Why Repetition Changes the Strategic Problem
In a one-shot game, each player chooses an action without expecting another round of interaction with the same players. In a repeated game, the stage game is played more than once. A player’s choice today can affect not only today’s payoff but also what other players choose in later rounds. The possibility of future consequences is the shadow of the future.
Repetition does not automatically produce cooperation or make a particular outcome an equilibrium. Its strategic effect depends on what players observe, how their strategies respond to what they observe, and how much they value future payoffs. The earlier material on Nash equilibrium and the Prisoner’s Dilemma provides the one-shot tools; here the central change is that a player chooses a plan for responding to different histories, not merely an action for one moment.
A strategy is therefore different from an action. “Choose C” names an action. “Choose C initially, and choose D next if the other player chose D last period” specifies a history-dependent strategy. To evaluate such a strategy, the model must say which actions are observable. In the examples below, actions are publicly observed, so both players can condition on the same history. If actions were hidden or observed with error, a different model would be needed.
Payoffs Across Periods
A repeated-game payoff accounts for the stream of stage-game payoffs, rather than only the payoff in one round. Let \(u_t\) be a player’s payoff in period \(t\), and let \(\beta\) be that player’s per-period discount factor, with \(0 \leq \beta < 1\). The discounted payoff over \(T\) periods is the period-1 present value of the payoff stream:
A period-1 payoff is not discounted; a period-2 payoff is multiplied by \(\beta\); a period-3 payoff is multiplied by \(\beta^2\). A higher \(\beta\) means future payoffs receive greater weight. Here \(\beta\) is a game-theory discount factor, not the actuarial force of interest \(\delta\) or the discount factor \(v\) used in interest calculations.
If a model assumes that the same payoff \(u\) is received in every period indefinitely, and \(0 \leq \beta < 1\), the discounted sum is \(u/(1-\beta)\). That calculation assumes the payoff continues forever and that the discount factor stays constant; it should not be applied to a finite stream or to payoffs that vary over time.
Worked Example: Discounting a Sequence of Outcomes
Two firms choose either C or D in each round. Their stage-game payoffs, listed as (Firm A, Firm B), are (4, 4) for (C, C), (0, 7) for (C, D), (7, 0) for (D, C), and (2, 2) for (D, D). For the first three observed rounds, the outcomes are (C, C), (D, C), and (D, D). If Firm A’s discount factor is \(\beta=0.9\), calculate its discounted payoff over these three rounds and compare it with receiving 4 in each round.
Method. Identify Firm A’s stage payoff in each round, then multiply the payoff in round \(t\) by \(\beta^{t-1}\). This is a three-round accounting comparison; it does not assume the relationship ends after round 3.
Firm A’s payoffs are 4, 7, and 2. The discounted payoff is
If Firm A instead received 4 in each of the three rounds, its discounted payoff would be
The difference is \(11.92-10.84=1.08\). By round, the difference is 0 in round 1, \(0.9(7-4)=2.7\) in round 2, and \(0.9^2(2-4)=-1.62\) in round 3. Thus the early gain exceeds the later loss by \(1.08\) in this observed window. The comparison does not establish that either outcome follows from an equilibrium strategy.
Common mistake avoided: adding the undiscounted payoffs \(4+7+2\) and comparing that sum with \(4+4+4\). That ignores the stated discount factor, even though the two paths have different payoffs in different periods.
How Strategies Use History
A history-dependent strategy can prescribe different actions after different observed sequences. One simple illustration is a copying rule: each firm chooses C in the first round and, in each later round, chooses the action its opponent chose in the immediately preceding round. The rule uses only the most recent action profile, but it is still a complete plan for what to do after every possible history.
In the payoff matrix above, (D, C) and (C, D) give the two firms different payoffs. Under the copying rule, one firm’s action can therefore affect the other firm’s next action. The next example follows this logic. It calculates a path under a specified rule; it does not claim that the rule is an equilibrium.
Worked Example: Following a History-Dependent Rule
Use the same payoff matrix as in the previous example. Both firms choose C in round 1. In later rounds, each chooses the action its opponent chose in the previous round. In round 2, Firm A chooses D instead of the action prescribed by its rule; Firm B follows its rule. With \(\beta=0.8\), compare Firm A’s discounted payoff over rounds 1 through 4 with the payoff from (C, C) in all four rounds.
Method. First use the observed history to determine each prescribed action after round 1. Then use the payoff matrix to obtain Firm A’s payoffs and discount them. This avoids treating an action in round 2 as if it were the firm’s entire strategy.
In round 1 the outcome is (C, C), so Firm A receives 4. In round 2, A chooses D and B copies A’s round-1 action C, giving (D, C); A receives 7. In round 3, A copies B’s round-2 C while B copies A’s round-2 D, giving (C, D); A receives 0. In round 4, A copies B’s round-3 D while B copies A’s round-3 C, giving (D, C); A receives 7.
With (C, C) in all four rounds, A receives 4 each round:
The difference is \(13.184-11.808=1.376\). The specified copying rule creates alternating outcomes after the round-2 deviation within this four-round calculation. That response pattern is a consequence of the rule, not a general property of repeated games.
