v2.0 to v2.0

29 added, 135 removed. Audit A to A.

---

name: foundations-team-theory

description: "Team-theory primitives for cooperative agents: what each subagent sees, when to communicate, aggregation, Shapley credit, coalition topology. Use when organizing subagents."

compatibility: Portable core only.

version: "2.0"

last_validated: 2026-08-14

---

# Team Theory Foundations

10 canonical team-theory primitives, plus an aggregation rule (#11: vote, debate, route, credit assignment, or coalition topology), for *cooperative* multi-agent decision problems — agents share a payoff but each sees a different slice of the world. Game theory handles strategic conflict; decision theory handles solo choice under uncertainty; team theory handles the regime in between, which is exactly where subagents and orchestrated AI agents operate.

## When to Apply

**Apply team-theory when:**

- Multiple agents with **shared payoff** but **partitioned observations** (the canonical subagent setting)

- Designing what each subagent sees vs. what is centralized

- Choosing between centralized orchestrator, decentralized swarm, and hierarchical structures

- Costing communication: is it worth the latency / token / coordination overhead?

- Building agent teams where role specialization matters (each agent owns a different observation lane)

- Multi-agent RL or Dec-POMDP problem framing

**Skip and use simpler alternatives when:**

- Agents have **divergent incentives** → use [foundations-game-theory](../foundations-game-theory/SKILL.md) (mechanism design, auctions, negotiation, collusion). Debate/vote/aggregation for a *shared*-payoff team lives here, in primitive #11 — game theory only if the payoff is genuinely contested.

- Single agent / single-shot decision under uncertainty → use [foundations-decision-theory](../foundations-decision-theory/SKILL.md)

- Communication itself is the bottleneck on a known structure → use [foundations-information-theory](../foundations-information-theory/SKILL.md) (channel capacity)

- The problem is recursive control of a viable system, not a one-shot team decision → use [foundations-cybernetics-vsm](../foundations-cybernetics-vsm/SKILL.md)

- - All agents see the same information — information partitioning offers no advantage by itself; compare a single decision-maker baseline while accounting for distinct capabilities and constraints

+ - All agents see the same information — information partitioning offers no advantage by itself; compare a single decision-maker baseline while accounting for distinct capabilities and constraints (rule 1 below)

+ - The delegate's hidden action or effort is the problem (moral hazard, monitoring, contracts) → see [principal-agent-delegation](../agents-subagents/references/principal-agent-delegation.md); team theory assumes a shared payoff

## When Multi-Agent Pays

Check this before reaching for any aggregation mechanism. The full decision table, the aggregation rows and their scope limits are in [references/aggregation-and-multi-agent-evidence.md](references/aggregation-and-multi-agent-evidence.md).

- - **Task structure first.** Decomposable, parallel-verifiable subtasks are multi-agent candidates; sequential or interdependent planning defaults to one strong agent. Kim et al. (arXiv:2512.08296) report relative changes from +80.8% to −70.0% against a single-agent baseline across task types.

- - **Vote before debate.** A matched-token-budget majority-vote / self-consistency run is the baseline debate must beat (Choi, Zhu & Li, arXiv:2508.17536).

- - **Equal budgets must be equal.** Count all tokens, including retries and verification; API budget controls can distort single- vs multi-agent comparisons (Tran & Kiela, arXiv:2604.02460).

- - **Route when one member is stronger.** If one participant is reliably stronger and identifiable, route or weight to it rather than pooling (Pappu et al., arXiv:2602.01011; see Fact-Checking for the exact figures).

- ---

- ## Contents

- - [Quick Reference](#quick-reference)

- - [Primitive Index](#primitive-index)

- - [Formal Supporting Theory](#formal-supporting-theory)

- - [Anti-Patterns](#anti-patterns)

- - [Decision Checklist](#decision-checklist)

- - [Composition Recipes](#composition-recipes)

- - [Workflow](#workflow)

- - [ASCII Flow](#ascii-flow)

- - [Navigation](#navigation)

- - [Related Skills](#related-skills)

- - [Fact-Checking](#fact-checking)

+ 1. **Information dominance.** Without new exogenous signals, a delegated multi-agent network is dominated by a centralized Bayes decision maker with the same information (Ao, Gao & Simchi-Levi, arXiv:2603.26993; abstract-level, theorems not read here). Delegate only when agents bring information the parent lacks (tools, retrieval, independent samples) or relieve a context limit; adding roles over the same evidence does not help. Tran & Kiela (arXiv:2604.02460, multi-hop reasoning only) find single agents match multi-agent systems at equal token budgets, and report multi-agent becoming competitive when single-agent context degrades or compute grows.

