Game Theory Foundations

22 applied game-theory primitives for strategic decision systems, backed by a formal theory map. Each applied primitive solves a specific incentive or coordination failure. Primitives are domain-agnostic: the same mechanism that prevents free-riding in agent teams prevents cost-shifting in partnership contracts; the same auction that routes tasks routes ad placements.

For the agent-team applied recipe layer (team.yaml manifest fields, agent-team anti-patterns, agent-team decision checklist, composition recipes for typical agent-team scenarios), see agents-subagents/references/game-theory-agent-teams.md.

Contents

• Quick Reference
• Primitive Index
• Formal Supporting Theory
• Expert Judgment: When the Model Helps vs Misleads
• Anti-Patterns
• Misuse Boundaries
• Decision Checklist
• Composition Recipes
• Workflow
• ASCII Flow
• Current Pattern Review
• Navigation
• Fact-Checking

---

Quick Reference

Primitive	Domain	Recipe Stub
Belief-Driven Coordination (ECON)	Multi-party teams, distributed analysis, agent teams	Members optimize against beliefs about co-members; reduces redundant work and inter-member chat
Adversarial Debate	Content moderation, risk review, audit	Two heterogeneous evaluators + reasoning-tree synthesis; no majority vote
Auction-Based Routing	Ad placement, task delegation, resource allocation	Sealed-bid truthful auction; highest-value-per-cost wins
Shapley Contribution	Attribution, revenue sharing, team composition	Marginal-contribution average across subsets
Reputation-Gated Autonomy	Supplier qualification, agent oversight, fraud gating	Tiered trust: proven → standard → probationary; oversight inversely proportional
Cooperation and Defection	Partnership design, incentive alignment, compliance	Iterated PD structure; payoff-scale to detect defection tendency
Mechanism Design for Synthesis	Decision aggregation, voting, policy-making	Vickrey truthful-revelation; dissent is a required section
Courtroom-Style Debate	Legal review, risk go/no-go, claim verification	Plaintiff/defense/court structure + progressive RAG + role-switching
Pareto-Nash Multi-Objective	Product tradeoffs, regulatory vs growth, pricing tiers	Map Pareto frontier; pick dominant options; flag non-dominated set
Evolutionary Coordination Search	Algorithm selection, prompt tuning, rule evolution	LLM-mutated program + fitness signal; ShinkaEvolve for sample efficiency
Prediction Market Confidence	Forecasting, risk calibration, hiring decisions	Stake-weighted confidence; CritiCal calibration step before stake
Negotiation ZOPA/BATNA	Pricing, partnership terms, resource contention	Map BATNA/ZOPA per party; target overlap zone; use interests not positions
Reasoning-Tree Audit	High-stakes synthesis, compliance review, claim checking	Trace claims to evidence at First Point of Disagreement; reject unsupported majority
Per-Claim Credibility Scoring	Misinformation detection, adversarial content, security	Evidence quality × corroboration weight per claim; isolate high-risk claims
Generative Social Choice	Multi-stakeholder policy, diverse-user product decisions	Maximin selection across candidate outputs; preserve minority-signal coverage
Meta-Debate Role Routing	Debate setup, role-fit selection, agent teams	Two-stage proposal + peer-review picks plaintiff/defense/judge from a pool
Online Shapley Prompt Evolution	High-frequency teams, prompt tuning over many runs	Per-member prompt mutation guided by Shapley contribution (HiveMind)
Beyond Majority Voting (BMV)	Best-of-N synthesis (discrete answer), ensemble selection	Optimal Weight (confidence × calibration) + Inverse Surprising Popularity
Radial Consensus Score (RCS)	Best-of-N synthesis (open-ended generation), self-consistency	Embedding-centroid selector for semantically clustered, lexically diverse answers
Conformal Social Choice	High-stakes debate verdicts, act/escalate gates	Calibrated prediction set: singleton acts, multi-answer set escalates
Attested Delegation Contracts	Cross-trust subagent routing, agent marketplaces, external tools	Route by verified capability and bounded contract, not self-claimed quality
Coalition Formation Routing	Large teams, departments, multi-workstream audits	Form stable subteams before synthesis; avoid flat-panel overload

---

When to Apply

Apply game-theory primitives when:

