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| Title: | Notes on the Economics of Game Theory - Part III |
| Author: | Sam Vaknin |
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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.
Von Neumann proved that there is a solution for every ZSG with 2
players, though it might require the implementation of mixed
strategies (strategies with probabilities attached to every move
and outcome). Together with the economist Morgenstern, he
developed an approach to coalitions (cooperative efforts of one
or more players – a coalition of one player is possible). Every
coalition has a value – a minimal amount that the coalition can
secure using solely its own efforts and resources. The function
describing this value is super-additive (the value of a
coalition which is comprised of two sub-coalitions equals, at
least, the sum of the values of the two sub-coalitions).
Coalitions can be epiphenomenal: their value can be higher than
the combined values of their constituents. The amounts paid to
the players equal the value of the coalition and each player
stands to get an amount no smaller than any amount that he would
have made on his own. A set of payments to the players,
describing the division of the coalition's value amongst them,
is the "imputation", a single outcome of a strategy. A strategy
is, therefore, dominant, if: (1) each player is getting more
under the strategy than under any other strategy and (2) the
players in the coalition receive a total payment that does not
exceed the value of the coalition. Rational players are likely
to prefer the dominant strategy and to enforce it. Thus, the
solution to an n-players game is a set of imputations. No single
imputation in the solution must be dominant (=better). They
should all lead to equally desirable results. On the other hand,
all the imputations outside the solution should be dominated.
Some games are without solution (Lucas, 1967).
Auman and Maschler tried to establish what is the right payoff
to the members of a coalition. They went about it by enlarging
upon the concept of bargaining (threats, bluffs, offers and
counter-offers). Every imputation was examined, separately,
whether it belongs in the solution (=yields the highest ranked
outcome) or not, regardless of the other imputations in the
solution. But in their theory, every member had the right to
"object" to the inclusion of other members in the coalition by
suggesting a different, exclusionary, coalition in which the
members stand to gain a larger payoff. The player about to be
excluded can "counter-argue" by demonstrating the existence of
yet another coalition in which the members will get at least as
much as in the first coalition and in the coalition proposed by
his adversary, the "objector". Each coalition has, at least, one
solution.
The Game in GT is an idealized concept. Some of the assumptions
can – and should be argued against. The number of agents in any
game is assumed to be finite and a finite number of steps is
mostly incorporated into the assumptions. Omissions are not
treated as acts (though negative ones). All agents are
negligible in their relationship to others (have no discernible
influence on them) – yet are influenced by them (their
strategies are not – but the specific moves that they select –
are). The comparison of utilities is not the result of any
ranking – because no universal ranking is possible. Actually, no
ranking common to two or n players is possible (rankings are
bound to differ among players). Many of the problems are linked
to the variant of rationality used in GT. It is comprised of a
clarity of preferences on behalf of the rational agent and
relies on the people's tendency to converge and cluster around
the right answer / move. This, however, is only a tendency. Some
of the time, players select the wrong moves. It would have been
much wiser to assume that there are no pure strategies, that all
of them are mixed. Game Theory would have done well to borrow
mathematical techniques from quantum mechanics. For instance:
strategies could have been described as wave functions with
probability distributions. The same treatment could be accorded
to the cardinal utility function. Obviously, the highest ranking
(smallest ordinal) preference should have had the biggest
probability attached to it – or could be treated as the collapse
event. But these are more or less known, even trivial,
objections. Some of them cannot be overcome. We must idealize
the world in order to be able to relate to it scientifically at
all. The idealization process entails the incorporation of gross
inaccuracies into the model and the ignorance of other elements.
The surprise is that the approximation yields results, which
tally closely with reality – in view of its mutilation, affected
by the model.
There are more serious problems, philosophical in nature.
It is generally agreed that "changing" the game can – and very
often does – move the players from a non-cooperative mode
(leading to Paretto-dominated results, which are never
desirable) – to a cooperative one. A government can force its
citizens to cooperate and to obey the law. It can enforce this
cooperation. This is often called a Hobbesian dilemma. It arises
even in a population made up entirely of altruists. Different
utility functions and the process of bargaining are likely to
drive these good souls to threaten to become egoists unless
other altruists adopt their utility function (their preferences,
their bundles). Nash proved that there is an allocation of
possible utility functions to these agents so that the
equilibrium strategy for each one of them will be this kind of
threat. This is a clear social Hobbesian dilemma: the
equilibrium is absolute egoism despite the fact that all the
players are altruists. This implies that we can learn very
little about the outcomes of competitive situations from
acquainting ourselves with the psychological facts pertaining to
the players. The agents, in this example, are not selfish or
irrational – and, still, they deteriorate in their behaviour, to
utter egotism. A complete set of utility functions – including
details regarding how much they know about one another's utility
functions – defines the available equilibrium strategies. The
altruists in our example are prisoners of the logic of the game.
