Nash Equilibrium Analyzer

Example 1 — Pricing game (unique pure equilibrium via successive elimination)

Situation: Two retailers are setting prices simultaneously in a range of $38–$42.

Analysis: 1. Build 5×5 payoff matrix 2. $42 is dominated by $41 for both firms (higher profit regardless of rival's price); eliminate 3. $38 is dominated by $39; eliminate 4. In the resulting 3×3 game, $40 is dominant for both firms 5. Nash equilibrium: Both price at $40 (40,000 profit each)

Insight: The $40 equilibrium is less profitable than the collusion price ($80 = 72,000 each), but neither firm can unilaterally raise price without losing customers. The equilibrium is stable even if both wish for a different outcome.

Example 2 — Penalty kick (no pure equilibrium, mixed strategy required)

Situation: Kicker vs. goalie, simultaneous choice of Left/Right. Payoffs (kicker success %):

| | Goalie: Left | Goalie: Right | |---|---|---| | Kicker: Left | 58 | 95 | | Kicker: Right | 93 | 70 |

Analysis: 1. No pure equilibrium — best-response arrows cycle 2. Rule 5 applies (zero-sum game) 3. Kicker indifference equation → p = 38.3% Left, 61.7% Right 4. Goalie indifference equation → y = 41.7% Left, 58.3% Right 5. Equilibrium success rate: 79.6% (minimax = maximin by von Neumann's theorem)

Recommendation: Kicker randomizes Left 38.3% using an objective device (page numbers, watch second hand). Goalie randomizes Left 41.7%. Any predictable pattern invites exploitation.

Example 3 — Coordination game with multiple equilibria (focal-point selection)

Situation: Two division managers must independently choose which cloud platform to deploy on (AWS or Azure). Payoffs: both benefit equally from coordinating (3 each), neither benefits from mismatching (0 each). Two Nash equilibria: (AWS, AWS) and (Azure, Azure).

Analysis: 1. Both equilibria are Nash — each is a mutual best response 2. No dominant strategy; game is pure coordination 3. Do not mix — independent randomization at 50/50 produces miscoordination 50% of the time, yielding expected payoff 1.5 < 3 4. Focal-point check: Does the company already use one platform? Is there an industry default? Is one option mathematically simpler? → Use whichever has the strongest salience as the focal point 5. If no focal point exists, establish one through explicit pre-game communication