Zero-Sum Game vs Repeated Game
Zero-Sum Game and Repeated Game are two Game Theory & Information concepts in AP Economics that students often mix up. A zero-sum game is a situation where one player's gain exactly equals another player's loss, so the total is unchanged. 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. Here is how they compare side by side.
Poker among friends is roughly zero-sum: winnings equal losses. Many real economic interactions, like voluntary trade, are positive-sum (both gain), which is why framing economics as zero-sum is usually a mistake.
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.
Zero-Sum Game vs Repeated Game: Payoff Structure Against Time Structure
| Zero-Sum Game | Repeated Game | |
|---|---|---|
| What the label classifies | How payoffs are split between the players | How many times the same game is played |
| Can one game be both | Yes, a zero-sum game can be played over and over | Yes, the stage game may or may not be zero-sum |
| Room for mutual gain | None, the joint total never changes | Depends on the stage game, not on the repetition |
| What repetition buys | Nothing, there are no joint gains to protect | Punishment threats that can hold up cooperation |
| Effect of tit-for-tat or a grim trigger | No gain, each player can already guarantee the game's value | Sustains cooperation when the players are patient enough |
| Effect of announcing a final round | Irrelevant, every round plays the same way | Can unravel cooperation all the way back to round one |
| Typical exam setup | Matching pennies, splitting a fixed prize | A prisoner's dilemma played round after round |
Repetition only pays when the single round leaves value unclaimed
Repeating a game adds nothing by itself. Repetition matters when one round has an outcome both players prefer to the equilibrium they reach on their own, because that gap is what a future punishment protects. Put numbers on the standard case. Two firms each earn 6 by holding output down, 3 each when both flood the market, and the one that cheats on the deal earns 10 while the honest firm gets 1. Flooding is a dominant move, so 3 each is the one-shot result even though 6 each is available. Now play it without a known end using a grim trigger, where one cheat means both flood forever after. Cooperating pays 6 every round, worth 6 divided by one minus d, where d is how heavily the next round counts. Cheating pays 10 now and 3 forever after, worth 10 plus 3d divided by one minus d. Cooperation holds when the first is at least the second. Multiply both sides by one minus d and the condition becomes 6 at least 10 minus 7d, so d must be at least 4 over 7, roughly 0.57. Above that threshold the deal survives; below it the one-shot outcome returns. Every step leans on the joint totals differing across cells: 12 when both cooperate, 6 when both cheat, 11 when one does. The collusion setup behind these numbers sits at /micro/oligopoly.
A repeated zero-sum game gives a punishment nothing to protect
Run that same machinery on a game whose two payoffs always add to the same total and it stalls at the first step. No outcome exists that both players prefer to the equilibrium, because any move lifting one player's payoff lowers the other's by exactly as much. A grim trigger has no cooperation to defend and tit-for-tat has nothing to reciprocate. The deeper reason is that each player in a zero-sum game can already guarantee the value of the game single-handedly by playing the equilibrium mix, whatever the opponent does. That guarantee does not rely on the opponent's goodwill, so no threat of future punishment can push a player below it and no promise of future reward can lift anyone above it. The folk theorem still applies, but the set of outcomes it can sustain has collapsed to one point: the value of the game. Repetition does change how people play a zero-sum game from round to round, since a pattern an opponent can read gets exploited, and it does not change what either of them can expect to earn.
Frequently asked questions
Can players cooperate in a repeated zero-sum game?
Cooperation has nothing to buy in a zero-sum game, because the two payoffs add to the same total in every cell and any gain for one player is an equal loss for the other. Repetition supports cooperation by threatening future punishment, and a threat is only worth something when there is a joint gain to protect. Players in a repeated zero-sum game do adapt, mainly by avoiding patterns an opponent could read, but their expected payoff stays at the value of the single-round game.
Is the prisoner's dilemma a zero-sum game?
The prisoner's dilemma is not zero-sum, and the joint totals prove it. With 6 each for mutual cooperation, 3 each for mutual defection, and 10 against 1 when one side cheats, the totals are 12, 6 and 11, so the size of the pie moves with the choices. That variation is the whole reason the game is interesting and the reason repetition can change its outcome. Calling it zero-sum is a common error, probably because the two players are described as adversaries.
Does a repeated game always support cooperation?
A repeated game supports cooperation only under conditions the question has to supply. Players must weight future rounds heavily enough, which in the output example means a discount weight of at least 4 over 7. The horizon usually has to be open-ended, since a known final round unravels the deal from the back. And the stage game must contain a joint gain worth protecting, which rules out every game with a fixed total.
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