+ 2. **Task structure and capability saturation.** Decomposable, parallel-verifiable subtasks are multi-agent candidates; sequential or interdependent planning defaults to one strong agent. Measure the single-agent baseline first: if it is already high, coordination gives diminishing or negative returns (Kim et al., arXiv:2512.08296, which reports the direction flipping by task type).

+ 3. **Vote before debate.** A matched-token-budget majority-vote / self-consistency run is the baseline debate must beat (Choi, Zhu & Li, arXiv:2508.17536).

+ 4. **Equal budgets must be equal.** Count all tokens, including retries and verification; API budget controls can distort single- vs multi-agent comparisons (Tran & Kiela).

+ 5. **Route when one member is stronger.** If one participant is reliably stronger and identifiable, route or weight to it rather than pooling (Pappu et al., arXiv:2602.01011). Trade-off: the consensus-seeking that dilutes experts also makes a team more robust to adversarial members, so keep an independent check.

+ 6. **Which regime each result covers.** Pappu: expert-relevant tasks with a stronger member, so route to the expert. Choi: comparable members on answerable tasks, so a vote is enough. Dochkina (arXiv:2603.28990): a single not-yet-accepted large-scale protocol comparison whose "drop the hierarchy" reading the other results contest, so self-organisation is a candidate to test, not a default.

---

## Quick Reference

| # | Primitive | When to Reach For It |

|---|-----------|----------------------|

| 1 | [Team Decision Problem](references/primitives-overview.md#1-team-decision-problem) | Frame any shared-goal multi-agent setup before designing it |

| 2 | [Information Structure](references/primitives-overview.md#2-information-structure) | Decide what each agent observes; classify centralized / decentralized / partial / nested |

| 3 | [Person-by-Person Optimality](references/primitives-overview.md#3-person-by-person-optimality) | Local optimality check — necessary but not sufficient for team optimum |

| 4 | [Value of Communication](references/primitives-overview.md#4-value-of-communication) | "Should agents share state?" — quantify expected payoff lift vs. cost |

| 5 | [Radner's LQG Theorem](references/primitives-overview.md#5-radners-lqg-theorem) | Classical convex LQG teams admit linear optimal rules under the theorem’s information assumptions |

| 6 | [Witsenhausen Counterexample](references/primitives-overview.md#6-witsenhausen-counterexample) | Why naive linearity fails when one agent's action signals to another |

| 7 | [Information Cost](references/primitives-overview.md#7-information-cost) | Bound observation/communication budgets; rationalize bounded rationality |

| 8 | [Organizational Forms](references/primitives-overview.md#8-organizational-forms) | Pick centralized vs. decentralized vs. hierarchical structure; see #8a (centralize/decentralize decision test) and #8b (team-size/coordination-cost curve) in `references/primitives-overview.md` |

| 9 | [Dec-POMDP / MARL Extension](references/primitives-overview.md#9-dec-pomdp--marl-extension) | Sequential team problems with state dynamics — modern computational form |

| 10 | [Common-Task Condition](references/primitives-overview.md#10-common-task-condition) | Boundary condition: when does shared payoff actually hold? |

- | 11 | [Aggregation Rule (Vote / Debate / Route)](references/aggregation-and-multi-agent-evidence.md) | How to combine multiple agents' outputs into one: vote, debate, route to the strongest, or a calibrated selection rule; also Shapley credit assignment (templates 04, 17) and coalition topology (22). Covers the shared-payoff aggregation templates formerly indexed in `foundations-game-theory` |

+ | 11 | [Aggregation Rule (Vote / Debate / Route)](references/aggregation-and-multi-agent-evidence.md) | Combine agents' outputs: vote, debate, route to the strongest, calibrated selection; Shapley credit (templates 04, 17); coalition topology (22) |

Full definitions, inputs, outputs, failure modes, and worked examples: [references/primitives-overview.md](references/primitives-overview.md). Primitive #11 (aggregation): [references/aggregation-and-multi-agent-evidence.md](references/aggregation-and-multi-agent-evidence.md).