Only an "outside" power can release them from their predicament
and permit them to materialize their true nature. Gauthier said
that morally-constrained agents are more likely to evade
Paretto-dominated outcomes in competitive games – than agents
who are constrained only rationally. But this is unconvincing
without the existence of an Hobesian enforcement mechanism (a
state is the most common one). Players would do better to avoid
Paretto dominated outcomes by imposing the constraints of such a
mechanism upon their available strategies. Paretto optimality is
defined as efficiency, when there is no state of things (a
different distribution of resources) in which at least one
player is better off – with all the other no worse off. "Better
off" read: "with his preference satisfied". This definitely
could lead to cooperation (to avoid a bad outcome) – but it
cannot be shown to lead to the formation of morality, however
basic. Criminals can achieve their goals in splendid cooperation
and be content, but that does not make it more moral. Game
theory is agent neutral, it is utilitarianism at its apex. It
does not prescribe to the agent what is "good" – only what is
"right". It is the ultimate proof that effort at reconciling
utilitarianism with more deontological, agent relative,
approaches are dubious, in the best of cases. Teleology, in
other words, in no guarantee of morality.
Acts are either means to an end or ends in themselves. This is
no infinite regression. There is bound to be an holy grail
(happiness?) in the role of the ultimate end. A more commonsense
view would be to regard acts as means and states of affairs as
ends. This, in turn, leads to a teleological outlook: acts are
right or wrong in accordance with their effectiveness at
securing the achievement of the right goals. Deontology (and its
stronger version, absolutism) constrain the means. It states
that there is a permitted subset of means, all the other being
immoral and, in effect, forbidden. Game Theory is out to shatter
both the notion of a finite chain of means and ends culminating
in an ultimate end – and of the deontological view. It is
consequentialist but devoid of any value judgement.
Game Theory pretends that human actions are breakable into much
smaller "molecules" called games. Human acts within these games
are means to achieving ends but the ends are improbable in their
finality. The means are segments of "strategies": prescient and
omniscient renditions of the possible moves of all the players.
Aside from the fact that it involves mnemic causation (direct
and deterministic influence by past events) and a similar
influence by the utility function (which really pertains to the
future) – it is highly implausible. Additionally, Game Theory is
mired in an internal contradiction: on the one hand it solemnly
teaches us that the psychology of the players is absolutely of
no consequence. On the other, it hastens to explicitly and
axiomatically postulate their rationality and implicitly (and no
less axiomatically) their benefit-seeking behaviour (though this
aspect is much more muted). This leads to absolutely outlandish
results: irrational behaviour leads to total cooperation,
bounded rationality leads to more realistic patterns of
cooperation and competition (coopetition) and an unmitigated
rational behaviour leads to disaster (also known as Paretto
dominated outcomes).
Moreover, Game Theory refuses to acknowledge that real games are
dynamic, not static. The very concepts of strategy, utility
function and extensive (tree like) representation are static.
The dynamic is retrospective, not prospective. To be dynamic,
the game must include all the information about all the actors,
all their strategies, all their utility functions. Each game is
a subset of a higher level game, a private case of an implicit
game which is constantly played in the background, so to say.
This is a hyper-game of which all games are but derivatives. It
incorporates all the physically possible moves of all the
players. An outside agency with enforcement powers (the state,
the police, the courts, the law) are introduced by the players.
In this sense, they are not really an outside event which has
the effect of altering the game fundamentally. They are part and
parcel of the strategies available to the players and cannot be
arbitrarily ruled out. On the contrary, their introduction as
part of a dominant strategy will simplify Game theory and make
it much more applicable. In other words: players can choose to
compete, to cooperate and to cooperate in the formation of an
outside agency. There is no logical or mathematical reason to
exclude the latter possibility. The ability to thus influence
the game is a legitimate part of any real life strategy. Game
Theory assumes that the game is a given – and the players have
to optimize their results within it. It should open itself to
the inclusion of game altering or redefining moves by the
players as an integral part of their strategies. After all,
games entail the existence of some agreement to play and this
means that the players accept some rules (this is the role of
the prosecutor in the Prisoners' Dilemma). If some outside rules
(of the game) are permissible – why not allow the "risk" that
all the players will agree to form an outside, lawfully binding,
arbitration and enforcement agency – as part of the game? Such
an agency will be nothing if not the embodiment, the
materialization of one of the rules, a move in the players'
strategies, leading them to more optimal or superior outcomes as
far as their utility functions are concerned. Bargaining
inevitably leads to an agreement regarding a decision making
procedure. An outside agency, which enforces cooperation and
some moral code, is such a decision making procedure. It is not
an "outside" agency in the true, physical, sense. It does not
"alter" the game (not to mention its rules). It IS the game, it
is a procedure, a way to resolve conflicts, an integral part of
any solution and imputation, the herald of cooperation, a
representative of some of the will of all the players and,
therefore, a part both of their utility functions and of their
strategies to obtain their preferred outcomes. Really, these
outside agencies ARE the desired outcomes. Once Game Theory
digests this observation, it could tackle reality rather than
its own idealized contraptions.
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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