Common mistake avoided: assuming that after A chooses D once, A must choose D in every later round. The rule instead says to copy the opponent’s last action, so each later action must be derived from the history.
Deriving the Value of Future Consequences
A useful way to study the shadow of the future is to compare an immediate deviation gain with the expected discounted payoff loss in later periods. The following calculation needs additional assumptions: the relationship continues from one period to the next with a constant probability \(q\), continuation events are independent across periods, and a specified history-dependent response causes a constant per-period payoff loss \(L\) for the player who deviated, whenever the relationship continues. The player’s discount factor is \(\beta\).
The probability that the relationship continues through \(k\) transitions is \(q^k\). A loss received \(k\) periods from now is also discounted by \(\beta^k\). Its expected discounted contribution is therefore \(L(q\beta)^k\). Summing the losses from all future periods gives
To justify the series, set \(x=q\beta\). The first \(n\) terms satisfy
When \(0 \leq x < 1\), \(x^n\) tends to 0 as \(n\) grows. The sum therefore tends to \(x/(1-x)\). Multiplying by \(L\) gives the expected discounted loss above. This argument requires a constant loss and continuation probability; it does not apply unchanged if future losses vary, continuation depends on history, or the relationship has a fixed end date.
The comparison is informative but not, by itself, a proof of equilibrium. To conclude that a strategy profile is an equilibrium, one must also check players’ incentives after the relevant histories and the strategic credibility of the prescribed responses.
Worked Example: Comparing a Deviation Gain with Future Losses
In a hypothetical ongoing business relationship, mutual cooperation gives a firm a payoff of 4 per period. A one-time deviation raises its current payoff to 7. Suppose the specified response to that deviation reduces the firm’s payoff by 2 in each future period that the relationship continues. The probability of continuing from one period to the next is \(q=0.8\), and the firm’s discount factor is \(\beta=0.875\). Compare the immediate gain with the expected discounted future loss.
Method. Calculate the gain in the current period, then use the geometric-series formula for the future loss. This treats continuation risk and discounting as distinct factors.
The current gain is \(7-4=3\). The combined factor for a loss one period ahead is
The expected discounted future loss is
to four decimal places. The net comparison is \(3-4.6667=-1.6667\), so the specified future loss exceeds the immediate gain by about \(1.6667\) payoff units. Under these assumptions, the deviation has a lower payoff when the full stream is counted than when only the current period is considered.
Common mistake avoided: counting only the loss in the next period, or applying the factor \(q\beta\) just once. Future periods contribute too, with successive factors \(0.7^2, 0.7^3,\) and so on.
What the Shadow of the Future Does—and Does Not—Show
The examples illustrate two distinct effects. First, a strategy can condition on history, so today’s actions can change tomorrow’s prescribed actions. Second, payoffs in later periods can matter when players value continued interaction. Both effects are needed to assess whether a particular history-dependent plan changes incentives.
A practical analysis should keep the following elements separate:
- Stage-game payoff: the payoff generated by the action profile in one period.
- History: the observed actions that the strategy is allowed to use.
- Strategy: the action prescribed after each relevant history.
- Continuation assumptions: how likely future periods are and how their payoffs are valued.
A common trap is to infer that repetition makes cooperation automatically rational. It does not. A future consequence matters only if a strategy actually prescribes a response after the observed history, the player’s future payoff is sufficiently important, and the response is credible in the relevant decision situation. Nor should a discounted payoff comparison be mistaken for a complete equilibrium analysis: the comparison evaluates the stipulated path and assumptions, while equilibrium requires checking incentives for all possible deviations that matter.
Check Your Understanding
For each problem, show how the history or payoff stream leads to the answer.
- A player’s payoffs over three observed rounds are 5, 3, and 1. If \(\beta=0.9\), calculate the discounted payoff over those rounds. (Answer: \(5+0.9(3)+0.9^2(1)=8.51\))
- Both firms choose C in round 1. In every later round, each copies its opponent’s action from the preceding round. If the round-2 outcome is (D, C), what outcomes are prescribed in rounds 3 and 4? (Answer: round 3 is (C, D); round 4 is (D, C))
- A deviation gives a player 4 extra payoff units now. Under a specified response, the player loses 1 unit in each of the next two periods. If \(\beta=0.8\), what is the net discounted gain over these three periods? (Answer: \(4-0.8(1)-0.8^2(1)=2.56\) units)
- A relationship continues each period with probability \(q=0.6\). A player’s discount factor is \(\beta=0.75\), and the specified future response imposes a loss of 3 units in each period the relationship continues. Find the expected discounted future loss. (Answer: \(3(0.6)(0.75)/[1-(0.6)(0.75)]=2.4545\) units, to four decimal places)
- A deviation increases the current payoff by 3 units. If it causes a constant future loss of 2 units per period, with continuation probability \(q=0.8\) and discount factor \(\beta=0.875\), does the discounted comparison favor the deviation? (Answer: no; the future loss is \(2(0.7)/(1-0.7)=4.6667\) units, exceeding the gain by \(1.6667\) units)