Check the game, the solution concept, and the evidence before acting on a model.
By the end: separate a valid incentive calculation from unsupported prediction or normative advice.
Start with the actual claim
A fictional maintenance committee says: "Repeating the relationship guarantees cooperation. The efficient outcome is the equilibrium, and a permanent sanction is fair because it supports that equilibrium." The one-shot utilities are 3,3 for mutual maintenance, 5,0 for unilateral skipping, and 1,1 for mutual skipping. The estimated effective future weight is 0.4.
First check the calculation under the supplied assumptions. With grim trigger, normalized cooperation gives 3 and deviation gives 3.4. Cooperation is not supported by that strategy at delta = 0.4. The one-shot efficient outcome is also not its Nash equilibrium. Two distinct incentive errors appear before any empirical debate.
Reconstruct the deviation before judging the recommendation
An audit starts by translating a sentence into a comparison that someone could actually make. The committee's word 'guarantees' claims more than the existence of a cooperative equilibrium. It claims that the proposed conditions rule out other behavior. To test even the weaker equilibrium claim, specify what each neighbor does after every relevant history. Maintaining while nobody has skipped, then skipping forever after any observed skip, is a complete grim-trigger rule. 'We intend to maintain' is not. It leaves the response to a deviation unspecified, although that response determines today's incentive.
Separate the current gain from the later loss. While Column maintains, Row gets 3 by maintaining and 5 by skipping, so the immediate temptation is 2 utility units. Under the proposed response, future mutual maintenance would give Row 3 per round, while future mutual skipping gives 1. The loss is therefore 2 in each later round. At delta 0.4, the next round's loss has current weight 0.4, the following round's weight is 0.16, and the next is 0.064. These are weights on utility, not probabilities measured by the displayed history.
Today's gain exceeds the future loss at delta 0.4
At delta 0.4, the full discounted loss is four thirds. The first three future losses account for 1.248 of that total.
The weighted future loss starts at 0.8, then adds 0.32, then 0.128. Its full value is 2 times 0.4 divided by 0.6, or four thirds. One way to check this sum without calculus is to call it L. Multiplying L by 0.4 shifts every term one round later. Subtracting leaves only the first term, so 0.6 times L equals 0.8. The future loss of four thirds is smaller than today's gain of 2. Row gains two thirds overall by deviating at the first cooperative history.
Now check the histories after punishment has started. If Column will skip forever regardless of subsequent actions, Row gets 1 by skipping today and 0 by maintaining. Maintaining does not restore cooperation under grim trigger. Row therefore loses 1 immediately and changes no later reward by trying it. The punishment continuation is credible under these stipulated preferences. That conclusion repairs no failure at the cooperative history. A strategy can specify a credible punishment that is still too weak, after discounting, to deter the initial deviation. Both comparisons belong in the audit.
A different proposed repair might reduce the temptation rather than increase patience. Consider a separate paper scenario, not an editable payoff in this lab, where unilateral skipping gives 4 instead of 5. Its immediate advantage over maintaining is now 1. At delta 0.4 the same future loss remains four thirds, which exceeds 1 by one third. Equivalently, normalized deviation utility is 4 minus 3 times 0.4, or 2.8, below cooperation's 3. This calculation supports the changed game, not a claim that the original committee already faces those incentives.
Reducing temptation changes the game being tested
Skipping payoff
Gain now
Loss later
Deviation minus cooperation
5, original
5 − 3 = 2
4/3
2 − 4/3 = 2/3
4, paper change
4 − 3 = 1
4/3
1 − 4/3 = −1/3
The second row is a separate paper scenario. Both use delta 0.4 and punishment utility 1; only the unilateral skipping payoff changes.
Changing a utility entry also requires an account of what changed outside the arithmetic. A penalty for skipping needs an observer, an enforcement rule, and a consequence the player values. A stronger preference for reliable service would require different evidence. Do not describe an assumed utility reduction as a measured cash fine. Nor should the analyst subtract a fine twice, once inside the utility entry and again after calculating expected utility. Preserve a record of the original assumptions so another reader can tell whether a repair corrects a calculation or substitutes a different game.
Repair a claim without overclaiming
If the effective weight were 0.75 and monitoring were perfect, the grim-trigger incentive comparison would pass. That establishes one supported strategy profile under a particular model. Always skipping remains another equilibrium. No theorem here says which equilibrium the committee will reach.
The estimate of delta, the utility entries, the accuracy of monitoring, and enforcement are separate evidence questions. A six-round illustration cannot validate an indefinite-horizon model. A pilot could examine missed observations, costly compliance, heterogeneous preferences, and participants' ability to leave. Consent and privacy constraints matter when collecting such evidence.
