Reference · Cheat Sheet 06
Backward Induction
Core idea: look forward, reason backward — solve sequential games by starting at the last decision nodes, finding what each player will rationally do there, then folding those choices back to the beginning. The surviving path is the prediction.
The routine (4 steps)
- Draw the tree in order of play. Root = first mover. Branches = their choices. Sub-branches = the next mover's choices. Leaves = final payoffs (you, them) for every terminal path.
- At each final decision node, pick the chooser's best payoff. The player at that node acts in their own interest when the moment arrives — prior threats, promises, or bluster do not change this.
- Prune the unchosen branch; fold the chosen payoff back. The parent node now "knows" what the sub-game delivers and can compare its own options.
- Repeat toward the root. The path that survives all pruning is the subgame perfect equilibrium — what rational players will actually do.
Entry game — worked example
YOU (entrant)
├── Stay Out ──────── You: 0 · Incumbent: 10
│
└── Enter
│
└── INCUMBENT
├── ~~Fight~~ ──── You: −2 · Incumbent: 2 ← pruned (2 < 5)
└── ★ Accommodate ★ You: 4 · Incumbent: 5 ← chosen
Backward induction:
- Step 1 — incumbent's node: Fight = 2 vs Accommodate = 5. Rational choice: Accommodate. Prune Fight.
- Step 2 — your node: Stay Out → 0 vs Enter → (Accommodate) → 4. Rational choice: Enter.
- Predicted path: Enter → Accommodate = (You 4, Incumbent 5).
Credibility — the most important corollary
A threat or promise is credible only if carrying it out is in the actor's best interest when that node is actually reached.
In the entry game, the incumbent may announce: "enter and we'll fight." Backward induction reveals this is non-credible — Fight pays 2, Accommodate pays 5. A rational incumbent will accommodate regardless of the announcement. You should enter.
Test every threat and promise: go to the actor's node, compare their payoffs. If following through is worse for them than the alternative, the threat is empty. Ignore it.
Quick reference: simultaneous vs sequential
When to use which tool
| Simultaneous | Sequential |
|---|
| Players choose | At the same time | In a defined order |
| Tool | Payoff matrix | Game tree |
| Solution | Nash equilibrium | Backward induction → SPE |
| Key question | Best response to every mix | What will they do when it's their turn? |
Glossary (quick)
- Sequential game — players move in order; later movers observe earlier moves.
- Game tree (extensive form) — diagram of nodes (decisions) and branches (options); leaves show final payoffs.
- Backward induction — solve from leaves to root; prune suboptimal branches at each node.
- Subgame perfect equilibrium (SPE) — the path where every player acts optimally at every node they reach.
- Credible threat — a threat the actor would actually follow through on because it's in their interest at the moment of decision.
- Non-credible threat — a threat the actor would not rationally carry out; backward induction prunes it automatically.
- First-mover advantage — benefit of committing early when your action shapes the follower's best reply in your favour.