Zero-Sum Game Definition in Finance, With Examples
By Markets & Investing DeskUpdated Nov 10, 2026
Definition
A zero-sum game is a situation where one participant's gain exactly equals another's loss, so all gains and losses add up to zero. Poker, chess and a single futures contract are classic examples.
Key takeaways
- In a zero-sum game, the payoffs of all players sum to zero in every possible outcome, so one side wins only what the other side loses.
- Derivative contracts such as futures and options are the closest real financial example, and fees push the traders' combined result below zero.
- Positive-sum games, such as voluntary trade or owning productive businesses, can leave every participant better off. Negative-sum (lose-lose) games leave the group worse off.
A zero-sum game is a situation in which one participant’s gain is exactly matched by another participant’s loss, so the total gains and losses of everyone involved add up to zero. Chess, poker and a single futures contract are standard examples. Most ordinary trade and long-term investing are not zero-sum, because both sides can end up better off.
What zero-sum means
The name comes from simple arithmetic. If you add the winnings of the winners to the losses of the losers (counting losses as negative numbers), the result is zero. A winner who gains +3 might be matched by two losers at −1 and −2: 3 − 1 − 2 = 0.
Three features define the situation:
- A fixed total. The amount at stake does not grow or shrink because of how the players act. A poker pot is redistributed, not enlarged.
- Strictly opposed interests. Any move that improves one player’s outcome worsens another’s by the same amount. There is no shared gain to bargain over.
- Redistribution, not creation. The game can change who holds the value, but the system as a whole ends where it started.
A zero-sum game can have many participants. A futures market may involve millions of traders, yet each contract still links one buyer to one seller, and the combined result across all contracts is a transfer among participants rather than an increase in total wealth.
The formula
For a game with players 1 through n, let uᵢ be the payoff to player i for a given outcome. The game is zero-sum if, for every possible outcome:
u₁ + u₂ + … + uₙ = 0
The condition must hold for every outcome, not just on average or in the result that actually happened. A game whose payoffs always add to the same nonzero number (for example, two players splitting a fixed $100 prize) is called constant-sum. Constant-sum games behave the same way strategically, because subtracting the constant from the payoffs turns them into a zero-sum game without changing anyone’s incentives.
Worked example: matching pennies
Matching pennies is the textbook zero-sum game. Players A and B each place a penny on the table at the same time, choosing heads or tails.
- If the pennies match (both heads or both tails), A takes B’s penny.
- If they do not match, B takes A’s penny.
| B plays heads | B plays tails | |
|---|---|---|
| A plays heads | A +1, B −1 | A −1, B +1 |
| A plays tails | A −1, B +1 | A +1, B −1 |
In each of the four cells, the two payoffs sum to zero. The best strategy for each player is to pick heads or tails at random with equal probability, since any predictable pattern can be exploited by the opponent. Played that way, each player’s expected payoff is zero.
Worked example: a futures contract
Hypothetical example for illustration only. Trader A buys (goes long) one crude oil futures contract covering 1,000 barrels at $70 per barrel. Trader B sells (goes short) the same contract at $70. At expiration the price is $74.
- Trader A gains ($74 − $70) × 1,000 = +$4,000
- Trader B loses ($70 − $74) × 1,000 = −$4,000
- Combined: $0
If the price had fallen to $66 instead, the signs would flip: B gains $4,000 and A loses $4,000. In every price scenario, the contract moves money between the two sides and nothing more.
Now add costs. Assume each trader pays $5 in commissions and exchange fees per side. The traders’ combined result becomes −$10: A nets +$3,990 and B nets −$4,010. The contract itself is zero-sum, but once the broker and exchange take their share, trading it is negative-sum for the traders as a group. The same applies to options and forward contracts, which are settled privately rather than through an exchange.
Zero-sum vs. positive-sum vs. negative-sum
| Type | Sum of payoffs | Typical example |
|---|---|---|
| Zero-sum | Exactly zero | Chess, poker, a single derivatives contract before fees |
| Positive-sum | Greater than zero | Voluntary trade, long-term ownership of productive companies |
| Negative-sum | Less than zero | Derivatives trading after fees, a war, a collapsed negotiation |
The opposite of a zero-sum game is usually taken to be a positive-sum (win-win) game. When you buy an apple for a dollar, you value the apple more than the dollar and the store values the dollar more than the apple. Both sides gain, and the total is above zero. Positive-sum games can still have losers, but the net result across everyone is positive.
