Source-linked AI summary
Collective Counterfactual Planning: Coordination, Consent, and Verification under Representational Constraints
Chainarong Amornbunchornvej
TL;DR
Teams can struggle to plan, execute, and verify shared goals when members represent different parts of a task. The paper formalizes this geometry and shows that relay enables collective reach, while dark requirements prevent valid completion.
Problem
Existing multi-agent planning models emphasize capability, knowledge, or observability, leaving representational limitations as a distinct modeling question.
Method
The paper models collective counterfactual planning as projections of a common task space with conception, consent, implementation, and verification gates.
Results
Iterated relay can enable solutions absent from one-shot plans, whereas requirements in the team’s dark subspace cannot be validly completed; consent notions are incomparable.
Takeaways & Limitations
Collective success depends on coordinated representation, execution, and qualified verification, with teamwork and sub-teaming sometimes forced by task geometry.
Takeaways & Limitations
Restricted practical searches are sound for returned plans but may miss existing solutions and therefore are not completeness certificates for general CCS.
Abstract
from arXiv · showhide
Groups routinely complete projects that no single member can plan, execute, or verify alone. We propose a formal model of this phenomenon, Collective Counterfactual Planning (CCP), in which the binding limitation on each agent is neither capability, knowledge, nor observability, but representational geometry: each agent perceives the state, conceives moves, consents to actions, and certifies goal requirements only through a projection onto an agent-specific subspace of a common task space. Four gates jointly determine whether a team can reach a conjunctive goal and legitimately recognize that it has done so: the exogenous implementation coalitions required to perform each action, together with three representational gates -- conception, consent, and task-relative verification qualification. We define the Collective Counterfactual Solvability (CCS) problem, separating geometric feasibility, executable attainment, and validated completion. The results expose a positive-negative duality. Iterated cross-agent relay can unlock a solution that no one-shot pooling of individual plans contains, but any goal requirement depending essentially on the subspace dark to the entire team is unverifiable and therefore not validly completable, even when the trajectory accidentally attains it. Memoryless and audited consent further constrain different objects -- action directions versus cumulative trajectory states -- and neither dominates the other. A four-step exhaustive horizon-bounded solvability scheme is sound and complete under exact representation of the relay closure; restricted implementations remain sound on returned plans but need not be complete. The model gives one geometry for sequential mutual enabling, competent execution of steps whose purpose is invisible to the executor, forced sub-teaming at expertise boundaries, and completion that cannot be validly declared.
1 Introduction
The paper models collective planning as a problem of representational geometry: agents perceive, conceive, consent to, and certify actions only through individual subspaces. It separates executable attainment from validated completion and derives results on relay, collective blindness, consent, sub-teaming, and exhaustive solvability.
- 1 Introduction: Each agent’s representational subspace bounds what it can perceive, conceive, agree to, and certify, while implementation coalitions remain exogenous task structure.The model separates implementation from conception, consent, and qualification; none of the four gates implies another.
- 1 Introduction: Collective Counterfactual Solvability separates geometric feasibility, executable attainment, and validated completion, so objective attainment and legitimate completion can diverge.The distinction is built into the problem formulation rather than treated as an incidental failure mode.
- 1 Introduction: Iterated cross-agent re-expansion can unlock solutions absent from every one-shot individual tree, whereas requirements depending essentially on the team’s dark subspace lack qualified verifiers.Relay requires the enabling signal to pass through a component visible to the next agent; dark-subspace requirements cannot be validly completed.
- 1 Introduction: Memoryless consent gates action directions, audited consent gates cumulative trajectory states, and their solvability notions are incomparable; the same geometry can force non-plenary sub-teaming.These claims are established as separate consent-geometry results and include a proposition on forced sub-teaming.
- 1 Introduction: A four-step horizon-bounded scheme is sound and complete with exact relay closure, while restricted practical search remains sound on returned plans but may miss existing solutions.The four steps are coverage preflight, relay closure, ratification, and terminal inspection.
