Zero sum game
In non-cooperative game theory, a zero-sum game describes a situation in which the gain or loss of one participant is exactly balanced by the gain or loss of the other participants.
It is called like this because if the total gains of the participants are added and the total losses are subtracted, the result is zero. Go, Pokémon, and the bear game are examples of zero-sum games. Zero sum is a special case of the more general case of constant sum where the wins and losses of all players add up to the same value, because exactly the amount the opponent loses is won. Cutting a pie is constant sum or zero because taking a larger piece reduces the amount of pie left for others. Situations where participants can gain or lose at the same time, such as the exchange of products between a nation that produces an excess of oranges and another that produces an excess of apples, in which both benefit from the transaction, are called "sum sum". not null”.
The concept was developed in game theory, which is why zero-sum situations are often called "zero-sum games." This is not to imply that the concept, or game theory itself, applies only to what is commonly known as games. Optimal strategies for two-player zero-sum games often employ minimax strategies.
In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a two-person zero-sum game, and that any game A nonzero-sum game for n players can be reduced to a zero-sum game for n + 1 players, where player (n + 1) represents the total gain or loss (can be thought of in banking certain games). This suggests that two-player zero-sum games form the essential core of game theory.
Treating a nonzero-sum situation as a zero-sum situation, or believing that all situations are zero-sum, is called the zero-sum fallacy.
In cooperative games, there is a type of game closely related to it, more commonly called deciding or auto-dual games.
Complexity and nonzero sum
Some authors, such as Robert Wright, have theorized about the evolution of society towards increasing forms of non-zero sum or additivity as it becomes more complex, specialized and interdependent. Bill Clinton, one of those who support this theory maintains:
The more complex societies become, and the more complex the networks of interdependence within and outside the limits of communities and nations are, the more people will be interested in finding solutions of a non-existent sum. This is, gain-winning solutions instead of profit-loss solutions... Because we discover that the more our interdependence grows, we generally prosper when others also prosper.Bill Clinton, interview in WiredDecember 2000
Example
| A | B | C | |
|---|---|---|---|
| 1 | 30, -30 | -10, 10 | 20, -20 |
| 2 | -10, 10 | 20, -20 | -20, 20 |
The reward matrix of a game is a convenient form of representation. Consider the zero-sum game example shown to the right.
The order of play is as follows: the first player secretly chooses one of the two actions 1 or 2; the second player, unaware of the first player's choice, secretly chooses one of the three actions A, B or C. Each player's choices are then revealed and the point total is affected according to the reward for those choices.
Example: The first player chooses 2 and the second chooses B. When the rewards are assigned, the first player wins 20 points and the second loses 20 points.
In this example, both players know the reward matrix and try to maximize their points; What should they do?
Player 1 might reason as follows: "with action 2, I can lose 20 points and gain only 20, while with action 1 I can lose only 10 but gain 30, so 1 seems much better." Using similar reasoning, 2 will choose C. If both players make those choices, the first player will win 20 points. But what if player 2 anticipates 1's reasoning, and chooses B, to earn 10 points, or if player 1 anticipates this trick, and chooses 2, to earn 20 points?
John von Neumann had the fundamental and surprising insight that probability provides a way out of this mess. Instead of settling on a final action, the two players assign probabilities to their actions, and then use a device that, based on those probabilities, chooses an action for them. Each player calculates the odds to minimize the maximum expected value of losses regardless of the opponent's strategy; this leads to a linear algebra problem with a unique solution for each player. This minimax method can compute optimal strategies for all two-player, zero-sum games.
For the example above, it turns out that the first player must choose 1 with probability 57%, and action 2 with probability 43%, while the second should assign the probabilities 0%, 57% and 43% to the three options A, B and C.
Player 1 will then earn 2.85 points on average per game.