Recognition: unknown
On Conservative Stable Standard of Behavior and Perfect Coalitional Equilibrium
Pith reviewed 2026-05-10 16:17 UTC · model grok-4.3
The pith
In coalitional repeated games, Perfect Coalitional Equilibrium paths form the largest nondiscriminating Conservative Stable Standard of Behavior.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In Greenberg (1989)'s coalitional repeated game situation, every nondiscriminating Conservative Stable Standard of Behavior is a subset of the set of Perfect Coalitional Equilibrium paths. Moreover, the set of Perfect Coalitional Equilibrium paths itself is a nondiscriminating Conservative Stable Standard of Behavior. The set of Perfect Coalitional Equilibrium paths is therefore the maximal nondiscriminating Conservative Stable Standard of Behavior.
What carries the argument
The set of Perfect Coalitional Equilibrium paths, which contains every nondiscriminating Conservative Stable Standard of Behavior and satisfies the nondiscriminating Conservative Stable Standard of Behavior property itself.
If this is right
- Any outcome stable under a nondiscriminating Conservative Stable Standard of Behavior must lie on a Perfect Coalitional Equilibrium path.
- The Perfect Coalitional Equilibrium paths supply a complete and largest collection of stable behaviors in the coalitional repeated setting.
- Smaller Conservative Stable Standards of Behavior cannot introduce stable paths beyond those already captured by Perfect Coalitional Equilibrium.
- Analyses of stability in repeated coalitional games can rely on Perfect Coalitional Equilibrium without loss of the nondiscriminating property.
Where Pith is reading between the lines
- The maximality result may simplify equilibrium selection by focusing attention on the broadest stable set rather than arbitrary subsets.
- The unification could extend to checking whether similar containment holds in non-repeated coalitional games or with different deviation protocols.
- Empirical tests in laboratory repeated games could examine whether observed coalition behaviors match the full Perfect Coalitional Equilibrium set or remain inside smaller standards.
- The finding links two stability notions, potentially aiding comparisons with other coalitional solution concepts like the core or bargaining sets in dynamic environments.
Load-bearing premise
The result depends on the specific coalitional repeated game situation and the exact definitions of nondiscriminating Conservative Stable Standard of Behavior and Perfect Coalitional Equilibrium matching the intended stability concepts.
What would settle it
A nondiscriminating Conservative Stable Standard of Behavior that includes a path outside the Perfect Coalitional Equilibrium set, or a Perfect Coalitional Equilibrium path that fails to qualify as a nondiscriminating Conservative Stable Standard of Behavior.
read the original abstract
We show that in Greenberg (1989)'s coalitional repeated game situation, every nondiscriminating Conservative Stable Standard of Behavior is a subset of the set of Perfect Coalitional Equilibrium (Ali and Liu 2026) paths. Moreover, the set of Perfect Coalitional Equilibrium paths itself is a nondiscriminating Conservative Stable Standard of Behavior. The set of Perfect Coalitional Equilibrium paths is therefore the maximal nondiscriminating Conservative Stable Standard of Behavior.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims that in Greenberg (1989)'s coalitional repeated game situation, every nondiscriminating Conservative Stable Standard of Behavior (CSSB) is a subset of the set of Perfect Coalitional Equilibrium (PCE) paths defined in Ali and Liu (2026). It further shows that the PCE path set itself satisfies the axioms of a nondiscriminating CSSB, implying that PCE paths constitute the maximal nondiscriminating CSSB.
