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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 (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
• Communication itself is the bottleneck on a known structure → use foundations-information-theory (channel capacity)
• The problem is recursive control of a viable system, not a one-shot team decision → use foundations-cybernetics-vsm
• 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; 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.

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.

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Quick Reference

#	Primitive	When to Reach For It
1	Team Decision Problem	Frame any shared-goal multi-agent setup before designing it
2	Information Structure	Decide what each agent observes; classify centralized / decentralized / partial / nested
3	Person-by-Person Optimality	Local optimality check — necessary but not sufficient for team optimum
4	Value of Communication	"Should agents share state?" — quantify expected payoff lift vs. cost
5	Radner's LQG Theorem	Classical convex LQG teams admit linear optimal rules under the theorem’s information assumptions
6	Witsenhausen Counterexample	Why naive linearity fails when one agent's action signals to another
7	Information Cost	Bound observation/communication budgets; rationalize bounded rationality
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	Sequential team problems with state dynamics — modern computational form
10	Common-Task Condition	Boundary condition: when does shared payoff actually hold?
11	Aggregation Rule (Vote / Debate / Route)	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. Primitive #11 (aggregation): references/aggregation-and-multi-agent-evidence.md.

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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; 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 (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)

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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. 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.

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Workflow

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).

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Local Validation Artifact

• Operational record, worked case and regression checks. 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
• Aggregation mechanism playbooks (moved from foundations-game-theory): debate and audit — adversarial debate, courtroom-style debate (PROClaim), meta-debate role routing, reasoning-tree audit; voting and consensus — beyond majority voting, radial consensus score, conformal social choice, prediction market; coordination and credit — belief-driven coordination (ECON), coalition formation routing, Shapley contribution scoring, online Shapley prompt evolution
• Domain-agnostic primitives overview: references/primitives-overview.md
• Formal theory map and production boundaries: references/formal-theory-map.md
• Patterns, scenarios, and traps for multi-agent / subagent design: 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

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Related Skills

• foundations-game-theory — 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 — single-agent uncertain choice
• foundations-information-theory — pricing the channel
• foundations-cybernetics-vsm — recursive viable-system control
• foundations-grounding-communication — how agents share state once you decide they should
• agents-subagents — applied subagent patterns
• agents-swarm-orchestration — applied multi-agent orchestration

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