/game-theory

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](#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](#multiple-equilibria-summary) below, then `references/multiple-equilibria.md` for selection rules and set identification
• **Identification argument?** See [Identification Summary](#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](#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/subst