Zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participant(s). It is so named because when the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Chess and Go are examples of a zero-sum game - it is impossible for both players to win. Zero-sum is a special case of a more general constant sum where the benefits and losses to all players sum to the same value. Cutting a cake is zero- or constant-sum because taking a larger piece reduces the amount of cake available for others. Chess is an abstract strategy board game and mental sport for two players. ... Go is a strategic, two-player board game originating in ancient China between 2000 BC and 200 BC. Go is a popular game in East Asia. ...

Situations where participants can all gain or suffer together, such as a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, are referred to as non-zero-sum. Other non-zero-sum games are games in which the sum of gains and losses by the players are always more or less than what they began with. For example, a game of poker played in a casino is a zero-sum game unless the pleasure of gambling or the cost of operating a casino is taken into account, making it a non-zero-sum game. Poker Room at the Trump Taj Mahal, Atlantic City, New Jersey Poker is a card game, the most popular of a class of games called vying games, in which players with fully or partially concealed cards make wagers into a central pot, which is awarded to the player or players... Mirage Hotel & Casino, Las Vegas. ...

The concept was first developed in game theory and consequently zero-sum situations are often called zero-sum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games. Optimal strategies for two-player zero-sum games can often be found using minimax strategies. Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... Minimax (sometimes minmax) is a method in decision theory for minimizing the maximum possible loss. ...

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalised form of a zero-sum game for two persons, and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players; the (n + 1) player representing the global profit or loss. This suggests that the zero-sum game for two players forms the essential core of mathematical game theory.^[1] John von Neumann in the 1940s. ... Oskar Morgenstern (January 24, 1902 - July 26, 1977) was an German- American economist who, working with John von Neumann, helped found the mathematical field of game theory. ...

Economics and non-zero-sum

Many economic situations are not zero-sum, since valuable goods and services can be created, destroyed, or badly allocated, and any of these will create a net gain or loss. Assuming the counterparties are acting rationally, any commercial exchange is a non-zero-sum activity, because each party must consider the good s/he is receiving as being at least fractionally more valuable to him/her than the good s/he is delivering (see also the law of comparative advantage) - to exchange in any other circumstances would not be rational. In economics, the theory of comparative advantage explains why it can be beneficial for two parties (countries, regions, individuals and so on) to trade, even though one of them may be able to produce every item more cheaply than the other. ...

In public policy, however, including the allocation of resources, tax policy and burden sharing, there are often broad economic winners and losers. There are defendable theories that posit many economic policies are zero sum, more often than classic economics typically acknowledges.

Psychology and non-zero-sum

The most common or simplistic example is from the subfield of Social Psychology is the concept of "Social Traps". In some cases we can enhance our collective well being by pursuing our personal interests or parties can pursue mutually destructive behavior as they choose their own ends. Mutually Assured Destruction (MAD) is another example. The quotation below is also apropos to this section.

Here is a link to a comprehensive text on the matter of Psychology and Game Theory. Game Theory & its Applications

Complexity and non-zero-sum

It has been theorized by Robert Wright, among others, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent. As former US President Bill Clinton states: Robert Wright. ...

The more complex societies get and the more complex the networks of interdependence within and beyond community and national borders get, the more people are forced in their own interests to find non-zero-sum solutions. That is, win-win solutions instead of win-lose solutions.... Because we find as our interdependence increases that, on the whole, we do better when other people do better as well - so we have to find ways that we can all win, we have to accommodate each other - Bill Clinton, Wired interview, December 2000.[1]

William Jefferson Bill Clinton (born William Jefferson Blythe III on August 19, 1946) was the 42nd President of the United States, serving from 1993 to 2001. ... Wired is a full-color monthly magazine and on-line periodical published in San Francisco, California since March 1993. ... This article is about the year 2000. ...

An example

A game's payoff matrix is a convenient way of representation. Consider for example the two-player zero-sum game pictured to the right. It has been suggested that this article or section be merged with normal form game. ...

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: the first player chooses action 2 and the second player chose action B. When the payoff is allocated the first player gains 20 points and the second player loses 20 points.

Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, player 2 would choose action C. If both players take these actions, the first player will win 20 points. But what happens if player 2 anticipates the first player's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if the first player in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games. John von Neumann in the 1940s. ... This article does not cite its references or sources. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ... Minimax (sometimes minmax) is a method in decision theory for minimizing the maximum possible loss. ...

For the example given above, it turns out that the first player should choose action 1 with probability 57% and action 2 with 43%, while the second player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Player one will then win 2.85 points on average per game.

See also

• Game
• Group-dynamic game
• Win-win game
• Double-entry book-keeping

Tug of war is an easily organized, impromptu game that requires little equipment. ... Group-dynamic games are experiential education exercises which help people to learn about themselves, interpersonal relationships, and how groups function from a group dynamics or social psychological point of view. ... A win-win game is a type of game which is designed in a way that all participants can profit from it in one way or the other. ... Double-entry book-keeping is the standard practice for recording financial transactions. ...

References

1. ^ This paragraph was translated from the French wikipedia article on this subject.

External links

• Play zero-sum games online by Elmer G. Wiens.