Entry Deterrence and Credible Commitment

Why Entry Deterrence Depends on Credibility

An established firm may say that it will cut prices aggressively if a new rival enters. That announcement deters entry only if the entrant expects the incumbent to carry it out. If accommodating entry would give the incumbent a better payoff once entry has occurred, the threat is not credible: a rational entrant anticipates accommodation and may enter.

The key distinction is between announcing a response and changing the incentives that determine the response. Sequential reasoning reveals whether a threatened action is optimal at the moment it must be chosen. An observable commitment, such as a costly investment that changes the firms’ payoffs, can sometimes make a formerly unattractive response optimal and thereby change the entrant’s decision.

Definition: In an entry-deterrence game, an entrant chooses whether to enter a market, and an incumbent may choose a response if entry occurs. A response is credible if it is optimal for the incumbent at the decision point where it would be taken. A commitment is an action taken before the entrant decides that is observable and changes the payoffs or available actions later in the game.

Analyze the Game from the Last Decision Backward

A standard entry game is sequential: the entrant decides whether to enter; if entry occurs, the incumbent chooses a response. Payoffs are often written as ordered pairs, with the entrant’s payoff first and the incumbent’s second. The entrant’s outside option from staying out is commonly normalized to zero, though the relevant comparison is always with the stated payoff from staying out.

The appropriate solution concept is subgame-perfect equilibrium. The method is to find the incumbent’s best response in the subgame after entry, then use that response to determine whether entry is worthwhile. This is backward induction, the sequential method used in the earlier material on sequential games and credible threats.

Solve the incumbent’s response.
Compare the incumbent’s payoff from each available action after entry. Keep an action in the predicted continuation only if it is optimal there.

Evaluate entry using that response.
The entrant compares the payoff from entering, given the incumbent’s optimal response, with the payoff from staying out.

Check the incumbent’s earlier choice.
If the incumbent can commit before the entry decision, compare its net payoff under commitment with its payoff without commitment.

Decision rule: Let \(\pi_I^F\) and \(\pi_I^A\) be the incumbent’s payoffs after entry from fighting and accommodating, respectively. The incumbent fights after entry only if \(\pi_I^F \geq \pi_I^A\). The entrant is deterred when its payoff from entering, given the incumbent’s optimal response, is less than its payoff from staying out. If the entrant is indifferent, whether it stays out depends on the specified tie-breaking assumption; strict deterrence requires a strict preference for staying out.

Why backward induction works: At the final decision node, the incumbent can compare the payoffs from its available actions and choose one that maximizes its payoff. The entrant, moving earlier, can anticipate that choice. Therefore, an entry decision based on a response the incumbent would not actually choose cannot be part of a sequentially rational prediction. Solving the last decision first and then moving backward enforces optimality at each decision point.

When a Threat Does Not Deter Entry

Worked Example: Testing a Price-Cutting Threat

A new supplier considers entering a regional market. If it stays out, the payoffs are (0, 10). If it enters and the incumbent accommodates, the payoffs are (2, 8). If the incumbent fights by cutting prices, the payoffs are (-2, 1). Determine the subgame-perfect outcome and assess the incumbent’s announced threat to fight.

Method. Solve the incumbent’s response after entry first, then use that response to evaluate the entrant’s decision. This avoids treating an announcement as if it were automatically credible.

Step 1: Find the incumbent’s best response. After entry, accommodation gives the incumbent 8, while fighting gives it 1. Since \(8 > 1\), the incumbent accommodates.

Step 2: Find the entrant’s best response. Anticipating accommodation, the entrant receives 2 from entering and 0 from staying out. Since \(2 > 0\), it enters.

Conclusion and check. The equilibrium outcome is entry followed by accommodation, with payoffs (2, 8). The announced fight threat is not credible: at the response node, the incumbent would gain 8 by accommodating rather than 1 by fighting. An entrant who anticipates the optimal response therefore enters.

The incumbent’s threat would appear to deter entry if the entrant took it at face value: fighting would leave the entrant with -2, which is worse than staying out for 0. But that calculation assumes the incumbent chooses to fight. Backward induction shows it does not. A threat can affect the entrant’s incentives only when the threatened action is optimal in the continuation game, or when an earlier commitment changes the relevant incentives.

