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".
Public-domain text, read in full here on John Shaqi.
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