Compare today's temptation with future consequences
Derive a repeated-game incentive condition and inspect what monitoring must support.
By the end: separate an indefinitely repeated equilibrium from a finite trace or a promise of cooperation.
A strategy can depend on history
Repeat the maintenance game with perfect observation of all previous actions. A grim-trigger strategy maintains until any Skip has occurred, then skips forever. If both use it and no deviation occurs, both receive 3 each round. After a deviation, mutual Skip gives 1 each round and is itself a stage-game equilibrium.
This punishment is credible under the displayed utilities because no player benefits from maintaining alone during punishment. Credibility does not make it forgiving, effective with noisy observations, or ethically desirable. A mistaken observation can activate the same permanent response as deliberate skipping.
A repeated strategy specifies responses to every history
The stage game is the single maintenance interaction that gets repeated. Within each round, both players choose without observing the other's current choice. Between rounds, they observe both previous actions perfectly. This distinction lets Row respond to Column's earlier Skip without allowing Row to wait and react to Column's current choice. Repetition changes the consequences of today's action through later behavior. It does not make Maintain a dominant action in the original matrix. Skip still gives more against either fixed current action when future consequences are left out of that matrix.
A history is the list of actions that have already occurred. A repeated-game strategy assigns an action after every possible history, including histories that would not occur if everyone followed the strategy. Grim trigger compresses this long instruction into a rule. Maintain if all previous actions were Maintain; otherwise Skip. It treats a player's own earlier Skip as a trigger too. Under two grim-trigger strategies, a single deviation changes both players' later instructions. The prescription to Skip forever belongs to the strategy even along the entirely cooperative path where it is never used.
The same current action can have different continuations
Column strategy
Row Maintains now
Row Skips now
Column in later rounds
Always Maintain
Row gets 3
Row gets 5
Maintain after either action
Grim trigger, clean history
Row gets 3
Row gets 5
Maintain after Maintain; Skip after Skip
The cooperative path alone does not distinguish these strategies. Their different responses after a Skip determine whether today's gain carries a later cost.
Distinguish that strategy from always Maintain. If Column maintains regardless of history, Row can Skip today for 5 instead of 3 without losing future cooperation. Row could also keep skipping against that unconditional strategy. A pleasant realized sequence is not enough to establish an equilibrium. The incentive comes from what the other player would do after a deviation. Grim trigger adds exactly that history dependence. It gives the same cooperative outcome as always Maintain before any deviation, but it assigns a different continuation after one, so its incentives can differ.
After a Skip has occurred, grim trigger prescribes mutual Skip. At that history, Row compares Skip's 1 with Maintain's 0 against Column's Skip. Skipping is better now, and neither action repairs the grim-trigger history. Punishment therefore passes the continuation test without a separate enforcer ordering either person to carry it out. Calling it punishment describes its lower utility relative to cooperation. It does not imply that a player willingly accepts a worse current response to help someone else impose a sanction on them.
The remaining question concerns a history with no previous Skip. Row can maintain and preserve future cooperation, or skip while Column still maintains this round. The immediate gain is 5 minus 3, or 2. Starting next round, mutual Skip replaces mutual Maintain, losing 3 minus 1, or 2 in each later round. Whether those future losses outweigh today's gain depends on their weight. A threat's severity alone cannot answer it without the timing and continuation weights.
The deviation gains two now and loses two per later round
Bars measure Row's utility difference, deviation minus continued cooperation, before discounting. Later negative terms continue indefinitely; four shown rounds do not define a final round.
The comparison also explains why delaying the first deviation does not create a new advantage when the future weight stays constant. After any cooperative history, the same current gain and same future loss stream remain available. At punishment histories, Skip remains a best response. These checks cover the different situations the rule creates. They rely on unchanged payoffs, perfect observation, and the same future weighting after each history. A relationship whose continuation chances change over time needs a new calculation rather than automatically inheriting the threshold derived for this stationary example.
Discount the continuation
Let delta be the weight on one additional future round, between zero and one. It can combine time discounting with an independent continuation probability. Maintaining forever has present utility 3 divided by 1 minus delta. Deviating once gives 5 now and then 1 per future round under punishment.
Multiplying both values by the positive number 1 minus delta gives a simpler comparison: 3 versus 5 minus 4 delta. Cooperation is at least as good when delta is at least one half. Check punishment histories too; there, Skip is already a best response. Under these assumptions, the condition supports a subgame-perfect grim-trigger profile, not a unique equilibrium.
