Reference · Cheat Sheet 04
Best Responses & Nash Equilibrium
Core idea: a Nash equilibrium is a strategy profile where every player is best-responding to everyone else. No one can do better by unilaterally changing their own choice. It's a self-enforcing standoff — stable even when it's not the best outcome for anyone.
The best-response method
- Scan each column. In each column, find the row player's highest payoff and mark it (underline, circle, bold). That's the row player's best response to that column strategy.
- Scan each row. In each row, find the column player's highest payoff and mark it. That's the column player's best response to that row strategy.
- Find doubly-marked cells. Any cell where both players' best replies coincide is a Nash equilibrium. Mark it clearly.
- Check for zero, one, or several. There may be no pure equilibrium, one unique equilibrium, or multiple. All are valid outcomes of the method.
Worked example 1 — keyword auction (unique equilibrium)
Monthly patient bookings (you, rival). Run dominates for both → unique Nash at (Run, Run).
| Rival: Run ads | Rival: Pause ads |
|---|
| You: Run ads | 6, 6 ← Nash ⬅ | 12, 4 |
|---|
| You: Pause ads | 4, 12 | 9, 9 |
|---|
- If rival Runs: Run gives 6 > Pause gives 4 → Run.
- If rival Pauses: Run gives 12 > Pause gives 9 → Run.
- Run dominates → (Run, Run) is the unique Nash equilibrium.
- (Pause, Pause) = (9, 9) is better for everyone but not an equilibrium — either player gains by switching to Run.
Worked example 2 — partner integration (two equilibria)
Coordination value (you, partner). No dominant strategy → two Nash equilibria.
| Partner: Adopt | Partner: Keep own |
|---|
| You: Adopt | 4, 4 ← Nash ⬅ | 0, 3 |
|---|
| You: Keep own | 3, 0 | 2, 2 ← Nash ⬅ |
|---|
(Adopt, Adopt) = Nash ✓
If you switch to Keep while partner Adopts: 4 → 3. Loss. You don't switch. Partner same. Nobody moves.
(Keep, Keep) = Nash ✓
If you switch to Adopt while partner Keeps: 2 → 0. Loss. You don't switch. Partner same. Nobody moves.
- (Adopt, Keep) and (Keep, Adopt) are not equilibria — one player can always gain by switching.
- Both-Adopt is better for everyone, yet neither is guaranteed. Which equilibrium is reached depends on expectations — that's Lesson 05.
Caution — equilibrium ≠ efficient: a Nash equilibrium is only guaranteed to be stable, not optimal. (Run, Run) at (6,6) and (Keep, Keep) at (2,2) are both stable standoffs — yet both are worse than an outcome that nobody will unilaterally move toward. Stability and efficiency are separate questions.
Note on mixed strategies: some games have no pure Nash equilibrium. When no single fixed-strategy profile forms a mutual best response, players must randomise — producing a mixed-strategy equilibrium. That's a story for a later lesson; always check for pure equilibria first using this method.
Key definitions
- Best response — the payoff-maximising strategy given one specific opponent choice. Conditional, not universal.
- Nash equilibrium — a profile where every player is simultaneously best-responding to all others.
- Pure strategy — a fixed, non-random choice.
- Dominant strategy — a best response to every opponent move. Every dominant-strategy outcome is automatically a Nash equilibrium.
- Coordination game / stag hunt — a game with two pure Nash equilibria: one payoff-dominant (better for all), one risk-dominant (safer). Lesson 05 is about which one you reach.