Test whether a promised response remains attractive when its decision point arrives.
By the end: explain why credibility depends on continuation incentives and enforcement.
A Nash threat can fail later
Return to the baseline entry game. The profile Out plus Fight-after-entry is a Nash equilibrium in its normal form. If the entrant expects Fight, entry loses 1. When the entrant stays out, the incumbent cannot improve its payoff by changing an unreached response.
Yet Fight is not optimal if entry actually occurs: the incumbent prefers Share. This profile is not subgame perfect. A verbal threat to fight leaves that continuation incentive unchanged. Calling a threat forceful or public is not a substitute for specifying its consequences.
Why an unreached response still needs checking
The entry game contains two different questions about a threat. Would staying out be sensible if Fight were the incumbent's response? Would Fight be sensible if entry actually occurred? The first question holds the response plan fixed. The second examines the incentives at the later decision. In the baseline game the answers are yes and no. This is possible because the incumbent receives 3 after Out regardless of its unused response plan. A plan can pass the whole-game Nash test even though its later instruction fails when examined alone.
To see this without a tree, write the incumbent's two strategies as Fight after In and Share after In. These are complete strategies because the incumbent has only one possible decision point. The entrant chooses Out or In. Both cells in the Out row contain the same pair, 0,3. The In row contains −1,−1 under Fight and 1,1 under Share. A column describes what would happen following entry, not an action the incumbent necessarily performs. Repeated numbers in the Out row encode the fact that no response is reached.
The Nash threat uses an unchanged Out row
Entrant strategy
Fight after In
Share after In
Out
0, 3 · Nash with Fight
0, 3 · entrant can gain 1 by In
In
−1, −1 · incumbent can gain 2 by Share
1, 1 · Nash and subgame perfect
Pairs list entrant, incumbent utility. At Out, changing the incumbent's response leaves its payoff unchanged. After actual entry, Share improves its utility by two.
Now test Out together with Fight after In. The entrant's unilateral switch gives −1 rather than 0, so it does not improve. The incumbent's unilateral switch changes an unused instruction and leaves its utility at 3. It also does not improve. Neither has a strictly profitable deviation while the other's complete strategy remains fixed. The profile is therefore Nash. This conclusion does not say that Fight is optimal after In. It says that changing Fight to Share cannot improve the incumbent's actual outcome when Out remains the entrant's strategy.
A subgame is a continuation that can be treated as a game in its own right without giving anyone information they lacked. Here the decision after observed entry is a subgame with one chooser. Once there, the incumbent compares its own utilities −1 and 1. Share wins by 2. Out plus Fight fails subgame perfection because that continuation is not a best response. The earlier Nash test was not calculated incorrectly. Subgame perfection asks an additional question at a decision point that the proposed outcome never reaches.
Contrast In together with Share after In. Given Share, the entrant prefers 1 to Out's 0. Given In, the incumbent prefers Share's 1 to Fight's −1. This profile passes Nash and the continuation test. The distinction explains why observing Out alone cannot establish credible deterrence. Out might reflect a mistaken expectation, omitted entry costs, or a different game. Even a perfectly recorded outcome does not reveal what the incumbent would have done after a choice the entrant did not make. Credibility concerns that unobserved conditional response.
The deviation check changes when entry is reached
Situation
Keep Fight
Switch to Share
Gain from switching
Entrant remains Out
3
3
0
Entry has occurred
−1
1
2
These are incumbent-only comparisons in the baseline game. Nash at Out examines an unused instruction. The continuation test instead compares actions after entry is observed.
Keep the previous lesson's distinction between analysis order and play order. We inspect the incumbent first to calculate what entry would lead to. We do not move the incumbent ahead of the entrant or allow it to choose before observing entry. Nor does representing the game as a matrix erase that observation. The matrix's columns are complete response strategies for the original sequential game. This lets the matrix expose Nash threats while the tree exposes their continuation incentives. The two representations organize the same stipulated utilities for different checks.
A commitment changes consequences
Suppose an enforceable bond charges the incumbent 3 utility units if it Shares after entry. It also costs 0.5 upfront in every outcome. The fee is already paid at the response node, so it subtracts equally from Fight and Share. Fight then gives the incumbent −1.5, compared with Share's −2.5.
Now Fight is credible under the stated preferences. The entrant anticipates −1 from entry and chooses Out, giving the incumbent 2.5 after the fee. The comparison with baseline Share's 1 can motivate buying the bond in this constructed game. The explorer compares installed rules; it does not solve a general contract-design problem.