How Commitment Can Change the Entry Decision

A commitment may involve a real cost, but cost alone does not make it effective. Its strategic role is to alter the choices or payoffs that follow. For example, an investment could change the incumbent’s position so that fighting after entry becomes more profitable than accommodating. The investment must be observable before the entrant decides; otherwise, the entrant cannot use it to predict the incumbent’s continuation action.

To evaluate a commitment, first solve the post-commitment game by backward induction. Then subtract the commitment cost from the incumbent’s payoff in every outcome on that branch. Because the cost is already incurred by the time the incumbent chooses its response, it does not affect the comparison between fighting and accommodating; it does affect whether the incumbent should commit in the first place.

Worked Example: An Investment That Makes Fighting Optimal

Suppose the game from the previous example is available without investment. Before the entrant decides, the incumbent can instead make an observable capacity investment costing 2. Without investment, staying out gives (0, 10); entry followed by accommodation gives (2, 6), and entry followed by fighting gives (-3, 1). After investment, the gross payoffs if entry occurs are (2, 5) under accommodation and (-2, 7) under fighting. If the entrant stays out after investment, the incumbent’s gross payoff is 10. Find the equilibrium and account for the investment cost.

Method. Solve the continuation game under each first-stage choice, then compare the incumbent’s net payoffs. The stated post-investment payoffs are gross of the cost, so subtract 2 from the incumbent’s payoff in each investment outcome.

Step 1: Solve the game without investment. If entry occurs, the incumbent prefers accommodation because \(6 > 1\). The entrant then prefers to enter because \(2 > 0\). The incumbent’s payoff in this outcome is 6.

Step 2: Solve the continuation after investment. If entry occurs, the incumbent prefers fighting because \(7 > 5\). The entrant anticipates a payoff of -2 from entering and 0 from staying out, so it stays out.

Step 3: Compare the incumbent’s first-stage payoffs. Investment deters entry, leaving the incumbent with a gross payoff of 10. Subtracting the sunk cost gives \(10 - 2 = 8\). Without investment, the equilibrium payoff is 6. Since \(8 > 6\), the incumbent invests.

Conclusion and check. The equilibrium is investment followed by the entrant staying out; the incumbent’s net payoff is 8. If entry nevertheless occurred after investment, fighting would be the incumbent’s optimal response because its gross payoff of 7 exceeds 5. The investment changes the entrant’s expectation because it changes the incumbent’s incentives, not simply because it is costly.

Conditions for Successful Deterrence

The logic of commitment can be expressed as a sequence of conditions. First, the commitment must lead to a continuation response that is optimal for the incumbent. Second, the entrant must prefer staying out to entering in light of that response. Third, the incumbent must prefer the net payoff from committing and deterring entry to the payoff it receives without commitment. A commitment that meets only the first two conditions may deter entry but still be too costly for the incumbent to choose.

\begin{aligned} \text{Credible response:} \quad & \pi_I^F \geq \pi_I^A, \\ \text{Entry deterred:} \quad & \pi_E^{\text{enter after response}} < \pi_E^{\text{stay out}}, \\ \text{Commitment worthwhile:} \quad & \pi_I^{\text{no entry}} - K > \pi_I^{\text{without commitment}}, \end{aligned}

Here \(K\) is the commitment cost, and the incumbent’s stated payoff from no entry is measured before subtracting that cost. If a commitment changes the entrant’s payoff directly as well as the incumbent’s incentives, the entrant’s actual payoff after the optimal response must still be used in the second condition. These comparisons are specific to the payoff structure: an investment that deters entry in one market need not do so in another.

Worked Example: Finding the Largest Profitable Commitment Cost

Without commitment, a potential entrant receives 4 from entry if the incumbent accommodates and -1 if it fights; staying out gives 0. The incumbent receives 11 from accommodation, 3 from fighting, and 11 in the resulting no-commitment equilibrium. The incumbent can make an observable commitment costing \(K\). After commitment, accommodation would give payoffs (2, 10), fighting would give (-3, 12), and staying out would give the incumbent a gross payoff of 14. For what values of \(K\) does the incumbent choose commitment and deter entry?

Method. Establish the no-commitment equilibrium, establish the post-commitment continuation, and then solve the incumbent’s cost inequality. This separates the entrant’s deterrence condition from the incumbent’s willingness to pay for deterrence.

