Collusion and Deviation in Repeated Games

Why Firms Compare Today’s Gain With Tomorrow’s Cost

In a one-period interaction, a firm may be able to increase its profit by undercutting a rival. If the firms meet repeatedly, however, a one-period gain can lead to lower profits in later periods: rivals may respond by abandoning cooperation. The strategic question is therefore not just whether undercutting raises this period’s profit. It is whether that gain is large enough to outweigh the discounted loss from the resulting punishment.

The analysis in this tutorial uses a repeated game with a collusive action, a one-period deviation, and a trigger punishment. This setup lets us compare the two relevant paths: maintain collusion every period, or deviate now and receive the punishment payoff thereafter. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: In an indefinitely repeated game, the same firms play a stage game in each period, with no predetermined final period. Let $\beta$, where $0<\beta<1$, be the common repeated-game discount factor. A payoff of $\pi$ received $t$ periods from now has present value $\beta^t\pi$. Let $\pi_C$ be a firm’s profit per period when all firms maintain the collusive actions, $\pi_D$ its profit in the current period from its best one-period deviation while rivals collude, and $\pi_N$ its per-period profit after the trigger punishment begins.

The trigger strategy considered here is straightforward: firms use the collusive actions as long as no deviation is observed; after a deviation, they switch to a punishment action forever. For the comparison to establish incentives in a repeated-game equilibrium, deviations must be observable, and the punishment action must be credible. A standard case is one in which the punishment repeats a Nash equilibrium of the stage game. The calculations also assume that payoffs remain the same from period to period and that firms evaluate future payoffs using the same $\beta$.

In this model, $\pi_D$ is the deviating firm’s payoff in the deviation period, not its payoff after punishment starts. The punishment-period payoff is $\pi_N$. Keeping those two payoffs separate is essential: the deviator may earn an unusually high profit now and then face a lower profit in each later period.

Two Paths: Comply or Deviate

If a firm complies in every period, it earns $\pi_C$ now, $\pi_C$ next period, and $\pi_C$ in every period thereafter. Its discounted value is the sum of that stream. If it deviates once, it earns $\pi_D$ now and $\pi_N$ in every later period under the trigger strategy.

V_C=\pi_C+\beta\pi_C+\beta^2\pi_C+\cdots =\frac{\pi_C}{1-\beta}

V_D=\pi_D+\beta\pi_N+\beta^2\pi_N+\cdots =\pi_D+\frac{\beta\pi_N}{1-\beta}

These expressions treat the current period as time zero. Thus, the deviation payoff $\pi_D$ is not multiplied by $\beta$, while the first punishment payoff, received next period, is multiplied by $\beta$.

Why the Deviation Test Works

The firm should maintain collusion rather than deviate if the value of compliance is at least as large as the value of deviating once. Subtract the compliance value from the deviation value to identify the incentive to deviate.

\begin{aligned} V_D-V_C &=\pi_D+\frac{\beta\pi_N}{1-\beta} -\frac{\pi_C}{1-\beta}\\ &=(\pi_D-\pi_C) -\frac{\beta(\pi_C-\pi_N)}{1-\beta}. \end{aligned}

The first term, $\pi_D-\pi_C$, is the immediate gain from deviation: the deviator gets $\pi_D$ instead of the current collusive profit $\pi_C$. The second term is the discounted future cost of punishment. In each future period, punishment reduces profit by $\pi_C-\pi_N$ relative to continued collusion; discounting and summing those losses gives $\beta(\pi_C-\pi_N)/(1-\beta)$. The firm prefers compliance when the future cost is at least the immediate gain.

This comparison checks a firm’s incentive to make a one-period deviation from the collusive path. Under the stated repeated-game assumptions and a credible trigger punishment, checking the best one-period deviation is the relevant incentive test: after any history of compliance, the same comparison applies, and after punishment begins, the firms’ repeated punishment actions must themselves be optimal.

Key test: Compare the two complete payoff streams: $V_C=\pi_C/(1-\beta)$ for compliance and $V_D=\pi_D+\beta\pi_N/(1-\beta)$ for a one-period deviation followed by punishment. Collusion passes this firm’s deviation test when $V_C\geq V_D$. Equality means the firm is indifferent; strict inequality means it strictly prefers compliance.

