Foundations Team Theory

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.

Quick Reference

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

Anti-Patterns

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.

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).

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

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

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.