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A Stable Strategy is similar to a Nash solution – though not identical mathematically. There is currently no comprehensive theory of Information Dynamics. Game Theory is limited to aspects of competition and exchange of information (cooperation). Strategies that lead to better results (independently of other agents) are dominant and where all agents have dominant strategies – a solution is established. Thus, Nash equilibrium is applicable to games that are repeated and wherein each agent reacts to acts of other agents. The agent is influenced by others – but does not influence them (he is negligible). The agent continues to adapt in this way – until no longer able to improve his position. The Nash solution is less available in cases of cooperation and is not unique as a solution. In most cases, players will adopt a minimax strategy (in zero-sum games) or maximin strategies (in nonzero-sum games). These strategies guarantee that loser will not lose more than value of game and that winner will gain at least this value. The solution is "Saddle Point".
The distinction between zero-sum games (ZSG) and nonzero-sum games (NZSG) is not trivial. A player playing a ZSG cannot gain if prohibited to use certain strategies. This is not case in NZSGs. In ZSG, player does not benefit from exposing his strategy to his rival and is never harmed by having foreknowledge of his rival's strategy. Not so in NZSGs: at times, a player stands to gain by revealing his plans to "enemy". A player can actually be harmed by NOT declaring his strategy or by gaining acquaintance with enemy's stratagems. The very ability to communicate, level of communication and order of communication – are important in cooperative cases. A Nash solution:
Is not dependent upon any utility function;
It is impossible for two players to improve Nash solution (=their position) simultaneously (=the Paretto optimality);
Is not influenced by introduction of irrelevant (not very gainful) alternatives;
Is symmetric (reversing roles of players does not affect solution).
Sam Vaknin is the author of Malignant Self Love - Narcissism Revisited and After the Rain - How the West Lost the East. He is a columnist for Central Europe Review, United Press International (UPI) and eBookWeb and the editor of mental health and Central East Europe categories in The Open Directory, Suite101 and searcheurope.com.
Visit Sam's Web site at http://samvak.tripod.com