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 implementation of mixed strategies (strategies with probabilities attached to every move and outcome). Together with 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 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, sum of values of two sub-coalitions). Coalitions can be epiphenomenal: their value can be higher than combined values of their constituents. The amounts paid to players equal value of 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 players, describing division of coalition's value amongst them, is "imputation", a single outcome of a strategy. A strategy is, therefore, dominant, if: (1) each player is getting more under strategy than under any other strategy and (2) players in coalition receive a total payment that does not exceed value of coalition. Rational players are likely to prefer dominant strategy and to enforce it. Thus, solution to an n-players game is a set of imputations. No single imputation in solution must be dominant (=better). They should all lead to equally desirable results. On other hand, all imputations outside solution should be dominated. Some games are without solution (Lucas, 1967).
Auman and Maschler tried to establish what is right payoff to members of a coalition. They went about it by enlarging upon concept of bargaining (threats, bluffs, offers and counter-offers). Every imputation was examined, separately, whether it belongs in solution (=yields highest ranked outcome) or not, regardless of other imputations in solution. But in their theory, every member had right to "object" to inclusion of other members in coalition by suggesting a different, exclusionary, coalition in which members stand to gain a larger payoff. The player about to be excluded can "counter-argue" by demonstrating existence of yet another coalition in which members will get at least as much as in first coalition and in coalition proposed by his adversary, "objector". Each coalition has, at least, one solution.
The Game in GT is an idealized concept. Some of 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 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 specific moves that they select – are). The comparison of utilities is not 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 problems are linked to variant of rationality used in GT. It is comprised of a clarity of preferences on behalf of rational agent and relies on people's tendency to converge and cluster around right answer / move. This, however, is only a tendency. Some of time, players select 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 cardinal utility function. Obviously, highest ranking (smallest ordinal) preference should have had biggest probability attached to it – or could be treated as collapse event. But these are more or less known, even trivial, objections. Some of them cannot be overcome. We must idealize world in order to be able to relate to it scientifically at all. The idealization process entails incorporation of gross inaccuracies into model and ignorance of other elements. The surprise is that approximation yields results, which tally closely with reality – in view of its mutilation, affected by model.
There are more serious problems, philosophical in nature.
It is generally agreed that "changing" game can – and very often does – move 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 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 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 equilibrium strategy for each one of them will be this kind of threat. This is a clear social Hobbesian dilemma: equilibrium is absolute egoism despite fact that all players are altruists. This implies that we can learn very little about outcomes of competitive situations from acquainting ourselves with psychological facts pertaining to 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 available equilibrium strategies. The altruists in our example are prisoners of logic of 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 existence of an Hobesian enforcement mechanism (a state is most common one). Players would do better to avoid Paretto dominated outcomes by imposing 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 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 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 agent what is "good" – only what is "right". It is ultimate proof that effort at reconciling utilitarianism with more deontological, agent relative, approaches are dubious, in best of cases. Teleology, in other words, in no guarantee of morality.