---

- ## Primitive Index

- | # | Primitive | Failure Mode It Addresses |

- |---|-----------|--------------------------|

- | 1 | Team Decision Problem | Modeling cooperative agents as either a single decision-maker (loses decentralization value) or as strategic players (overstates conflict) |

- | 2 | Information Structure | Agents observing redundant signals; missing observations; uncosted communication assumed free |

- | 3 | Person-by-Person Optimality | Local-optimum trap: each agent optimal given others' fixed rules, but the joint policy is suboptimal |

- | 4 | Value of Communication | "Add a Slack channel between agents" with no expected-payoff justification; channel may cost more than it earns; inter-agent message routing also degrades information efficiency via Data Processing Inequality even at zero explicit channel cost |

- | 5 | Radner's LQG Theorem | Reinventing optimization for LQG teams when a closed-form linear rule exists |

- | 6 | Witsenhausen Counterexample | Assuming linear policies are optimal when an agent's action carries signaling content |

- | 7 | Information Cost | Designing perfect-observation agents in a world where observation has latency, token, and money cost |

- | 8 | Organizational Forms | Defaulting to a central orchestrator when a decentralized or hierarchical form would dominate |

- | 9 | Dec-POMDP / MARL Extension | Treating each timestep as a fresh team problem; ignoring history and state |

- | 10 | Common-Task Condition | Calling a system a "team" when payoffs actually diverge — game theory regime, not team regime |

- | 11 | Aggregation Rule (Vote / Debate / Route) | Unweighted vote over a correlated (same-family) pool; debate used without a baseline it must beat; pooling when routing to the strongest agent would do better; no credit signal for which member contributed; a team topology fixed by habit rather than fit |

- ---

- ## Formal Supporting Theory

- | Theory Area | Use When | Applied Primitives It Grounds |

- |---|---|---|

- | Marschak–Radner team theory | Need normative framing of cooperative multi-agent decisions | #1, #2, #3, #4, #5, #8 |

- | Decentralized stochastic control | Action of one agent affects information of another | #6, #9 |

- | Information economics | Need to cost observation and communication | #4, #7 |

- | Organization economics | Compare centralized hierarchy, decentralized markets, polyarchy | #8 |

- | Dec-POMDP and MARL | Sequential team problems with state dynamics | #9 |

- | Mechanism design boundary | Where shared payoff breaks down | #10 (then exit to game-theory) |

- | Social choice and forecast aggregation | Combine member outputs into one answer: vote, weight, route, or calibrated selection | #11 |

- | Cooperative game theory (Shapley value) | Credit assignment for a shared payoff; exact cost grows as 2^N, so approximate for larger teams | #11 (templates 04, 17, 22) |

- See [references/primitives-overview.md](references/primitives-overview.md) for the formal-theory map and theorem statements.

- ---

## Anti-Patterns

| Anti-Pattern | Team Theory Diagnosis | Fix |

|---|---|---|

| One central orchestrator reads everything, decides everything | Centralization may duplicate observations and communication cost | Compare centralized and decentralized policies under the actual information structure; invoke Radner-style linear optimality only after verifying the theorem's information and convexity assumptions |

| Each subagent given full system context "just in case" | Information cost (#7) ignored; redundant observations dominate token budget | Partition observations to what each agent's action depends on; redundancy must justify itself |

| Local optimization per subagent declared "team-optimal" | Person-by-person optimality (#3) confused with team optimum; PBPO is necessary, not sufficient | Search unilateral, pairwise and broader deviations to falsify candidate optimality; failed limited searches cannot certify a joint optimum |

| "Add a comms channel between agents" added without payoff analysis | Value of communication (#4) not computed; channel may cost more than it earns | Compute expected payoff with vs. without the channel; if delta < cost, drop it |

| "Multi-agent by default for complex tasks" | Coordination edges can grow quadratically under all-to-all coupling, but realized cost depends on topology; DPI only says post-processing cannot increase information about a target in a Markov chain, not that every routed message loses useful information | Measure the actual communication graph and compare against a single-agent baseline; use DPI only when its random variables and Markov condition are explicit |

| "Assume every text handoff is lossless" | Summaries may omit decision-relevant detail; LatentMAS is a specialized architecture, not a portable guarantee | Verify text/structured-artifact fidelity. Consider latent exchange only with embedding access, compatible representations and task-specific evidence; compare costs and outcomes locally |

| "Default to all-to-all communication between agents" | Dense graphs can increase cost and error propagation on evaluated tasks; no universal optimal sparsity follows | Compare sparse, dense and adaptive candidates under actual dependencies and matched budgets; preserve necessary channels |

- | "Put the expert on the team and the team will use them" | Pappu et al. find expertise dilution in controlled tasks; their ML benchmark synergy gaps use an oracle bound, distinct from the best fixed individual | Compare competence routing and pooling against the best individual locally; also test robustness to unreliable contributions |