2 Related work
The paper situates Collective Counterfactual Planning among multi-agent, epistemic, coordination, organizational, and distributed-cognition research traditions. It distinguishes its within-basis prospective use of “counterfactual” from causal and basis-changing interpretations.
- Multi-agent planning: Multi-agent planning models heterogeneity through operator ownership, while epistemic planning locates limitations in knowledge and implicit coordination uses perspective shifts.These traditions provide the closest related planning and relay context identified by the paper.
- Coordination and collective agency: Coordination research addresses salience, equilibrium structure, common knowledge, group agency, cooperative problem solving, and convergence on time-varying interaction networks.The paper presents these accounts as complementary perspectives on coordination and collective agency.
- The phenomenon: Distributed cognition and epistemic dependence document groups accomplishing what no individual represents, while transactive memory addresses who knows what as a team.Organizational research further anticipates sub-teaming, verification coverage, and representationally undetectable failure.
- Terminology: Here, “counterfactual” means within-basis prospective rollout from the current state, distinct from Pearl’s and Lewis’s causal senses and from models where the representational basis changes.The terminology also connects forward simulation to prospection and treats actions as vectors in conceptual spaces for single events rather than multi-agent planning.
3 The model
CCP models team planning in a common task space where each agent can perceive, propose, consent to, and verify actions only through an agent-specific subspace. Executable, valid completion requires coalition implementation plus conception, consent, and task-relative verification gates, with the team’s relay closure capturing state-dependent plan extension.
- Representational geometry: Agents represent directions through subspaces Vi of an ambient task space V, while the group span S contains everything somebody can see and S⊥ is dark to everyone.Each agent also has an interpretability threshold τi, determining the angular consent bound θi.
- Implementation and consent: Actions are unit directions with exogenous required coalitions R(u), and a move executes only when every necessary implementer participates and interprets it.The coalition requirement is independent of agent geometry, while interpretability governs participation.
- Representational gates: Conception requires projected apparent progress, whereas consent requires sufficient interpretability, so neither gate implies the other.An agent cannot conceive a direction invisible in its subspace; memoryless consent evaluates the direction, while audited consent evaluates the new cumulative displacement.
- Verification: A requirement is verifiable only by a task-relative qualified agent whose projection completely determines membership, and different requirements may have different qualified verifiers.Qualification establishes competence to decide a requirement but does not establish that the requirement holds.
- Verification: The verification cover requires at least one qualified verifier for every requirement; under cover, STOP(x) iff x∈O, but without it objective attainment is not validated completion.The cover is state-independent and belongs to the team–task pair.
- Planning closure: Plan extension is state-dependent because availability depends on the prefix-created state, making relay iteration essential rather than reducible to distributed bookkeeping.With exogenously state-independent availability, order-irrelevance returns for the memoryless model and T≤d suffices.
4 The problem
Section 4 defines Collective Counterfactual Solvability as the existence of a horizon-bounded plan whose steps execute and whose terminal state satisfies STOP. It distinguishes geometric feasibility, executable attainment, and validated completion, with the final level requiring a verification cover.
- Collective Counterfactual Solvability: CCS asks whether a plan p ∈ L_h exists such that every step executes and STOP(x_T) holds, returning p or ⊥ in the synthesis version.The problem is parameterized by the task structure, consent model, and horizon h.
- Three semantic levels: The model separates geometric feasibility, executable attainment, and validated completion, with validation requiring attainment plus a verification cover.Geometric feasibility asks whether the goal displacement intersects cone(U); attainment requires an executable relay plan to land in O.
- Three semantic levels: Conception and consent create the gap between available physical moves and an executable route, while verification creates the gap between attainment and legitimate recognition.These representational gates distinguish what could physically add up, what the group can execute, and what it can validly declare complete.
5 Results
The results show that collective solvability depends on representational visibility, relay-enabled conception, consent, and qualified verification. They establish darkness-based unverifiability, distinguish memoryless from audited consent, and show that adaptive consent can strictly expand solvability.
