Nash equilibria (solutions) are the most stable (it is where the system "settles down", to borrow from Chaos Theory) - but they are not guaranteed to be the most desirable.
Consider the famous "Prisoners' Dilemma" in which both players play rationally and reach the Nash equilibrium only to discover that they could have done much better by collaborating (that is, by playing irrationally). Instead, they adopt the "Paretto-dominated", or the "Paretto- optimal", sub-optimal solution. Any outside interference with the game (for instance, legislation) will be construed as creating a NEW game, not as pushing the players to adopt a "Paretto-superior" solution.
The behaviour of the players reveals to us their order of preferences. This is called "Preference Ordering" or "Revealed Preference Theory". Agents are faced with sets of possible states of the world (=allocations of resources, to be more economically inclined). These are called "Bundles". In certain cases they can trade their bundles, swap them with others. The evidence of these swaps will inevitably reveal to us the order of priorities of the agent.
All the bundles that enjoy the same ranking by a given agent - are this agent's "Indifference Sets". The construction of an Ordinal Utility Function is, thus, made simple. The indifference sets are numbered from 1 to n.
These ordinals do not reveal the INTENSITY or the RELATIVE INTENSITY of a preference - merely its location in a list. However, techniques are available to transform the ordinal utility function - into a cardinal one.
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 the aspects of competition and exchange of information (cooperation). Strategies that lead to better results (independently of other agents) are dominant and where all the agents have dominant strategies - a solution is established. Thus, the Nash equilibrium is applicable to games that are repeated and wherein each agent reacts to the 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, the players will adopt a minimax strategy (in zero-sum games) or maximin strategies (in nonzero-sum games). These strategies guarantee that the loser will not lose more than the value of the game and that the winner will gain at least this value. The solution is the "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 the case in NZSGs. In ZSG, the 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 the "enemy". A player can actually be harmed by NOT declaring his strategy or by gaining acquaintance with the enemy's stratagems. The very ability to communicate, the level of communication and the order of communication - are important in cooperative cases. A Nash solution: 1. Is not dependent upon any utility function; 2. It is impossible for two players to improve the Nash solution (=their position) simultaneously (=the Paretto optimality); 3. Is not influenced by the introduction of irrelevant (not very gainful) alternatives; and 4. Is symmetric (reversing the roles of the players does not affect the solution).
The limitations of this approach are immediately evident.
It is definitely not geared to cope well with more complex, multi-player, semi-cooperative (semi-competitive), imperfect information situations.