Zero-sum game

A zero-sum game is a mathematical model used in game theory and economic theory. It describes a situation involving two or more competing entities. In these scenarios, the result is an advantage for one side and an equivalent loss for the other. This means one player's gain is exactly equal to another player's loss. If you add all the gains and subtract all the losses, the net result is zero. Because the total benefit remains unchanged, these are often called strictly competitive games.

To understand the mechanism, imagine a payoff matrix. This is a chart used to represent the possible outcomes of a game. In a two-player game, the first player might choose an action in secret. The second player also chooses an action without knowing the first player's choice. Once both choices are revealed, points are allocated according to the matrix. For example, if Red chooses action 2 and Blue chooses action B, Red might gain 20 points while Blue loses 20 points. The players act based on these values, attempting to maximize their own profit while the opponent tries to minimize it.

Zero-sum games are a specific type of constant sum game. In these games, the sum of each outcome is always zero. They are considered distributive rather than integrative. This means the "pie" cannot be enlarged through negotiation. This is a key distinction from non-zero-sum games. In a non-zero-sum situation, the aggregate gains and losses can be more or less than zero. For instance, if one country trades excess bananas for another country's excess apples, both sides benefit. This trade creates a situation where the total benefit is greater than zero.

History shows that mathematicians found clever ways to solve these competitive puzzles. Émile Borel and John von Neumann provided fundamental insights using probability. They realized that instead of choosing one definite action, players could use a mixed strategy. This involves assigning probabilities to different actions and using a random device to choose between them. This approach leads to the minimax theorem. The minimax method helps players compute optimal strategies by minimizing the maximum expected point loss.

Another important concept is the Nash equilibrium. For two-player finite zero-sum games using mixed strategies, an equilibrium solution always exists. In these cases, the different solution concepts of Nash equilibrium, minimax, and maximin all yield the same result. This is not necessarily true for pure strategies. Finding a Nash equilibrium can be done by solving a linear programming problem. This mathematical method uses a vector to find the probability that a player will choose a specific strategy. If a payoff matrix contains negative numbers, a constant can be added to make all elements positive without changing the equilibrium strategies.

Zero-sum dynamics appear in many real-world sectors. In financial markets, derivatives trading is often seen as a zero-sum game. Every dollar gained by one party in a transaction is lost by the other. Options contracts are a specific example of this wealth transfer. Even in aviation, market entries can create zero-sum effects. In Hong Kong, the entry of low-cost airlines brought in $671 million in revenue but resulted in an outflow of $294 million. If the number of arriving and departing flights remains the same, the economic contribution might become a zero-sum game due to the displacement of existing models.

Complexity increases when more than two players are involved. In a zero-sum three-person game, there is an absolute antagonism of interests. A single move by one player might benefit them while disbenefiting both other players. It is also possible for two players to find a parallelism of interests. This means they might cooperate to the detriment of the third player.

For example, if Player 1 chooses defense while Players 2 and 3 choose offense, both attackers gain a point while the defender loses two. This shows how players can adjust their conduct to build temporary alliances within a competitive framework.