Worked Example: A Large Future Loss Makes Undercutting Unattractive

Two hypothetical suppliers compete for weekly contracts. If both maintain their collusive actions, each earns $\$12$ thousand per week. A supplier that undercuts while its rival continues to collude earns $\$19$ thousand that week. After an observed deviation, the trigger punishment gives each supplier $\$5$ thousand per week indefinitely. Let $\beta=0.80$. Does a supplier prefer to maintain collusion or deviate once?

Method. Calculate the present value of the compliance stream and the deviation-then-punishment stream, using thousands of dollars throughout.

1. Compliance value. The firm receives $\$12$ thousand each period:

V_C=\frac{12}{1-0.80} =\frac{12}{0.20} =60

Compliance is worth $\$60$ thousand in present value.

2. Deviation value. The firm receives $\$19$ thousand immediately. Beginning next week, it receives $\$5$ thousand per week:

V_D=19+\frac{0.80(5)}{1-0.80} =19+\frac{4}{0.20} =19+20 =39

The deviation path is worth $\$39$ thousand in present value.

3. Decision. Since $60>39$, the firm prefers compliance. The current gain from deviation is $19-12=7$ thousand. The discounted future loss is $0.80(12-5)/0.20=28$ thousand, so the net advantage of deviating is $7-28=-21$ thousand. The two methods agree: deviation lowers present value by $\$21$ thousand.

Common mistake avoided: Counting the deviation payoff as if it were earned every week. The $\$19$ thousand applies only in the deviation period; subsequent periods pay the punishment amount.

Separate the One-Time Gain From the Continuing Loss

A useful way to organize the comparison is to calculate the immediate gain and future loss separately. This makes clear why a high deviation payoff alone does not determine whether a firm will undercut. A punishment matters only to the extent that it changes future payoffs, and future losses matter less when they are heavily discounted.

The future cost begins in the period after deviation. It is therefore $\beta$ times a stream of per-period losses, not the undiscounted value of those losses. If $\pi_N=\pi_C$, the specified punishment creates no profit loss relative to continued collusion. If $\pi_D\leq\pi_C$, there is no immediate profit gain from the proposed deviation, though a firm could still compare the full payoff streams if other consequences are present.

Worked Example: A Firm Is Indifferent at the Given Payoffs

Two hypothetical regional distributors receive $\$10$ thousand each per month when they maintain coordinated pricing. One distributor can raise its current-month profit to $\$16$ thousand by secretly offering a lower price; this is detected and triggers permanent competition, under which each earns $\$4$ thousand per month. Let $\beta=0.50$. Compare the two paths and state what the result implies.

Method. Use the discounted-value comparison, with the current deviation payoff received immediately and punishment starting next month.

1. Compliance value.

V_C=\frac{10}{1-0.50} =\frac{10}{0.50} =20

2. Deviation value.

V_D=16+\frac{0.50(4)}{1-0.50} =16+\frac{2}{0.50} =16+4 =20

3. Incentive comparison. The values are equal, so the distributor is indifferent between compliance and deviating once. The immediate deviation gain is $16-10=6$ thousand. The discounted future cost is $0.50(10-4)/0.50=6$ thousand. The gain and cost exactly offset.

Indifference does not mean that deviation is strictly unprofitable. It means this firm has no strict preference between the two paths under these assumptions. Other strategic considerations or small changes in payoffs could change its choice.

Common mistake avoided: Discounting the current $\$16$ thousand deviation payoff. It is received now, so only payoffs beginning next month are multiplied by $\beta$.

Test Each Firm’s Incentive

Firms in the same market need not have the same collusive profit, deviation opportunity, or punishment profit. A favorable comparison for one firm does not establish that all firms will comply. Each firm must compare its own value from following the agreed actions with its value from its best deviation.

Worked Example: One Firm’s Incentive Can Be the Weak Link

Two hypothetical producers have different costs. When both maintain their collusive actions, producer A earns $\$20$ thousand per quarter and producer B earns $\$14$ thousand. If A deviates while B complies, A earns $\$29$ thousand that quarter; if B deviates while A complies, B earns $\$23$ thousand. Once either deviation triggers the punishment, A earns $\$8$ thousand and B earns $\$6$ thousand each quarter. Let $\beta=0.75$. Determine whether each firm prefers compliance to its own one-period deviation.

Method. Calculate $V_C$ and $V_D$ separately for A and B. Use each firm’s own payoff numbers; the punishment is triggered for both firms after a deviation.