+ | "Put the expert on the team and the team will use them" | Pappu et al. find expertise dilution; consensus-seeking also buys robustness to adversarial members | Compare competence routing and pooling against the best individual locally; also test robustness to unreliable contributions |

| Assume linear/affine policies optimal in non-LQG team | Witsenhausen (#6) shows nonlinear policies can dominate when signaling exists | Map action-dependent observations and information nestedness; consider nonlinear candidates when theorem premises fail, rather than excluding all signaling cases |

| Treat all multi-agent setups as game-theoretic | Common-task condition (#10) holds — agents share a goal — so mechanism-design overhead is wasted | Skip incentive compatibility; design for joint optimization directly |

| Treat all multi-agent setups as team problems | Common-task condition (#10) fails — agents have divergent goals — team theory underestimates conflict | Switch to game theory (auctions, mechanism design) |

| Importing human-team findings (c-factor, psychological-safety inverted-U) as agent-team design rules | These are contested human-team results, not team-theory primitives; they do not transfer to LLM agents without a local test | Use the observation partition (#2) and the aggregation rule (#11). Evidence and hedges live in `startup-operating-system/references/people/team-effectiveness-evidence.md` |

- | Hierarchical orchestrator-worker with no information flow back up | Polyarchy/hierarchy structure (#8) chosen by default; loses bottom-up signal that decentralized form preserves | Add upward observation channel or switch to decentralized form; for tasks with parallelizable subtasks at suitable task scale, consider emergent self-organization protocols where agents self-assign roles — reported advantages in particular evaluated settings need matched local validation (Dochkina 2026) |

- | "Just add more agents/people, coordination scales for free" | Team-size/coordination-cost curve (#8b): pairwise coordination channels grow ~n(n-1)/2 with team size under high coupling; the marginal member's coordination cost can exceed their contribution (Brooks's Law) | Before adding a member/agent, count how many existing members it must (not could) coordinate with; if most, cut coupling first (modularize, assign a single owner for shared state) rather than adding capacity |

+ | Hierarchical orchestrator-worker with no information flow back up | Polyarchy/hierarchy structure (#8) chosen by default; loses bottom-up signal that decentralized form preserves | Add upward observation channel or switch to decentralized form; for tasks with parallelizable subtasks at suitable task scale, consider emergent self-organization protocols where agents self-assign roles (contested single preprint, Dochkina 2026; validate locally) |

+ | "Just add more agents/people, coordination scales for free" | Team-size/coordination-cost curve (#8b): pairwise channels grow ~n(n-1)/2 under high coupling (#8b) | Count how many existing members the new one must (not could) coordinate with; if most, cut coupling first (single owner for shared state) |

| "We are a centralized [org/agent system]" or "we are a decentralized one" applied blanket-wide | Centralize-vs-decentralize is a per-decision judgment (#8a: information compressibility, reversibility, coupling, blast radius of a wrong local call), not a company-wide or system-wide constant | Apply the four-question test per decision class; expect most real orgs and agent systems to be centralized on some decisions (shared state, brand, safety) and decentralized on others (local execution) |

---

## Decision Checklist

+ - [ ] **Does each agent bring information or capacity the single-agent baseline lacks?** If not, do not split (When Multi-Agent Pays, rules 1-2)

- [ ] **Shared payoff?** If no, exit to [foundations-game-theory](../foundations-game-theory/SKILL.md). If yes, continue (#10)

- [ ] **What does each agent observe?** Map the information structure explicitly (#2)

- [ ] **Are observations redundant?** Trim — observation has cost (#7)

- [ ] **Is communication channel justified?** Compute expected payoff lift; reject if < cost (#4)

- [ ] **Is the communication topology justified by actual dependencies and a matched-budget local comparison?** Moderate sparsity is a tested candidate, not a universal requirement.

