Game Theory

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