Zero-Sum Game
What is Zero-Sum Game?
A zero-sum game is a situation where one player's gain exactly equals another player's loss, so the total is unchanged.
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.
Zero-Sum Game: a worked example
Two firms split a fixed pool of 500,000 customers, 250,000 each if they behave identically. Each can run an ad campaign, and if one advertises while the other does not, the advertiser pulls 80,000 customers across. In customers gained the payoffs are (0, 0) when both advertise, (0, 0) when neither does, and (plus 80,000, minus 80,000) or (minus 80,000, plus 80,000) in the mixed cells. Every cell sums to zero, so the fight over customers is zero sum. Now attach money. A customer is worth $30 of profit and a campaign costs $2 million. The advertiser in a mixed cell nets 80,000 times $30, or $2.4 million, minus the campaign, leaving plus $0.4 million, while the rival loses $2.4 million, so that cell sums to minus $2 million. Advertising is dominant for both, so both advertise and each burns $2 million for nothing. Customers were zero sum. Profits were not.
The mistake students make with zero-sum game
The habit is to call any competitive situation zero sum, and international trade is where it bites hardest. Because one country's surplus is another's deficit, the accounting looks like a fixed pie, so students conclude that an exporter's gain must be an importer's loss. Balances net out by construction, but gains from trade do not. Comparative advantage lets both countries consume beyond their own production possibilities, so the combined bundle of goods available is larger after trade than before. A trade balance records who shipped what, not who came out ahead.
Zero-Sum Game questions
How do you tell if a game is zero-sum?
Add the two players' payoffs cell by cell across the whole matrix. If every cell totals zero, the game is zero sum, and whatever one player gains the other loses exactly. If the totals differ from cell to cell, the size of the pie depends on the choices the players make, so the game is not zero sum. Run the test on the payoffs the players actually care about, since a fight over market share can be zero sum in customers while the profits attached to it are not.
Is the prisoner's dilemma a zero-sum game?
The prisoner's dilemma is not zero sum. Mutual cooperation produces a larger combined payoff than mutual defection, so the total changes depending on what the players choose. That variation is the whole point of the game, since both players end up worse off together when each follows a dominant strategy. A zero sum game has no shared outcome to lose, because any improvement for one player is subtracted from the other.
Is a zero-sum game the same as a constant-sum game?
Constant sum is the general case and zero sum is the special one. A game is constant sum when every cell's payoffs add to the same total, say a market of 100 units split between two firms, and zero sum when that shared total happens to be zero. Adding a fixed number to one player's payoff in every cell changes no best response, so the two types are strategically identical. Rewriting a constant sum game as deviations from an even split turns it into a zero sum one.
Related terms
Common comparisons
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