When nobody has a dominant strategy, the solution is a Nash equilibrium — a standoff of mutual best replies.
Without scrolling back: in iterated dominance, whose options do you cross off first?
Lessons 1–3 gave you a clean elimination machine: find dominated strategies, delete them, repeat until one profile survives. That works beautifully when it works. But many real strategic situations — coordination with a referrer, platform pricing against a rival with a different cost structure, deciding how aggressively to bid at the same hospital — have no dominant move to delete.
Your best move genuinely depends on theirs. If they advertise heavily, maybe you should too. If they pull back, maybe you should as well. Neither option dominates the other across the board. So the iterated-dominance toolkit hits a wall.
The answer comes in two tightly linked definitions:
That second definition is the core insight: a Nash equilibrium is a self-enforcing standoff. Once you're there, nobody has a profitable reason to leave on their own.
In each column, underline the row player's best reply. In each row, underline the column player's best reply. A cell marked by both is a Nash equilibrium.
Let's ground this in the ad-auction standoff from Lesson 1. You and a rival clinic each choose whether to keep running Google/Meta ads or pause them. Payoffs are new-patient bookings per month (your bookings, their bookings):
| Rival: Run ads | Rival: Pause ads | |
|---|---|---|
| You: Run ads | 6, 6 ← Nash ⬅ |
12, 4 |
| You: Pause ads | 4, 12 | 9, 9 |
Best-response trace:
If rival Runs → your options are 6 (Run) vs 4 (Pause) → Run (6 > 4).
If rival Pauses → your options are 12 (Run) vs 9 (Pause) → Run (12 > 9).
Run is your best response to everything — that's a dominant strategy. Same logic applies symmetrically to the rival.
So (Run, Run) = (6, 6) is the unique Nash equilibrium.
Notice: every dominant-strategy outcome is automatically a Nash equilibrium. If Run dominates for both players, each is already best-responding when both Run — neither can do better by switching. Nash just handles the cases where dominance didn't give us an answer.
Also notice the trap: (Pause, Pause) = (9, 9) is better for everyone, but it's not a Nash equilibrium — either player can gain by secretly switching to Run while the other pauses. That unilateral temptation is why (9,9) doesn't hold.
The keyword-auction structure is a classic Prisoner's Dilemma: individually rational play leads to a mutually worse outcome. See SEP — Prisoner's Dilemma for the formal treatment.
Now consider a genuinely harder decision. You and a referring partner (say, an orthopaedic surgeon who sends you spine cases) each independently decide whether to adopt a shared digital booking system or keep your existing separate systems. Neither of you gets a unilateral win from adopting — you only benefit if the other side does too.
Payoffs (your coordination value, partner's coordination value) — higher is better for each:
| Partner: Adopt | Partner: Keep own | |
|---|---|---|
| You: Adopt | 4, 4 ← Nash ⬅ |
0, 3 |
| You: Keep own | 3, 0 | 2, 2 ← Nash ⬅ |
Best-response trace (your moves):
If partner Adopts → 4 (Adopt) vs 3 (Keep) → best response: Adopt.
If partner Keeps → 0 (Adopt) vs 2 (Keep) → best response: Keep.
Your best response switches depending on what they do. No dominant strategy for either player.
Finding equilibria:
At (Adopt, Adopt): you'd get 3 by switching to Keep while they Adopt — but 3 < 4, so you don't switch. Partner symmetric. Equilibrium ✓
At (Keep, Keep): you'd get 0 by switching to Adopt while they Keep — but 0 < 2, so you don't switch. Partner symmetric. Equilibrium ✓
At (Adopt, Keep): you get 0 but could get 2 by switching to Keep. Profitable deviation → not an equilibrium ✗
At (Keep, Adopt): partner could get 2 by switching to Keep. Profitable deviation → not an equilibrium ✗
This structure — two equilibria, one better for everyone, no guarantee you reach the good one — is called a stag hunt or coordination game. It's the skeleton behind many real decisions: switching software platforms, aligning on referral protocols, jointly investing in a shared capability.
A Nash equilibrium is not necessarily the best outcome — only one no one can improve on alone. (Run, Run) at (6,6) and (Keep, Keep) at (2,2) are both equilibria, both worse than an alternative that exists — yet both are stable standoffs once reached.
One more thing worth naming: some games have no pure Nash equilibrium at all — no single fixed strategy for each player that forms a mutual best response. In those games, players need to randomise (choose strategies with probabilities), which creates what we call a mixed-strategy equilibrium. That's a full story for a later lesson — for now, know it exists.
A Nash equilibrium is a powerful prediction tool — but it comes with real fine print. Know these before you lean on it for a live decision:
Bring it back to me. Take a real decision where you and another party each have two options and neither of you has an obviously dominant move — a referral protocol, a shared tool adoption, a pricing standoff with a nearby clinic, a co-marketing deal. Build the 2×2. Find the cell(s) where neither of you would want to unilaterally change. Is there one equilibrium or two? Is the equilibrium you're at the good one or the bad one? I'll stress-test your matrix and your read of their objective with you.
Even better — run it on the decision already live in DECISIONS.md: D3 — the keyword-auction prisoner's dilemma with the rival clinic (no dominant move; your best bid depends on theirs). Copy learning-records/REP-TEMPLATE.md to a new REP-D3-*.md and fill Phase 1: the four outcomes, your ranking, their objective, their ranking, and your predicted outcome.
REP-*.md exists with Phase 1 filled. Delivered ≠ learned. One honest rep beats reading the next three lessons.Primary sources: SEP — Game Theory · SEP — Prisoner's Dilemma · Open Yale ECON 159 (Polak) · Dixit & Nalebuff, The Art of Strategy (Norton, 2008).