• Multiple agents/teams/users with potentially divergent incentives
• Synthesis where minority-correct outcomes matter (high-stakes, irreversible)
• Auctions, bidding, mechanism design, or pricing where strategic behaviour exists
• Repeated interactions where reputation, cooperation, or trust evolves
• Best-of-N selection across 5+ candidates (BMV/RCS)
• Cross-trust delegation, dynamic agent pools, or high-stakes act/escalate decisions

Skip and use simpler alternatives when:

• Single agent / single-shot task — game theory is about interactions, not solo work
• Routine task with high majority-correct rate — a deterministic check or oracle is cheaper
• Hard verification exists (test suite, schema, calculator) — use the oracle, not voting
• Team < 3 members on a low-stakes call — overhead exceeds diversity gain
• Information-only retrieval / pure compression — use foundations-information-theory instead
• Single-system reliability/SLO question — use foundations-reliability-theory or queueing-theory

---

Primitive Index

Each primitive has a full playbook (problem, solution, how-it-works, launch-prompt template, domain applications, citations).

#	Mechanism	Failure Mode It Addresses
1	Belief-Driven Coordination (ECON)	Pooling equilibrium — members read same context, produce same analysis
2	Adversarial Debate	Confabulation consensus, correlated bias
3	Auction-Based Task Routing	Static routing, ambiguous selection
4	Shapley Contribution Scoring	Free-riding, unverifiable attribution
5	Reputation-Gated Autonomy	Uniform trust regardless of track record
6	Cooperation and Defection	Shallow output, scope dumping, echo chambers
7	Mechanism Design for Synthesis	Loudest-wins aggregation, suppressed dissent
8	Courtroom-Style Debate (PROClaim)	Evidence stagnation, position-anchored reasoning
9	Pareto-Nash Multi-Objective	Single-objective optimization on multi-objective problems
10	Evolutionary Coordination Search	Hand-tuned rules are sub-optimal vs. measured fitness
11	Prediction Market / Confidence Betting	Verbose output dominates synthesis
12	Negotiation Protocol (ZOPA/BATNA)	Adversarial framing on genuine compromise situations
13	Reasoning-Tree Audit	Confident-but-wrong consensus; majority vote unsafe
14	Per-Claim Credibility Scoring	Single-claim failure modes reputation gating misses
15	Generative Social Choice	Multi-stakeholder buy-in; averaging erases minority evidence
16	Meta-Debate Role Routing	Wrong specialist gets the wrong debate role; static role assignment
17	Online Shapley Prompt Evolution	Weak team members never improve; static prompts under-utilize Shapley signal
18	Beyond Majority Voting (BMV)	Majority vote on best-of-N erases minority-correct answers (calibration ignored)
19	Radial Consensus Score (RCS)	Lexical-overlap voting fails on semantically clustered open-ended generations
20	Conformal Social Choice Act/Escalate	Wrong consensus turns into irreversible action
21	Attested Delegation Contracts	Self-claimed quality corrupts routing across trust boundaries
22	Coalition Formation Routing	Large flat panels duplicate work and produce unstable synthesis

---

Formal Supporting Theory

The 22 primitives are the applied layer, not the whole field. Use references/formal-theory-map.md when the task needs formal assumptions, proof obligations, or classical theory coverage.

Theory Area	Use When	Applied Primitives It Grounds
Game forms	Need to classify normal-form, extensive-form, Bayesian, repeated, stochastic, or cooperative structure	#1, #6, #8, #9, #10, #12
Solution concepts	Need dominance, minimax, Nash, Bayesian Nash, subgame-perfect, perfect Bayesian, or correlated equilibrium	#1, #2, #6, #8, #9, #10, #18
Mechanism and auction design	Need incentive compatibility, individual rationality, revelation principle, VCG, Myerson, reserves, or bid shading	#3, #7, #11, #20, #21
Information economics	Need signaling, screening, adverse selection, moral hazard, principal-agent framing, or attestation	#5, #7, #12, #14, #21
Cooperative game theory	Need Shapley, core, nucleolus, Banzhaf, coalition formation, or surplus sharing	#4, #6, #15, #17, #22
Market design and matching	Need stable matching, deferred acceptance, matching with contracts, or allocation without prices	#3, #7, #12, #15
Bargaining theory	Need Nash bargaining, Rubinstein bargaining, BATNA/ZOPA, outside options, or alternating offers	#12
Learning in games	Need no-regret, fictitious play, CFR, PSRO, self-play, or empirical game-theoretic analysis — including no-regret Nash policy convergence in RLHF (INPO, ICLR 2025 Oral) and smooth RM+ last-iterate convergence [NeurIPS 2025]	#6, #10, #11, #17
Strategic failure analysis	Need collusion, equilibrium selection, Goodharting, manipulation, or off-equilibrium threats	all primitives

