Nash Equilibrium

The Process

Step 1: Identify the Players and Their Available Strategies

Map all decision-makers and their realistic strategic options. Be exhaustive - missing a key player or strategy invalidates the analysis.

Player identification:

• Direct competitors (obvious)
• Substitutes and complements (often overlooked)
• Customers with bargaining power (can defect)
• Regulators who might intervene

Strategy mapping:

• Price levels (high/medium/low)
• Quality tiers (premium/standard/budget)
• Market segments (geographic, demographic)
• Feature sets (full-featured/specialized/minimal)

Step 2: Build the Payoff Matrix

For each combination of strategies, determine the outcome (profit, market share, utility) for each player. This requires market research, financial modeling, or informed estimation.

Example: Two-player pricing game

Step 3: Find Best Responses for Each Player

For each strategy your opponent might play, identify your best response. Circle or highlight these cells.

Analysis:

• If they price high: your best response is low ($12M > $10M)
• If they price low: your best response is low ($5M > $2M)
• Symmetrically, their logic mirrors yours

Step 4: Identify Nash Equilibrium (Mutual Best Responses)

The Nash Equilibrium occurs where each player's strategy is a best response to the other's. Both players choosing "Low Price" is the equilibrium - neither can improve by unilaterally switching.

Interpretation:

• Equilibrium (Low, Low) yields $5M each
• Both would prefer (High, High) at $10M each, but it's unstable
• Any unilateral defection from (High, High) is profitable

Step 5: Assess Stability and Strategic Implications

Is this equilibrium desirable? If not, what changes the game structure?

Breaking undesirable equilibria:

• Repeated games (future retaliation changes payoffs)
• Binding commitments (contracts, public announcements)
• Changing the game (differentiation removes direct competition)
• Signaling (price leadership, capacity investment)

Example Application

Situation: Mobile telecom market with two major carriers, both considering price cuts.

Application:

• Players: Carrier A, Carrier B
• Strategies: Maintain rates vs. Cut rates 10%
• Payoffs modeled: Price cut gains customers short-term but triggers retaliation
• Equilibrium: Both maintain rates (neither benefits from solo cut if rival matches)
• Real-world observation: Telecom prices remain stable until new entrant disrupts

Outcome: Company understands price competition is lose-lose. Differentiates on service quality instead, changing game structure rather than competing on equilibrium price.

Example Application 2

Situation: Two coffee shops deciding where to locate on a beach (Hotelling model).

Application:

• Players: Shop A, Shop B
• Strategies: Position anywhere along 1-mile beach
• Customer behavior: Each customer walks to nearest shop
• Nash Equilibrium: Both position at center (50% point)
• Analysis: Any deviation from center loses more customers than gained

Outcome: Explains why competitors cluster - gas stations at intersections, fast food restaurants in same strip mall. Positioning at center minimizes maximum distance to any customer.

Anti-Patterns

• Assuming only one equilibrium exists (many games have multiple equilibria)
• Ignoring dynamic games (repeated interactions change equilibrium logic)
• Treating equilibrium as "optimal" (prisoners dilemma equilibrium is suboptimal for all)
• Assuming rational actors (behavioral biases shift real-world outcomes)
• Forgetting asymmetric information (players may have different data about payoffs)
• Analyzing one-shot when game is actually repeated (changes everything)

Related

• prisoners-dilemma (famous example where Nash Equilibrium is collectively suboptimal)
• dominant-strategy-analysis (finding strategies that win regardless of opponent)
• game-theory (broader field encompassing strategic interaction analysis)
• porters-five-forces (competitive dynamics framework)
• incentives (understanding what drives player behavior in equilibrium)