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The most well-known mention of non-cooperative game theory was made in John Nash's article in the Annals of Mathematics journal in 1951.

According to Tamer Başar in his Lecture Notes on Non-Cooperative Game Theory, a non-cooperative game requires specification of:


 * 1) the number of players;
 * 2) the possible actions available to each player, and any constraints that may be imposed on them;
 * 3) the objective function of each player which she attempts to optimise;
 * 4) any time ordering of the execution of the actions if the players are allowed to act more than once;
 * 5) any information acquisition that takes place and how the information available to a player at each point in time depends on the past actions of other players, and;
 * 6) whether there is a player (nature) whose action is the outcome of a probabilistic event with a fixed (known) distribution.

Further, it has been supposed that non-cooperative game theory is purported to analyse the effect of independent decisions on society as a whole. In comparison, cooperative game theory focuses only on the effects of participants in a certain coalition, when the coalition attempts to improve the collective welfare.

In the game of rock-paper-scissors, there is no cooperative option between the two players available: if Player 1 plays "rock", it is in Player 2's interest to play "paper"'; if Player 2 plays "paper", it is in Player 1's interest to play "scissors"; if Player 1 plays "scissors", it is in Player 2's interest to play "rock". The preference of the players is cyclical, and no cooperative outcome can be reached. This fails the transitive preference property.

Prisoner's Dilemma
Another example of a non-cooperative game is the well-known Prisoner's Dilemma game. The game involves two players, or defendants, who are kept in separate rooms and thus are unable to communicate. Players must decide whether to cooperate amongst themselves, or betray the other player and confess to law authorities. As shown in the diagram, both players will receive a lesser payoff (in the form of a higher jail sentence) if they both remain silent. If both confess, they receive a higher payoff in the form of a lesser jail sentence. If one player confesses and the other remain silent and cooperates, the confessor will receive a higher payoff, while the silent player will receive a lower payoff than if both players cooperated with each other.

The Nash equilibrium therefore lies where players both betray each other, in the players protecting oneself from being punished more.