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In game theory and economic theory, zerosum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participant(s). If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Zerosum can be thought of more generally as constant sum where the benefits and losses to all players sum to the same value of money (or utility). Cutting a cake is zero or constantsum, because taking a larger piece reduces the amount of cake available for others. In contrast, nonzerosum describes a situation in which the interacting parties' aggregate gains and losses is either less than or more than zero. Zerosum games are also called strictly competitive.
Contents
Definition
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).^{[1]}
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 concept was first developed in game theory and consequently zerosum situations are often called zerosum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games.
Solution
For 2player finite zerosum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. In the solution, players play a mixed strategy.
Example
A game's payoff matrix is a convenient representation. Consider for example the twoplayer zerosum game pictured at right.
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.
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