On the Simplification of Nash and Satoshi's Research Projects From 42 Variables to 21
In the writing Research Studies Approaching Cooperative Games with New Methods Nash explains there is a limitation in solutions plausible or approachable when two players have too strong of 'control' (here we use control as a loose reference to whatever coalition gravitates to as there could be different applications to Nash's work):
Effectively, this phenomenon of alternative modes of cooperation, for the players in a game, was noticed mathematically when the smooth graph describing the cooperative behavior of the players in a model for cooperation of three players via “agencies” and “acceptances” failed to be continuable when one or two of the coalitions having only two members became too strong in comparison with the “grand coalition” (of all three of the players in the game).
Nash's observations of the difficulties of some formations of 3 player games being effectively irre-solvable is well exemplified by our base case representation of the the Generalization and Possible Transformations of the Byzantine Generals Problem which observes:
no solution exists for three generals that works in the presence of a single traitor.
In the paper THE AGENCIES METHOD FOR MODELINGCOALITIONS AND COOPERATION IN GAMES Nash notes in a section called "Observed Market Clearing Phenomena":
Relatively late in the period of the work on the calculation of numerical solutions we found, empirically (by observation), that some of the parameter values in calculated solutions were coming out the same. This was first noticed, but not understood, when we initially solved for solutions of entirely symmetric games (where b1 = b2 = b3). We found that of 4 distinct quantities describing a player’s choice for the allocation of utility (if the player would become the “final agent”) that two of these quantities were very nearly the same (according to the numerical calculations done with many decimal places of accuracy using Mathematica).
He continues through the rigorous game theoretical explanation to conclude:
This suggests the economic concept of a “market price” which is associated with the “market clearing” concept.
Satoshi - Sirius emails 2009-2011
Inflation issues were superseded by changes I made later to support transaction fees and the limited circulation plan.
this inflation discussion was before the transaction fee mechanism and fixed plan of 21 million coins was posted, so it may not be as applicable anymore
The Equations for the Equilibrium Solutions From the quantities above, not including the demand numbers, the patterns of the actual (steady) behavior of the three players is fully described. These numbers, 18 a-numbers and 24 u-numbers, or 42 parameters in all, therefore describe the directly observed behavior of the players.
When the game has symmetries the equation set can be much reduced. If b1 = b2 = b3 then all of the two player coalitions have the same strength and then we can look for solutions involving the same behavior for all players. Then the equations reduced to merely 7 in number (and this was a good basis for finding the first solutions!). If merely b1 = b2 then the coalitions (1, 3) and (2, 3) have the same strength and we can look for solutions with P1 and P2 in symmetrically patterned behavior. This leads to a reduction to 21 equations, and we did most of our work on calculations with these 21 equations since that level of symmetry was enough to yield differentiation among the various value concepts that could be compared.
...we can remark that FULL symmetry reduces the number of descriptive variables needed down to 7 (from 42 in general or 21 with two players being symmetrically situated in the game)
The forthcoming book of E. Maskin, which expands on his Presidential Address to the Econometric Society, has a theme that connects with our idea of “ProCooperative Games”. This is the theme of “externalities” as realistic considerations that are not included in the formal description of a game (say as a “CF game” in particular) and which COULD act, for example, to (effectively) prevent the formation of the grand coalition. We began to see that in our games studied by our modeling method (with agencies) that if the strengths of all of the 2-player coalitions were quite large (and comparable to v = (1, 2, 3) = +1) then that it would be quite reasonable, in a repeated game context, for there to be various stable equilibria. Thus any two of the players could be seen as being able to “learn” that they are natural allies and then, through an alliance, gain the lion’s share of all the possible benefits from the game.
In THE AGENCIES METHOD FOR MODELINGCOALITIONS AND COOPERATION IN GAMES Nash notes a specification of the model:
In the experiments it was found that it was possible to preserve and utilize “the method of acceptances”, in a general sense, so that coalitions were formed always by a process in which one player or one leader of an established coalition (or alliance) would elect to “accept” the leadership of another player or coalition leader.
We feel this maps well to the Byzantine Generals Problem specification as outlined here where:
"A small number of traitors cannot cause the loyal generals to adopt a bad plan."
In THE AGENCIES METHOD FOR MODELING COALITIONS AND COOPERATION IN GAMES Nash coins and explains the Nash Program:
...“Nash program” (which was the suggestion that the study of cooperative games should, somehow, be reduced to that of non-cooperative games).
The Nash Program considers games of players with respect to symmetrical simplifications:
There are types of CF games of 4 players that are essentially different from games of three or two players, yet which, because of symmetries of the game, would involve a much smaller number of strategy parameters to be determined than would be needed for a plausible model for a general sort of 4-person CF game.
Or another variety of very simple 4-person game for computational study would be, instead, a 4-person game where one player is different and three are isomorphic in form
The Methods for Finding Solutions and Calculating Data Points
This work has in two ways an experimental character. First, the actual design of a model is like a matter of artistic discretion, and it is simply an ATTEMPT to provide for the possibility of naturally reactive behavior so that the phenomenon of “the evolution of cooperation” may occur and may be revealed through the actual calculation of equilibrium solutions Our present work, in its nature, does not attempt to find the ultimately ideal form of reactive behavior (of an individual player) so that, with that behavior, that the resulting equilibrium in the context of a repeated game of interactive behavior would be optimized as far as the interests of that player are concerned. In principle, we feel, the issue of the optimization of the form of the reactive behavior pattern of an individual player is what would be done in Nature by selective evolution. In game-theoretical studies the parallel achievement might be realized by comparison of alternative models. (It is certainly rather straightforward to compare various programs, say for playing chess or Go by simply letting the programs compete in playing the game.)
