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A chain opens stores in one town after another, and in each a local rival decides whether to compete. The chain cannot beat them all on the merits. But if it crushes the first few challengers, the rest may not dare try. A reputation for ferocity does the work that force cannot.

Reputation

Backward induction said a threat to fight entry is empty: once entry happens, fighting only hurts you. That logic breaks when rivals are not sure who they are dealing with. A sliver of doubt that you might be the type who always fights, defended by fighting early, can deter challengers for a very long time.

From earlier lessons

Two quick ones before we start.

In the one-shot entry game, the incumbent's threat to fight is...

A finitely repeated Prisoner's Dilemma unravels because of...

Predict, then play

You are a sequence of entrants, one per town, facing the same incumbent. You cannot see its type: with a small chance it is tough and always fights, otherwise it is a normal firm that would rather accommodate. You watch its history and decide each town.

Why did that happen?

Take a guess, then reveal the reason

A normal incumbent would rather accommodate any single entrant. But accommodating once tells every future entrant that it is normal, and they all pour in. Fighting, by contrast, keeps them guessing and keeps them out. As long as there are enough towns left, the future deterrence is worth more than the one-time cost of a fight, so even a normal firm fights, and the small chance it is genuinely tough is enough to make entrants believe the fight will come. The doubt is the weapon.

The math

Let the incumbent be tough with small prior probability $\mu$ (it always fights, perhaps because fighting is how it is wired), and normal otherwise (payoffs as in the entry game: accommodating an entrant beats fighting in a single town). There are $N$ towns.

Solve from the end. In the last town the normal incumbent has no future to protect, so it accommodates; entrants there can safely enter. But earlier, a normal incumbent compares two paths. Accommodate now: gain $1$ this town, but reveal normality and face entry in every remaining town. Fight now: lose the one-period cost, but keep entrants out for many towns to come. When enough towns remain, the deterrence value swamps the cost, so the normal type fights.

Definition, reputation effect (Kreps-Wilson, Milgrom-Roberts)

With a small probability that a long-lived player is a committed type, the normal type mimics that type to build a reputation. In the chain-store game, a tiny doubt about toughness supports deterrence across almost all towns; the reputation only unravels in the final rounds, where mimicry no longer pays. Entrants stay out precisely because they cannot tell a tough incumbent from a normal one defending its name.

This resolves Selten’s chain-store paradox. Under complete information, backward induction says the incumbent cannot deter anyone: fighting is never credible, so entry occurs everywhere. Add an ounce of incomplete information and deterrence returns across the whole chain. The paradox was an artifact of assuming the type was known.

Work it out

Why a normal incumbent fights early

Take the entrant payoffs: accommodate gives the entrant $+2$ , a fight gives $-1$ , staying out gives $0$ . An entrant enters only if the expected payoff beats $0$ , that is, if a fight is unlikely enough.

If the total probability of a fight is $f$ , entry is worth $2(1-f)-f=2-3f$ . The entrant stays out when $f\ge 2/3$ . The game’s initial 20% tough prior and adjustable normal-type fight probability determine this total. The default 85% is an illustration, not a probability derived from the incumbent’s missing payoff parameters.

Last town. The normal incumbent accommodates (no future to protect). Knowing this, and that the incumbent is tough only with small probability, entry pays, so entrants enter.

Earlier towns. If the normal incumbent ever accommodates, it is revealed normal and every later entrant enters, costing it the monopoly profit in all those towns. Fighting instead preserves the doubt and keeps entrants out. With many towns left, the preserved profits exceed the one-period cost of fighting (accommodate $1$ versus fight $0$ ), so the normal type fights. Because both the tough and normal types then fight, an entrant who sees fights cannot rule out toughness, so the expected payoff to entering stays negative and they stay out. Deterrence holds until the horizon comes into view.

Key idea

A small doubt about your type, worth defending, lets you build a reputation that deters rivals. It holds across a long horizon and cracks only near the end, when protecting the reputation no longer pays.

Check yourself

Check yourselfapply

Does building a deterring reputation require the incumbent to actually be the tough type?

Check yourselfanalyse

Under complete information, backward induction says the incumbent cannot deter entry anywhere. What changes with a little incomplete information?

Check yourselfbrings back: backward induction

Why does the reputation eventually collapse in the final towns?

Sources

Kreps, D. M., and Wilson, R. (1982). Reputation and imperfect information. Journal of Economic Theory, 27(2), 253-279.
Milgrom, P., and Roberts, J. (1982). Predation, reputation, and entry deterrence. Journal of Economic Theory, 27(2), 280-312.