Distinguish a compatible observation from evidence for a mechanism
Several explanations can produce the same observed cooperation. Neighbors might follow grim trigger, maintain regardless of history, or value the other neighbor's service enough to prefer maintenance in the one-shot game. The first and second plans give the same actions until someone skips, yet their responses afterward differ. The third explanation changes the utility model. An observation identifies a mechanism only if it helps distinguish it from such alternatives. A cooperative path alone does not reveal which conditional plan participants would use at a history that never occurred during observation.
A separate paper example shows why a material payoff table is not enough to establish utilities. Suppose a player values their own material points plus half the other player's points. Against maintenance, maintaining then gives 3 plus half of 3, or 4.5 utility units. Skipping gives 5 plus half of 0, or 5. The temptation falls from 2 to 0.5. Against skipping, maintaining gives 0 plus half of 5, or 2.5, while skipping gives 1 plus half of 1, or 1.5. Skip no longer dominates Maintain in this changed preference model.
A preference change reverses one best response
Row action
Column maintains
Column skips
Maintain
3 + 0.5 × 3 = 4.5
0 + 0.5 × 5 = 2.5, best
Skip
5 + 0.5 × 0 = 5, best
1 + 0.5 × 1 = 1.5
Paper utilities equal own material points plus half the neighbor's. Holding Column fixed shows that neither Row action dominates under these assumed preferences.
This example is an explicit assumption about one person's preferences, not permission to add different people's measured welfare. It also does not establish altruism in the fictional committee. CORE's discussion of social preferences motivates considering alternatives to own-money maximization, but the exact half-weight here is invented. An analyst would need evidence about relevant choices to support that weight. If the committee's original entries already represent full preferences, adding another concern for the neighbor would count that concern again. The meaning of an entry must be settled before revising its value.
Observation quality creates another competing explanation. A record marked Skip could represent an actual skipped action or a missed maintenance report. The original lab assumes perfect observation, so those possibilities coincide there. They need not coincide in a field record. In a separate paper monitoring example, suppose there are 100 maintenance events and the recording process misses 5 of them. The 5% missed-record rate describes this stipulated recording process. It does not establish how frequently neighbors deliberately skip, how observers respond, or how the grim-trigger equilibrium changes under imperfect information.
To calculate a mistaken-sanction rate, the denominator and the decision rule must both be explicit. Consider another paper count of 100 rounds, with 90 actual maintenance actions and 10 actual skips. Suppose observers flag 9 skips correctly but also flag 9 maintenance actions. There are 18 flags, of which 9 concern actual skips. Only half the flags then identify a skip, despite the detector catching 9 out of 10 skips. A rule that punishes every flag would punish 9 maintenance actions. The lesson's perfect-monitoring formula does not include this distinction.
A flag is not the same event as an actual skip
Actual action
Flagged
Not flagged
Total
Skip
9
1
10
Maintain
9
81
90
Total
18
82
100
These stipulated paper counts describe 100 rounds. Nine of eighteen flags concern maintenance, so catching most skips does not make every flag reliable.
A useful investigation would compare records with an independently checked sample, with participants' consent and clear limits on collection. It would also distinguish what players actually observe from what a later analyst can reconstruct. A private inspection available only afterward cannot justify a player's earlier type-dependent or history-dependent choice. Conversely, an observed departure might reflect an omitted exit option rather than impatience. Evidence about actions, observation, and available alternatives answers different questions. Combining them into a single label such as 'trust' hides which assumption failed and which model would need revision.
Compare incentives and outcomes
A continuation condition and a separate history
Plan
Present utility
Normalized utility
Maintain forever
5
3
Deviate now, then punishment
5.667
3.4
Grim-trigger cooperation incentive passes: No. Equality permits indifference; passing does not imply uniqueness.
Compare complete discounted plans
The first round receives weight one. Subsequent round weights are delta, delta squared, and so on. Payoffs are utility units.
Normalize and rearrange
Multiplication by 1 minus delta preserves the inequality because delta is below one.
Round 1Row CCol C3, 3
Round 2Row DCol C5, 0
Round 3Row DCol D1, 1
Round 4Row DCol D1, 1
Round 5Row DCol D1, 1
Round 6Row DCol D1, 1
C means Maintain; D means Skip. The last line in each round gives Row, Column utility. A permanent response begins after the observed deviation.
Discount presets and the slider change delta only. Trace presets change only the scripted action history. Six displayed rounds explain the response rule; the formulas assume indefinitely repeated play, perfect monitoring, and credible mutual-Skip punishment. Normalized values multiply both present values by 1 minus delta.
Layer
Committee claim
Audit
Arithmetic
Repetition suffices at delta 0.4
Deviation pays more
Solution concept
Efficiency equals equilibrium
Different tests
Empirical prediction
Everyone will cooperate
No selection or calibration evidence
Normative judgment
Permanent punishment is fair
Requires independent justification
Reject each unsupported inference at the layer where it occurs.
