npbuilds Game Solver — The Analyst
Every strategic situation is a game, whether the participants realize it or not. This skill takes informal descriptions of strategic interactions — pricing decisions, negotiations, platform competition, political dynamics, fictional faction conflicts — and produces rigorous game-theoretic analysis with actionable interpretation.
The core capability is formalization: bridging the gap between "I'm trying to decide whether to enter this market" and a well-specified entry deterrence game with equilibrium predictions.
How to Run
Input
The user provides one or more of:
- A situation description — informal narrative of who's involved, what they can do, and what they want
- A specific question — "should I cooperate?", "what's the likely outcome?", "is this agreement stable?"
- A formal game — already specified with players, strategies, and payoffs (skip to Step 3)
- A real-world scenario — business, political, technological, biological, or fictional context needing strategic analysis
Steps
Step 1 — Extract the Strategic Elements
From the user's description, identify:
| Element | Question to Answer | Example |
|---|---|---|
| Players | Who are the decision-makers? | Firm A, Firm B, Regulator |
| Strategies | What can each player choose? | Enter/Stay Out, High Price/Low Price |
| Timing | Simultaneous or sequential? Repeated? | Firm A moves first (Stackelberg) |
| Information | What does each player know? Private info? | Firm A knows its costs, B doesn't |
| Payoffs | What are they optimizing? | Profit, market share, welfare |
| Commitments | Can they make binding agreements? | Contract possible / no enforcement |
If elements are ambiguous, ask the user to clarify rather than assuming. The quality of the analysis depends on the quality of the formalization.
Step 2 — Classify the Game
Map to the appropriate game type using the classification tree:
Is there private information?
├── No → Complete information
│ ├── Simultaneous → Normal-form game
│ │ ├── One-shot → Standard Nash analysis
│ │ └── Repeated → Folk theorem / repeated game analysis
│ └── Sequential → Extensive-form game
│ └── Subgame perfect equilibrium (backward induction)
│
└── Yes → Incomplete information
├── Simultaneous → Bayesian game
│ └── Bayesian Nash equilibrium
└── Sequential → Signaling / screening game
└── Perfect Bayesian equilibrium + refinements
Can players make binding agreements?
├── Yes → Also analyze as cooperative game
│ └── Core, Shapley value, bargaining solutions
└── No → Non-cooperative analysis only
Read references/formalization-patterns.md for common real-world-to-game mappings.
Step 3 — Formalize the Game
Produce the formal game specification:
For normal-form games:
Players: {P1, P2, ...}
Strategies:
P1: {s1a, s1b, ...}
P2: {s2a, s2b, ...}
Payoff matrix:
P2: s2a P2: s2b
P1: s1a (a,b) (c,d)
P1: s1b (e,f) (g,h)
For extensive-form games: Describe the game tree with decision nodes, information sets, actions, and terminal payoffs. Use indentation to show the tree structure.
For cooperative games: Specify the characteristic function v(S) for relevant coalitions.
Step 4 — Solve
Apply the appropriate solution concept:
Recipe — Zero-Sum Value
For a zero-sum game the value of the game is the saddle point: take each row's minimum and each column's maximum; when the maximin equals the minimax, that common number is the value to the row player.
Recipe — Strictly Dominant Strategy
A strategy is strictly dominant when it yields a strictly higher payoff against each of the opponent's actions (it strictly dominates every alternative). Check each of your strategies against every opponent action; if none clears that bar for every action, answer "none".
Recipe — Pareto-Optimal Outcomes
An outcome is Pareto-optimal when no other outcome makes one player better off without making the other worse off. Enumerate the cells and discard any cell weakly dominated in both payoffs by another cell.
Recipe — Mixed-Strategy Nash (2x2)
To find a mixed equilibrium, make the opponent indifferent: choose your mixing probability so the opponent's expected payoff is the same across their actions (set expected payoffs equal and solve). Report p (prob the row player plays its first strategy) and q (prob the column player plays its first strategy).
Recipe — Maximin / Security Strategy
A player's maximin (security level) strategy maximizes their guaranteed worst-case payoff: for each of your strategies take the minimum payoff over the opponent's actions, then pick the strategy with the largest such minimum.
| Game Type | Primary Solution | Also Check |
|---|---|---|
| Simultaneous, complete info | Nash Equilibrium (pure + mixed) | Dominated strategy elimination, correlated equilibrium |
| Sequential, perfect info | Subgame Perfect Equilibrium | Nash equilibria (to show which are non-credible) |
| Bayesian game | Bayesian Nash Equilibrium | Information value, posterior beliefs |
| Signaling game | PBE + Intuitive Criterion | Pooling vs. separating equilibria |
| Cooperative | Core + Shapley Value | Nucleolus, Nash bargaining |
| Repeated | Folk theorem bounds | Trigger strategies, renegotiation-proofness |
Show your work. For small games, enumerate strategies and check each cell. For larger games, use best-response analysis or dominance reasoning.
Step 5 — Interpret
Translate the formal results into plain-language strategic insight:
- Prediction: What does the equilibrium say will happen?
- Recommendation: Given the prediction, what should the user do?
- Robustness: What assumptions drive the result? What changes if they shift?
- Comparison: How does this compare to the cooperative optimum? Is there a Pareto improvement available through commitment or communication?
- Real-world caveats: Where might the model oversimplify? Behavioral biases, institutional constraints, enforcement problems?
Step 6 — Sensitivity Analysis
Identify the key parameters and how the equilibrium changes:
- What if a player gets a new strategy option?
- What if information structure changes (player learns something)?
- What if the game is repeated instead of one-shot?
- What if players are boundedly rational?
Output
A structured analysis document containing:
## Strategic Analysis: [Situation Name]
### Game Specification
[Formal game: players, strategies, timing, information, payoffs]
### Classification
[Game type and justification]
### Solution
[Equilibrium analysis with work shown]
### Interpretation
[Plain-language prediction and recommendation]
### Sensitivity
[Key assumptions and how results change]
### Caveats
[Where the model simplifies reality]
Error Handling
Situation is too vague: Ask the user to specify players and what they can choose. "A competitive situation" needs more structure before analysis.
Multiple equilibria: Present all equilibria with selection criteria (risk-dominance, payoff-dominance, focal points, evolutionary stability). Don't arbitrarily pick one.
No pure-strategy equilibrium: Compute the mixed-strategy equilibrium and explain what randomization means in context (e.g., "the auditor should randomly select 30% of returns for review").
Game is too large to solve by hand: Identify dominant strategies and use iterated elimination to reduce the game. Note the computational complexity and suggest algorithmic approaches if needed.
Cooperative and non-cooperative analyses disagree: Present both. The non-cooperative analysis predicts what happens without enforcement; the cooperative analysis shows what's achievable with binding agreements. The gap between them quantifies the "value of commitment."