- [ ] **Does the team meet the precise Gaussian/quadratic, convexity and information-structure assumptions?** Approximate resemblance alone does not license Radner’s linear-policy result (#5)

- [ ] **Does an action affect later observations, and what information is nested?** Classify the information sets before invoking non-classical results; action dependence alone is insufficient (#6)

- [ ] **Sequential decisions over time with state?** Frame as Dec-POMDP (#9)

- [ ] **Picked an organizational form?** Centralized / decentralized / hierarchical — justify choice against the alternatives (#8)

- [ ] **Tested joint alternatives, not just person-by-person tuning?** Vary joint policies; finite comparisons do not certify a global optimum (#3)

- [ ] **Did the aggregation beat a matched-budget majority-vote / self-consistency baseline?** If not, use the vote; debate or audit only when it beats that baseline or the audit trail is the product (#11)

- [ ] **Does the aggregation rule weight by competence, or does it pool?** If agents differ in task-relevant skill, opinion pooling destroys the expert's advantage; route or weight instead (#11, Pappu et al. 2026). Check the single-best-member baseline before shipping any team design

- [ ] **Human team rather than an agent team?** Out of scope; evidence lives in `startup-operating-system/references/people/team-effectiveness-evidence.md`. Human media findings do not establish LLM transport choices.

---

- ## Composition Recipes

- ### Designing a subagent team for a research task

- _Context_: Orchestrator needs to dispatch N subagents on a multi-source research task with a shared answer goal.

- 1. Verify common-task condition (#10): all subagents and orchestrator share the payoff "produce one accurate answer." If not, exit to game-theory.

- 2. Map information structure (#2): which sources/files/tools does each subagent observe? Avoid redundant assignment. For human-in-the-loop teams: the temporal ordering of observation matters — simultaneous presentation of human and AI outputs outperforms sequential (human-first then AI) on average (npj Artificial Intelligence 2025, N=52 clinical studies). Design observation lanes with this moderator in mind. **CTP conditions (Hemmer et al. 2024, EJIS 2025)**: human-AI complementarity — performance exceeding either agent or human alone — requires either (a) information asymmetry (human has local context AI lacks) or (b) capability asymmetry (human provides judgment on novel cases AI cannot handle). These are candidate performance-complementarity conditions, not permission to remove oversight: human participation may be required for authority, rights, accountability or safety even without an accuracy gain. Test local performance and preserve those duties.

- 3. Cost observations (#7): each tool call has token + latency cost. Compare the joint value and cost of observation bundles. Diminishing marginal value requires assumptions; complementary observations can have increasing value.

- 4. Compute value of communication (#4): does mid-task chatter between subagents lift expected payoff more than its cost? Test communication when observations, constraints or intermediate decisions interact; late synthesis is a candidate for low coupling, not a theorem-backed default.

- 5. Pick organizational form (#8): compare single-agent, split-and-merge, centralized and hierarchical candidates at matched total budgets. Use split-and-merge for separable work only when aggregation and shared constraints remain adequate; LQG analogy supplies no default.

- 6. Compare bounded joint-policy candidates (#3). An improving pairwise deviation disproves team optimality but does not establish prior person-by-person optimality; report the best tested candidate and uncertainty, with no global certificate.

- ### Choosing between orchestrator-led and swarm forms

- _Context_: A product team must pick between a single planner agent that delegates everything (centralized) and a peer-to-peer agent network (decentralized).

- 1. Invoke #5 only for an explicitly modeled problem satisfying the exact quadratic, Gaussian, convexity and information-structure premises of the applicable theorem. Similarity to LQG gives no upper bound for LLM architectures; otherwise compare matched-budget candidates.

- 2. Cost the communication channel: orchestrator-led requires every observation to flow up and every decision to flow down. Decentralized requires only local observation but needs convergence on the synthesis.

- 3. Apply #8: high information cost and low coupling motivate a decentralized candidate; low information cost and high coupling motivate a centralized candidate. Compare them locally; these heuristics do not prove dominance. For parallelizable subtasks where feasible, add a fourth candidate: emergent self-organization with minimal structure (sequential protocol + autonomous role assignment). Reported scaling results are architecture-, benchmark- and budget-dependent; do not infer a universal four-agent threshold or hierarchy disadvantage. Justified by Dochkina (2026, arXiv:2603.28990) and CARD (Wu et al., ICLR 2026).

- 4. Worked example: code review across 5 files. Files are independent (no signaling, low coupling). A decentralized candidate (5 parallel reviewers + final aggregator) may reduce duplicated observations; compare total cost and verification quality against an orchestrator-led baseline. Code review across one tightly-coupled module: signaling is high (one agent's finding changes another's interpretation); test centralized or hierarchical coordination rather than assuming a winner.

- ### Sizing the orchestrator's context budget

- _Context_: How much should the orchestrator observe vs. push down to subagents?

- 1. Apply #7: every token in the orchestrator's context has cost. Give the orchestrator the information needed for assignment, synthesis and verification, including domain evidence when its conclusions depend on that evidence.