Separate the response penalty from the upfront fee
The enforced bond changes one conditional consequence and adds one unconditional cost. Call a conditional consequence a cost charged only when a specified event occurs. Sharing after entry triggers the penalty of 3. The upfront fee of 0.5 applies even if entry never occurs. Keeping these charges separate matters because only the conditional difference can reverse the response ranking. These are utility deductions stipulated by the example. Calling the rule a bond does not establish that the numbers describe cash, a legally valid agreement, or anyone's measured preferences over actual contracts.
Start with all terminal pairs before choosing branches. Out changes from 0,3 to 0,2.5 because the incumbent pays the fee. Fight changes from −1,−1 to −1,−1.5. Share changes from 1,1 to 1,−2.5 because both the penalty and fee apply to the incumbent. The entrant's coordinates do not change. After entry, compare only the incumbent coordinates −1.5 and −2.5. Fight now wins by 1. Anticipating that response, the entrant compares entry's −1 with Out's 0 and stays out.
Charge the fee everywhere and the penalty only at Share
Terminal action
Base incumbent
Fee
Share penalty
Final pair
Out
3
−0.5
0
0, 2.5
In, Fight
−1
−0.5
0
−1, −1.5
In, Share
1
−0.5
−3
1, −2.5
Each deduction applies to incumbent utility only. The response ranking changes because Share loses three more units than Fight, not because both incur the fee.
The fee is sunk at the response node, meaning that it has already been incurred and cannot be recovered by either available response. Subtracting 0.5 from both alternatives leaves their difference unchanged. Before the penalty, Share beat Fight by 2. The penalty removes 3 from Share alone, so Fight instead beats Share by 1. An argument that fighting is credible because the incumbent has already spent money misses this difference. Prior expenditure affects a future decision only through consequences that still differ between the choices available at that decision.
For a paper comparison, not an additional lab control, keep the fee at 0.5 and reduce the Share penalty to 1. Fight still gives −1.5, while Share gives 1 minus 1 minus 0.5, or −0.5. Share wins by 1 and entry occurs. The incumbent finishes with −0.5 rather than the baseline 1. The rule is costly without deterring entry. Raising a penalty can therefore have a threshold effect rather than a smooth benefit. A small change below the credibility threshold may lower the outcome without changing either selected action.
A second paper comparison sets the penalty to 2 with the same fee. Both responses give −1.5. Our incumbent tie convention selects Share, so the entrant still enters and receives 1. Equality makes Fight a best response too, but it does not justify reporting uniquely credible deterrence. With penalty 3, Fight is strictly better. The exact boundary follows by comparing baseline Fight's −1 with Share's 1 minus the penalty. The fee cancels. The penalty must exceed 2 to make Fight strictly optimal under these baseline response utilities.
The fee leaves the penalty threshold at two
Paper comparisons at fee 0.5. Horizontal distance measures penalty; vertical distance measures incumbent utility. At penalty two, Share wins the stipulated tie, so entry continues.
This separates two decisions made at different times. At the later response node, the fee is irrelevant to the ranking. Before installation, its cost matters to whether the rule is attractive at all. With deterrence, the incumbent receives 2.5 rather than the no-bond solution's 1, a gain of 1.5. That comparison uses the outcome selected by backward reasoning in each installed game. It does not compare against the noncredible Nash threat's payoff of 3 as if that threat were already an enforceable alternative available at no cost.
Compare incentives and outcomes
Compare a statement with an enforced rule
No-bond reference
In, Share. Payoffs 1, 1.
Selected rule
Out; response if In: Fight. Outcome 0, 2.5.
Complete continuation calculation
Decision
Own alternatives
Selected
Incumbent after In
Fight -1.5; Share -2.5
Fight
Entrant
Out 0; In -1
Out
The reference has no bond. Announcing Fight changes no payoff. Only the enforced-bond preset changes the Share penalty and charges the fee in all outcomes. Payoff order is entrant, incumbent. Blue thick branches show the selected contingent plan, including an unreached response.
Installed rule
Incumbent Fight
Incumbent Share
No bond
−1
1
Bond 3 and fee 0.5
−1.5
−2.5
The fee subtracts from both responses; the contingent penalty changes their ranking.