Build the discounted totals without an infinite tally
Discounting means assigning smaller weight to utility received farther in the future. Delta is the weight multiplier for moving one round later. At delta 0.5, a utility unit now has weight 1, next round has weight 0.5, and the round after that has weight 0.25. Each later weight multiplies the previous weight by 0.5. At delta 0, only today's utility counts. These are weights for comparing one player's complete plans, not exchange rates between different people's utilities. The lab restricts delta below 1 so the totals used here remain finite.
At delta 0.5, maintaining forever begins with contributions 3, 1.5, 0.75, and 0.375. Adding four terms gives 5.625, not the full value. The remaining contributions are positive but keep halving. There is an arithmetic shortcut that avoids calculating endlessly. Call the complete value V. From next round onward, the plan is another copy of the same maintenance stream, weighted by delta. Therefore V equals today's 3 plus delta times V. Moving that weighted copy to the other side leaves one minus delta copies of V equal to 3.
Halving weights leaves a positive tail
At delta one half, bar lengths show discounted contributions in Row utility units. Four maintenance terms total 5.625; the remaining positive tail brings the value to six.
At delta 0.5, half of V therefore equals 3, so V is 6. The same rearrangement at delta 0.75 says that one quarter of V equals 3, so V is 12. In general, divide 3 by one minus delta. This is the geometric-sum result, where geometric means successive terms have a constant multiplier. We use it only for the stated indefinitely continuing stream with discount below 1. A known final round would remove the identical full continuation on which this shortcut depends and would also change the strategic problem.
A deviation earns 5 today, then 1 in each punishment round. At delta 0.5, the future punishment contributions are 0.5, 0.25, 0.125, and so on. Their total is 1, making the deviation value 6. At delta 0.75, the punishment stream starting today would total 4 because one quarter of its value equals 1. Our punishment begins next round, so multiply that 4 by 0.75 to get 3. Add today's 5 and the deviation value is 8. Keeping the starting round explicit prevents accidentally counting punishment today.
Normalize means multiply both plan values by the same positive factor to compare them on a different scale. Here the factor is one minus delta. Maintenance becomes 3. Deviation becomes 5 times one minus delta, plus delta, because the denominator in its future punishment value cancels. Expanding gives 5 minus 4 delta. At delta 0.5, both normalized values equal 3. At delta 0.75, deviation gives 2 instead of maintenance's 3. These are transformed plan values, not the utility actually received in a particular round of the six-round trace.
The normalized plans cross at delta one half
Both values use the same positive normalization. The solid cooperation line stays at three; the dashed deviation line falls as future losses receive more weight.
The threshold can also be derived directly from gains and losses. Today's gain of 2 must be no larger than the weighted future loss of 2 each round, starting next round. Divide that future loss stream by its per-round amount to get delta divided by one minus delta. It must be at least 1. Multiplying by positive one minus delta gives delta at least one minus delta. Hence twice delta must reach 1. This derives one half without assuming that repetition automatically supports cooperation, and it keeps the first affected future round visible.
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.
Delta
Normalized maintain
Normalized deviate
0.25
3
4
0.5
3
3
0.75
3
2
Normalization preserves the comparison; delta is dimensionless.
Use the discount and trace controls for different questions
Select Low future weight: 25% in the lab above and read both present values. Maintenance gives 3 divided by 0.75, or 4. Deviation gives 5 plus 0.25 divided by 0.75, or five and one third. Its advantage is one and one third. The normalized values are 3 and 4. Normalization changes the size of the difference but not which plan wins. Predicting only that low patience is bad would miss the useful calculation. The deviation gains 2 today while the future loss has present value only two thirds.
Next select Break-even: 50%, then Patient: 75%. At break-even both complete values are 6, so the test passes weakly. This means following grim trigger is a best response, not that maintaining is strictly preferred. At 75%, maintaining gives 12 while deviation gives 8, so maintaining wins by 4. Move the slider one step below and above break-even, to 45% and 55%. The normalized deviation values are 3.2 and 2.8. Comparing each with 3 shows the direction of the incentive without relying on rounded present-value decimals.