- Unverifiability from darkness: A requirement depending essentially on the team’s dark subspace is unverifiable, so CCS is no even when the physical trajectory reaches the goal.No agent can distinguish satisfying from violating states along the invisible component, leaving the required verification qualification empty.
- Conception and visibility: A move wholly in the dark subspace is neither conceivable nor appendable; deliberate dark motion can occur only as a side-component of a visible direction.Thus unsolvability and undetectability coincide on the team-invisible subspace.
- Relay: Iterated relay can succeed when one-shot pooling cannot: agent 1 reaches x1 ≈ (1, 0.364), making agent 2’s previously inconceivable move executable and verifiably completing the goal.The relay requires the preceding action to alter a component visible to the next agent; here P2u1 = sin α ≠ 0.
- Consent models: Memoryless consent gates action directions, whereas audited consent gates cumulative trajectory states and constrains only the required auditor’s steps.Audited consent imposes ∥P⊥j Dt∥≤ tan θj ∥PjDt∥; memoryless consent is scale-invariant and imposes no magnitude constraint.
- Consent-model incomparability: Neither consent model contains the other: E1 is memoryless-solvable but audited-unsolvable, while E2 is audited-solvable but memoryless-unsolvable.The separation depends on the progress-realizing conception clause and holds for witness angles satisfying tan α ≤ cot θ2.
- Adaptive consent: Adaptive consent is strictly more permissive than either pure regime, with an orthogonal-block instance solvable under an adaptive sequence combining audited and memoryless consent.The constructed endpoint is (1, 0, 1, 0.3), and all consent checks pass under σ = (A, A, M, M).
6 Exhaustive solvability scheme, correctness, and complexity
The section gives an exhaustive four-step, horizon-bounded scheme that is sound and complete for CCS when qualification sets, relay closure, consent, and inspection tests are represented and evaluated exactly. Restricted searches preserve soundness for returned plans but lose completeness unless they explore the full closure.
- Exhaustive solvability scheme: For fixed horizon h, exhaustive generation ranges over every instantiated plan in Lh, including every duration allowed by the projected-progress clause.This extensional formulation separates mathematical exhaustiveness from whether Lh has an efficient finite representation.
- Exhaustive solvability scheme: The four-step scheme preflights verifier coverage, generates the complete relay closure, ratifies candidate plans, and performs terminal inspection.It rejects immediately when any qualification set is empty, considers closure endpoints in O, removes refused candidates, and accepts only plans passing every qualified requirement check.
- Correctness: Under exact representation and evaluation assumptions, the scheme returns a plan if and only if CCS is yes for horizon h.Every returned plan is a valid CCS witness.
- Restricted search: A restricted search over bLh ⊆ Lh remains sound for returned plans, but returning ⊥ certifies CCS = no only when bLh = Lh.Completeness can fail when a valid witness lies outside the explored subset.
- Complexity and representation: Algorithmic and complexity statements restrict requirements Ok to effectively represented sets whose membership, projected distance, and qualification tests are computable.Affine or polyhedral requirements are identified as the principal example, making the input finite.
7 Computational illustration
The computational illustration provides a reference implementation for affine requirements and uses structured examples and figures to instantiate the formal results. Its restricted search is sound for returned plans but is not claimed complete, and the illustrations are not empirical validation.
- Reference implementation: The reference implementation searches a restricted subset by choosing the projected-optimal improving λ at each extension, while exact ratification and terminal predicates make every returned plan sound.The implementation targets affine requirements and preserves Definition 10 for generated tokens.
- Computational illustrations: Figure 3 illustrates an E1 two-step relay that is memoryless-solvable but audited-unsolvable, because agent 2 sees a closed loop despite goal-relevant progress in its dark dimension.Agent 1 changes the state so agent 2 can conceive a continuation unavailable initially.
- Reference implementation: The implementation is not complete for general CCP because a valid solution may require a different improving duration.Numerical claims in Examples E1 and E2 are executable assertions in the test suite.