Read the comparison as two installed games
In the comparison above, first select No bond. Both sides then represent the baseline rule, so both show In followed by Share. Next select Announce Fight only. Before checking the displayed answer, compare the terminal pairs with the reference. They have not changed. The selected response must still be Share because its incumbent utility is larger. This isolates the difference between describing an intended action and changing the incentive to perform it. A statement could affect beliefs in another model, but this lab does not assign statements a separate belief-changing effect.
Now select Enforced bond: 3, fee 0.5. Keep the reference visible while reading the selected rule. Compare the two Share endpoints first, then the two Fight endpoints, and finally Out. The unequal changes identify the mechanism. Share falls by 3.5, Fight by 0.5, and Out by 0.5 for the incumbent. Open Complete continuation calculation to check the two own-payoff comparisons in order. The lab supplies installed-rule presets, not an editor for arbitrary fees or penalties. The earlier boundary calculations are paper examples rather than hidden slider settings.
A selected response can remain unreached
Schematic trees reproduce the two installed games. Thick blue branches mark complete selected plans. Under the bond, Out occurs; the marked Fight response remains conditional.
Notice the thick blue Fight branch even though the selected entrant action is Out. It marks the incumbent's selected contingent plan, meaning the instruction to use if entry occurred. It is not evidence that a fight happens on the selected outcome path. Only Out is realized under that profile. This distinction prevents a misleading interpretation of deterrence as actual conflict. Conversely, leaving Fight unmarked merely because it is unreached would remove the very response on which the entrant's decision depends. The full plan and its realized outcome must remain distinguishable.
The comparison does not add a preliminary Buy bond choice to the tree. It shows the no-bond game and whichever rule you select. To model voluntary purchase, add that earlier decision on paper and specify who observes it before entry. If installation is observed, known to be irrevocable, and otherwise has the displayed consequences, the incumbent compares 1 without installation against 2.5 with installation. It prefers installation. This extra comparison is a simple extension under explicit assumptions, not another decision that the current lab searches or animates for you.
Observability matters even when enforcement works perfectly. In a paper scenario where the entrant cannot learn whether installation occurred, the entrant cannot use a different action merely because the analyst can see two trees. A hidden bond can change what happens after entry without producing the earlier deterrent choice assumed in the visible-installation extension. Solving that uncertainty would require beliefs about installation and strategies consistent with what each player knows. We do not need that later machinery to identify the error. The observed-installation calculation cannot be copied into an unobserved-installation game.
Installation matters only through a changed anticipated response
Rule before entry
Anticipated response
Entrant choice
Incumbent outcome
Announcement only
Share
In
1
Enforced bond
Fight
Out
2.5
Paper installation choice
Compare both solutions
Observed rule
2.5 − 1 = 1.5 gain
The first two rows are lab presets. The final comparison is a paper extension with an observed installation choice, not an extra decision in the lab.
Return to Announce Fight only after inspecting the bond. The reference and selected outcomes should coincide again. This reversal is useful because it holds the baseline utilities fixed while removing the institution that changed the continuation comparison. Nothing in the control sequence establishes how persuasive a real announcement would be. The experiment is a calculation under known rules, not a study of intimidation or trust. Its transferable result is narrower and testable within the model. Identify which consequence differs across future actions before claiming that an intervention changes their ranking.
Check reversibility and external costs
The bond works only if the forfeiture is enforceable and cannot be costlessly renegotiated or canceled after entry. A refundable deposit with no relevant penalty is not the same mechanism. We treat any forfeiture as a payment to an outside administrator, whose welfare is not in the two-player payoff pair.
A commitment can benefit one player by removing an option or changing a later incentive. It can also be costly, harmful, or illegal in real contexts. Distinguish a positive model of strategic advantage from a recommendation to use threats, exclusion, or coercion. The arithmetic does not supply that recommendation.
Test cancellation before crediting deterrence
Irrevocable means that the incumbent cannot remove the penalty after seeing entry. This assumption does real work. Consider a separate paper extension where cancellation after entry is free, while the upfront fee remains sunk. Keeping the rule makes Fight optimal and gives the incumbent −1.5. Canceling removes the Share penalty, so the incumbent can Share for 0.5 instead. Canceling and sharing improves its continuation utility by 2. An entrant who anticipates this extension should not treat the original enforced response as the solution. The available actions have changed.
