Lecture 16: Reputation, the Chain Store Paradox, and Duels

What this lecture covers

This lecture tackles a puzzle left over from the previous class: backward induction says a monopolist facing entrants in many separate markets should never fight, yet real monopolists do fight, and it can work. Using a classroom simulation with ten "markets," Polak shows that introducing even a tiny probability that the incumbent is irrationally aggressive can make fighting rational, a result known as the chain-store paradox, and connects it to hostage negotiation policy and professional reputations for honesty.

The second half shifts to duels and games of timing, using a sponge-throwing contest to model situations like duels in literature, breakaways in cycling, and product launches. You'll come away able to explain why building a reputation can be rational even for a "sane" player, and how to combine dominance arguments with backward induction to solve exactly when a pre-emptive action should occur.

Key ideas

• Chain-store paradox: backward induction predicts a monopolist should never fight entrants across multiple markets, yet fighting early can be rational once uncertainty about the incumbent's type is introduced.
• Reputation under uncertainty: even a small probability that a player is an irrational "type" who enjoys fighting can make it worthwhile for a rational player to imitate that type early on.
• Selten's chain-store model: the formal setting, due to Reinhard Selten, where a firm with monopolies in several sequential markets can seem to justify fighting through reputation.
• Games of timing (duels): strategic problems where the decision is not what to do but when to do it, as in duels, cycling breakaways, or racing to launch a product.
• Dominance plus backward induction: the method used to solve the duel, first ruling out early shots with a dominance argument, then finding the exact critical distance by backward induction from the closest possible range.
• Pro-active bias: Polak's term for a tendency, which he argues is especially strong in American culture, to act too early rather than wait for a better opportunity.

Walkthrough

The chain-store paradox in action (0:00)

Ten students represent separate markets that a monopolist can either let in or fight. Even though backward induction (worked out in the previous lecture) says the monopolist should never fight, the classroom monopolist fights several early entrants and only concedes near the end, deterring many potential entrants along the way.

Formalizing reputation with a "crazy type" (0:00)

Polak resolves the paradox by introducing a small probability, such as 1%, that the incumbent is genuinely aggressive rather than profit-maximizing. Once an incumbent fights early, entrants update their belief that they are facing the aggressive type, deterring later entrants; the argument goes further to show that even a "sane" incumbent may rationally fight early to build this same reputation, though he notes the full equilibrium requires mixed strategies, which he does not work out in detail.

Reputation beyond business (20:56)

The lecture connects reputation-building to hostage negotiation policy, where refusing to negotiate deters future hostage-taking, and to professions like medicine and accounting, where a reputation for honesty is valuable enough to maintain even at a cost, citing Arthur Andersen's collapse after the Enron scandal as an example of reputation loss ending a business.

Setting up the duel (25:18)

Two volunteers each hold one wet sponge and alternate turns, either throwing at their opponent or stepping closer; missing ends the game in the opponent's favor since the thrower has no sponge left. Polak connects this format to literary duels, cycling breakaways from the pack, and firms racing to launch a product before a rival locks in a market standard, framing all of them as games where the key decision is timing rather than action.

Solving the duel: dominance first (34:04)

Defining Pi[d] as the probability that a player hits at distance d, the class derives two facts: if a player believes their opponent will not shoot on the next turn, they should wait, and if they believe the opponent will shoot next, they should shoot now only if their hit probability exceeds the opponent's probability of missing. This produces a threshold distance, called d*, the first point where a player's hit chance plus the opponent's next-turn hit chance exceeds 1.

Solving the duel: backward induction from distance zero (45:42)

Starting backward induction at distance zero, where hitting is certain, the class works outward step by step and confirms that no one should shoot before d*, but at d* and beyond, shooting is correct. The result holds regardless of whether the opponent is equally sophisticated, since not shooting early is a dominant strategy independent of the opponent's skill or rationality.

Why people shoot too early (1:11:24)

Polak notes that in repeated classroom trials, players consistently shoot earlier than the model predicts and misses are common. He attributes this partly to overconfidence and partly to what he calls a cultural bias toward being pro-active, arguing that the goal in these games is not to "go down swinging" but to win, which sometimes means waiting.

Before you watch

• This lecture depends directly on the market-entry backward induction result from Lecture 15, which is revisited and then overturned by adding uncertainty.
• Comfort with basic probability, such as comparing a probability to one minus another probability, is needed to follow the duel's math.

Check your understanding

1. Why does adding a small probability that the incumbent is "crazy" change the outcome of the chain-store game so dramatically?
2. Why might even a rational, non-aggressive incumbent choose to fight early entrants?
3. In the duel, why is it a dominant strategy not to shoot before distance d*, regardless of the opponent's skill?
4. What connects reputation-building in the chain-store game to a policy of not negotiating with hostage-takers?