A zero-sum game is a game in which the players' payoffs add to zero for every possible outcome. In the two-player case, one player's gain is exactly the other's loss. For a quant, this distinction determines whether protecting against a rival's best response is the same objective as protecting against the worst payoff the rival can impose.
The definition concerns the specified payoffs, not whether the interaction feels competitive. A contest for a fixed prize can be constant-sum; a productive partnership can change the total available. Yale's penalty-kick and partnership lecture, chapters 1 and 4, contrasts these payoff structures.
A guarantee that pure choices cannot achieve
Consider an original fictional one-round game. Row chooses A or B; Column simultaneously chooses L or R. The entries below are tokens paid to Row; Column's payoff is their negative. Both maximise expected tokens, know the payoff table and cannot observe the other's current choice.
| Row chooses | Column: L | Column: R | Row's worst payoff |
|---|---|---|---|
| A | 3 | 0 | 0 |
| B | 1 | 2 | 1 |
| Highest Row payoff in column | 3 | 2 |
If Row must use a pure strategy, B gives the better guarantee: at least 1 token. This is the pure maximin payoff: take each row's minimum, then maximise those minima.
Looking from Column's side, choosing R caps Row's payoff at 2 tokens, better than L's cap of 3. This is the pure minimax bound on Row: take each column's maximum, then minimise those maxima. The gap between 1 and 2 means no pure pair meets both security objectives.
Now let Row choose A with probability , otherwise B. Against the two fixed Column choices, Row's expected payoffs are
Row's guaranteed expected payoff is the lower of these two lines. Their intersection gives
Before , the lower line rises; after it, the lower line falls. Thus Row maximises the floor by choosing A 25% and B 75%. Either pure Column choice then gives Row 1.5 expected tokens; any mixture of them does too. Equal randomisation would guarantee only 1 token.
Can Column hold Row to 1.5? Write for Column's probability of L. Row's fixed-action payoffs are
Column minimises the upper of these lines. They meet at , with payoff 1.5. Column's 50%/50% mix caps every Row strategy at 1.5 expected tokens. The two guarantees agree, certifying a mixed strategy equilibrium with payoffs .
Original fictional game. The thick lines show worst-response expected payoffs: Row raises a floor, while Column lowers a ceiling. They are not realised paths or market returns.
The guarantee is in expectation. Row can still receive 0 in an individual round. The random draw must remain unobserved until Column commits: knowing Row's probabilities differs from seeing Row's realised action. If Column sees that action first, it can choose R after A and L after B, bringing Row's expected payoff under this mix down to token. That is a different game.
What minimax says in general
For a finite two-player zero-sum game, let be Row's payoff matrix, and the players' probability vectors, and under independent randomisation. The minimax theorem gives
The optimisations include all mixed strategies, including pure ones. Optimal strategies satisfy the saddle inequalities
Row cannot improve against ; Column cannot reduce Row's payoff against . This is precisely the Nash equilibrium condition here. The value is unique; the optimal strategies need not be. Echenique's notes, §§6.1–6.2, develop this result. The finite theorem does not by itself cover arbitrary infinite action sets or establish convergence of learning dynamics.
For larger matrices, the floor has a useful computational form. Row solves a linear programme:
Each inequality protects the proposed floor against one pure Column response. Linearity then protects it against every mixture. Column has the dual ceiling problem. MIT's treatment of zero-sum equilibria, §L3.1.1, explains this connection; the toy example's two linear programmes both return 1.5.
Zero-sum money need not mean zero benefit
For a fully performed forward on one unit, the terminal payoffs satisfy
This accounting identity does not say that hedging has no benefit. A participant with an existing business exposure may value transferring price risk. Nor does it say the players' utilities sum to zero: different initial wealth and risk preferences can break that implication. SelfMadeQuant's Forwards and Futures lesson connects these opposite payoffs to an input-price hedge and daily settlement. The full lesson is available with free sign-in; the linked page provides its public preview.
State the accounting boundary. With costs paid outside the two-player group, their net payoffs add to . If these costs are fixed across all action pairs, the game is still constant-sum and strategically equivalent to a zero-sum game. More generally, if in every outcome, subtracting from each player's payoffs makes the sum zero without changing best responses; Echenique's chapter 6 discusses constant-sum equivalence. Action-dependent total costs need not preserve it.
A statement that “trading is zero-sum” therefore needs specified participants, payoff measures, horizon and costs. A bilateral contract's gross transfer identity alone does not classify the welfare effects of risk transfer, the returns from owning productive assets, or an entire market as a two-player zero-sum game.
Sources
- Federico Echenique, Game Theory Lecture Notes, UC Berkeley Econ 201B, May 2026, chapter 6, especially §§6.1–6.2: zero-sum and constant-sum payoffs, minimax value and saddle inequalities.
- Constantinos Daskalakis and Gabriele Farina, Properties of Nash equilibrium, MIT 6.7980, 22 September 2026, §L3.1.1: minimax strategies, equilibrium and linear programming.
- Ben Polak, Best Responses in Soccer and Business Partnerships, Yale ECON 159, chapters 1, 2 and 4: opposed payoffs, omitted strategic choices and productive interaction.