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

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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

Information Structure	Timing	Concept	Refinement	Key Reference
Complete	Simultaneous	Nash equilibrium	Dominant strategy, trembling-hand perfect	—
Complete	Sequential	Subgame perfect equilibrium (SPE)	Backward induction	—
Complete	Repeated	SPE with trigger strategies	Folk theorem, Nash reversion	Green-Porter (1984)
Complete	Dynamic (states)	Markov perfect equilibrium (MPE)	Strategies depend only on payoff-relevant state	Ericson-Pakes (1995)
Incomplete (private types)	Simultaneous	Bayesian Nash equilibrium (BNE)	Monotone strategies, threshold equilibria	—
Incomplete	Sequential	Perfect Bayesian equilibrium (PBE)	Sequential rationality + Bayesian updating	Kreps-Wilson

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.

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Multiple Equilibria Summary

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

Strategy	Approach	Trade-off	Key Reference
Impose selection rule	Order firms by profitability; pick unique NE	Point identification, but selection rule is an assumption	Berry (1992)
Set identification	Accept all NE-consistent parameters	No selection assumption, but wider confidence regions	Ciliberto-Tamer (2009)
Exploit multiplicity	Use correlates of equilibrium selection as instruments	Point identification with weaker assumptions	Sweeting (2009)
QRE	Bounded rationality generates unique equilibrium	Testable, but imposes logistic choice structure	McKelvey-Palfrey

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

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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.

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Structural IO Applications: Routing

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

Application	Model Class	Estimation	Reference File
Market structure (symmetric firms)	Bresnahan-Reiss ordered probit	MLE	io-applications.md
Entry (asymmetric firms)	Berry ordered equilibrium	MLE with equilibrium constraints	io-applications.md
Entry (multiple equilibria)	Ciliberto-Tamer partial identification	Moment inequalities	io-applications.md
Conduct testing	BLP supply side + markup equation	GMM + Rivers-Vuong test	io-applications.md
Vertical bargaining	Generalized Nash bargaining (Horn-Wolinsky)	GMM with outside option instruments	io-applications.md
Procurement/first-price auctions	BNE bidding + GPV inversion	Nonparametric	io-applications.md
Dynamic oligopoly	MPE (Ericson-Pakes)	CCP two-step (Bajari-Benkard-Levin)	estimation-diagnostics.md
Collusion sustainability	Repeated game + trigger strategies	Threshold discount factor	equilibrium-concepts.md

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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

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Common Anti-Patterns

Anti-Pattern	Problem	Better Approach
Assuming unique equilibrium without verification	Model may have multiple equilibria; point estimates are identification-assumption-dependent	Enumerate all Nash equilibria at estimated parameters; verify uniqueness or state selection rule
Using complete-information entry model when firms have private information	Equilibrium concept is wrong; identification fails	Use incomplete-information model (Seim 2006, Bajari-Hong-Ryan 2010) or test for information structure
Ignoring the multiple equilibria problem in partial identification	Inference is invalid under point identification when set identification is required	Use Ciliberto-Tamer bounds or impose and justify a selection rule
Conduct test with weak instruments	Low power to reject Bertrand; cannot distinguish conduct	Report first-stage relevance; use optimal instruments (BLP supply side)
Treating equilibrium prices as exogenous regressors in demand	Prices are endogenous (set in equilibrium); OLS demand estimates are biased	Instrument with cost shifters; use BLP/IV approach
Estimating bargaining weight without outside option variation	β is not identified without variation in outside options	Find instruments for outside options (market-level variation in alternatives)
Nash reversion assumption in collusion test without threshold test	Assumes away the inference problem	Estimate threshold discount factor; test whether δ* is plausible given observed interest rates
Not reporting equilibrium verification	Referees cannot assess model validity	Always report that estimated parameters support equilibrium existence

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Method Selection Guide

Setting	Model	Equilibrium Concept	Estimation Approach	Key Reference
Oligopoly market structure	Complete information entry	Nash (ordered selection)	Ordered probit MLE	Bresnahan-Reiss (1991)
Asymmetric firm entry	Complete information entry	Nash (ordered selection)	MLE with equilibrium constraints	Berry (1992)
Entry with multiple equilibria	Partial identification	Nash (all equilibria)	Moment inequalities	Ciliberto-Tamer (2009)
Entry with private cost info	Bayesian game	Bayesian Nash (threshold)	MLE / two-step	Seim (2006)
Conduct: competitive vs. collusive	Oligopoly pricing	Nash in prices/quantities	BLP supply + Rivers-Vuong test	Berry-Levinsohn-Pakes (1995)
Vertical bargaining	Nash bargaining	Generalized Nash solution	GMM with outside option instruments	Horn-Wolinsky (1988), Crawford-Yurukoglu (2012)
Procurement auctions	First-price sealed-bid	Bayesian Nash (bidding)	GPV nonparametric inversion	Guerre-Perrigne-Vuong (2000)
Takeover/merger auctions	Ascending auction	Dominant strategy (IPV)	Order statistics / MLE	Athey-Haile (2002)
Common value auctions	Affiliated values	BNE (affiliated)	Parametric MLE	Li-Perrigne-Vuong (2002)
Dynamic oligopoly	Markov perfect equilibrium	MPE	CCP two-step (Bajari-Benkard-Levin)	Pakes-McGuire (1994), Bajari et al. (2007)
Collusion sustainability	Repeated game	Subgame perfect	Threshold discount factor estimation	Green-Porter (1984), Porter (1983)
Matching markets	Stable matching	Stable (Gale-Shapley)	Revealed preference from match outcomes	Fox (2010), Choo-Siow (2006)
Small 2-player game (theory)	Normal form	Nash (all equilibria)	nashpy / gambit computation	—

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

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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