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SKILL.md
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 for selection rules and set identification
references/multiple-equilibria.md
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
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:
Do players have private information? → Yes: BNE framework. No: Nash/SPE.
Is the game sequential? → Yes: SPE (backward induction) or MPE (dynamic states). No: simultaneous Nash.
Is the game repeated? → Yes: folk theorem applies; collusion may be sustainable.
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:
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.
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.
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
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
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
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:
Is the game static or dynamic?
Dynamic → Markov perfect equilibrium; use CCP two-step (Bajari-Benkard-Levin 2007)
Static → proceed below
Is information complete or incomplete?
Incomplete (private types) → BNE; use threshold strategy estimation or GPV for auctions
Complete → Nash equilibrium; proceed below
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
Is the question about conduct?
Use BLP supply side + Rivers-Vuong test, or Rotemberg-Saloner markup test
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
