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A commander lands an army and burns the ships behind it. Retreat is now impossible, and that is the point: the enemy knows there will be no retreat, and plans around a foe who cannot run. Throwing away an option made the army stronger.

Subgame perfection and commitment

Some Nash equilibria are held up by threats no one would carry out. Subgame perfection throws those away and keeps only the credible ones. The same idea, run forward, explains why moving first and committing, giving up your own flexibility, can leave you better off.

From earlier lessons

Two quick ones before we start.

Backward induction solves a game tree by starting from...

In Cournot with P = 120 - Q and zero cost, each firm produces...

Predict, then play

Back to the two Cournot firms. This time you move first and commit to a quantity the rival then sees before choosing its own. You are giving up the chance to react. Does that help you or hurt you?

Why did that happen?

Take a guess, then reveal the reason

When both firms move together, each hedges against the other. When you commit first, the rival can no longer hope you will back down: your large quantity is a fact, and its best reply is to make less. Because your commitment is visible and irreversible, it changes what the rival does. The same logic that exposes an empty threat, looking ahead to what each player would really do, is what makes a credible commitment valuable.

The math

Run the entry game from the last lesson in normal form instead of as a tree. It has two pure Nash equilibria. Because the incumbent is indifferent between its actions after Stay out, it also has mixed equilibria supported by a sufficiently high probability of Fight. Show the best responses to locate the pure equilibria.

Two gold cells: $(\text{In},\text{Accommodate})$ and $(\text{Out},\text{Fight})$ . Both are pure Nash equilibria. But $(\text{Out},\text{Fight})$ survives only because the incumbent’s threat to fight is never tested when the entrant stays out.

Definition, subgame-perfect equilibrium

A strategy profile is subgame perfect if it is a Nash equilibrium in every subgame, not just in the game as a whole. Equivalently, it prescribes optimal play at every decision node, which is exactly what backward induction finds. Here only $(\text{In},\text{Accommodate})$ qualifies; the fight is not optimal in the subgame that follows entry.

Now commitment, through Stackelberg. The leader picks $q_1$ , foreseeing the follower’s reaction $q_2 = (120 - q_1)/2$ . Substituting into the leader’s profit,

\pi_1 = \left(120 - q_1 - \tfrac{120 - q_1}{2}\right)q_1 = \tfrac{1}{2}(120 - q_1)\,q_1,

which is maximized at $q_1 = 60$ . Then $q_2 = 30$ , $P = 30$ , and the leader earns $1800$ , above the Cournot $1600$ . The ability to commit, not react, is worth $200$ .

A note from the real world

Reality check

Credibility has limits that are not on the blackboard. In the ultimatum game one player offers a split of $10 and the other accepts or rejects; reject and both get nothing. Subgame perfection says offer a single penny and it will be taken, since a penny beats zero. Real responders routinely reject anything they see as unfair, walking away from real money to punish a lopsided offer. The Behavioral unit takes up why, and what it means for the theory of credible commitment.

Work it out

Prune the empty threat, then solve Stackelberg

Credibility. In the table, $(\text{Out},\text{Fight})$ is a Nash equilibrium: given the incumbent would fight, staying out is the entrant’s best reply, and given the entrant stays out, the incumbent is indifferent, so fighting is a best reply too. But look at the subgame after entry: there, fighting pays $0$ and accommodating pays $1$ . Fighting is not optimal in that subgame, so the profile is not subgame perfect. Only $(\text{In},\text{Accommodate})$ is.

Commitment. Substitute the follower’s reaction $q_2 = (120 - q_1)/2$ into the leader’s profit to get $\tfrac12(120 - q_1)q_1$ . The first-order condition $60 - q_1 = 0$ gives $q_1 = 60$ , then $q_2 = 30$ , $P = 30$ , leader profit $1800$ . Moving first and committing beats the $1600$ of simultaneous Cournot, because the follower must adjust to a quantity it cannot undo.

Key idea

Subgame perfection keeps only equilibria that are credible at every point in the game. Its forward-looking twin is commitment: a visible, irreversible move can beat keeping your options open.

Check yourself

Check yourselfapply

The entry game has two pure Nash equilibria in normal form. Which is subgame perfect, and why?

Check yourselfanalyse

Why does committing first to a large quantity raise the Stackelberg leader's profit above Cournot?

Check yourselfbrings back: backward induction

How is finding a subgame-perfect equilibrium related to backward induction?

Sources

Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory, 4(1), 25-55.
Schelling, T. C. (1960). The Strategy of Conflict. Harvard University Press.
von Stackelberg, H. (1934). Marktform und Gleichgewicht. Springer.