The entry game had two Nash equilibria on the grid and only one survived the tree, and the difference between them is the difference between a threat that would be carried out and a threat that would not.
The equilibrium the tree rejects
Recall the game. An entrant chooses In or Out; if it enters, the incumbent chooses Fight or Accommodate. Staying out pays the entrant 0 and the incumbent 10. Fighting pays -2 and 2. Accommodating pays 2 and 5.
| Entrant | Incumbent fights | Incumbent accommodates |
|---|---|---|
| In | -2, 2 | 2, 5 |
| Out | 0, 10 | 0, 10 |
Both (In, Accommodate) and (Out, Fight) are Nash equilibria. Check the second one against the definition: given that the incumbent's plan is to fight, the entrant's best reply is to stay out, since 0 beats -2. Given that the entrant stays out, the incumbent's payoff is 10 whatever it plans to do, so planning to fight is a best response too. Nothing in the definition of Nash equilibrium is violated.
And yet the plan is nonsense. If the entrant did come in, the incumbent would compare 2 with 5 and accommodate. The threat works only for as long as it is never tested, and the reason Nash equilibrium tolerates it is that the definition only asks whether a player could gain by deviating given the others' strategies, and the entrant's staying out means the incumbent's promise is never priced.
Subgame perfection
Reinhard Selten's repair, introduced in 1965, is to require the strategies to be an equilibrium not just in the game as a whole but in every part of it that could be reached.
A subgame is any node of the tree together with everything that follows it, provided no information set is cut in half by taking it. The whole game is a subgame of itself, and in a game of perfect information every node starts one. A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in every subgame.
The entry game has two subgames: the whole thing, and the small one that begins after entry in which the incumbent alone chooses. In that small subgame, the only equilibrium is Accommodate, because the incumbent is the only player and 5 beats 2. So (Out, Fight) is not subgame perfect: it fails in the subgame after entry. Only (In, Accommodate) survives, and it is exactly what backward induction gave.
For finite games of perfect information the two ideas coincide: backward induction is subgame perfection carried out mechanically. The general definition earns its keep when information sets are not all singletons, because then the tree contains simultaneous-move subgames that must be solved with the earlier machinery and cannot be settled by comparing two numbers.
The concept refines rather than replaces. Every subgame perfect equilibrium is a Nash equilibrium, and some Nash equilibria are not subgame perfect. What is discarded is precisely the set of profiles held up by threats and promises that would not be honoured.
Example. A supplier and a buyer. The buyer chooses whether to invest £4 in equipment that only this supplier's parts fit. If it does not invest, both earn 0. If it does, the supplier chooses whether to charge a fair price, paying the buyer 6 and the supplier 4, or to exploit the lock-in, paying the buyer -2 and the supplier 8. Find the Nash equilibria and the subgame perfect one.
In the subgame after investment the supplier compares 4 with 8 and exploits. So the buyer, anticipating -2, does not invest, and the unique subgame perfect equilibrium is "do not invest, exploit if invested", paying nothing to either party. The grid also holds a Nash equilibrium in which the buyer does not invest and the supplier plans to charge fairly, which survives only because the plan is never tested, and which subgame perfection discards. What matters is the £10 of joint surplus that vanishes because a promise cannot be made binding, and that is the standard argument for why contracts exist.
Now you. Change the supplier's exploitation payoff from 8 to 3, leaving everything else. What is the subgame perfect outcome now?
Answer
In the subgame after investment the supplier now compares 4 from fair dealing with 3 from exploiting, and deals fairly. The buyer, anticipating 6, invests. The subgame perfect outcome is investment and fair dealing, paying 6 and 4. Nothing about the buyer's problem changed: the investment became safe because the other party's temptation shrank, which is what a reputation, a repeat relationship or a penalty clause is for.
Buying credibility by destroying options
If a threat is not credible because carrying it out would be costly, one repair is to make carrying it out cheap, or to make not carrying it out impossible. The striking thing is that this is worth paying for.
Give the incumbent a prior move. Before the entrant decides, the incumbent may install excess capacity at a cost of 4. The capacity is useless in normal trading but makes a price war cheap to sustain, adding 5 to the payoff from fighting. With the investment made, the payoffs become: staying out pays the incumbent ; fighting pays ; accommodating pays .
Now solve the enlarged game backwards. If the investment has been made, the incumbent facing entry compares 3 with 1 and fights, so the entrant facing an equipped incumbent expects -2 and stays out, leaving the incumbent 6. If the investment has not been made, the earlier analysis applies: the entrant enters, the incumbent accommodates, and the incumbent gets 5. Since 6 beats 5, the incumbent invests, and the market is never entered.
Two features of that calculation are worth isolating.
The investment is a sunk cost by the time entry is decided, and it therefore does not affect the fight-or-accommodate comparison at all: the 4 is subtracted from both branches. What makes the threat credible is the change in the difference between the branches, not the money spent. A commitment that costs a great deal and does not alter any comparison at a future node buys nothing.
And the incumbent is made better off by an action that lowers its payoff in every cell of the table. Before the investment its best outcome was 10 and after it is 6, and it is glad to have made it. That is the central paradox of commitment, and Thomas Schelling's formulation is still the best: the power to constrain an adversary may depend on the power to bind oneself.
