Solving Games

Analyzing games and finding Nash equilibria.

The solve Statement

solve GameName;

This analyzes the specified game and outputs information.

Nash Equilibrium Solver

Tenet supports finding Pure Strategy Nash Equilibria.

solve PrisonersDilemma;

Output:

═══════════════════════════════════════
Game: PrisonersDilemma
Players: Alice, Bob
Strategies: Cooperate, Defect
───────────────────────────────────────
Nash Equilibria (Pure Strategy):
  -> (Defect, Defect) with payoffs (1, 1)
═══════════════════════════════════════

How it Works

The solver iterates through every possible strategy profile (combination of strategies). For each profile, it checks if any player has a unilateral incentive to deviate (switch strategies) to improve their payoff. If no player wants to switch, it is a Nash Equilibrium.

Multiple Equilibria

If a game has multiple pure strategy equilibria, like the Battle of the Sexes, Tenet will list them all:

Nash Equilibria (Pure Strategy):
  -> (Opera, Opera) with payoffs (3, 2)
  -> (Football, Football) with payoffs (2, 3)

No Pure Equilibrium

Some games, like Matching Pennies, have no pure strategy equilibrium. In this case, Tenet will inform you:

No Pure Strategy Nash Equilibrium found.
Try mixed strategies (coming soon).

Algorithm: Check all strategy profiles for mutual best responses.

Phase 3: Mixed Strategy Nash Equilibria

solve MatchingPennies;
// Expected output:
// Nash Equilibrium (Mixed):
//   Matcher: 50% Heads, 50% Tails
//   Mismatcher: 50% Heads, 50% Tails
//   Expected payoffs: 0, 0

Algorithm: Lemke-Howson for 2-player bimatrix games.

Solve Options (Planned)

solve GameName using "pure";      // Only pure strategy NE
solve GameName using "mixed";     // Include mixed strategies
solve GameName using "dominant";  // Find dominant strategies

Understanding Nash Equilibrium

A Nash Equilibrium is a strategy profile where no player can improve their payoff by unilaterally changing their strategy.

In Prisoner's Dilemma:

• At (Defect, Defect):
  • Alice gets 1. If she switches to Cooperate, she gets 0. (Worse)
  • Bob gets 1. If he switches to Cooperate, he gets 0. (Worse)
  • Neither wants to deviate → Nash Equilibrium

Roadmap

Next Steps

• Classic Games → — See solved examples