What is a non-zero-sum game theory

Non-zero-sum game

The zero-sum game devised in game theory has not proven to be a suitable model for interactions in real life. Therefore game theory has been extended to non-zero-sum games and especially to those of a partially competitive nature.

The non-zero-sum games have ideally been divided into negotiable and non-negotiable games; The difference lies in the way the game is conducted: In the case of negotiable games (cooperation game), an agreement between the opponents is permitted before the game, in the course of which the players may come to an agreement on their respective behavior - which is then binding. The agreement is comparable to a business arrangement or a contract. In the case of non-negotiable games (non-cooperation games), the player must make his decision without prior agreement with the other player or prior knowledge of his intentions.
In the strategy pair diagram, each strategy pair arbj can be represented as a point: The abscissa represents the payout or expected payout for PA for this strategy pair, the ordinate represents the corresponding consequence for PB.
The matrix comes from Luce & Raiffa, who they interpret as representative of the “battle of the sexes”: PA and PB are a man and a woman who both have to decide whether they want to go to a boxing match or to ballet. The man prefers the boxing match, the woman the ballet. However, each of the two would rather go to the non-preferred event together than to the preferred event alone. The strategy pairs a1b2 and a2b1 lead to the same payouts and are therefore represented in the diagram by the same point; such pairs are said to be equivalent. Pairs of randomized mixed strategies can also be shown in the diagram, but it does not have to be a convex set. In the illustration, the dotted area with its boundary lines contains all the pure and all randomly controlled mixed strategies that are available to the players PA and PB in a non-negotiable game.

Any mixed strategy that can be chosen in a non-negotiable game is also available in a negotiable game - players can agree to play any pure or mixed strategy pair that they could have played without consultation. In addition, negotiation enables the consideration of pairs of strategies that would be impossible under the condition of non-negotiability. These new pairs of strategies represent - correlated mixed strategies. They are mixed because a player's strategy involves choosing more than one alternative as a possibility, and they are correlated because both players' decisions are not made independently in one round.
The best known and most intensively studied partially competitive game is the Prisoner's Dilemma Game (PDG).

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