Example. What is the most the incumbent would pay for the capacity, and does a higher price make the threat less credible?
Credibility does not depend on the price at all. Whatever costs, fighting pays and accommodating pays after entry, so fighting always wins once the capacity exists. What the price bounds is whether deterrence is worth buying: investing yields and not investing yields 5, so the incumbent invests when . At it gains 1. At it would deter entry and earn 4, which is worse than the 5 it gets by accommodating an entrant, so it does not invest.
Now you. Suppose instead the capacity costs 4 but adds only 2 to the payoff from fighting rather than 5. Does the incumbent invest?
Answer
After entry, fighting would pay and accommodating , so the incumbent would still accommodate. The threat is not made credible, the entrant enters anyway, and the incumbent ends up with 1 instead of 5. The investment is worse than useless: a commitment device that does not reverse the future comparison is money burned. The condition to check is always whether the ranking at the later node has flipped, never how much was spent.
The forms commitment takes
Once the mechanism is clear, the same move can be recognised in very different clothes.
Removing the retreat. Xiang Yu, crossing the river before the battle of Julu in 207 BC, sank the boats and broke the cooking pots. Cortés, at Veracruz in 1519, had his ships scuttled and run aground, a story that later became "burned his boats". Both destroyed their own option to withdraw, which is only rational if the enemy's decision depends on whether withdrawal is possible.
Contracts and penalties. A supply contract with a large penalty for buying elsewhere, a most-favoured-customer clause that forces a firm to refund past buyers if it ever cuts prices, a mortgage that makes walking away expensive. Each turns a preference into an obligation that the other side can verify.
Delegation. Sending an agent whose interests differ from yours: a union negotiator with a mandate, a lawyer paid a contingency fee, a central bank with an inflation target and no instruction to care about employment. A negotiator who genuinely cannot accept less than the mandate is more credible than a principal who could.
Reputation. Doing something costly now so that a later opponent expects the same, which is the subject of the next two lessons.
Brinkmanship. Schelling's insight that a threat too terrible to carry out can be replaced by an action that raises the probability of the terrible outcome without anyone choosing it: massing troops on a border, letting a crisis run, keeping missiles on alert. A commitment that is certain may be incredible while the same commitment made probabilistic is not.
All of these need two properties, and applied analysis should check both. The commitment must be observable, since a commitment the other side cannot verify changes nothing about their reasoning. And it must be irreversible, since one that can be quietly undone is cheap talk with extra steps. Announcements of a pricing policy in a trade journal are commitment because they are both; a private resolution to fight the next entrant is neither.
The chain store paradox
Selten produced the sharpest objection to his own concept in 1978. A chain store operates in twenty towns and faces a potential entrant in each, one after another. The per-town payoffs are those of the entry game.
Solve it backwards. In town twenty there is no future to protect, so the chain accommodates, and the twentieth entrant enters. Knowing that, fighting in town nineteen buys nothing, since the twentieth entrant's decision does not depend on it, so the chain accommodates there too. The argument runs all the way back: the unique subgame perfect equilibrium has entry in all twenty towns and accommodation every time, and the chain earns rather than the it would earn by deterring everyone.
Selten's own view was that he would not play this way, and neither would anyone else. Real chains do fight early entrants, and the deterrence appears to work. The paradox is not that the mathematics is wrong; it is that the conclusion depends on twenty layers of confident reasoning about a player who has just been observed doing something the reasoning said was impossible.
Two repairs exist and both matter later. Repetition without a known end changes the arithmetic completely, because there is no last town from which to unravel, and that is the next lesson. Alternatively, keep the finite horizon and add a small doubt: if the entrants think there is even a one per cent chance the chain is the kind that simply enjoys fighting, the chain will fight in the early towns to keep that doubt alive, and deterrence returns. That argument, published in 1982 by David Kreps, Paul Milgrom, John Roberts and Robert Wilson, is one of the most cited results in the subject, and it needs the machinery of private information that arrives near the end of this course.
Example. A chain faces entrants in two towns in sequence, with the entry game payoffs in each. What is the subgame perfect outcome and what does the chain earn in total?
In town two the chain accommodates, so the entrant enters and the chain earns 5 there. In town one, fighting costs the chain 3 (it earns 2 rather than 5) and cannot change what happens in town two, because the town-two subgame has a unique equilibrium regardless of history. So the chain accommodates in town one as well. Both towns are entered, both are accommodated, and the chain earns 10 in total against the 20 it would earn if both entrants stayed out.
Now you. Suppose the entrant in town two is unusual: entering costs it so much that staying out is a dominant strategy there. What does the chain do in town one?
Answer
Town two is settled whatever happens, so it still contributes nothing to the town-one decision, and the chain accommodates in town one exactly as before. The chain earns . The point is that deterrence only pays when the later player's decision actually depends on what is observed, and a later player who is going to stay out anyway is worth nothing to impress.
Where this leaves us
Subgame perfection makes threats accountable, and the price it charges is visible in the chain store: a finite sequence of identical games unravels from the last one, so nothing can ever be sustained by the fear of what comes after.
Real relationships do not have a known last round. A cartel, a marriage, a trade relationship and a border dispute all continue with some probability into a future the participants cannot see the end of. The next lesson puts a number on that future, and finds that the whole conclusion of the prisoner's dilemma reverses once the number is large enough.