Zero-Sum Game

A zero-sum game is a situation in which one participant's gain is exactly balanced by the losses of the others, so the total of all gains and losses is zero. Whatever one side wins, another side must lose. Poker among friends, a bet between two people and, before costs, a futures or options contract are standard examples. The term comes from game theory but is widely used in economics, finance and politics.

The opposite is a positive-sum game, in which total benefits grow and all parties can come out ahead, as in voluntary trade, or a negative-sum game, in which the total shrinks, as in war or a gamble where the house takes a cut. Much economic debate turns on whether an activity is zero-sum: mercantilism treated trade largely as a contest over a fixed stock of wealth, whereas classical economics argued that exchange and specialization create value for both sides. "Zero-sum thinking", the belief that one group can only gain at another's expense, is studied by psychologists and economists as a common way people view the economy.

Imagine a pizza already on the table. However it is cut, an extra slice for one person means one fewer for someone else. Arguing over the slices does not make the pizza bigger. That is a zero-sum situation.

Now imagine two neighbors: one bakes bread, the other grows tomatoes. If they swap, both end up with a better lunch than either could make alone. Nothing was taken from anyone; the total went up. Most everyday buying and selling looks more like the second case. The difference matters because people who see the world as a fixed pizza tend to support different policies from people who think the pizza can grow.

Formally, in a game with players $i = 1, \ldots, n$, every possible outcome $o$ satisfies:

• $u_i(o)$ is the payoff to player $i$ when outcome $o$ occurs.
• $n$ is the number of players.

For example, if Alice and Bob bet $20 on a coin toss, the outcomes are $(+20, -20)$ or $(-20, +20)$, and each pair sums to zero. A game where payoffs always sum to the same constant (such as two firms splitting a fixed market of 100%) is called constant-sum and behaves the same way strategically.

John von Neumann proved the minimax theorem in 1928: in every two-player zero-sum game with finitely many strategies, there is a value $V$ such that one player can guarantee winning at least $V$ on average and the other can guarantee losing no more than $V$, usually by randomizing. Each player minimizes the maximum loss the opponent can inflict. This was the starting point for von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior (1944). In two-player zero-sum games, minimax strategies are the same as a Nash equilibrium.

• Derivatives. A futures contract or options contract is a zero-sum transfer between buyer and seller before fees: what one side gains as prices move, the other loses. After brokerage fees, exchange fees and spreads, it becomes slightly negative-sum for traders taken together. Derivatives can still create value overall by letting businesses transfer risk through hedging.
• Active versus passive investing. The economist William Sharpe argued in 1991 that, before costs, the average actively managed dollar must earn the same return as the average passively managed dollar, because together they hold the market. Active management is therefore zero-sum relative to the market before costs and negative-sum after them, which is a core argument for passive management.
• Short-term trading. Currency speculation and day trading are often described as close to zero-sum: profits for one trader are losses for others. High-frequency trading firms earn part of their revenue from spreads paid by slower traders.
• Stock ownership over time. Holding shares is not zero-sum in the long run, because companies generate profits and dividends; the total value of the stock market can grow as the economy does.
• Crypto. Critics argue that tokens with no cash flows are zero-sum or negative-sum for holders as a group, since gains for early sellers come from later buyers; supporters reply that networks can create real uses that add value.

The question of whether trade is zero-sum has run through economics for centuries. Mercantilist writers in the 16th–18th centuries treated national wealth as stocks of gold and silver and saw a trade surplus as a win and a deficit as a loss. Adam Smith and David Ricardo argued that the division of labor and comparative advantage make trade positive-sum. Modern economists broadly agree that trade raises total income, while also noting that gains are unevenly shared and some workers and regions lose, which is why trade policy remains contested.

Some economic situations really are close to zero-sum: dividing a fixed budget, competing for a fixed number of contracts, or, in the short run, bargaining over a price. Rent-seeking, spending resources to capture existing wealth rather than create new wealth, is often negative-sum; see rent-seeking. Researchers including Stefanie Stantcheva and co-authors studied zero-sum beliefs in the United States in a 2023 paper, finding that people who see the economy as zero-sum tend to favor more government redistribution and tighter immigration rules.

Labelling a situation zero-sum depends on what is counted and over what period. A takeover battle may be zero-sum between two bidders but positive-sum if the company is run better afterwards. A tariff dispute can look zero-sum to negotiators but negative-sum for both economies. Critics of zero-sum framing warn that it can justify protectionism or conflict; critics of the opposite view warn that "everyone gains" claims can ignore real distributional losers. Most economists treat the concept as a precise tool for certain games and markets, and as a rough metaphor everywhere else.

• Stanford Encyclopedia of Philosophy: Game Theory
• NBER: Zero-sum Thinking and the Roots of US Political Differences (Chinoy, Nunn, Sequeira and Stantcheva)
• William F. Sharpe: The Arithmetic of Active Management