Game Theory · Lesson 06 · ← Lesson 05

Look forward, reason backward

When players move in sequence, solve the game from the end.

~14 min · one sitting Skill: sequential games, game trees, backward induction Builds on: best response & Nash equilibrium

First — 20-second recall from Lesson 05

Without scrolling back: a focal point is…?

01 · SIMULTANEOUS VS SEQUENTIALMatrices assume everyone moves at once

Lessons 01–05 built your toolkit for simultaneous games: you and your rival both pick a strategy, neither knowing the other's choice, and we read the outcome from the cell at the intersection. The payoff matrix captures this beautifully.

But many of the decisions that matter most are sequential: one player acts first, the other observes that move and then reacts. Think about:

You announce a new skull-base course; a competing centre decides whether to run a rival event.
You price your SaaS tier; a customer decides whether to subscribe, negotiate, or leave.
You enter a new sub-specialty; an established clinic decides whether to accommodate you or fight.

In each of these, order of play changes everything. Knowing the other side will see your move — and react to it — turns the whole game into a different strategic animal. A matrix can't capture this. We need a game tree.

In a sequential game, you're not just choosing a strategy — you're choosing a signal that shapes what the responder will do.

The distinction between simultaneous (normal/strategic form) and sequential (extensive form) games is foundational — see SEP — Game Theory and Lectures 6–7 of Open Yale ECON 159 (Polak).

02 · THE ENTRY GAMEA worked example: enter the new service area?

You're considering expanding into a sub-specialty — say, functional spine — where an established clinic already operates. They have referral networks, sunk costs, and a reputation. You're deciding whether to enter or stay out.

Here's what we know about the payoffs:

If you stay out: the incumbent keeps the market to themselves. You earn 0 (no change), they earn 10.
If you enter and the incumbent accommodates (shares the market): you earn 4, they earn 5.
If you enter and the incumbent fights (price war, aggressive referral poaching): you earn −2, they earn 2.

Notice: this cannot be a simple matrix, because the incumbent only chooses after they see whether you've entered. Their move is contingent on yours. The right representation is a game tree:

YOU (entrant) ├── Stay Out ──────────────── You: 0 · Incumbent: 10 │ └── Enter │ └── INCUMBENT ├── Fight ──────── You: −2 · Incumbent: 2 └── Accommodate ── You: 4 · Incumbent: 5

Read it left to right: you move first at the root. One branch ends the game immediately (Stay Out). The other branch passes the move to the incumbent, who then picks Fight or Accommodate. Only then do we reach a leaf with final payoffs.

The game tree makes the order of play explicit in a way no matrix can.

Before we solve it formally — write your own guess: if you Enter, what will the incumbent rationally do, Fight or Accommodate, and why?

Model read: look at the incumbent's own payoffs at that node, not their reputation for toughness — Fight pays them 2, Accommodate pays them 5. A rational incumbent takes the 5. Section 03 turns this same instinct into a repeatable method: backward induction.

03 · BACKWARD INDUCTIONSolve from the end, fold back to the start

Backward induction is the central technique for sequential games. The logic: a rational player at a later node will choose whatever is best for them at that moment. So you — moving earlier — can anticipate their choice and plan accordingly.

Work through the entry game step by step:

Start at the final decision — the incumbent's node. The incumbent is choosing between Fight (pays them 2) and Accommodate (pays them 5). Rational choice: Accommodate. 5 > 2.
Prune the Fight branch — it won't happen. We cross out the Fight branch. A rational incumbent will not choose it. The only thing that will happen if you Enter is that the incumbent Accommodates.
Fold back to your node (the root). Now your two real options are: Stay Out → 0, or Enter → (incumbent Accommodates) → 4. You compare 4 vs 0. You choose Enter.
The prediction: Enter → Accommodate = (You 4, Incumbent 5). This is the subgame perfect equilibrium: the path rational players will actually follow.

The backward induction logic

You look forward to what the other player will rationally do at the end, then reason backward to what you should do now. The incumbent's payoff at their node — not their threat — determines their action.

Same reflex you already have Planning a staged operation, you don't sequence forward from today — you reason backward from the desired final state: fix where the patient must end up, then work out the earliest step that gets you there. Backward induction is that exact habit, in game form — start at the last node, find what's optimal there, then fold back to what you should do first.

Visually, the solved tree looks like this — the predicted path is highlighted in gold:

YOU (entrant) ├── Stay Out ──────────────── You: 0 · Incumbent: 10 │ └── ★ Enter ★ │ └── INCUMBENT ├── ~~Fight~~ ───── You: −2 · Incumbent: 2 ← pruned └── ★ Accommodate ★ ── You: 4 · Incumbent: 5 ← predicted path

Backward induction is introduced formally in Open Yale ECON 159, Lectures 6–7. The solution concept it produces (subgame perfect equilibrium) is discussed in SEP — Game Theory. See also Dixit & Nalebuff, The Art of Strategy (Norton, 2008), Ch. 2–3.

04 · CREDIBILITYDoes the threat actually stop you?

