Whose interests are linked
One of the first questions to ask about any game is how the players’ interests relate. Do they want opposite things, the same thing, or some of each? The answer decides what kind of reasoning will work. Advice built for a game of pure rivalry can fail badly in a team game, and the reverse is also true.
The cleanest way to describe the relationship is to add up the payoffs of all players in each outcome and see whether that total changes.
Zero-sum games
In a zero-sum game, the payoffs always add up to zero (or to the same constant). Whatever one player gains, the others lose in total. If you score +1, your rival scores -1. There is no way for both sides to be better off together.
Chess is the standard example: a win, a loss and a draw are scored in a way that leaves the total fixed. Many two-player contests with a single winner fit the pattern, as do games like rock paper scissors, where one player’s win is exactly the other’s loss.
Strategic consequences of zero-sum structure:
- Interests are directly opposed. Information that helps you hurts your rival, so you hide it.
- Communication has little value. Anything your rival says is meant to help them, so it is usually discounted.
- Equilibrium has a clean meaning. In two-player zero-sum games there is a guaranteed average result you can secure by playing a safe strategy, even if you are completely open about it. This is the idea behind mixed strategies in simple matching games.
- Unpredictability matters. Because the rival gains whenever you lose, any exploitable pattern is dangerous.
Strictly speaking, constant-sum is the more general term, and zero-sum is that same idea with the constant set to zero. In practice both are used loosely.
Cooperative and positive-sum games
At the other end are games where players share a common outcome. In a fully cooperative board game, all players win or lose together against the game itself. Their payoffs are the same, so what is good for one is good for all, and the total is not fixed. Cooperation can create value that did not exist.
The strategic consequences are different:
- Interests are aligned. Sharing information helps.
- The challenge is coordination. The difficulty is no longer an opponent but working out who does what. Even with identical goals, groups can fail by duplicating effort or by leaving a task to someone else, much as in a coordination game.
- Dominance and equilibrium can look different. With identical payoffs the best joint plan is simply the plan with the highest shared payoff, and a player who deviates alone usually hurts the whole group.
- Leadership problems appear. In a group where one strong player tells the others what to do, the others may lose the sense that their choices matter. That is a human problem rather than a mathematical one, but it is real.
A word of caution about terminology: in game theory, cooperative game theory refers specifically to settings where players can make binding agreements, and it studies how groups form coalitions and divide gains. The board-game sense of cooperative, a game where players win together against the system, is related but not the same.
The middle ground
Most real games sit between the poles. These are often called mixed-motive games. Players have partly shared and partly opposed interests, so the total is not fixed, but the shares still matter.
Examples:
- Trading. In a trade where both sides get something they value more than what they give, both can gain. The total value rises, which is positive-sum. But the split can be argued over, and a lopsided deal leaves one side with less. Resource trading in settlement games works this way, and each deal can be good for both traders while still changing who is likely to win the game.
- Multiplayer games with one winner. Even a friendly trade matters only for the final ranking. A trade that helps two players equally may leave a third player further behind. Because the game still has only one winner, one player’s gain is at least partly the others’ loss.
- Alliances. Temporary teams of players sharing an interest in stopping a leader are cooperative while the threat lasts. Once the leader is stopped, the allies become rivals again. Players who forget this are often surprised.
- Negative-sum situations. Sometimes the total shrinks. If two players waste resources on a fight, both lose and a third may gain. Real-world arrangements like fees for running a game can also make the total slightly negative for the players taken together.
The prisoner dilemma is a classic mixed-motive case. Both players prefer mutual cooperation to mutual defection, which is the cooperative element, but each gains by defecting on the other, which is the competitive element.
Money and odds
In casino-style games, money moves between players and the operator, so in terms of cash the totals are fixed, with the operator on the other side. But a built-in house edge means the average player’s expected result is negative, which differs from the symmetrical rivalry of chess. In games between players with a fee taken by the host, the players together lose a small amount on average, so the game is slightly negative-sum for them, even though it still feels like pure rivalry between them. Outcomes in either case are uncertain, and nothing about the structure guarantees a result in any single session.
How to tell which kind you are in
Try these checks:
- Add up the payoffs. If they always sum to the same total, the game is constant-sum.
- Ask whether everyone can win. If there is a possible outcome where every player is better off than in another outcome, the game is not zero-sum.
- Ask whether you can talk. If sharing information helps you, the game has some cooperative element.
- Watch for the temptation to defect. If cooperating is good for the group but costly to you personally, you are in mixed-motive territory.
Why the distinction matters in practice
Misreading the type of game leads to predictable mistakes.
- Treating a mixed-motive game as zero-sum can make you refuse deals that would help everyone, including you.
- Treating a zero-sum contest as friendly can lead you to share information with someone who benefits when you lose.
- In team games, playing as if you were competing with teammates wastes the structure that is supposed to help you.
- In rating systems that assume wins and losses balance out, such as an Elo rating, the underlying model treats each match as zero-sum, which is a reasonable approximation for head-to-head games but a rougher one for team games with shared results.
Start every new game by asking whose interests line up and whose do not. The answer will tell you whether to hide information or share it, whether to keep your options unpredictable or to build a plan with others, and whether a gain for another player is a threat or a benefit to you.