- 2. Apply #2: start with task graph, roster and status, then add decision-relevant object-level evidence and provenance needed to check worker claims.

- 3. Anti-pattern: orchestrator reads all source files and then asks subagents to summarize them — observation paid for twice. Fix: delegate first-pass source reading, then return claims with source locations, assumptions and counterevidence; the orchestrator reads underlying contents where synthesis or verification needs them.

- ### Designing subagent context as an information-structure problem (AI app builders)

- _Context_: You are building a multi-agent AI application and must decide what context (system prompt, retrieved docs, tool permissions, conversation history) each subagent receives.

- 1. **Name the information structure first (#2).** For each subagent, write a one-row entry: `agent | observation sources | action produced`. This is the `η` partition — make it explicit before writing any code.

- 2. **Apply the task-type rule.** For separable subtasks, consider partitioned observations with shared goal, acceptance criteria, constraints and necessary interfaces. For coupled subtasks, compare centralized, hierarchical or communicating candidates; pass decision outputs with evidence, uncertainty and provenance needed downstream. A raw reasoning trace is not required, but verified source evidence may be.

- 3. **Strip context to decision-relevant signals (#7).** Each token in a subagent's context window is an observation cost. Give the subagent only what changes its action. If a document section never alters the subagent's output, remove it from its context.

- 4. **Check for signaling (#6).** If A’s output affects B’s context, inspect whether B also knows A’s relevant information; output dependence alone does not establish non-classical territory. Do not assume A's optimal policy is the same as if it were working in isolation — A should write with B's downstream use in mind (nonlinear policy).

- 5. **Compute value of communication before adding agent-to-agent messaging (#4).** Does mid-task chatter between subagents increase the joint answer quality by more than the latency + token cost? Use task coupling and missing information to motivate communication candidates; compare costs and quality locally rather than imposing a blanket no-communication rule.

- 6. **Compare decentralized + late synthesis with a central baseline at matched budgets.** Count total observation, communication and synthesis costs; O(1) per-agent work does not imply O(1) total system cost. No universal four-agent threshold is established.

- ---

## Workflow

- 1. Confirm common-task condition (#10). If payoffs diverge, exit to game-theory.

- 2. Map the information structure (#2): one agent per row, one observation source per column, fill the matrix.

- 3. Cost the observations (#7) and the communication channels (#4).

- 4. Pick organizational form (#8) — centralized, decentralized, hierarchical — justified against alternatives.

- 5. Verify the exact theorem premises before applying a linear-policy result (#5). Inspect information nestedness when actions affect observations (#6); dependence alone does not disqualify partially nested cases. Otherwise evaluate bounded policy candidates.

- 6. Report best tested joint policy (#3), searched deviations and uncertainty. Do not certify global optimality from unilateral or pairwise search.

- 7. For sequential problems with state, frame as Dec-POMDP (#9) and pick an MARL solver.

- ---

- ## ASCII Flow

- ```text

- Cooperative multi-agent decision

- -> Confirm shared payoff and common-task condition

- +-- payoff diverges -> exit to game theory

- +-- shared payoff holds -> map information structure

- -> Cost observation and communication channels

- -> Choose centralized, decentralized, or hierarchical form

- -> Compare bounded joint policies and sequential-state needs

- -> Return team design, information lanes, communication rule, and solver path

- ```

+ 1. Run When Multi-Agent Pays: does a second agent add information or capacity? If not, stop at one agent.

+ 2. Confirm the common-task condition (#10; exit to game theory if payoffs diverge), then map and cost the information structure (#2, #7) and channels (#4).

+ 3. Pick the organizational form (#8) and the aggregation rule (#11) against a matched-budget single-agent and majority-vote baseline.

+ 4. Check theorem premises before using #5 or #6; otherwise report the best tested joint policy (#3) with searched deviations and uncertainty.

+ 5. For sequential problems with state, frame as Dec-POMDP (#9).

---

## Local Validation Artifact

- [Operational record, worked case and regression checks](references/local-team-comparison.md). Read before translating a mechanism into a deployment recommendation.

