---
name: foundations-game-theory
description: Game-theory primitives for strategic decision systems, auctions, mechanism design, incentives, attribution, negotiation, debate, and trust. Use when modeling strategic play.
compatibility: Claude Code + Codex. Portable core — primitives apply across domains.
version: "1.2"
last_validated: 2026-08-14
---

# 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`](../agents-subagents/references/game-theory-agent-teams.md).

## Contents

- [Quick Reference](#quick-reference)
- [Primitive Index](#primitive-index)
- [Formal Supporting Theory](#formal-supporting-theory)
- [Expert Judgment: When the Model Helps vs Misleads](#expert-judgment-when-the-model-helps-vs-misleads)
- [Anti-Patterns](#anti-patterns)
- [Misuse Boundaries](#misuse-boundaries)
- [Decision Checklist](#decision-checklist)
- [Composition Recipes](#composition-recipes)
- [Workflow](#workflow)
- [ASCII Flow](#ascii-flow)
- [Current Pattern Review](#current-pattern-review)
- [Navigation](#navigation)
- [Fact-Checking](#fact-checking)

---

## Quick Reference

| Primitive | Domain | Recipe Stub |
|-----------|--------|-------------|
| [Belief-Driven Coordination (ECON)](#1-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](#2-adversarial-debate) | Content moderation, risk review, audit | Two heterogeneous evaluators + reasoning-tree synthesis; no majority vote |
| [Auction-Based Routing](#3-auction-based-task-routing) | Ad placement, task delegation, resource allocation | Sealed-bid truthful auction; highest-value-per-cost wins |
| [Shapley Contribution](#4-shapley-contribution-scoring) | Attribution, revenue sharing, team composition | Marginal-contribution average across subsets |
| [Reputation-Gated Autonomy](#5-reputation-gated-autonomy) | Supplier qualification, agent oversight, fraud gating | Tiered trust: proven → standard → probationary; oversight inversely proportional |
| [Cooperation and Defection](#6-cooperation-and-defection) | Partnership design, incentive alignment, compliance | Iterated PD structure; payoff-scale to detect defection tendency |
| [Mechanism Design for Synthesis](#7-mechanism-design-for-synthesis) | Decision aggregation, voting, policy-making | Vickrey truthful-revelation; dissent is a required section |
| [Courtroom-Style Debate](#8-courtroom-style-debate) | Legal review, risk go/no-go, claim verification | Plaintiff/defense/court structure + progressive RAG + role-switching |
| [Pareto-Nash Multi-Objective](#9-pareto-nash-multi-objective) | Product tradeoffs, regulatory vs growth, pricing tiers | Map Pareto frontier; pick dominant options; flag non-dominated set |
| [Evolutionary Coordination Search](#10-evolutionary-coordination-search) | Algorithm selection, prompt tuning, rule evolution | LLM-mutated program + fitness signal; ShinkaEvolve for sample efficiency |
| [Prediction Market Confidence](#11-prediction-market--confidence-betting) | Forecasting, risk calibration, hiring decisions | Stake-weighted confidence; CritiCal calibration step before stake |
| [Negotiation ZOPA/BATNA](#12-negotiation-protocol-zopabatna) | Pricing, partnership terms, resource contention | Map BATNA/ZOPA per party; target overlap zone; use interests not positions |
| [Reasoning-Tree Audit](#13-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](#14-per-claim-credibility-scoring) | Misinformation detection, adversarial content, security | Evidence quality × corroboration weight per claim; isolate high-risk claims |
| [Generative Social Choice](#15-generative-social-choice) | Multi-stakeholder policy, diverse-user product decisions | Maximin selection across candidate outputs; preserve minority-signal coverage |
| [Meta-Debate Role Routing](#16-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](#17-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)](#18-beyond-majority-voting) | Best-of-N synthesis (discrete answer), ensemble selection | Optimal Weight (confidence × calibration) + Inverse Surprising Popularity |
| [Radial Consensus Score (RCS)](#19-radial-consensus-score) | Best-of-N synthesis (open-ended generation), self-consistency | Embedding-centroid selector for semantically clustered, lexically diverse answers |