Suppose cancellation instead costs the incumbent 1 additional utility unit after entry. This is also a paper extension, not a lab preset. Canceling and sharing yields 1 minus the sunk fee 0.5 minus cancellation cost 1, or −0.5. Keeping the rule and fighting yields −1.5. Cancellation still improves utility by 1. At cancellation cost 2 the two plans tie; above 2 keeping and fighting is better. These comparisons show why the label costly commitment is insufficient. The relevant escape cost must be compared with the gain from escaping at the later node.
An escape option can reverse the continuation choice
Plan after entry
Incumbent arithmetic
Utility
Versus keep and Fight
Keep bond, Fight
−1 − 0.5
−1.5
Reference
Cancel free, Share
1 − 0.5
0.5
Gain 2
Cancel for 1, Share
1 − 0.5 − 1
−0.5
Gain 1
Cancel for 2, Share
1 − 0.5 − 2
−1.5
Tie
Cancel for 3, Share
1 − 0.5 − 3
−2.5
Lose 1
Paper extension after entry. All rows retain the sunk fee. Cancellation removes only the Share penalty; its separate cost determines whether escape is attractive.
Renegotiation means the parties can revise an agreement together rather than one party canceling it alone. If revision can erase the penalty, the revised continuation may differ from the installed tree. Who must consent, what they receive, and what the outside administrator can enforce become part of the game. The original lesson assumes no such revision. We cannot conclude that renegotiation necessarily happens, or never happens, from the two displayed payoff pairs. We can identify precisely which missing decision would have to be modeled before relying on irreversible deterrence.
The recipient of forfeiture also matters. In the displayed model the penalty goes outside the entrant-incumbent pair, so the entrant still receives 1 after Share. Imagine instead a paper rule that pays the 3 to the entrant and stipulates an equal utility increase for that receipt. Share would then give entrant utility 4, while its incumbent coordinate remains −2.5 after the fee. Fight remains optimal for the incumbent, so this particular solution still deters entry. Yet the off-path entrant consequence differs. That difference would matter in extensions with uncertain enforcement or revised responses.
Do not add the two utility coordinates to declare the bond socially beneficial. The incumbent's selected utility increases from 1 to 2.5, while the entrant's decreases from 1 to 0. Neither selected outcome Pareto-dominates the other because each makes a different player better off. The administrator is also outside the reported pair. Even if these numbers represented money under an extra assumption, judging welfare would require deciding how to value the changed opportunities and external consequences. The strategic calculation alone establishes an incentive change, not fairness or a justified policy.
The two selected outcomes do not Pareto-rank each other
Axes show each player's own stipulated utility, not a common welfare scale. Moving to the bond raises incumbent utility but lowers entrant utility. Neither outcome Pareto-dominates.
For another proposed commitment, reconstruct the same comparisons with that proposal's actual actions. Record the response utilities before and after the rule, include costs charged even when nothing happens, and ask whether cancellation or revision remains available. Then calculate the earlier player's choice using only information available before that choice. Evidence for enforcement, preferences, and observation belongs beside the calculation rather than inside an unexplained claim of credibility. The linked Yale lectures on commitment and credible threats explain this method through conditional collateral and noncredible threats. Our bond numbers remain constructed teaching assumptions, not measured contract outcomes.
Exercises
A sunk fee alone
Suppose the incumbent pays an irreversible fee of 0.5 but no contingent penalty. Does the fee make Fight credible?
Hint
Subtract the same fee from both continuation payoffs.
Worked answer
Response
Base payoff
After fee
Fight
−1
−1.5
Share
1
0.5
An identical subtraction preserves the response ranking.
No. Fight becomes −1.5 and Share becomes 0.5, so Share remains better. A sunk fee can reduce total utility without changing the ranking of future choices. The penalty for Share, not the mere fact of prior expenditure, changed the earlier example.
The boundary penalty
With no fee, what penalty on Share makes the incumbent exactly indifferent? Does our tie convention then deter entry?
Hint
Compare Fight's −1 with Share's 1 minus the penalty.
Worked answer
Share penalty
Fight payoff
Share payoff
Selected reply
2
−1
−1
Share by convention
3
−1
−2
Fight strictly
Declare tie behavior rather than rounding it away.
A penalty of 2 makes both payoffs −1. Under the stated convention, the incumbent selects Share at a tie. Entry then gives the entrant 1 and still occurs. A penalty strictly above 2 makes Fight strictly preferred in this model. Equality alone does not justify claiming a unique deterrent response.
Sources and further reading
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Selected transcript on solving continuation decisions and how enforceable collateral changes incentives. No actual loan or enforcement terms are inferred.