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