| [Conformal Social Choice](#20-conformal-social-choice-actescalate) | High-stakes debate verdicts, act/escalate gates | Calibrated prediction set: singleton acts, multi-answer set escalates |
| [Attested Delegation Contracts](#21-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](#22-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)](assets/templates/game-theory/01-econ-belief-driven.md) | Pooling equilibrium — members read same context, produce same analysis |
| 2 | [Adversarial Debate](assets/templates/game-theory/02-adversarial-debate.md) | Confabulation consensus, correlated bias |
| 3 | [Auction-Based Task Routing](assets/templates/game-theory/03-auction-task-routing.md) | Static routing, ambiguous selection |
| 4 | [Shapley Contribution Scoring](assets/templates/game-theory/04-shapley-contribution.md) | Free-riding, unverifiable attribution |
| 5 | [Reputation-Gated Autonomy](assets/templates/game-theory/05-reputation-gating.md) | Uniform trust regardless of track record |
| 6 | [Cooperation and Defection](assets/templates/game-theory/06-cooperation-defection.md) | Shallow output, scope dumping, echo chambers |
| 7 | [Mechanism Design for Synthesis](assets/templates/game-theory/07-mechanism-design-synthesis.md) | Loudest-wins aggregation, suppressed dissent |
| 8 | [Courtroom-Style Debate (PROClaim)](assets/templates/game-theory/08-courtroom-proclaim.md) | Evidence stagnation, position-anchored reasoning |
| 9 | [Pareto-Nash Multi-Objective](assets/templates/game-theory/09-pareto-nash.md) | Single-objective optimization on multi-objective problems |
| 10 | [Evolutionary Coordination Search](assets/templates/game-theory/10-alphaevolve.md) | Hand-tuned rules are sub-optimal vs. measured fitness |
| 11 | [Prediction Market / Confidence Betting](assets/templates/game-theory/11-prediction-market.md) | Verbose output dominates synthesis |
| 12 | [Negotiation Protocol (ZOPA/BATNA)](assets/templates/game-theory/12-negotiation-zopa-batna.md) | Adversarial framing on genuine compromise situations |
| 13 | [Reasoning-Tree Audit](assets/templates/game-theory/13-reasoning-tree-audit.md) | Confident-but-wrong consensus; majority vote unsafe |
| 14 | [Per-Claim Credibility Scoring](assets/templates/game-theory/14-credibility-scoring.md) | Single-claim failure modes reputation gating misses |
| 15 | [Generative Social Choice](assets/templates/game-theory/15-generative-social-choice.md) | Multi-stakeholder buy-in; averaging erases minority evidence |
| 16 | [Meta-Debate Role Routing](assets/templates/game-theory/16-meta-debate-routing.md) | Wrong specialist gets the wrong debate role; static role assignment |
| 17 | [Online Shapley Prompt Evolution](assets/templates/game-theory/17-online-shapley-prompt-evolution.md) | Weak team members never improve; static prompts under-utilize Shapley signal |
| 18 | [Beyond Majority Voting (BMV)](assets/templates/game-theory/18-beyond-majority-voting.md) | Majority vote on best-of-N erases minority-correct answers (calibration ignored) |
| 19 | [Radial Consensus Score (RCS)](assets/templates/game-theory/19-radial-consensus-score.md) | Lexical-overlap voting fails on semantically clustered open-ended generations |
| 20 | [Conformal Social Choice Act/Escalate](assets/templates/game-theory/20-conformal-social-choice.md) | Wrong consensus turns into irreversible action |
| 21 | [Attested Delegation Contracts](assets/templates/game-theory/21-attested-delegation-contracts.md) | Self-claimed quality corrupts routing across trust boundaries |
| 22 | [Coalition Formation Routing](assets/templates/game-theory/22-coalition-formation-routing.md) | 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`](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`](references/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`](references/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](#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`](references/patterns-scenarios-traps.md)); 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, n