Adjacent slider settings reverse the incentive
Delta
Maintain
Deviate
Maintain minus deviate
0.25
3
4
−1
0.45
3
3.2
−0.2
0.50
3
3
0
0.55
3
2.8
0.2
0.75
3
2
1
Exact normalized comparisons avoid rounding present values. Passing at equality permits indifference. Neither equality nor a strict advantage establishes which repeated-game equilibrium players will select.
Leave delta at 75% and select No deviation. Every displayed round has mutual Maintain and utilities 3,3. Now select Row skips in round 2 without changing delta. Round 1 remains 3,3, round 2 becomes 5,0, and rounds 3 through 6 become 1,1. Column did not observe Row's current action before choosing in round 2. Its response begins in round 3, after the Skip becomes part of the public history. The trace describes one scripted deviation and subsequent instructions. It does not generate choices by optimizing or sampling a strategy.
While that deviation trace remains selected, return to Low future weight: 25%. The actions do not change. Only the valuation and incentive comparison change. Conversely, switching the trace does not replace delta. This separation lets you examine a rule's consequences independently of whether adhering to it is currently optimal. The six-round display is not a simulation that becomes more cooperative when delta rises. Its purpose is to make the observation delay and permanent response inspectable while the formulas answer a separate indefinite-horizon question.
If you add the six displayed utilities without discounting, Row receives 3 plus 5 plus four copies of 1, or 12. Column receives 3 plus 0 plus four copies of 1, or 7. Without deviation each receives 18 over those six displayed rounds. These are descriptive sums of the shown illustration, not the values used in the incentive test. The comparison neither stops punishment at round 6 nor proves anything about a truly six-round game. Truncating an illustration is different from announcing a final round to the players before they choose.
The displayed trace has six rounds, not a final-round incentive
No deviation
Row
Column
Round 1
3
3
Round 2
3
3
Rounds 3–6
12
12
Visible sum
18
18
Skip in round 2
Row
Column
Round 1
3
3
Round 2
5
0
Rounds 3–6
4
4
Visible sum
12
7
These undiscounted sums describe only the visible illustration. The permanent response continues beyond its display; the indefinite incentive calculation includes weighted utility after round six.
Interpret delta carefully if translating the model to a claim about a relationship. In a paper example, a 90% chance of reaching the next meeting and a 20% reduction in the weight on utility at that meeting produce effective delta 0.9 times 0.8, or 0.72. This assumes continuation is independent of the actions and applies repeatedly. The normalized deviation value would be 2.12, below 3. That arithmetic is not an estimate of anyone's patience. The lab allows effective weights but does not separately infer continuation probabilities, personal discounting, or how either depends on behavior.
A visible six-round trace is not an infinite proof
The timeline below illustrates six actions after an optional one-time deviation. It does not approximate the infinite incentive calculation or forecast six real meetings. The slider changes the discount factor; the trace selector changes only which scripted deviation appears.
If a known finite final round exists and the stage game has a unique Nash equilibrium, backward reasoning selects that equilibrium for subgame-perfect play at every round under the usual complete-information assumptions. This does not extend unchanged to stage games with multiple equilibria, uncertain types, or an unknown end. Always specify the horizon and observation rules.
Identify which assumption supports the future consequence
A known six-round maintenance game is a different model. In its final round there is no further utility to change, so Skip strictly dominates Maintain. Both skip at every final-round history under subgame-perfect play. In round 5, tomorrow's mutual Skip gives the same continuation whatever happens today. It therefore cannot reward today's maintenance or punish today's skip. Skip remains optimal. Repeat the reasoning backward to round 1. This argument assumes the stated complete information and unique stage-game equilibrium. It is not a theorem that all finite relationships must show no cooperation.
The unique-equilibrium condition matters because different credible final-round outcomes could otherwise supply different continuation rewards. To see the missing possibility, consider a paper game with two stage equilibria giving a player 3 or 1. If earlier compliance selects the former and deviation selects the latter, tomorrow's equilibrium payoff can depend on today's history even though tomorrow is final. Whether such a construction actually works requires all players' deviation checks in that new game. Our maintenance game has no such pair of stage equilibria, so this escape from the backward argument is unavailable here.
A known final round supplies no conditional reward
Decision date
Opponent Maintains
Opponent Skips
Continuation after today
Final round: Row Maintain
3
0
None
Final round: Row Skip
5
1
None
Penultimate: Row Maintain
3 + δ
0 + δ
Final-round Row utility 1
Penultimate: Row Skip
5 + δ
1 + δ
Final-round Row utility 1
Paper finite-horizon game, not the lab trace. Tomorrow's unique equilibrium adds the same discounted utility to every current alternative, leaving today's strict dominance comparison unchanged.
