How to Calculate Whether Cooperation Pays in a Repeated Game
Cooperation pays when the total from holding the agreement over the rounds still to come beats the one-off cheating payoff plus the punishment payoffs that follow it.
The Repeated Game Payoffs formula
Calculator
Enter the cooperate, cheat and punishment payoffs plus the rounds still to come to see which path pays more.
What each side earns per round while the agreement holds.
The one-off reward for undercutting while the rival still cooperates.
What each side earns per round after retaliation starts.
Count the current round. This is the shadow of the future.
Cooperating beats cheating by $9,000, so the threat of punishment is doing its job.
- Payoff from holding the agreement
- $45,000
- Payoff from cheating now
- $36,000
- Break-even rounds
- 2
- Critical discount factor
- 0.5
- Verdict
- Cooperation holds
Staying with the agreement for every remaining round is worth $45,000.
Cheating this round and then living with retaliation is worth $36,000.
Cooperation only pays with at least 2 rounds still to come, rounding up when the answer is not a whole number.
With no known end date, cooperation survives when each side values the next round at least 0.5 times as much as this one.
The stream of future losses outweighs the one-off gain, which is why rivals can hold a price without any written agreement.
How to calculate Repeated Game Payoffs, step by step
- 1Write down the three payoffs. You need the payoff per round while both sides hold the agreement, the one-off payoff from cheating while the rival still cooperates, and the payoff per round once the rival retaliates.
- 2Count the rounds still to come. Include the current round. What matters is how many rounds each side expects to keep playing, not how many have already been played.
- 3Total up the cooperating path. Multiply the cooperate payoff by the rounds still to come, since nothing changes while both sides hold the line.
- 4Total up the cheating path. Take the cheat payoff for this round, then the punishment payoff in every round after it: cheat payoff + punishment payoff × (rounds still to come − 1).
- 5Compare the two totals. Cooperation holds when its total is larger. The gap measures how much the relationship is worth against a quick one-off win.
- 6Find the break-even horizon. Rounds needed = (cheat payoff − punishment payoff) ÷ (cooperate payoff − punishment payoff). With fewer rounds left than that, cheating pays.
Worked example: Repeated Game Payoffs
Two suppliers each earn $9,000 a round while both hold a high price. Undercutting wins the whole market for one round and pays $12,000, after which the rival matches the low price and both settle at $6,000. With five rounds still to come, holding the price pays 5 × $9,000 = $45,000 while cheating pays $12,000 + 4 × $6,000 = $36,000, so cooperation wins by $9,000. The break-even horizon is (12,000 − 6,000) ÷ (9,000 − 6,000) = 2 rounds, and with exactly two rounds left both paths pay $18,000. In the open-ended version the one-off gain is 12,000 − 9,000 = $3,000 and the loss per round of punishment is 9,000 − 6,000 = $3,000, so the critical discount factor is 3,000 ÷ 6,000 = 0.5. Cooperation survives when each side values the next round at least half as much as this one.
Repeated Game Payoffs questions
Does cooperation still hold if both players know the final round?
No. In the last round there is no future left to protect, so both defect. Knowing that, the round before it carries no threat either, and the reasoning unravels back to the first round. Cooperation needs an open-ended horizon, or real uncertainty about when the relationship ends.
What is a grim trigger strategy?
Cooperate until the other side cheats once, then punish in every round that follows and never return to the agreement. It is the harshest credible threat, which makes it the standard benchmark. Tit-for-tat is the forgiving alternative: punish once, then cooperate again if the rival does.
Why does the cheating path use rounds minus one?
Because the cheat payoff already covers the current round. Retaliation starts in the next round, so punishment runs for the rounds still to come minus the one you cheated in.
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