Step 1: Find the no-commitment outcome. After entry, the incumbent accommodates because \(11 > 3\). The entrant then enters because \(4 > 0\). The incumbent’s equilibrium payoff without commitment is 11.

Step 2: Test the post-commitment response and entry decision. After commitment, the incumbent fights if entry occurs because \(12 > 10\). The entrant compares -3 from entry with 0 from staying out and therefore stays out. Commitment successfully deters entry.

Step 3: Find the cost condition. With entry deterred, commitment gives the incumbent a net payoff of \(14-K\). It chooses commitment over the no-commitment outcome when

14-K > 11 \quad\Longleftrightarrow\quad K < 3.

Conclusion and check. For \(K < 3\), the incumbent commits and the entrant stays out. At \(K = 3\), the incumbent is indifferent between commitment and no commitment; above 3, it does not commit. The post-commitment response remains credible because 12 exceeds 10, independently of the sunk cost.

Common Exam Traps and Economic Significance

Taking a threat at face value. A statement that the incumbent will fight does not establish that fighting is optimal. Compare the incumbent’s payoffs at the post-entry decision node.
Letting the entrant ignore the response. The entrant compares staying out with entry followed by the incumbent’s optimal response, not with an arbitrary action the incumbent has announced.
Subtracting a sunk cost from only one continuation action. Once incurred, the commitment cost is common to the incumbent’s post-investment response alternatives. It affects the earlier investment decision, not which response maximizes the incumbent’s continuation payoff.
Assuming every commitment deters entry. Verify that the entrant’s payoff from entering after the optimal incumbent response is below its outside option.
Ignoring the cost-benefit test. Even if commitment deters entry, the incumbent chooses it only if its net payoff exceeds its payoff without commitment. At equality, the incumbent is indifferent.

Entry deterrence matters because it shows how market outcomes can depend on the timing and observability of decisions, not just on a firm’s announced intentions. The central test is sequential rationality: the incumbent’s response must make sense when it is chosen, the entrant must rationally anticipate it, and any earlier commitment must be worthwhile after its cost is counted.

This approach is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It provides a disciplined way to analyze strategic entry without assuming that a threat is binding merely because it has been stated.

Check Your Understanding

For each game, solve the continuation decisions before deciding whether entry or commitment occurs.

A firm may enter a market. Staying out gives payoffs (0, 9). After entry, accommodation gives (3, 7), while fighting gives (-2, 2). Find the incumbent’s optimal response, the entrant’s choice, and the equilibrium payoffs. (Answer: The incumbent accommodates; the entrant enters; payoffs are (3, 7).)
Staying out gives the entrant 0. If it enters, the incumbent can accommodate, giving payoffs (1, 8), or fight, giving (-4, 5). Find the sequential outcome and explain why the fight threat does or does not deter entry. (Answer: The incumbent accommodates because \(8 > 5\); the entrant enters because \(1 > 0\); payoffs are (1, 8), so the fight threat is not credible.)
Without commitment, the incumbent’s equilibrium payoff is 9. A commitment deters entry if chosen and gives the incumbent a gross no-entry payoff of 13 before a cost \(K\). After commitment, fighting is optimal if entry occurs, and the entrant would receive -1 from entry versus 0 from staying out. For what values of \(K\) is commitment strictly preferred, and what happens at the boundary? (Answer: Commitment is strictly preferred for \(K < 4\); at \(K = 4\), the incumbent is indifferent.)
Without investment, entry followed by the incumbent’s best response gives the incumbent 12. An observable investment costs 3 and changes the post-entry incumbent payoffs to 10 from accommodation and 11 from fighting; if entry occurs after investment, the entrant gets -1 from fighting, compared with 0 from staying out. The incumbent’s gross payoff if investment deters entry is 16. Determine the equilibrium investment decision and the net incumbent payoff. (Answer: The incumbent invests; fighting would be optimal after entry, so the entrant stays out; the incumbent’s net payoff is \(16-3=13\), greater than 12.)
After entry, the incumbent earns 7 from accommodating and 4 from fighting. The entrant would receive 2 if accommodated, -1 if fought, and 0 from staying out. Find the equilibrium and assess a stated fight threat. (Answer: The incumbent accommodates because \(7 > 4\); the entrant enters because \(2 > 0\); the fight threat is not credible, and the payoffs are (2, 7).)

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