Use the controls to test a conditional claim
Use the repeated-game lab above to isolate what its calculation actually tests. Select Audit case at 40% and No deviation. The six displayed rounds now show mutual maintenance, even though the incentive check fails. This is not a contradiction. The trace selector displays a specified action history, not behavior generated by optimizing players. Present utility for maintaining forever is 3 divided by 0.6, or 5. Deviating immediately and then receiving punishment gives 5 plus 0.4 divided by 0.6, or seventeen thirds. The trace cannot erase that profitable deviation.
Keep the no-deviation trace and select Patient at 75%. Maintaining forever now gives 3 divided by 0.25, or 12. Immediate deviation gives 5 plus 0.75 divided by 0.25, or 8. Cooperation has an advantage of 4 in present utility. Normalized values express the same ranking as 3 against 2, because multiplying both totals by 0.25 preserves their order. The six actions remain unchanged. This control-guided comparison separates a changed incentive condition from changed observed behavior without pretending that the lab estimates how frequently cooperation occurs.
The same scripted actions can accompany opposite incentive results
Quantity
Audit case, delta 0.4
Patient, delta 0.75
Maintain forever
3 / 0.6 = 5
3 / 0.25 = 12
Deviate immediately
5 + 0.4 / 0.6 = 17/3
5 + 0.75 / 0.25 = 8
Cooperate minus deviate
−2/3
4
Normalized comparison
3 < 3.4
3 > 2
These are indefinite-horizon values from the lab's formulas. Selecting No deviation fixes the displayed six-round actions at mutual maintenance in either column.
Next select Row skips in round 2 while leaving delta at 75%. Round 1 still gives 3 to each player. In round 2, Row skips while Column maintains, giving 5 to Row and 0 to Column. Mutual skipping begins in round 3, giving 1 each thereafter in the displayed trace. The selector scripts the deviation even when it is unprofitable. It therefore illustrates the consequence of breaking the rule, not the claim that a rational Row would choose that deviation under the current assumptions. The discount setting remains unchanged.
The timing also allows an independent check of a delayed deviation. Relative to permanent cooperation at delta 0.75, a deviation at the first round gives Row 8 minus 12, or negative 4. Moving that same deviation one round later leaves the first cooperative round identical and multiplies the later difference by 0.75. The disadvantage becomes 3. Directly, Row receives 3 now, then 0.75 times the deviation continuation value of 8, for a total of 9. Permanent cooperation still gives 12. These are indefinite totals, not sums of the six visible rounds.
Finally select Break-even at 50%, retaining the round-two deviation trace. The totals become 6 for both immediate plans. Cooperation and the specified deviation tie, so the incentive test passes weakly. The word 'passes' means no profitable deviation under this strategy and these assumptions, not that maintenance is strictly better. Move the slider to 45% and then 55%, its adjacent five-percentage-point settings. Normalized deviation values are 3.2 and 2.8 respectively. A threshold at 50% separates failure from success, but no estimate of actual participants' delta comes from moving this control.
A passing boundary includes indifference
The lines use the lab's normalized utilities on exact linear scales. At 50% they tie; adjacent slider settings reverse the strict preference.
The lab's limits help locate the next analysis rather than merely adding warnings. Its slider cannot change payoffs, observation errors, the punishment rule, or participants' exit options. A proposed one-round punishment requires a new continuation analysis, including incentives to end punishment when promised. A known final round requires finite-horizon reasoning instead. The displayed six-round endpoint supplies neither feature. If a recommendation relies on any such change, write a separate model with its own assumptions. Reusing an unchanged passing incentive result would answer the old question rather than test the revised institution.
Keep mechanism and judgment separate
A sanction can change incentives and still be disproportionate, unlawful, or harmful. Efficiency does not compare every distribution fairly, and an equilibrium is not a moral permission. State the policy objective, whose welfare counts, the outside options, and the constraints on acceptable means.
A useful recommendation therefore has a conditional form: under these actions, information, preferences, and enforcement, this deviation is or is not profitable. Then list which assumptions need evidence and what changes if they fail. The purpose of game theory is disciplined strategic reasoning, not replacing missing facts with confidence.
Separate supportable incentives from an acceptable policy
An incentive result answers whether a player can gain by changing their plan alone. A policy judgment must also say whose interests count and what means are permitted. In the maintenance matrix, mutual maintenance gives each player more utility than mutual skipping. That within-person comparison establishes a Pareto improvement between those outcomes. It does not rank every possible institution, because monitoring, administration, and restrictions on leaving may introduce consequences absent from the four cells. A policy cannot inherit the matrix's efficiency label while adding unmodeled costs and omitting the people who bear them.