## Navigation

- Primitive #11, aggregation rule and the multi-agent evidence layer (moved from `foundations-game-theory`): [references/aggregation-and-multi-agent-evidence.md](references/aggregation-and-multi-agent-evidence.md)

- Aggregation mechanism playbooks (moved from `foundations-game-theory`): debate and audit — [adversarial debate](assets/templates/team-theory/02-adversarial-debate.md), [courtroom-style debate (PROClaim)](assets/templates/team-theory/08-courtroom-proclaim.md), [meta-debate role routing](assets/templates/team-theory/16-meta-debate-routing.md), [reasoning-tree audit](assets/templates/team-theory/13-reasoning-tree-audit.md); voting and consensus — [beyond majority voting](assets/templates/team-theory/18-beyond-majority-voting.md), [radial consensus score](assets/templates/team-theory/19-radial-consensus-score.md), [conformal social choice](assets/templates/team-theory/20-conformal-social-choice.md), [prediction market](assets/templates/team-theory/11-prediction-market.md); coordination and credit — [belief-driven coordination (ECON)](assets/templates/team-theory/01-econ-belief-driven.md), [coalition formation routing](assets/templates/team-theory/22-coalition-formation-routing.md), [Shapley contribution scoring](assets/templates/team-theory/04-shapley-contribution.md), [online Shapley prompt evolution](assets/templates/team-theory/17-online-shapley-prompt-evolution.md)

- Domain-agnostic primitives overview: [references/primitives-overview.md](references/primitives-overview.md)

- Formal theory map and production boundaries: [references/formal-theory-map.md](references/formal-theory-map.md)

- Patterns, scenarios, and traps for multi-agent / subagent design: [references/patterns-scenarios-traps.md](references/patterns-scenarios-traps.md)

- Human-team evidence (owned by `startup-operating-system`; Westrum typology in `software-ux-research`): `team-effectiveness-evidence.md`

- Sources: [`data/sources.json`](data/sources.json)

---

## Related Skills

- [foundations-game-theory](../foundations-game-theory/SKILL.md) — exit here when payoffs diverge (incentives, auctions, collusion, bargaining); it also holds the Shapley worked example and `scripts/game_artifacts.py` self-test that template 04 relies on

- [foundations-decision-theory](../foundations-decision-theory/SKILL.md) — single-agent uncertain choice

- [foundations-information-theory](../foundations-information-theory/SKILL.md) — pricing the channel

- [foundations-cybernetics-vsm](../foundations-cybernetics-vsm/SKILL.md) — recursive viable-system control

- - [foundations-grounding-communication](../foundations-grounding-communication/SKILL.md) — *how* agents share state once you decide they should (see also: temporal information structure in human-AI teams — primitive #2, npj AI 2025)

+ - [foundations-grounding-communication](../foundations-grounding-communication/SKILL.md) — *how* agents share state once you decide they should

- [agents-subagents](../agents-subagents/SKILL.md) — applied subagent patterns

- [agents-swarm-orchestration](../agents-swarm-orchestration/SKILL.md) — applied multi-agent orchestration

---

## Citation Notes

- Marschak (1955) "Elements for a Theory of Teams." *Management Science* 1(2). Founded the field.

- Marschak and Radner (1972) *Economic Theory of Teams*. Yale. Canonical text.

- Radner (1962) "Team Decision Problems." *Annals of Mathematical Statistics* 33(3). LQG theorem.

+ - Ho and Chu (1972) "Team Decision Theory and Information Structures in Optimal Control Problems, Part I." *IEEE Trans. Automatic Control*. Partially nested information structures.

- Witsenhausen (1968) "A Counterexample in Stochastic Optimum Control." *SIAM J. Control* 6(1). Nonlinear-dominance result.

- - Bernstein, Givan, Immerman, Zilberstein (2002) "The Complexity of Decentralized Control of Markov Decision Processes." *Math. of Operations Research*. Dec-POMDP framing — NEXP-complete.

+ - Bernstein, Givan, Immerman, Zilberstein (2002) "The Complexity of Decentralized Control of Markov Decision Processes." *Math. of Operations Research*. Dec-POMDP framing, NEXP-complete.

- Oliehoek and Amato (2016) *A Concise Introduction to Decentralized POMDPs*. Springer.

- - Numeric thresholds (cost-of-communication, observation budgets) are domain-specific. Verify against primary references before using as decision authority.

- - Brooks, F.P. (1975/1995) *The Mythical Man-Month*. Addison-Wesley. Source for the team-size / coordination-cost curve (#8b): pairwise coordination links grow combinatorially with team size, used here as a qualitative mechanism, not a quantitative team-theory result.

- - Human-team citations and their source entries are owned by `startup-operating-system`: `team-effectiveness-evidence.md`.