Now the critical strategic payoff. Before you decided to enter, suppose the incumbent publishes an editorial in a local medical journal: "Any new entrant into functional spine in this city will face an immediate, sustained price war."

Should that change your decision?

Quiz — does the incumbent's threat make you stay out?

This is the concept of a credible threat. Backward induction cuts through rhetoric instantly: go to the incumbent's node, look at their payoffs. Fight = 2, Accommodate = 5. If they're rational, they will not fight — regardless of what they said beforehand. The threat is non-credible.

A non-credible threat is just a bluff. Backward induction is your bluff-detector.

This is one of the most valuable things you can learn as a strategist. When a competitor, hospital, insurer, or investor makes a threat or a promise, the right question is not "do they sound serious?" but "is carrying this out in their interest when the moment arrives?" Only if the answer is yes should you let it change your behaviour.

05 · FIRST-MOVER ADVANTAGEMoving first can shape the responder's best reply

Sequential games also reveal when it pays to move first. By committing to a visible action early, you can narrow the follower's rational options — sometimes to only the one you want.

In the entry game, you move first and it works in your favour: entering puts the incumbent into a position where their best response (accommodating) benefits you more than staying out. You seized the move.

But first-mover advantage is not universal. Sometimes moving first exposes your hand. The question — which we explore more in Lesson 07 — is whether committing early binds the follower to a better response or a worse one.

Launching publicly before a competitor — announces your position, forces them to react. Good if their best reaction helps you; dangerous if it triggers an attack you'd rather avoid.
Making the first offer in a negotiation — anchors the conversation, but removes flexibility.
Publicly announcing a new service or course — locks you in and signals to referrers, but also signals to rivals who now know exactly what to compete with.

The key insight: commitment changes the game. Lesson 07 shows how to make your own threats credible by commitment — burning the boats so the follower knows you will carry through.

06 · KNOW THE LIMITSWhen not to draw a tree

A game tree is a scalpel, not a hammer. Reach for a different tool — or no tool — when:

Contraindications — don't force the tree here

Moves are genuinely simultaneous. If neither side sees the other's move first — you and a rival both quietly set this month's price at the same time, with no observation in between — there's no order of play to draw. Use a matrix (Lessons 01–05), not a tree.
The tree runs deep. Backward induction assumes every player has perfect foresight and reasons flawlessly all the way to the last leaf. Very deep trees strain that assumption hard — real players satisfice and miscount, they don't compute every branch. The further out you predict, the more that assumption is quietly doing the work.
The prescribed move isn't believable. Backward induction can output an off-the-path action that nobody would actually take if that branch were ever reached. That's not a flaw in the method — it's the credibility problem, and it's the whole subject of Lesson 07.

Naming when backward induction doesn't apply is what keeps it from becoming a party trick. Diagnose the shape — simultaneous or sequential, shallow or deep, on-path or off — before you solve it.

07 · LOCK IT INThree steps, every sequential game

Draw the tree. Map every decision in order — who moves when, what options they have at each node.
Solve from the leaves. At every final decision node, determine what the player there will rationally do. Mark it. Prune the unchosen branches.
Fold back and drop non-credible threats. Work back toward the root. Any threat or promise that would require a player to act against their own interest at the moment of decision is non-credible — ignore it in your planning.

Bring it back to me. Pick one real sequential decision from your work right now:

You act, then someone reacts — a competitor, partner, hospital, patient. Draw the two-step tree (your branches, their branches, the leaf payoffs). Guess their rational reaction at the end — not what they've said they'll do, but what's actually in their interest. Fold back: which of your options wins? Which of their past warnings are non-credible?

Candidates: skull-base course launch vs rival centre; SaaS pricing tier announcement vs early adopter; a negotiation with a hospital for OR time. Share the tree and I'll pressure-test the payoffs with you.

Prefer a decision that's already live over a hypothetical? Open DECISIONS.md and use D2 — the particular pricing floor (hold cranial ≥ R$30k; spine floor pending Sami). It's genuinely sequential: you set the floor and hold it, prospective patients & referrers then see that price and decide whether to book, negotiate, or shop elsewhere — a real you-move-they-respond tree, not a matrix.

Copy learning-records/REP-TEMPLATE.md and fill Phase 1, reframed for a tree: ① draw your two branches (hold the floor / discount it) and their branches off each ② name the leaf payoffs, for you and for them ③ say what patients/referrers are actually optimising (price as a quality signal? convenience? pure price-shopping?) ④ solve from the last node backward — what do they rationally do off each of your branches? ⑤ your predicted path ⑥ the one payoff you're least sure of ⑦ the contraindication check — is this truly sequential, or could it collapse back into simultaneous?

The gate: this lesson isn't "done" when you finish reading — it's done when one REP-*.md exists with Phase 1 filled. Delivered ≠ learned. One honest rep beats reading the next three lessons.

Primary sources: Open Yale ECON 159 (Polak), Lectures 6–7 · SEP — Game Theory. Named reference: Dixit & Nalebuff, The Art of Strategy (Norton, 2008).