Distribution can remain unresolved even when a total quantity is fixed. Revisit a paper bargaining comparison using the earlier twelve-unit model with outside options 2 for A and 4 for B. Offering B 4 leaves A 8, giving gains over disagreement of 6 and 0 respectively. Offering B 7 leaves A 5, giving gains of 3 each. Both splits use all twelve units and both weakly satisfy the outside-option tests. Neither split Pareto-dominates the other because shifting units toward B reduces A's stipulated utility. A fairness criterion must make an additional comparison.
Equal gains and equal shares are different allocations
Paper allocations reuse the twelve-unit bargaining assumptions. Bars measure each person's gain above their own outside option, without adding utilities across people.
The equality of gains in the second split is an explicit possible criterion, not a theorem selecting the bargain. Equal final shares would instead give 6 each, with gains of 4 and 2. A one-shot proposer rule with A proposing and B accepting indifference selects a different point, 8 for A and 4 for B. These conclusions disagree because they answer different questions about fairness and strategic procedure. CORE distinguishes judgments about allocations from judgments about how they arose. The analyst should name the selected criterion instead of calling every preferred allocation 'efficient'.
Commitment provides an equally concrete separation between an individual advantage and a policy endorsement. In the earlier enforced-bond example, deterrence gives the incumbent 2.5 rather than the no-bond continuation outcome of 1. The entrant receives 0 instead of 1. One player improves and the other loses, so the change is not a Pareto improvement for those two players. These utility numbers do not share a justified interpersonal scale merely because they appear in one pair. Subtracting the entrant's loss from the incumbent's gain would introduce an extra welfare assumption.
Even an observed cooperative period has two unresolved causal questions. Did the proposed rule cause cooperation, and would cooperation persist under a different history? Comparing participants who volunteered for a scheme with people who refused it does not isolate the rule's effect, because their preferences or outside options may differ beforehand. A carefully bounded pilot could compare voluntary alternatives and document the available choices, while avoiding deliberate harmful sanctions. A causal conclusion would still need a defensible comparison. An equilibrium calculation supplies neither that comparison nor permission to collect whatever evidence seems convenient.
A credible commitment can help one player and hurt another
Selected outcome
Entrant utility
Incumbent utility
No bond: In, Share
1
1
Enforced bond: Out
0
3 − 0.5 = 2.5
Change with bond
−1
+1.5
These are the earlier entry model's selected outcomes, not new controls. The incumbent improves, but the entrant loses, so neither outcome Pareto-dominates the other.
A defensible final recommendation can therefore be precise without promising a unique forecast. For the committee's stated utilities and perfect-monitoring grim trigger, delta 0.4 leaves a profitable deviation. Raising delta to 0.75 supports this cooperative profile while always skipping remains an equilibrium. Whether either value describes participants needs evidence, and whether permanent punishment is acceptable needs an independent judgment about rights, costs, and alternatives. Keep those conclusions separate in the final report. Then a changed estimate, an observation error, or a rejected sanction tells the next analyst exactly which part to reconsider.
Exercises
A more tempting deviation
Keep cooperation at 3 and punishment at 1, but raise the one-round deviation payoff to 7. Derive the new grim-trigger threshold.
Hint
Compare 3 with 7 minus 6 delta after normalization.
Worked answer
Deviation payoff
Normalized deviation
Required delta
5
5 − 4 delta
At least 1/2
7
7 − 6 delta
At least 2/3
A changed payoff requires a changed threshold, not reuse of the old one.
The normalized deviation value is 7 − 6 delta. Cooperation requires 3 ≥ 7 − 6 delta, so delta must be at least two thirds. Raising the temptation makes the incentive condition harder to satisfy. This still assumes perfect monitoring, credible punishment, and the indefinite horizon.
Write a defensible conclusion
A committee observes cooperation in one trial. It claims to have proved participants are in a unique repeated-game equilibrium. Replace that conclusion with one the evidence supports.
Hint
An observed path is not a complete contingent strategy or a counterfactual test.
Worked answer
Observed
Still unknown
Cooperation on one path
Off-path plans and incentives
One trial's conditions
Transfer to other groups or horizons
A joint outcome
Unique equilibrium or causal mechanism
A compatible observation is weaker than identification of the model.
The trial observed cooperation under its particular conditions. It did not reveal every off-path response, prove correct beliefs, establish uniqueness, or distinguish equilibrium incentives from norms or other preferences. A defensible next step is to specify competing explanations and ethically gather evidence about preferences, observation errors, continuation expectations, and alternative choices.
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 passages on players, strategies, preferences, and how alternative payoffs change the game. Examples here are original, not observations of learner behavior.
Selected grim-trigger incentive comparisons and credible continuation behavior. Perfect monitoring and an indefinite horizon are assumptions, not facts about relationships.
Selected reservation-option, bargaining-power, Pareto, and fairness distinctions. Distributional and ethical claims are not settled by equilibrium arithmetic.