In game theory, a non-cooperative game is a game in which there are no external rules or binding agreements that enforce the cooperation of the players. A non-cooperative game is typically used to model a competitive environment. This is stated in various accounts most prominent being John Nash's 1951 paper in the journal Annals of Mathematics. Counterintuitively, non-cooperative game models can be used to model cooperation as well, and vice versa, cooperative game theory can be used to model competition. Some examples of this would be the use of non-cooperative game models in determining the stability and sustainability of cartels and coalitions.
The difference between cooperative and non-cooperative game theory The terminology of cooperative and non-cooperative game theory does not imply that the players can only "cooperate" in one case and not another. Rather, the distinction between the two branches of game theory lies in the assumption regarding the institutional structure. The distinction between the two branches of game theory was introduced by John Nash, who wrote
[Cooperative game theory] is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game. Our [non-cooperative game theory], in contradistinction, is based on the absence of coalitions in that it is assumed that each participant acts independently, without collaboration or communication with any of the others. Non-cooperative game theory models different situations in which agents are unable to reach a resolution to a conflict that enforces some action on one another. This form of game theory pays close attention to the individuals involved and their rational decision making. There are winners and losers in each case, and yet agents may end up in Pareto-inferior outcomes, where every agent is worse off and there is a potential outcome for every agent to be better off. Agents will have the ability to predict what their opponents will do. Cooperative game theory models situations in which a binding agreement is possible. In other words, the cooperative game theory implies that agents cooperate to achieve a common goal and they are not necessarily referred to as a team because the correct term is the coalition. Each agent has its skills or contributions that provide strength to the coalition. 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. Many results or solutions proposed by the agents involved in Game Theory are important in understanding the rivalry between these agents under a set of conditions that are strategic.
Elements of a non-cooperative game To specify a non-cooperative game completely, one must specify
The number of players, The actions available to each player at any given state of the game, The function that each player is attempting to maximize, The time ordering of actions (if needed), How information is acquired by the players. Whether there is any randomness in the game. The following assumptions are commonly made:
Perfect recall: each player remembers their decisions and known information. Self-interest: each player does not consider the effect of actions on the others but only on their own. Rational: each player is interested to maximise their utility or payoff. Complete information: each player knows the preferences and strategies of the other players. Each player has the same understanding of how the game works.
Examples Strategic games are a form of non-cooperative game, where only the available strategies and combinations of options are listed to produce outcomes.
Rock paper scissors
In the game of rock-paper-scissors, if Player 1 decides to play "rock", it is in Player 2's interest to play "paper"; if Player 2 chooses to play "paper", it is in Player 1's interest to play "scissors"; and if Player 1 plays "scissors", Player 2 will, in their own interests, play "rock".
Prisoner's dilemma
The prisoner's dilemma game is another well-known example of a non-cooperative game. The game involves two players, or defendants, who are kept in separate rooms and thus are unable to communicate. Players must decide, by themselves in isolation, whether to cooperate with the other player or to betray them and confess to law authorities. As shown in the diagram, both players will receive a higher payoff in the form of a lower jail sentence if they both remain silent. If both confess, they receive a lower payoff in the form of a higher 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.
The battle of the sexes
The game involves two players, boy and girl, deciding either going to a football game or going to an opera for their date, which respectively represent boy's and girl's preferred activity (i.e. boy prefers football game and girl prefers opera). This example is a two-person non-cooperative non-zero sum (TNNC) game with opposite payoffs or conflicting preferences. Because there are two Nash equilibria, this case is a pure coordination problem with no possibility of refinement or selection. Thus, the two players will try to maximise their own payoff or to sacrifice for the other and yet these strategies without coordination will lead to two outcomes with even worse payoffs for both if they disagree on what to do on their date.
Matching pennies game
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