Significance. If the result holds, it supplies a maximality characterization that identifies the PCE path set as the largest nondiscriminating CSSB within the fixed coalitional repeated-game framework. This connects an established stability concept (Greenberg 1989) to the authors' PCE definition via explicit inclusion and self-verification steps, potentially clarifying equilibrium selection when coalitions can deviate. The direct comparison between the two solution concepts, grounded in the external CSSB axioms rather than ad-hoc restrictions, is a constructive contribution to the literature on repeated games with coalitional moves.
major comments (1)
- The inclusion direction (every nondiscriminating CSSB ⊆ PCE paths) is load-bearing for the maximality claim. The manuscript must explicitly map each CSSB axiom (conservative stability and nondiscrimination) to the one-shot deviation principle and coalitional perfection conditions that define PCE paths; without this mapping, it is unclear whether the containment follows directly or requires additional restrictions on the strategy space.
minor comments (2)
- The introduction would benefit from a short self-contained recap of the nondiscriminating CSSB definition and the coalitional repeated-game situation from Greenberg (1989), so that readers need not consult the 1989 reference to follow the proof.
- Notation for paths, standards of behavior, and the nondiscrimination property should be introduced once and used consistently; occasional shifts between 'path' and 'outcome' reduce readability.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our contribution and for the constructive comment on improving the clarity of the inclusion argument. We address the point below.
read point-by-point responses
-
Referee: The inclusion direction (every nondiscriminating CSSB ⊆ PCE paths) is load-bearing for the maximality claim. The manuscript must explicitly map each CSSB axiom (conservative stability and nondiscrimination) to the one-shot deviation principle and coalitional perfection conditions that define PCE paths; without this mapping, it is unclear whether the containment follows directly or requires additional restrictions on the strategy space.
Authors: We agree that an explicit mapping would strengthen the exposition. Our proof of the inclusion proceeds by showing that any path violating the one-shot deviation principle or coalitional perfection cannot satisfy conservative stability (or nondiscrimination) in Greenberg's coalitional repeated-game situation; the argument relies only on the definitions and does not impose extra restrictions on the strategy space. In the revised version we will insert a dedicated paragraph immediately after the statement of the main theorem that tabulates the correspondence: conservative stability maps to the one-shot deviation principle (via the conservative deviation test), and nondiscrimination maps to coalitional perfection (via the requirement that all coalition members receive the same continuation value). This makes the logical steps direct without altering the result. revision: yes
Circularity Check
No significant circularity; standard comparison of independent solution concepts
full rationale
The paper establishes two directions of inclusion between nondiscriminating CSSB (defined externally in Greenberg 1989) and PCE paths (defined in the authors' prior 2026 paper) within the fixed coalitional repeated game situation. The inclusion every nondiscriminating CSSB ⊆ PCE paths follows from verifying that CSSB must obey the one-shot deviation and coalitional perfection conditions that characterize PCE; maximality follows from directly checking that the PCE path set itself satisfies Greenberg's conservative stability and nondiscrimination axioms. These verifications are performed in the present manuscript against the external CSSB definition and do not reduce by construction to a self-definition, fitted parameter, or unverified self-citation chain. The self-citation merely supplies the PCE definition; the load-bearing arguments are the independent proofs supplied here.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Coalitional repeated game situation as defined in Greenberg (1989)
- domain assumption Definitions of Conservative Stable Standard of Behavior (nondiscriminating) and Perfect Coalitional Equilibrium
Reference graph
Works this paper leans on
-
[1]
On the Theory of Infinitely Repeated Games with Discounting
Abreu, Dilip. 1988. “On the Theory of Infinitely Repeated Games with Discounting.” Econometrica56 (2):383–396
1988
-
[2]
Toward A Theory of Dis- counted Repeated Games with Imperfect Monitoring
Abreu, Dilip, David Pearce, and Ennio Stacchetti. 1990. “Toward A Theory of Dis- counted Repeated Games with Imperfect Monitoring.”Econometrica:1041–1063
1990
-
[3]
Coalitions in Repeated Games
Ali, S. Nageeb and Ce Liu. 2026. “Coalitions in Repeated Games.” Working Paper
2026
-
[4]
An application of the theory of social situations to repeated games
Greenberg, Joseph. 1989. “An application of the theory of social situations to repeated games.”Journal of Economic Theory49 (2):278–293. 12
1989
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.