---

Expert Judgment: When the Model Helps vs Misleads

Applying a primitive correctly is mechanical. Knowing whether the game-theoretic frame is the right frame at all — and which game — is the actual expert skill. This section is judgment, not a lookup table.

The equilibrium selection problem

Most interesting games (repeated games especially — see the Folk Theorem in formal-theory-map.md) have many equilibria, not one. A non-expert computes an equilibrium and reports it as "the" prediction. An expert checks multiplicity first and asks what actually selects among the candidates in this specific situation — precedent, an explicit contract, a public commitment, a focal point, or repeated-play reputation. Reporting "the Nash equilibrium is X" without naming the selection mechanism is a tell that the analysis stopped one step too early.

Common-knowledge assumptions failing in practice

Nash equilibrium, Bayesian Nash equilibrium, and most mechanism-design proofs assume common knowledge of rationality, of payoffs (or their distribution), and of the rules of the game itself. Real organizations violate all three routinely:

• A "competitor" may be a satisficer bound by an internal OKR or a legacy contract, not a profit-maximizing best-responder — modeling them as rational invites a confidently wrong prediction.
• Bidders or negotiating parties often do not share a common prior on value — private information about downstream use, not risk attitude, is driving the gap.
• LLM agents do not reliably best-respond at all: pro-social bias, framing sensitivity, and authority compliance are documented, repeated deviations from Nash play (see the LLM rationality trap in patterns-scenarios-traps.md). Any incentive-compatibility argument built on "agents best-respond" needs a held-out behavioral check before it is trusted for LLM participants.
• The deviation runs in both directions, and over-truthfulness breaks proofs the same way strategic misreporting does. LLM agents in matching markets reveal preferences truthfully at higher rates than human subjects, but truth-telling does not track strategy-proofness — a strategy-proof mechanism did not elicit more truthful reports than a manipulable one (Hoshino, Kitadai & Nishino, arXiv:2606.03030, June 2026). Mechanism-based markets still beat free negotiation on stability and efficiency; the conclusion is that matching theory is a useful but incomplete guide for LLM-agent institutions, not that the guarantees transfer.

Self-assessment is the binding constraint on agent markets. Auctions, task routing (#3), and confidence staking (#11) all consume agent self-reports of cost and success probability. MarketBench (Fradkin & Krishnan, arXiv:2604.23897, April 2026) measured six recent models on 93 SWE-bench Lite tasks and found them poorly calibrated on both success rate and token consumption; auctions built from those self-reports diverged from the full-information allocation, and supplying prior-capability context improved calibration only modestly. Before routing real work by agent bids, measure calibration on held-out tasks — an incentive-compatible mechanism fed miscalibrated valuations allocates badly without anyone misreporting strategically.

Habit: before invoking a solution concept, ask "would every party recognize this as the same game I do?" If not, either model it explicitly as a game of incomplete information (Bayesian game) or drop equilibrium language and use the frame as a heuristic only.

Mapping a business situation to the right game

Non-experts reach for "prisoner's dilemma" or "Nash equilibrium" as a generic label for any tense multi-party situation. An expert asks a short sequence of diagnostic questions before naming a game form or picking a primitive:

1. Who are the real strategic actors? Not every interested party is a strategic player — a regulator reacting on a multi-year lag is closer to an exogenous constraint than a player in a weekly pricing game.
2. One-shot or repeated — do the players expect to meet again? A single vendor negotiation is a bargaining problem (#12); an ongoing supplier relationship is a repeated game where reputation and folk-theorem-style cooperation are available — analyzing it as one-shot recommends defection that is actually irrational given the relationship's shadow of the future.
3. Simultaneous or sequential, and who commits first? Prices set quarterly and observed by competitors before they respond is closer to Stackelberg (sequential, first-mover) than Cournot/Bertrand (simultaneous) — the right model changes the recommendation from "best-respond" to "commit and signal."
4. Is value created cooperatively or contested? Cooperative-game tools (Shapley, core) fit attribution and surplus-sharing (#4); competitive tools (auctions, Nash) fit contested allocation (#3, #9). Applying auction logic to a joint-venture split, or Shapley logic to a zero-sum negotiation, produces answers that are precise and wrong.
5. Is there a credible commitment device? A threat or promise only constrains behavior if the counterparty believes it will be carried out even against the threatener's own later interest. A pricing "war" threat with no sunk cost or public commitment behind it is cheap talk — treat it as information about intent, not as a binding constraint on the game tree.
6. Is this actually a game, or an oracle-verifiable fact? The most common non-expert error in this whole domain is running a debate, auction, or negotiation protocol over a question that has a deterministic answer — a test suite, a contract clause, a calculator. See Misuse Boundaries.

Mechanism-design failure modes that only surface in production

Textbook mechanism design proves existence of a truthful, efficient, individually rational mechanism under an idealized participant model. Each row below is a normal way real deployments break that idealization — not an edge case to footnote.

Failure Mode	What Breaks	Real-World Trigger	Mitigation
Collusion / bidder rings	Dominant-strategy truthfulness assumes independent bidders; a ring that agrees off-mechanism to suppress bids and split the surplus defeats VCG and second-price auctions alike	Repeated auctions with a small, stable, identifiable bidder pool	Reserve prices, bidder-pool rotation, anti-collusion monitoring (AntiCollusionAI); detect via markup-over-marginal-cost drift over many rounds, not spot price
False-name bids	A single bidder submits multiple identities; VCG is provably not false-name-proof in combinatorial auctions, and no false-name-proof mechanism is Pareto efficient in general (Yokoo, Sakurai & Matsubara, Games and Economic Behavior, 2004)	Any auction where identity is cheap to fabricate — email-based registration, sybil-able agent pools, unverified marketplace accounts	Require attested identity before bidding (mirrors #21 Attested Delegation Contracts) — price identity verification into the mechanism, not as an afterthought
Participation constraints failing	Individual rationality assumes the average outside option; when the highest-value participants have the best outside options, they opt out first and adversely select the remaining pool	A mechanism designed around expected participants, not the marginal one who is deciding whether to walk	Check IR against the highest-value participant's outside option; Myerson & Satterthwaite (1983) show no mechanism for private-value bilateral trade can be simultaneously efficient, budget-balanced, and individually rational — some efficiency loss or subsidy is structurally unavoidable
Budget imbalance	VCG is efficient and truthful but generally runs a deficit or surplus that must land somewhere	Multi-sided mechanisms with no natural residual claimant	Decide upfront who absorbs the imbalance (platform take-rate, budget-neutral variant, or accept the inefficiency) rather than discovering it at settlement
Computational infeasibility	Exact VCG for combinatorial allocation requires solving an often NP-hard optimization for the winning allocation and every counterfactual-without-bidder-i allocation	Task/resource routing over bundles, not single-item slots	Use approximate/greedy VCG variants and disclose the resulting efficiency loss, or restrict to single-item/separable settings where exact VCG is tractable
Rules stated only in the prompt	A policy the participants can read but nothing enforces is cheap talk; under optimization pressure agents route around it	LLM participants told "do not collude" in a system prompt, with no state machine, sanction, or audit log behind it	Enforce in the orchestration layer, not the prompt — declare legal states, transitions, and sanctions outside the agents and have a controller apply them (Institutional AI, arXiv:2601.11369, Jan 2026: prompt-only constitutional policy was ineffective; governance-graph enforcement cut severe-collusion incidence from 50% to 5.6% across 90 runs)

Practical tell: if a mechanism is called "truthful" or "incentive-compatible" but nobody can name (a) the participation constraint being satisfied, (b) how false identities are prevented, (c) who absorbs budget imbalance, and (d) what enforces the rules other than the prompt, the claim has not actually been checked.

Communication channels are a collusion dial, not a neutral feature. Direct seller-to-seller messaging raises collusive tendency in simulated continuous double auctions, with the effect varying by model and modulated by oversight and authority pressure (Agrawal et al., arXiv:2507.01413, 2025). The same channel that reduces conflict in coordination games raises coordinated overpricing in market games — decide which game you are actually running before granting agents a side channel.

---

Anti-Patterns