Zero-sum and non-zero-sum games: what the payoff totals establish

Specify players, actions and preferences

Consider two abstract players called Row and Column. Row selects A or B, and Column selects L or R. A payoff pair lists Row’s value first and Column’s value second. The Yale introduction to the ingredients of a game emphasises that outcomes alone are insufficient: a model also needs preferences represented by payoffs. The numbers used here are stipulated teaching values. They do not measure a real population, establish moral priorities or imply that either player has a financial stake.

Check the definition across every outcome

In the first model, payoffs are (2,−2), (−1,1), (0,0) and (3,−3) for the four possible action pairs. Each pair sums to zero. Row preferring a larger first number exactly corresponds to Column preferring a smaller first number in this particular representation. To call a table zero-sum, check all its entries, including outcomes thought unlikely or unattractive. A single observed exchange in which one side gains and another loses does not establish that every feasible outcome has the same property.

Compare an opposing-interest table with coordination

In the coordination table, both players prefer either matching arrangement to the mismatched outcomes, and both prefer (A,L) to (B,R). Its payoff totals are six, zero, zero and four in the displayed representation. The arithmetic distinguishes the models; it does not by itself supply a social welfare measure.

Understand constant-sum representations

The University of Manchester game-theory notes develop zero-sum games and their mathematical structure. If two-player payoffs always add to a fixed constant rather than zero, subtracting that constant from one player’s payoffs produces a zero-sum representation without changing that player’s rankings. For example, pairs summing to five can be written with the second payoff reduced by five. This is an algebraic change of origin, not a change in available actions. Explain such normalisation so readers do not confuse transformed utility labels with physical quantities.

Keep mathematical totals separate from welfare

Utilities ordinarily represent each player’s preferences within the model. Adding different people’s numerical utilities is not automatically a meaningful measurement of collective wellbeing. Rescaling one person’s utility can change a displayed sum while preserving that person’s choices. Claims about total welfare therefore need additional assumptions about comparability and the relevant objective. The coordination example supports a simpler statement that does not require adding utilities: both players prefer one specified outcome to another. That is a comparison of individual rankings, not an unexplained interpersonal arithmetic rule.

Do not confuse the payoff class with an equilibrium

A zero-sum classification describes the relationship between payoffs. A Nash equilibrium instead concerns whether any player can improve by changing their own strategy while others keep theirs fixed. Nash’s original dissertation connects the broader equilibrium concept with the earlier two-person zero-sum framework. Identifying a table’s payoff class does not complete its equilibrium analysis. Likewise, calling an interaction non-zero-sum does not prove that cooperation will occur, that an agreement is enforceable or that all players prefer the same outcome.

Identify boundaries before applying the model

A real interaction can appear zero-sum if the analyst includes only one scarce resource and omits other consequences. Conversely, a nominally cooperative project can involve distributional conflict over effort, credit or access. State whose payoffs are included, what outcomes are feasible and which costs or benefits are outside the model. Changing that boundary changes the question. The abstract tables deliberately have only two players and four outcomes; they are useful for learning definitions but should not be presented as evidence about a particular institution or community.

Payoff-classification checklist

• List all players and available actions.
• State the ordering and meaning of each payoff pair.
• Check totals across every action combination.
• Explain any constant-sum normalisation.
• Keep individual preferences distinct from welfare aggregation.
• Analyse equilibrium as a separate question.
• Identify excluded participants and consequences.

A useful exercise is to alter one coordination payoff and ask which conclusions survive. The action set remains the same, while shared preferences or equilibrium incentives may change. This shows why labels such as competitive or cooperative are insufficient substitutes for an explicit model with inspectable assumptions and complete payoff entries.

Sources and verification

• Game Theory, Lecture 2: formal ingredients, payoffs and common knowledge — Open Yale Courses — 2026-10-23
• Games, Games, Games: notes for the game theory course — Andrea Schalk, University of Manchester — 2026-10-23
• Non-Cooperative Games, doctoral dissertation, 1950 — John Nash; Princeton University Library archive — 2026-10-23