In game theory and economic theory, a zerosum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zerosum game if all participants value each unit of cake equally (see marginal utility).
In contrast, nonzerosum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zerosum game is also called a strictly competitive game while nonzerosum games can be either competitive or noncompetitive. Zerosum games are most often solved with the minimax theorem which is closely related to linear programming duality,^{[1]} or with Nash equilibrium.
Contents
Definition
Choice 1  Choice 2  
Choice 1  −A, A  B, −B 
Choice 2  C, −C  −D, D 
Generic zerosum game 
The zerosum property (if one gains, another loses) means that any result of a zerosum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game.^{[2]}
Zerosum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.
Situations where participants can all gain or suffer together are referred to as nonzerosum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a nonzerosum situation. Other nonzerosum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.
The idea of Pareto optimal payoff in a zerosum game gives rise to a generalized relative selfish rationality standard, the punishingtheopponent standard, where both players always seek to minimize the opponent's payoff at a favorable cost to himself rather to prefer more than less. The punishingtheopponent standard can be used in both zerosum games (e.g. warfare game, chess) and nonzerosum games (e.g. pooling selection games).^{[3]}
Solution
For twoplayer finite zerosum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.
Example
Blue Red

A  B  C 

1  −30 30

10 −10

−20 20

2  10 −10

−20 20

20 −20

A game's payoff matrix is a convenient representation. Consider for example the twoplayer zerosum game pictured at right or above.
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.
Émile Borel and John von Neumann had the fundamental 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 minimize the maximum expected pointloss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute probably optimal strategies for all twoplayer zerosum games.
For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, and Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.
Solving
The Nash equilibrium for a twoplayer, zerosum game can be found by solving a linear programming problem. Suppose a zerosum game has a payoff matrix M where element M_{i,j} is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of M is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector u:
 Minimize:
 Subject to the constraints:
 u ≥ 0
 M u ≥ 1.
The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the M u vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.
The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of M (adding a constant so it's positive), then solving the resulting game.
If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a twoplayer, zerosum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general.^{[citation needed]}
Universal solution
If avoiding a zerosum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zerosum game. For any two players zerosum game where a zerozero draw is impossible or noncredible after the play is started, such as Poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zerozero draw after a zerosum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find rewardasyougo in optimal choice computation shall prevail over all two players zerosum games with regard to starting the game or not.^{[4]}
Nonzerosum
Psychology
The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing our personal interests can enhance our collective wellbeing, but in others personal interest results in mutually destructive behavior.
Complexity
It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly nonzerosum as it becomes more complex, specialized, and interdependent.
Extensions
In 1944, John von Neumann and Oskar Morgenstern proved that any nonzerosum game for n players is equivalent to a zerosum game with n + 1 players; the (n + 1)th player representing the global profit or loss.^{[5]}
Misunderstandings
Zerosum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games.^{[1]}
Politics is sometimes called zero sum.^{[6]}^{[7]}^{[8]}
Zerosum thinking
In psychology, zerosum thinking refers to the perception that a situation is like a zerosum game, where one person's gain is another's loss.
See also
References
 ^ ^{a} ^{b} Ken Binmore (2007). Playing for real: a text on game theory. Oxford University Press US. ISBN 9780195300574., chapters 1 & 7
 ^ Bowles, Samuel (2004). Microeconomics: Behavior, Institutions, and Evolution. Princeton University Press. pp. 33–36. ISBN 0691091633.
 ^ Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 9781507658246. Chapter 1 and Chapter 4.
 ^ Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 9781507658246. Chapter 4.
 ^ "Theory of Games and Economic Behavior". Princeton University Press (1953). June 25, 2005. Retrieved 20180225.
 ^ Rubin, Jennifer (20131004). "The flaw in zero sum politics". The Washington Post. Retrieved 20170308.
 ^ "Lexington: Zerosum politics". The Economist. 20140208. Retrieved 20170308.
 ^ "Zerosum game  Define Zerosum game at". Dictionary.com. Retrieved 20170308.
Further reading
 Misstating the Concept of ZeroSum Games within the Context of Professional Sports Trading Strategies, series Pardon the Interruption (20100923) ESPN, created by Tony Kornheiser and Michael Wilbon, performance by Bill Simmons
 Handbook of Game Theory  volume 2, chapter Zerosum twoperson games, (1994) Elsevier Amsterdam, by Raghavan, T. E. S., Edited by Aumann and Hart, pp. 735–759, ISBN 0444894276
 Power:Its Forms, Bases and Uses (1997) Transaction Publishers, by Dennis Wrong
External links
 Play zerosum games online by Elmer G. Wiens.
 Game Theory & its Applications  comprehensive text on psychology and game theory. (Contents and Preface to Second Edition.)
 A playable zerosum game and its mixed strategy Nash equilibrium.