- Computational illustrations: Figures 3–5 provide structured illustrations of the formal results, including relay geometry, sweeps over span coverage and consent thresholds, and variation of the shared representational core.The paper explicitly distinguishes these illustrations from empirical validation.
8 Discussion
The discussion frames the results as a geometry of sequential mutual enabling and competent execution without globally visible purpose, while delimiting the model through fixed, truthful representations and sovereign execution. It also identifies general CCS hardness and minimum-team synthesis as future problems.
- What the results formalize: Sequential mutual enabling lets repeated reconsideration succeed where one-shot pooling fails, because one agent’s move can make another agent’s next move conceivable.This is identified as the first formalized phenomenon, linked to Theorem 2 and Corollary 1.
- What the results formalize: An executor can fully understand the step performed even when the reason it matters lies outside the executor’s projection.The discussion characterizes this as competence without global purpose, formalized by Proposition 3.
- What the results formalize: Audited consent helps at lenient thresholds but hurts at strict thresholds, demonstrating incomparability with memoryless consent.Figure 4 reports this contrast alongside the finding that greater representational coverage increases attainment and validated completion.
- Scope: The model excludes strategy and learning, assumes truthful agents sharing O, fixes representational bases, and uses sovereign execution that bars performing steps the executor does not understand.These exclusions define the intended regime as a cognitive skeleton of cooperation and professionalized collaboration on timescales shorter than teaching.
- Future work: General CCS is hard, and future work includes constructing the minimum team that makes CCS yes from tasks, candidate agents, subspaces, and thresholds.This is presented as the representational successor to team formation with required skills.
9 Conclusion
Collective Counterfactual Planning derives relay, consent, sub-teaming, and sign-off from representational geometry interacting with task implementation structure. It explains both how teams can achieve goals no member can plan alone and why some requirements cannot be validly completed, while providing reproducible implementations and experiments.
- Conclusion: Collective Counterfactual Planning derives relay, consent, sub-teaming, and sign-off from projections of a common task space combined with implementation structure.The model’s positive result concerns group achievement beyond any single member’s planning ability; its negative result concerns requirements the team cannot validly finish.
- Conclusion: The exhaustive horizon-bounded solvability scheme is correct for the model’s stated setting.The supplied conclusion identifies this as a central theoretical result, though the passage is truncated before stating the full qualification.
- Conclusion: As the common representational core grows, successful plans use larger coalitions and more whole-team steps, while smaller subteams carry progress outside that core.All instances shown in Figure 5 remain solvable.
- Reproducibility: The accompanying code and reproduction materials make the numerical claims of Examples E1 and E2 and Proposition 1 executable assertions in a seeded test and experiment suite.The implementation uses the restricted duration policy from Section 7 to check witnesses and illustrate the theory, rather than establish completeness.
- Disclosure: The author used Claude and ChatGPT to edit and polish the draft, then reviewed and edited the content and accepted full responsibility for the publication.This disclosure appears in the supplied conclusion materials.
A Additional results
The additional results show that interpretability can enable an action without revealing its purpose, while state-independent availability yields a horizon-bounded solvability guarantee through additive cone structure. Together, they clarify the relay mechanism and isolate the role of prefix-dependent enabling.
- Loop inversion: In the E1 relay plan, executor 2 fully interprets his step, yet sees a closed-loop trajectory whose goal-relevant progress is invisible to him.His projected trajectory is 0 →tan α →0, with P2(xT −x0) = 0 and P2e1 = 0; interpretability gates motion, never purpose.
- State-independent comparison variant: Under state-independent availability, memoryless consent, and verification cover, any reachable goal direction in cone(U∗) has a witnessing plan with T ≤d.Here U∗ contains directions conceived by every required relay agent and executable by at least one, while ∆∩cone(U∗) ≠ ∅.
- State-independent comparison variant: The T ≤d bound follows because prefix-independent availability and consent make token appendability depend only on direction, and conic Carathéodory reduces combinations to at most d independent directions.The resulting guarantee fits every horizon h ≥d and isolates prefix-dependent enabling as the source of order effects.