name: game-theory argument-hint: "<equilibrium concept or game type>" description: >- This skill covers game-theoretic methods in structural econometrics and industrial organization. Use when the user is working with strategic interactions, equilibrium analysis, or game-theoretic structural models — including entry games, conduct testing, auction models with strategic bidding, bargaining, or matching markets. Triggers on "Nash equilibrium", "subgame perfect", "best response", "strategic interaction", "entry game", "conduct testing", "auction", "mechanism design", "matching market", "bargaining", "BNE", "Bayesian Nash", "static game", "dynamic game", "repeated game", "multiple equilibria", "equilibrium selection", "discrete game", "oligopoly", "game-theoretic", "player", "payoff", "strategy", "dominant strategy", "Bresnahan-Reiss", "Ciliberto-Tamer", "partial identification", "set identification", or markup test.

Game Theory

Reference for game-theoretic methods in applied structural econometrics and industrial organization. Covers equilibrium concepts, computational methods, structural IO applications, and the identification challenges unique to game-theoretic models.

When to Use This Skill

Use when the user is:

• Estimating a structural model where agents interact strategically (oligopoly, entry, bargaining, auctions)
• Deriving or computing Nash equilibria, BNE, or subgame perfect equilibria
• Handling the multiple equilibria problem in empirical games
• Testing firm conduct (competitive vs. collusive vs. oligopolistic)
• Estimating entry models, matching models, or bargaining models
• Formalizing an identification argument for a game-theoretic model

Skip when:

• The model is single-agent (use structural-modeling skill for dynamic discrete choice, demand estimation)
• The task is standard causal inference without strategic interaction (use causal-inference skill)
• The game is a well-known IO model with standard estimation code (pyblp covers BLP demand; see structural-modeling)

Quick reference only — for full implementation code, see references/ subdirectory.

Where to Start

• Choosing equilibrium concept? See Equilibrium Concept Routing below, then references/equilibrium-concepts.md for definitions and formulas
• Computing equilibria? See references/equilibrium-computation.md
• Estimating an IO model? See references/io-applications.md
• Estimation code and diagnostics? See references/estimation-diagnostics.md
• Facing multiple equilibria? See Multiple Equilibria Summary below, then references/multiple-equilibria.md for selection rules and set identification
• Identification argument? See Identification Summary below, then references/identification-in-games.md for exclusion restrictions and rank conditions

Quick Start: Nash Equilibrium Computation

import nashpy as nash
import numpy as np

# Define a 2-player game: row player payoffs A, column player payoffs B
A = np.array([[3, 0], [5, 1]])  # e.g., Prisoner's Dilemma
B = A.T                          # Symmetric game
game = nash.Game(A, B)

# Find ALL Nash equilibria via support enumeration
for i, (sr, sc) in enumerate(game.support_enumeration()):
    print(f"NE {i+1}: row={sr.round(3)}, col={sc.round(3)}")

For larger games, extensive-form games, or QRE computation, see references/equilibrium-computation.md.

Equilibrium Concept Routing

Decision tree:

1. Do players have private information? → Yes: BNE framework. No: Nash/SPE.
2. Is the game sequential? → Yes: SPE (backward induction) or MPE (dynamic states). No: simultaneous Nash.
3. Is the game repeated? → Yes: folk theorem applies; collusion may be sustainable.
4. Are there multiple equilibria? → See Multiple Equilibria Summary.

For detailed definitions, formulas, and the complete-vs-incomplete information comparison table, see references/equilibrium-concepts.md.

Multiple Equilibria Summary

The central identification challenge in empirical games. Three resolution strategies:

For the full selection rule comparison table, QRE implementation code, and Ciliberto-Tamer bounds procedure, see references/multiple-equilibria.md.

Identification Summary

Two sources of endogeneity distinguish games from single-agent models: (1) strategic complementarities/substitutes create simultaneity, and (2) correlated unobservables create spurious correlation in actions.

Resolution: Firm-specific instruments Z_i (cost, distance, regulatory history) excluded from rival j's profit equation. Variation in Z_i shifts firm i's entry, which instruments for j's strategic response.

Rank condition (Bajari-Hong-Ryan 2010): The Jacobian of the best-response system w.r.t. exogenous variables must have full rank. Fails when all firms share the same instruments, competitive effects are zero, or instruments are weak.

Conduct parameter identification: Cost shifters must shift supply independently of demand (standard simultaneous equations condition). The conduct parameter θ is identified from the curvature of the markup-quantity relationship.

For the full treatment — exclusion restriction formulas, two-step estimation logic, competitive effect identification, and conduct rank condition failure modes — see references/identification-in-games.md.

Structural IO Applications: Routing

For full model specifications, estimation code, and references, see references/io-applications.md and references/estimation-diagnostics.md.

Integration with compound-science

• Use identification-critic agent to verify equilibrium existence, uniqueness, and stability properties before reporting results
• Use structural-modeling skill for the estimation machinery (GMM, MLE, NFXP, MPEC) when the game-theoretic structure is already set up
• Use identification-critic agent to stress-test the game-theoretic identification argument — exclusion restrictions, rank conditions, separability assumptions
• Use the identification-critic agent (or identification-proofs skill) to formalize the full identification argument: target parameter → model → equilibrium concept → moment conditions → rank condition
• Use numerical-auditor agent to design Monte Carlo studies verifying identification and estimator performance in your specific game

Common Anti-Patterns

Method Selection Guide

Decision heuristic:

1. Is the game static or dynamic?
   • Dynamic → Markov perfect equilibrium; use CCP two-step (Bajari-Benkard-Levin 2007)
   • Static → proceed below
2. Is information complete or incomplete?
   • Incomplete (private types) → BNE; use threshold strategy estimation or GPV for auctions
   • Complete → Nash equilibrium; proceed below
3. Are there multiple equilibria at plausible parameter values?
   • Yes, and willing to impose selection → ordered probit / Berry (1992)
   • Yes, and not willing to impose selection → moment inequalities / Ciliberto-Tamer (2009)
   • No → standard MLE or GMM
4. Is the question about conduct?
   • Use BLP supply side + Rivers-Vuong test, or Rotemberg-Saloner markup test
5. Is bargaining the mechanism?
   • Use generalized Nash bargaining with outside option instruments

Reference Files

Read these when implementing a specific model type:

• references/equilibrium-concepts.md — Detailed definitions and formulas for Nash, BNE, SPE, MPE, mixed strategies, repeated games, folk theorem, complete-vs-incomplete information comparison
• references/equilibrium-computation.md — Computing Nash, BNE, and SPE: best response iteration, support enumeration, Gambit solver integration, linear complementarity, dynamic programming
• references/multiple-equilibria.md — Selection rules (risk dominance, QRE, ordered equilibrium), QRE implementation code, Ciliberto-Tamer set identification procedure, multiplicity-as-variation
• references/identification-in-games.md — Exclusion restrictions, rank conditions (Bajari-Hong-Ryan), two-step estimation logic, competitive effect identification, conduct parameter identification and failure modes
• references/io-applications.md — Entry models (Bresnahan-Reiss, Berry, Ciliberto-Tamer), conduct testing (BLP supply, Rivers-Vuong), bargaining (Nash, Rubinstein), auction foundations
• references/estimation-diagnostics.md — Estimation code (MLE, two-step, MPEC, moment inequalities), convergence diagnostics, model fit tests, equilibrium verification