Repeated Game vs Sequential Game
Repeated Game and Sequential Game are two Game Theory & Information concepts in AP Economics that students often mix up. A repeated game is the same game played again and again by the same players, so cheating today can be punished later and cooperation becomes possible. A sequential game is one where players move in turns and later movers see what came before, so it is drawn as a game tree and solved backward from the end. Here is how they compare side by side.
In a one-shot game, players only weigh today's payoff, so a tempting defection carries no cost. Repeat the game and each player's move becomes a signal about how the rest of the relationship will go, which lets strategies such as tit-for-tat or a grim trigger punish a cheat in every later round. Cooperation holds when the stream of future losses from being punished outweighs the one-time gain from cheating, so it depends on players being patient and on the game not having a known final round. This is how oligopolists sustain high prices without a written agreement, and it explains why the same firms behave differently in a one-time deal than in an ongoing supply relationship.
Because the second player observes the first player's move before choosing, a sequential game is drawn as a tree of decision nodes rather than a table. You solve it by backward induction: start at the last decision, pick the move that pays that player most, cross out the branches they would never take, then step back one node and repeat with those later choices treated as settled. The result is a path through the tree that no player wants to leave, which is why order of play can change the outcome completely. This is the difference from a simultaneous game shown in a payoff matrix, where neither side knows the other's move; here the first mover can act on the knowledge that their choice will be seen.
Repeated Game vs Sequential Game: Two Different Ways a Game Gets a Timeline
| Repeated Game | Sequential Game | |
|---|---|---|
| What unfolds over time | The same whole game is played again and again | One game in which players take their turns in order |
| What you can see when you move | The history of earlier rounds, but not this round's move | The moves already made inside this same game |
| How it is drawn | A single payoff matrix, played repeatedly | A tree with branches for each decision |
| How it is solved | Weigh the one-off gain from cheating against the punishment that follows | Backward induction, starting from the final decision |
| What a strategy is | A rule for reacting to what the rival did before | A plan naming a move at every point you might be asked to move |
| What it explains | Why rivals can sustain cooperation without a contract | Why moving first can pay and which threats a rival believes |
| Why the ending matters | A known last round unravels cooperation backwards | The last node is where the solution begins |
Repetition changes the incentives; sequence changes the information
Start with a matrix, payoffs written as (row player, column player). If both cooperate, each gets 3. If both defect, each gets 1. If one defects while the other cooperates, the defector gets 5 and the cooperator gets 0. Played once, defecting wins whatever the rival does, since 5 beats 3 against a cooperator and 1 beats 0 against a defector. Both defect and both collect 1, and neither gains by switching alone, which is what makes that pair an equilibrium. Now play the same matrix every week against the same rival. Cheating still gains 2 today, the gap between 5 and 3. But if the rival answers by defecting from then on, the cheat loses 2 every week after, the gap between 3 and 1. Weigh a single gain of 2 against 2 a week forever, and cooperation survives so long as the next round carries at least half the weight of this one. The matrix never changed. Repetition changed what a strategy is, because a strategy is now a rule for responding to history, such as the one described at /glossary/tit-for-tat. A sequential game does something else entirely: it lets one player watch the other move inside the same game.
A known final round eats cooperation from the end backwards
Sequence is solved from the end. In a tree, you find the last player's best move at each final decision, replace that decision with the payoffs it produces, and step back one level with the game now shorter. Repeat until you reach the opening move. That is /glossary/backward-induction, and it is what separates threats a rival believes from threats a rival ignores, since a threat only survives if carrying it out is still the best move once the moment arrives. The same backwards reasoning has a bite for repeated games with a fixed and known number of rounds. In the final round there is no future left to protect, so both players defect. Knowing that the last round is lost anyway, the round before it has nothing left to protect either, so both defect there too. The argument runs all the way back to round one, and cooperation never gets started. This is why the interesting repeated games are the ones with no known ending, or with a chance of continuing each round. Uncertainty about the end is what gives the future enough weight to hold a bargain together.
Frequently asked questions
Is a repeated game the same as a sequential game?
No. A repeated game is the same game played over and over by the same players, usually with simultaneous moves inside each round, while a sequential game is one game in which players move in turn and later movers see what came before. A game can be both, such as a bargaining game with turns that the same two firms replay every year.
Why does cooperation break down when players know which round is the last?
Because the last round has no future to protect, so both players defect in it. That makes the second to last round effectively the last one that matters, so they defect there too, and the reasoning unwinds all the way back to the first round.
How do you solve a sequential game?
Use backward induction: start at the final decisions, pick the best move for whoever is choosing there, replace each of those branches with the payoffs it delivers, and work back one stage at a time to the opening move. The path you trace out is the predicted play, and it automatically drops any threat the threatener would not actually want to carry out.
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