- - Expertise aggregation failure: Pappu et al. (2026), "Multi-Agent Teams Hold Experts Back," arXiv:2602.01011. In [Table 2 and §4.2](https://arxiv.org/html/2602.01011#S4.SS2), the 41.1% relative synergy gap is HLE Text-Only, Expert Not Mentioned: team accuracy 28.0% versus the At Least One Correct oracle bound 47.5%. It is not a 41.1% loss against the best fixed individual (29.0%); the Reveal Expert team scores 36.0%. Controlled psychology tasks separately show expertise dilution and team-size effects. Do not transfer the abstract's broad comparator wording into a benchmark claim.

- - DPI argument for single-agent dominance: Tran & Kiela (2026, arXiv:2604.02460) demonstrate empirically across **three** model families (Qwen3, DeepSeek-R1-Distill-Llama, Gemini 2.5) and five MAS architectures (Sequential, Subtask-parallel, Parallel-roles, Debate, Ensemble) that single-agent systems match or exceed multi-agent on multi-hop reasoning under equal token budgets, with theoretical grounding in the Data Processing Inequality. Preprint; the paper itself flags that "artifacts in API-based budget control ... distort effective computation," which can inflate apparent MAS gains — control for this before citing the comparison as decisive.

- - Latent-space inter-agent communication: LatentMAS (Zou et al. 2025, arXiv:2511.20639). [Section 4.1](https://arxiv.org/html/2511.20639#S4.SS1) reports an average 14.6% accuracy improvement over the single-model baseline in its sequential setting; the reported sequential gain over text-based MAS is 2.8%, with 4× average inference speedup against sequential TextMAS. These are architecture- and benchmark-specific results, not a portable guarantee for arbitrary agents. Preserve the authors' units rather than converting to percentage points without table-level calculation.

- - Emergent self-organization at scale: Dochkina (2026, arXiv:2603.28990, **submitted to IEEE Access**, not yet accepted) demonstrates across 25,000 tasks, 8 models, and 4-256 agents that a sequential hybrid protocol outperforms **centralized coordination** by 14% (p<0.001); a separate statistic — a 44% quality spread between protocols — carries Cohen's d=1.86. Do not attach d=1.86 to the 14% figure; they measure different things. "Corroboration" by CARD (Wu et al., ICLR 2026) and MegaAgent (ACL 2025) is unverified — treat as a lead, not a confirmed citation.

- - Human-AI teaming mode as information structure: Systematic review (npj AI, Nature portfolio, December 2025, N=52 clinical studies) finds simultaneous human-AI observation outperforms sequential mode; corroborated by PRISMA review of 104 studies (Kargarnovin et al., Frontiers in Robotics and AI, 2026) and PNAS Nexus complementarity framework (Gonzalez et al., 2026).

+ - Brooks (1975/1995) *The Mythical Man-Month*: qualitative source for the team-size / coordination-cost curve (#8b).

+ - Human-team citations are owned by `startup-operating-system`: `team-effectiveness-evidence.md`.

+ - Pappu et al. (2026), "Multi-Agent Teams Hold Experts Back," arXiv:2602.01011. The abstract reports team losses on ML benchmarks and teams failing to match their expert agent. The comparator behind each loss figure sits in the paper's tables; read the table before quoting a magnitude or calling it an oracle gap or a best-member gap.

+ - Tran & Kiela (2026, arXiv:2604.02460): equal-token-budget comparison of single-agent and multi-agent architectures on multi-hop reasoning, with a Data Processing Inequality argument. Preprint; the paper flags that API budget-control artifacts can inflate apparent multi-agent gains.

+ - LatentMAS (Zou et al. 2025, arXiv:2511.20639): latent-space inter-agent exchange. Architecture- and benchmark-specific; not a portable guarantee for hosted agents.

+ - Dochkina (2026, arXiv:2603.28990): emergent self-organisation at scale. Submitted to IEEE Access, not accepted; contested, see rule 6.

+ - Ao, Gao & Simchi-Levi (2026, arXiv:2603.26993): decision-network formalisation of delegation. Abstract-level reading only.

+ - Numeric thresholds (cost of communication, observation budgets) are domain-specific. Verify against primary references before using as decision authority.

## Learnings Loop

When prior decisions or pitfalls are relevant, consult `learnings.consolidated.md` if present; use `learnings.md` only for needed history or as the available fallback. Otherwise skip both.

After applying it, if you encountered a pattern worth remembering, a mistake worth preventing, or a domain fact that surprised you, append one dated bullet to `learnings.md` via `agents-skills-feedback-loop/scripts/append_learning.py`. Do not modify `SKILL.md` itself.