A zero-sum game is not the same as a lose-lose situation. In zero-sum, someone wins what someone else loses. In a lose-lose (negative-sum) game, such as diplomatic talks that break down and leave both countries worse off, the total is below zero and there may be no winner at all.
Many of the best-known games in game theory are non-zero-sum, including the prisoner’s dilemma, Cournot competition, the centipede game and deadlock. Players’ interests partly overlap, so cooperation, trust and credible commitments become central questions.
Role in game theory
Game theory studies decision-making among two or more rational parties whose outcomes depend on each other’s choices. Zero-sum games were its first major subject. John von Neumann proved the minimax theorem in 1928, and in 1944 he and Oskar Morgenstern published Theory of Games and Economic Behavior, the founding text of the field.
The minimax theorem says that in a two-player zero-sum game with finite strategies, each player has a strategy (possibly randomized) that guarantees a certain value. One player can secure at least that much, and the other can hold losses to no more than that much. Because the players’ interests are directly opposed, playing to minimize your maximum possible loss is the same as playing to maximize your guaranteed gain.
John Nash extended the analysis to games of any kind in his 1951 paper “Non-Cooperative Games.” A Nash equilibrium is a set of strategies in which no player can do better by changing their own choice while the others keep theirs. In two-player zero-sum games, Nash equilibrium strategies coincide with minimax strategies, so the earlier result fits inside Nash’s broader framework.
Is the stock market a zero-sum game?
Not as a whole. Companies use capital to produce goods and services, earn profits and pay dividends, so long-term owners of stocks share in value that did not exist before. That makes long-term investing positive-sum. Growth in total output, measured by gross domestic product, is the reason the economy overall is not zero-sum.
Short-term trading is closer to zero-sum. When you sell a share to another investor, any gain or loss from that point is measured against what the buyer receives. Relative to the market’s own return, one trader’s outperformance is another’s underperformance. Economist William Sharpe set out this “arithmetic of active management” in 1991: before costs, the average actively managed dollar must earn the market return, because active and passive investors together hold the market. After fees, the average active dollar must trail it. The same holds for hedge funds and other active managers: excess returns (alpha) across all participants sum to zero before costs.
Even a trade that is zero-sum in dollars can benefit both sides in other ways. A wheat farmer who sells futures to lock in a price may lose money on the contract if prices rise, but gains certainty about income. The speculator on the other side takes on that risk for a possible profit. Because their risk needs differ, each may be better off even though the cash flows net to zero.
Zero-sum situations in daily life
Daily life contains plenty of zero-sum moments: two people competing for the last taxi, roommates dividing limited fridge space, colleagues vying for a single promotion.
Treating every situation this way is a common reasoning error. The belief that “life is a zero-sum game” overlooks how trade, specialization and cooperation create value. Many disputes that look like a fight over a fixed pie involve parties who value different things and can both gain from a deal.
Limitations of the concept
- Gains are hard to measure. Real outcomes include risk reduction, time, satisfaction and information, so a trade can be zero-sum in dollars and positive-sum in welfare.
- Costs break the symmetry. Commissions, spreads, fees and taxes make most real “zero-sum” markets negative-sum for participants.
- Strong assumptions. Classical analysis assumes rational players with full information about the rules and payoffs. Real traders have different information, beliefs and constraints.
- Time horizon matters. A single bet may be zero-sum while owning businesses over decades is positive-sum.
FAQs
Does zero-sum mean all or nothing?
Not necessarily, though the two phrases are often used interchangeably. A game is zero-sum if total gains equal total losses. That allows partial outcomes: in poker, several players can win and lose different amounts. "All or nothing" describes a winner-take-all result, which is one kind of zero-sum outcome.
Is money itself zero-sum?
No. New money can be created by central banks and commercial bank lending, and the goods and services money buys can grow as the economy grows. A specific transfer of existing money between two people is zero-sum, but the money supply and the economy are not fixed totals.
The Bottom Line
The distinction matters most to traders and investors deciding where returns come from. Derivatives and short-term trading mostly move money between participants, and fees make them negative-sum for the group, so the average active dollar trails the market after costs. Long-term ownership of productive companies is positive-sum, because businesses create profits and dividends that did not exist before.
