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A startup weighs entering a market that one big incumbent already owns. The incumbent could slash prices to punish the newcomer, or share the market and keep its margins. The startup has to guess which, before it spends a cent.

Extensive form and backward induction

Until now everyone chose at once. Many games are not like that: one player moves, the other sees it and responds. Drawing the game as a tree, and reasoning from the final move back to the first, tells you what will actually happen, and which threats are just talk.

From earlier lessons

Two quick ones before we start.

In Battle of the Sexes, both choose Opera. If one changes to Football alone, what do they lose?

A best response is a move that is optimal...

Predict, then play

Here is the entry game as a tree. You are the entrant and move first. Walk down a path to read a payoff, then hit “Solve backward” to see what rational play folds it to.

Why did that happen?

Take a guess, then reveal the reason

The incumbent would love you to believe it will fight, because fear keeps you out. But suppose you enter anyway. Now fighting costs the incumbent (0 instead of 1), so it will not actually do it. A rational entrant looks ahead to that moment, sees the threat evaporate, and enters. Working backward from the last decision strips out every move a player would never really make.

The math

An extensive-form game is a tree: nodes are decision points labeled with whose turn it is, edges are the moves, and leaves carry a payoff for each player. A subgame is any node together with everything below it.

Definition, backward induction

Start at the last decision nodes, where the payoffs are known. At each, the mover picks the branch best for them, and you replace the node with that branch’s payoff. Repeat up the tree. The path of choices this produces is the backward-induction solution.

Solve the incumbent’s node first: Accommodate pays $1$ , Fight pays $0$ , so it accommodates, and the node is worth $(2,1)$ . Now the entrant compares entering, worth $2$ , against staying out, worth $0$ , and enters. The solution is $(\text{In},\text{Accommodate})$ with payoff $(2,1)$ .

Notice what backward induction discards. “Stay out because the incumbent will fight” would keep the entrant at $0$ , but it rests on a move the incumbent would never make once tested. That is a non-credible threat, and the next lesson gives it a name.

A note from the real world

Reality check

Backward induction can predict strange things. In the centipede game, two players alternately choose to grab a growing pot or pass; backward induction says grab immediately, on the very first move. Real players pass for several rounds, growing the pot and splitting more than the theory allows. The Behavioral unit returns to why people depart from the backward-induction path, and when it costs them.

Work it out

Fold the tree, then change a leaf

Solve the incumbent’s node: $\text{Accommodate} = 1 > \text{Fight} = 0$ , so replace the node with its value $(2,1)$ . Then the entrant compares $\text{In} = 2$ against $\text{Out} = 0$ and enters. Outcome $(\text{In},\text{Accommodate})$ , payoff $(2,1)$ .

Faded. Suppose fighting were cheap for the incumbent, so In-then-Fight paid $(-1, 2)$ instead of $(-1, 0)$ . Now the incumbent’s node compares Accommodate $1$ against Fight $2$ : it fights. Foreseeing $-1$ from entering, the entrant stays out for $0$ . Changing one leaf flips the whole prediction, because it made the threat credible.

Solo. Take a three-move centipede and fold it from the last node back. You will find the grab-now result, and can check it against what you would actually do.

Key idea

Solve sequential games from the end backward. Doing so keeps only the moves players would truly make, and throws out threats that no one would carry through.

Check yourself

Check yourselfapply

Why does the incumbent accommodate rather than fight, once the entrant has entered?

Check yourselfanalyse

The incumbent announces it will fight any entrant. Should a rational entrant believe it?

Check yourselfbrings back: nash equilibrium

The normal form of this game has a Nash equilibrium where the entrant stays out and the incumbent would fight. Why is backward induction stricter?

Sources

Selten, R. (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit. Zeitschrift fur die gesamte Staatswissenschaft, 121, 301-324.
Rosenthal, R. W. (1981). Games of perfect information, predatory pricing and the chain-store paradox. Journal of Economic Theory, 25(1), 92-100.