Strong Nash equilibrium¶
A strategy profile from which no coalition can jointly deviate so that every coalition member becomes strictly better off.
Core Idea¶
A strong Nash equilibrium is a strategy profile immune to every coordinated deviation that makes all members of the deviating coalition strictly better off while outsiders keep their strategies. It therefore implies, but is stronger than, ordinary Nash equilibrium. Singleton coalitions are included, so every SNE is a Nash equilibrium. Singleton coalitions are included, so every SNE is a Nash equilibrium.
Scope of Application¶
SNE applies to strategic-form games where coalition deviations and individual payoffs are well defined. Use it in games with explicit strategies, individual payoffs, coalitions, fixed-outsider counterfactuals, and strict all-member improvement.
- Voting games. Filters outcomes vulnerable to coordinated voter changes.
- Public-goods games. Tests group deviations beyond unilateral free riding.
- Network formation. Examines joint rewiring under fixed outsiders.
- Mechanism design. Evaluates coalition resistance of implemented outcomes.
- Equilibrium comparison. Contrasts Nash, strong, and coalition-proof stability.
Clarity¶
The definition resolves three quantifiers: every coalition, some joint deviation, and strict gain for every member. Writing them explicitly prevents aggregate coalition gain or a single member's improvement from being mistaken for a blocking deviation. The closest near miss sets the boundary: A coalition-proof Nash equilibrium is the closest near miss: it screens multilateral deviations for internal credibility, while SNE blocks every all-member-benefiting coalition deviation whether self-enforcing or not. A positive case must satisfy this test: Include a profile only if every coalition lacks a feasible joint deviation that strictly improves each member's payoff with outsiders fixed.
Manages Complexity¶
A game with n players has exponentially many coalitions and potentially many joint deviations. SNE compresses that large search into a robustness label, while an actual proof still needs structural arguments or exhaustive bounds. The central robustness–existence tradeoff is this: Testing all coalitions gives a powerful stability guarantee but can eliminate every profile. A second coalition power–deviation credibility tension matters because SNE blocks even deviations that might unravel internally; coalition-proof equilibrium screens credibility.
Abstract Reasoning¶
Use three linked moves: specify players, feasible strategies, and payoff functions; fix the candidate strategy profile; enumerate or characterize each nonempty coalition while holding outsiders fixed. As a collapse test, the case exits when one coalition has a joint deviation giving every member a strict payoff gain. A fourth check is to search for a joint deviation that strictly raises every member's payoff. A final check is to accept strong equilibrium only if no coalition passes that blocking test.
Knowledge Transfer¶
The test transfers literally across games with comparable coalition and payoff structure. Organizational uses of 'strong agreement' are analogy unless strategies, outsiders, feasible joint deviations, and individual strict benefits are specified. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Singleton coalitions recover unilateral stability. It applies an additional self-enforcement filter.
Relationships to Other Abstractions¶
Current abstraction Strong Nash equilibrium Domain-specific
Parents (1) — more general patterns this builds on
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Strong Nash equilibrium is a kind of Equilibrium Prime
A strong Nash equilibrium is an equilibrium robust against jointly beneficial coalition deviations.
Hierarchy path (1) — routes to 1 parentless root
- Strong Nash equilibrium → Equilibrium → Fixed Point
Neighborhood in Abstraction Space¶
Strong Nash equilibrium sits in a crowded region of the domain-specific corpus (28th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Allocation, Ranking & Bargaining Models (11 abstractions)
Nearest neighbors
- Rubinstein bargaining model — 0.92
- Max-dominated strategy — 0.91
- Correlated equilibrium — 0.91
- Evaluation function — 0.89
- Game — 0.88
Computed from structural-signature embeddings · 2026-10-08