UP - The Zero-Sum Game Lesson

The Zero-Sum Game Lesson

Adapted from Wikipedia contributors. "Zero-sum game." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 19 Aug. 2022. Web. 20 Sep. 2022.

Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, therefore the net improvement in benefit of the game is zero.

Many card games are zero-sum games. Chess is also a zero-sum game.  Zero-sum games can have 2 players or millions of players.  There are also real world applications in financial markets, economics, and sports. 

Example

A game's payoff matrix is a convenient representation. Consider these situations as an example, the two-player zero-sum game pictured.

A matrix representing a two-person zero-sum game. The matrix has three columns labeled A, B, and C, and two rows labeled 1 and 2. Each cell contains a pair of numbers: the top number in blue and the bottom number in red. The pairs are as follows: Column A has -30/30 for row 1 and 10/-10 for row 2; Column B has 10/-10 for row 1 and -20/20 for row 2; Column C has -20/20 for row 1 and 20/-20 for row 2. The text below the matrix reads ‘A zero-sum game (Two person).

The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.

Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.

In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.

Zero-sum three-person games

It is clear that there are manifold relationships between players in a zero-sum three-person game, in a zero-sum two-person game, anything one player wins is necessarily lost by the other and vice versa; therefore, there is always an absolute antagonism of interests, and that is similar in the three-person game. A particular move of a player in a zero-sum three-person game would be assumed to be clearly beneficial to him and may disbenefits to both other players, or benefits to one and disbenefits to the other opponent. Particularly, parallelism of interests between two players makes a cooperation desirable; it may happen that a player has a choice among various policies: Get into a parallelism interest with another player by adjusting his conduct, or the opposite; that he can choose with which of other two players he prefers to build such parallelism, and to what extent. This picture shows that a typical example of a zero-sum three-person game. If Player 1 chooses to defense, but Player 2 & 3 chooses to offense, both of them will gain one point. At the same time, Player 2 will lose two-point because points are taken away by other players, and it is evident that Player 2 & 3 has parallelism of interests.

A matrix for a zero-sum three-person game. It consists of two tables side by side, each representing the possible outcomes for Player 1’s strategies against Player 2 and Player 3. The left table is labeled ‘Player 2’ at the top and ‘Player 1’ on the left side, with rows labeled ‘Offense’ and ‘Defence,’ and columns labeled the same. The right table is similarly labeled but represents Player 3’s interaction with Player 1. Each cell within the tables contains a set of numbers representing the outcomes for the players.

View the videos below to learn more about zero-sum games.

IMAGES CREATED BY GAVS