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| Title: | Notes on the Economics of Game Theory - Part II |
| Author: | Sam Vaknin |
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Games, naturally, can consist of one player, two players and
more than two players (n-players). They can be zero (or fixed) -
sum (the sum of benefits is fixed and whatever gains made by one
of the players are lost by the others). They can be nonzero-sum
(the amount of benefits to all players can increase or
decrease). Games can be cooperative (where some of the players
or all of them form coalitions) – or non-cooperative
(competitive). For some of the games, the solutions are called
Nash equilibria. They are sets of strategies constructed so that
an agent which adopts them (and, as a result, secures a certain
outcome) will have no incentive to switch over to other
strategies (given the strategies of all other players). 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:
Is not dependent upon any utility function;
It is impossible for two players to improve the Nash solution
(=their position) simultaneously (=the Paretto optimality);
Is not influenced by the introduction of irrelevant (not very
gainful) alternatives;
and
Is symmetric (reversing the roles of the players does not affect
the solution).
(continued)
About the author:
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
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