Solution To An N Players Game example essay topic

Sam Vaknin's Psychology, Philosophy, Economics and Foreign Affairs Web Sites Consider this: Could Western management techniques be successfully implemented in the countries of Central and Eastern Europe (CEE) Granted, they have to be adapted, modified and cannot be imported in their entirety. But their crux, their inalienable nucleus can this be transported and transplanted in CEE Theory provides us with a positive answer. Human agents are the same everywhere and are mostly rational. Practice begs to differ. Basic concepts such as the money value of time or the moral and legal meaning of property are non existent. The legal, political and economic environments are all unpredictable.

As a result, economic players will prefer to maximize their utility immediately (steal from the workplace, for instance) than to wait for longer term (potentially, larger) benefits. Warrants (stock options) convertible to the companys shares constitute a strong workplace incentive in the West (because there is an horizon and they increase the employees welfare in the long term). Where the future is speculation speculation withers. Stock options or a small stake in his firm, will only encourage the employee to blackmail the other shareholders by paralyzing the firm, to abuse his new position and will be interpreted as immunity, conferred from above, from the consequences of illegal activities.

The very allocation of options or shares will be interpreted as a sign of weakness, dependence and need, to be exploited. Hierarchy is equated with slavery and employees will rather harm their long term interests than follow instructions or be subjected to criticism never mind how constructive. The employees in CEE regard the corporate environment as a conflict zone, a zero sum game (in which the gains by some equal the losses to others). In the West, the employees participate in the increase in the firms value.

The different between these attitudes is irreconcilable. Now, let us consider this: An entrepreneur is a person who is gifted at identifying the unsatisfied needs of a market, at mobilizing and organizing the resources required to satisfy those needs and at defining a long-term strategy of development and marketing. As the enterprise grows, two processes combine to denude the entrepreneur of some of his initial functions. The firm has ever growing needs for capital: financial, human, assets and so on.

Additionally, the company begins (or should begin) to interface and interact with older, better established firms. Thus, the company is forced to create its first management team: a general manager with the right doses of respectability, connections and skills, a chief financial officer, a host of consultants and so on. In theory if all our properly motivated financially all these players (entrepreneurs and managers) will seek to maximize the value of the firm. What happens, in reality, is that both work to minimize it, each for its own reasons. The managers seek to maximize their short-term utility by securing enormous pay packages and other forms of company-dilapidating compensation.

The entrepreneurs feel that they are strangled, shackled, held back by bureaucracy and they rebel. They oust the management, or undermine it, turning it into an ineffective representative relic. They assume real, though informal, control of the firm. They do so by defining a new set of strategic goals for the firm, which call for the institution of an entrepreneurial rather than a bureaucratic type of management. These cycles of initiative-consolidation-new initiative-revolution-consolidation are the dynamos of company growth. Growth leads to maximization of value.

However, the players dont know or do not fully believe that they are in the process of maximizing the company's worth. On the contrary, consciously, the managers say: lets maximize the benefits that we derive from this company, as long as we are still here. The entrepreneurs-owners say: we cannot tolerate this stifling bureaucracy any longer. We prefer to have a smaller company but all ours. The growth cycles forces the entrepreneurs to dilute their holdings (in order to raise the capital necessary to finance their initiatives). This dilution (the fracturing of the ownership structure) is what brings the last cycle to its end.

The holdings of the entrepreneurs are too small to materialize a coup against the management. The management then prevails and the entrepreneurs are neutralized and move on to establish another start-up. The only thing that they leave behind them is their names and their heirs. We can use Game Theory methods to analyse both these situations. Wherever we have economic players bargaining for the allocation of scarce resources in order to attain their utility functions, to secure the outcomes and consequences (the value, the preference, that the player attaches to his outcomes) which are right for them we can use Game Theory (GT).

A short recap of the basic tenets of the theory might be in order. GT deals with interactions between agents, whether conscious and intelligent or Dennettic. A Dennettic Agent (DA) is an agent that acts so as to influence the future allocation of resources, but does not need to be either conscious or deliberative to do so. A Game is the set of acts committed by 1 to n rational DA and one a-rational (not irrational but devoid of rationality) DA (nature, a random mechanism). At least 1 DA in a Game must control the result of the set of acts and the DAs must be (at least potentially) at conflict, whole or partial. This is not to say that all the DAs aspire to the same things.

They have different priorities and preferences. They rank the likely outcomes of their acts differently. They engage Strategies to obtain their highest ranked outcome. A Strategy is a vector, which details the acts, with which the DA will react in response to all the (possible) acts by the other DAs. An agent is said to be rational if his Strategy does guarantee the attainment of his most preferred goal.

Nature is involved by assigning probabilities to the outcomes. An outcome, therefore, is an allocation of resources resulting from the acts of the agents. An agent is said to control the situation if its acts matter to others to the extent that at least one of them is forced to alter at least one vector (Strategy). The Consequence to the agent is the value of a function that assigns real numbers to each of the outcomes. The consequence represents a list of outcomes, prioritized, ranked.

It is also known as an ordinal utility function. If the function includes relative numerical importance measures (not only real numbers) we call it a Cardinal Utility Function. 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 agents 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 (ZSG) 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 rivals 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: (1) is not dependent upon any utility function (2) it is impossible for two players to improve the Nash solution ( = their position) simultaneously ( = the Paretto optimality) (3) is not influenced by the introduction of irrelevant (not very gainful) alternatives and (4) is symmetric (reversing the roles of the players does not affect the solution).

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 epiphenomena l: 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 coalitions 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). Human and Masc hler 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 peoples 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 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 Hobbesian 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 consequential ist 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 mnemoc 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 (competition) 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.

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