Passing the indefinite-horizon threshold still does not select a unique forecast. Consider both players using always Skip. Against an opponent who skips regardless of history, maintaining gives 0 rather than 1 today and creates no later benefit. Skipping every round is therefore a best response at any allowed delta, including 75%. At that weight, both mutual grim trigger and mutual always Skip are equilibria with different outcomes. The incentive calculation establishes that cooperation can be supported by the specified profile. It does not establish that players coordinate on that profile or believe others will follow it.
Perfect observation is also part of the mechanism. As a paper accounting example, suppose mutual maintenance occurs in round 1 but both players mistakenly receive a public report of Skip afterward. If their rule reacts to that report, mutual Skip begins in round 2. At delta 0.75, that conditional path gives each 3 now plus 3 in discounted future punishment, totaling 6 rather than 12. This is not an equilibrium solution for noisy monitoring. It shows how a mistaken trigger can reduce utility when permanent punishment responds to reports rather than correctly observed actions.
A shorter punishment needs a fresh strategy and fresh continuation checks. Suppose, as a paper incentive comparison only, deviating gains 2 now and causes exactly one future round's loss of 2 before cooperation resumes. The discounted loss is just 2 times delta. For every delta allowed by the lab, that amount is below 2, so this particular comparison cannot deter the deviation. Permanent punishment accumulates losses over many rounds; replacing it with one round is not a cosmetic change. This calculation alone does not define or verify a complete forgiving equilibrium strategy.
One lost round cannot replace an indefinite loss stream here
Comparison at delta 0.75
Arithmetic
Present utility difference
Immediate deviation gain
5 − 3
2
One punishment round loss
2 × 0.75
1.5
Permanent punishment loss
2 × 0.75 ÷ 0.25
6
Gain minus one-round loss
2 − 1.5
0.5 profitable
Gain minus permanent loss
2 − 6
−4 unprofitable
Paper deviation comparison at delta 0.75. Both punishments lose two per affected round. One delayed loss totals 1.5; permanent losses total six, exceeding today's gain.
Finally, two players might jointly prefer restarting cooperation after punishment begins, although neither benefits from maintaining alone against Skip. Joint revision is different from a unilateral deviation, so subgame perfection does not automatically resolve renegotiation. A real recommendation would need evidence about observation, beliefs, options to leave, and what future behavior participants can sustain. It would also need an independent judgment about acceptable sanctions. The Yale repeated-game passages support checking credible continuations and distinguishing finite from indefinite horizons. Their illustrative payoffs differ from ours, so their numerical thresholds cannot replace the calculations for this maintenance game.
Exercises
Check the threshold without a slogan
At delta = 0.4, compare maintaining forever with deviating now and then receiving punishment.
Hint
Use 3 / 0.6 and 5 + 0.4 / 0.6.
Worked answer
Plan
Current utility
Discounted future
Total
Maintain
3
2
5
Deviate
5
2/3
17/3
The current gain 2 exceeds the future loss 4/3.
Maintaining gives 5. Deviating gives 5 plus two thirds, about 5.667. The deviation is profitable, so the proposed grim-trigger cooperation is not an equilibrium at this delta. Calling the relationship repeated is insufficient; the continuation must matter enough.
Does the threshold guarantee cooperation?
At delta = 0.75 the condition passes. A report concludes that everyone must cooperate and permanent punishment is safe with noisy monitoring. What is wrong?
Hint
Look for a second equilibrium and an omitted observation assumption.
Worked answer
Claim
What the calculation establishes
Cooperation is possible as this equilibrium
Yes, under the stated assumptions
Cooperation is the only equilibrium
No
Noise causes no problem
Not analyzed
Feasibility of one equilibrium is not a behavioral guarantee.
Always Skip remains an equilibrium because Skip best responds to Skip in every round. Passing the threshold therefore does not select cooperation uniquely. The derivation assumes perfect monitoring. Observation errors can trigger punishment without deliberate deviation; a noisy-monitoring analysis needs new strategies and incentive checks.
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 grim-trigger incentive comparisons and credible continuation behavior. Perfect monitoring and an indefinite horizon are assumptions, not facts about relationships.