When you have no dominant move, predict by stripping out their bad options first.
Before you write the OTHER player's payoffs, what must you do first?
Lessons 1 and 2 gave you a clean path: rank your outcomes, find the strategy that beats all others, done. But what if no such strategy exists? Your best move in column A is different from your best move in column B — it depends on what they do.
This happens constantly. Think about setting your SaaS tiers when a competitor also has multiple offerings. Or choosing which surgeon referral network to court when rival clinics are courting the same ones. Your best action depends on theirs, and theirs on yours — a genuine interdependence.
This is the core insight: even without a dominant strategy of your own, you can start with their side of the matrix. Eliminate what a rational player would never do — then re-examine your options in the shrunken game. Repeat. This is iterated elimination of dominated strategies (IEDS).
One caveat worth naming now: the whole procedure rests on common knowledge of rationality. You assume they won't play dominated strategies; they assume you won't; both of you know the other assumes that; and so on. When that assumption is shaky — say, a rival who is new or erratic — lean on IEDS lightly.
The formal treatment of IEDS and common knowledge lives in SEP — Game Theory. The intuition is developed in depth in The Art of Strategy — Dixit & Nalebuff (Norton, 2008), ch. 3.
Here is a 2×3 matrix. You choose a row (Up or Down); your opponent chooses a column (Left, Middle, Right). Each cell shows (your payoff, their payoff).
| Left | Middle | Right | |
|---|---|---|---|
| Up | 1, 0 | 1, 2 | 0, 1 |
| Down | 0, 3 | 0, 1 | 2, 0 |
Your payoffs are shown in gold. Work through the steps below — each one reveals a new elimination.
Look only at the second number (their payoff) in each column pair. Compare Middle vs Right: if you play Up, they get 2 (Middle) vs 1 (Right). If you play Down, they get 1 (Middle) vs 0 (Right). Middle beats Right for the opponent in every row. Right is strictly dominated — a rational opponent will never play it.
Right is gone. The game is now 2×2: {Up, Down} vs {Left, Middle}. Now look at your payoffs (gold). Up vs Left: 1 > 0. Up vs Middle: 1 > 0. Up strictly dominates Down in every remaining column. Down is eliminated.
You play Up. The opponent compares Left (0) vs Middle (2). Middle is better. The rational opponent plays Middle.
You earn 1; they earn 2. No dominant strategy existed for you at the start — but by deleting their bad options first, your decision clarified, and theirs followed.
You started with their side even though it's your strategy you ultimately care about. IEDS often makes your own move obvious only after you've cleared away the things a rational opponent would never do.
Open Yale ECON 159 (Polak), lectures 2–3, covers IEDS and common knowledge of rationality with additional worked examples.
Back to the same matrix. No peeking at the steps above yet.
IEDS assumes everyone eliminates dominated options simultaneously and infinitely. A vivid illustration: the "2/3 of the average" game.
Every player in a large group picks a number from 0 to 100. The winner is whoever is closest to 2/3 of the group's average. What should you pick?
The reasoning — if everyone is fully rational and knows everyone else is — goes like this: no rational player should pick above 67 (2/3 of 100), because even if everyone else picked 100, the target would be 67. Delete all numbers above 67. Now no rational player should pick above 44 (2/3 of 67). Delete above 44. And so on. The logic iterates until it collapses to a single point.
Real experimental data — from central bank economists, finance students, and management consultants — routinely average around 20–35. Why? Because people typically think only 1–2 iterations ahead. "Everyone will pick around 50, so 2/3 of that is 33, so I'll pick 33." That's one step. This is called level-k thinking.
When setting your SaaS Pro tier or your ad budget against a rival clinic, you aren't just choosing a price — you're choosing how many levels deep to reason about how they reason about you. A competitor who prices by gut instinct is level-0. You can win consistently by being level-1. But if they're level-1 already, you need to be level-2. Depth of reasoning is itself a strategic variable.
The Art of Strategy — Dixit & Nalebuff (Norton, 2008) covers the beauty-contest game and level-k thinking in ch. 7–8. See also SEP — Game Theory for formal treatment of common knowledge and iterated elimination.
IEDS is a scalpel, not a universal solvent. It cut cleanly in the worked example above because the deletions kept cascading until one cell survived. Real situations don't always cooperate.
Naming where the method runs out is what keeps it a tool instead of a reflex. Half of good strategy is diagnosing which lesson actually applies before you start eliminating.
Bring it back to me. Find a real situation where you don't have an obvious dominant move — your best tier price when the competitor also has several options, your ad budget when rival clinics can match you, which referral network to prioritise when two hospitals both court you.
① List their options as columns. ② Cross out any they would never rationally choose (and say why — what are they optimising?). ③ Does your decision simplify once those are removed? Tell me what you eliminated and why. That's the rep that makes this skill sticky.
Make it a real rep, not a hypothetical. Open DECISIONS.md and use row D3 — the rival clinic bidding on your shared keywords: a real competitor with several possible responses, some of them clearly irrational for a volume-paid clinic to make.
Copy learning-records/REP-TEMPLATE.md and fill Phase 1: list their column of options, cross out what a rational version of them would never play, and say why — then bring your elimination to me.
REP-*.md exists with Phase 1 filled. Delivered ≠ learned.Primary sources: SEP — Game Theory (IEDS, common knowledge) · Open Yale ECON 159 (Polak). Concept of level-k thinking from The Art of Strategy — Dixit & Nalebuff (Norton, 2008).