game-theory

Files (7)

• SKILL.md
• equilibrium-computation.md · references/
• equilibrium-concepts.md · references/
• estimation-diagnostics.md · references/
• identification-in-games.md · references/
• io-applications.md · references/
• multiple-equilibria.md · references/

SKILL.md (first lines)

# 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

---

## Quick Start: Nash Equilibrium Computation

```python
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`.

---

## 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

From James-Traina/compound-science, MIT.