1. Producer A’s compliance value.

V_{C,A}=\frac{20}{1-0.75} =\frac{20}{0.25} =80

2. Producer A’s deviation value.

V_{D,A}=29+\frac{0.75(8)}{0.25} =29+\frac{6}{0.25} =29+24 =53

A prefers compliance because $80>53$. Its current gain from deviating is $29-20=9$ thousand, while its discounted future loss is $0.75(20-8)/0.25=36$ thousand.

3. Producer B’s compliance value.

V_{C,B}=\frac{14}{0.25}=56

4. Producer B’s deviation value.

V_{D,B}=23+\frac{0.75(6)}{0.25} =23+\frac{4.5}{0.25} =23+18 =41

B also prefers compliance because $56>41$. Its immediate gain is $23-14=9$ thousand, and its discounted future loss is $0.75(14-6)/0.25=24$ thousand.

Both firms pass their own deviation tests. In another set of payoffs, one firm might pass while the other fails; then the collusive plan would not be stable against unilateral deviation, even if the firm that passes has no desire to undercut.

Common mistake avoided: Using one firm’s punishment loss or collusive profit for both firms. With asymmetric payoffs, the incentive calculation is firm-specific.

Common Traps in the Comparison

Discounting the current-period deviation payoff. The deviator receives $\pi_D$ now. The first punishment payoff is received next period, so its present value begins with $\beta\pi_N$.
Comparing only current profits. A deviation may raise current profit and still reduce the full value of the payoff stream once punishment is included.
Using the deviation payoff as the punishment payoff. These are different parts of the path: $\pi_D$ is earned once, while $\pi_N$ is earned after punishment starts.
Ignoring credibility. A threatened punishment cannot support the comparison if rivals would not actually carry it out. The punishment action must be a credible continuation of the game.
Checking only one firm. Each firm’s own collusive, deviation, and punishment profits matter. A firm with a particularly attractive deviation may be the one that breaks cooperation.
Interpreting indifference as strict compliance. If $V_C=V_D$, the firm is indifferent; compliance is not strictly better on the basis of these payoffs alone.

The comparison is useful because it separates the incentive problem into two concrete components: the extra profit from deviating now and the future profit sacrificed if punishment follows. In a market application, those components depend on the firms’ payoff assumptions and on whether deviations can be observed and punished credibly. The calculation is a model of strategic incentives, not a claim about any particular firm or an assessment of the legality of a business practice.

Key takeaway: A firm has an incentive to deviate when its immediate gain exceeds the discounted loss from the punishment that follows. To assess collusion in a repeated game, compare the complete payoff streams and perform the calculation separately for every firm whose compliance matters.

Check Your Understanding

For each problem, compare the value of compliance with the value of deviating once and then receiving the stated punishment payoff. Payoffs are in thousands of dollars per period unless noted otherwise.

A firm earns $9$ per period under collusion, can earn $15$ in the current period by deviating, and then earns $3$ per period under permanent punishment. If $\beta=0.75$, which path has the higher value? (Answer: $V_C=36$, $V_D=24$; compliance is better by $12$.)
A supplier’s collusive payoff is $11$ per period, its one-period deviation payoff is $17$, and its punishment payoff is $5$. With $\beta=0.50$, is the supplier strictly better off complying? (Answer: $V_C=22$, $V_D=22$; it is indifferent, so compliance is not strictly better.)
A firm receives $14$ per period under collusion, $22$ in the deviation period, and $6$ per period after punishment. If $\beta=0.60$, calculate both values and identify the preferred path. (Answer: $V_C=35$, $V_D=31$; compliance is better by $4$.)
Two firms share $\beta=0.50$. Firm A’s collusive, deviation, and punishment payoffs are $18$, $27$, and $9$. Firm B’s are $12$, $21$, and $5$. Which firms prefer compliance, and does the collusive plan pass both firms’ deviation tests? (Answer: A has $V_C=36$ and $V_D=36$, so it is indifferent; B has $V_C=24$ and $V_D=26$, so it prefers deviation. The plan fails B’s test.)
A firm earns $20$ per period under collusion, $26$ in the deviation period, and $10$ per period under punishment. Let $\beta=0.80$. Calculate the immediate gain and discounted future cost, then compare the complete values. (Answer: The immediate gain is $6$; the discounted future cost is $40$. $V_C=100$ and $V_D=66$, so compliance is better by $34$.)

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