $\varepsilon \to 0$.

All SE and closed cycles are stochastically stable: they persist even in the presence of rare mistakes, and are selected in the long run by the dynamics. This generalized Young’s and Kandori’s unilateral selection results to the full coalitional setting.

3. Computational Complexity and Structural Characterization

Finding SE, especially in mixed strategies, is computationally challenging. The presence of coalitional deviations complicates verification and computation compared to NE. Key results include:

• Verification of a given strategy profile as SE is in $\mathcal{P}$ for fixed $n$ (Gatti et al., 2017). It decomposes into checking NE conditions and weak Pareto efficiency for every coalition, feasibly via support enumeration and minmax analysis.
• Finding an SE is $\mathcal{NP}$-complete (fixed $n$) (Gatti et al., 2013, Braggion et al., 2015). The hardness is solely due to mixed-strategy SEs; all pure-strategy SEs can be enumerated in polynomial time.
• Smoothed complexity: After independent perturbations to payoffs, mixed-strategy SEs generically vanish (payoff alignment is destroyed), leaving only easily-verifiable pure-strategy SEs (Gatti et al., 2013, Braggion et al., 2015). Thus, SE-finding algorithms are in smoothed-$\mathcal{P}$, in contrast to NE, which remains $\mathcal{PPAD}$-hard even after perturbations.

Structural theorems detail necessary conditions for mixed-strategy SEs:

• For two-agent games, payoffs in support must be collinear; for $n$ agents, they must lie on an $(n-1)$-dimensional hyperplane (Gatti et al., 2013, Braggion et al., 2015).
• In two-player games, the existence of a full-support SE implies the game is strictly competitive within its support; every supported outcome is (weakly) Pareto efficient (Braggion et al., 2015).
• The set of games admitting mixed-strategy SE is measure zero; almost all SEs are pure, except in special, knife-edge instances.

4. Extensions: Stochastic Evolution, Time-Inconsistent Control, and Restricted Coalitions

Stochastic Nash Equilibria

In evolutionary games with environmental noise, the notion of strong stochastic Nash equilibrium (SNE) extends SE by requiring geometric mean payoffs to strictly favor the equilibrium over all other strategies (Li et al., 2023). The hierarchy is:

$$\text{Strong SNE} \implies \text{Stochastically Evolutionarily Stable (SES)} \implies \text{SNE}$$

Noise can induce new strong SNEs, possibly in mixed strategies and with coexistence, unlike in classical NE. Multiple, completely mixed strong SNEs may arise as stochastic fluctuations increase, fundamentally altering evolutionary outcome landscapes.

Time-Inconsistent Control and Stopping

Strong equilibrium is also essential in continuous-time stochastic control with non-exponential discounting (Bayraktar et al., 2019, Huang et al., 2018). Here, SE corresponds to subgame perfect equilibrium: for every state and infinitesimal time window, there is no profitable deviation (not merely a zero-rate incentive as in weak equilibrium). This refinement ensures genuine dynamic consistency and incentive compatibility in time-inconsistent environments.

New iterative algorithms enable direct construction of strong equilibria in continuous-time Markov stopping problems, provided the discount function is log sub-additive (e.g., decreasing impatience) (Bayraktar et al., 2019).

Social Coalition Structure Restrictions

Intermediate equilibrium concepts arise by restricting which coalitions are viable, inspired by partition, laminar, contiguous, and centralized social structures (Caskurlu et al., 2019). Super strong equilibrium (no coalition can weakly benefit), partition equilibrium (coalitions fixed), and further relaxations yield a spectrum of solution concepts. Existence results and counterexamples delineate which coalition structures admit equilibrium in resource selection games.

5. Applications: Network Formation, Job Scheduling, Distributed Systems

SE has direct relevance in systems where coalitional stability is necessary:

• Network Creation Games: SE characterizations depend on edge cost parameter $\alpha$ (Janus et al., 2017). For all $\alpha \geq 2$, every star graph is an SE (star conjecture proven); for large enough $\alpha$, non-star trees may also be SE. The strong price of anarchy is tightly bounded, and coalitional improvement dynamics may or may not converge (c-FIP, weak acyclicity conditions).
• Job Scheduling Games: NE fails to prevent coalitional deviations, but their benefits are bounded; NE and LPT schedules can be viewed as approximate SE via quantitative measures (minimum improvement ratio IR_min, maximum IR_max, maximum damage ratio DR_max) (Feldman et al., 2014). A PTAS achieves schedules arbitrarily close to SE for makespan minimization.
• Resource Selection Games (RSGs): Existence of SE and its relaxations depends on resource homogeneity, coalition restrictions, and social structure (Caskurlu et al., 2019).

6. Implications, Limitations, and Open Directions

Strong equilibrium imposes the most robust coalition-proof stability available among static solution concepts. However, its existence is rare, computationally demanding, and sensitive to game structure and perturbations. In practical and stochastic environments, the interplay between robustness, tractability, and dynamic stability must be carefully balanced.

Essential areas of ongoing research include:

• Algorithm design for SE in multi-agent games, especially for mixed strategies and smoothed analysis (Gatti et al., 2017, Gatti et al., 2013).
• Dynamic selection and learning processes under coalitional deviations and environmental stochasticity (Avrachenkov et al., 2015, Li et al., 2023).
• Applications in time-inconsistent agency, control, and social decision systems where credible commitment and group coordination are fundamental (Bayraktar et al., 2019, Huang et al., 2018, Caskurlu et al., 2019).
• Structural classification of equilibria under realistic coalition formation constraints.

Summary Table: Key Properties of Strong Equilibrium

Dimension	Strong Equilibrium (SE)	Nash Equilibrium (NE)
Stability against coalitions	Yes (all coalitions, strict)	No (only unilateral)
Existence	Rare, structure-dependent	Always (finite games)
Computational complexity	NP-complete (mixed), P (pure/verification)	PPAD-complete
Smoothed complexity	Polynomial (pure SE generically)	Still hard
Stochastic stability (CBR dyn.)	All SE and cycles stochastically stable	Payoff/risk dominance dynamics
Pareto efficiency of outcomes	Required for coalitions	Not required

Strong equilibrium thus occupies a foundational role in coalitional game theory, bridging classical static analysis, stochastic dynamic selection, computational tractability, and robust stability in complex multi-agent environments.