Because the whole concept of our idea of modeling the attainment of cooperation by the players in a 3-party game situation in terms of a process involving a sequence of elections of agencies was inspired by thinking of the analogy to evolution in Nature, it is logical to consider that when evolution has already arrived at SOME DEGREE of success in improving the cooperativeness of the behavior of players (that initially were entirely independent in their interests, and with regard to their utility functions); because of this, we should consequently realize that a found modeling that favors effectively cooperative behavior by the players in a form that derives from their independently motivated actions (like actions in a non-cooperative game) may not PERFECTLY model the NATURAL possibilities for the attainment of cooperative behavior by the route of a form of evolution. This is because evolution in Nature is generally viewed as an ONGOING process and thus it cannot be expected, presumably, at any particular time, to have arrived at final perfection.
... from the general viewpoint of evolutionary theory, we DO NOT KNOW, from the apparent success of the model, that we have found an ultimately PERFECT form of modeling for the natural process of cooperation. Therefore, in a logical or philosophical sense, we should think that we don’t know that a specific model (which allows for a natural mode of cooperative behavior in a repeated game context) is perfect and gives the final answer until we have made some further explorations
Recall our framework for The Bohmian Rheomode experiment in which Bohm prescribes the dualistic nature of observation:
..it is possible merely to observe language as it is, and has been, in various differing social groups and periods of history, but what we wish to do in this chapter is to experiment with changes in the structure of the common language. In this experimentation our aim is not to produce a well-defined alternative to present language structures. Rather, it is to see what happens to the language function as we change it, and thus perhaps to make possible a certain insight into how language contributes to the general fragmentation
There is the language "experimenter" that perhaps uses a rheomode etc. but also the overall evolution of language the experiment inspires (for example maybe there can be other language experiments beyond rheomodes):
… one of the best ways of learning how one is conditioned by a habit (such as the common usage of language is, to a large extent) is to give careful and sustained attention to one’s overall reaction when one ‘makes the test’ of seeing what takes place when one is doing something significantly different from the automatic and accustomed function.
Nash gives a metaphorical framework that suggests we consider such evolution of games and with regard to the evolution of the characteristic interactions of the players:
In Nature an example is given by lichen species. In a complex case there will be combined (1) a fungus, (2) a green alga species, and (3) a cyanobacterial species, with all of the three contributing distinct and essential functions. Over a period of time, like, say, 100 million years, a complex lichen variety existing today might naturally evolve into a form exhibiting changes in the two or three component species and changes in how the components would effectively cooperate.
The analogy to this is that a found formula (as it were) for the cooperation of three payers in a three-player game context might not be perfect (and a final answer) if a better formula might appear as a natural evolution of a search for reasonable models (that naturally enable cooperation in repeated game contexts).
Thus, in principle, game theorists working with this sort of modeling, where cooperation results from a repeated game equilibrium, need to consider to what extent they can justifiably think that perfection has been achieved with regard to how each player’s behavior can be considered to optimize in relation to his/her interaction with the others.
The Nash Program is as if an evolving experiment in and of itself to be observed as it also makes its own internal observations:
THE AGENCIES METHOD FOR MODELING COALITIONS AND COOPERATION IN GAMES
I feel, personally, that the study of experimental games is the proper route of travel for finding “the ultimate truth” in relation to games as played by human players. But in practical game theory the players can be corporations or states; so the problem of usefully analyzing a game does not, in a practical sense, reduce to a problem only of the analysis of human behavior
It is apparent that actual human behavior is guided by complex human instincts promoting cooperation among individuals and that moreover there are also the various cultural influences acting to modify individual human behavior and at least often to influence the behavior of humans toward enhanced cooperativeness.
This work has in two ways an experimental character. First, the actual design of a model is like a matter of artistic discretion, and it is simply an ATTEMPT to provide for the possibility of naturally reactive behavior so that the phenomenon of “the evolution of cooperation” may occur and may be revealed through the actual calculation of equilibrium solutions.
Our study has the character of an experiment, but rather than working directly with human subjects we computationally discover the evolutionarily of a triad of bargaining or negotiative players. And these players are, as far as the experimental science is concerned, equivalent to a set of three robots. So whether or not the experiment can be carried out successfully becomes simply a matter of the mathematics
Whatever choices we make at first, with regard to how the players are to reactively behave, there is, a priori, the possibility that some other design might have each player (or an individual player) behaving more effectively in terms of effectively inducing desirably cooperative behavior on the part of the other players.
Our present work, in its nature, does not attempt to find the ultimately ideal form of reactive behavior (of an individual player) so that, with that behavior, that the resulting equilibrium in the context of a repeated game of interactive behavior would be optimized as far as the interests of that player are concerned.
In principle, we feel, the issue of the optimization of the form of the reactive behavior pattern of an individual player is what would be done in Nature by selective evolution.
In game-theoretical studies the parallel achievement might be realized by comparison of alternative models. (It is certainly rather straightforward to compare various programs, say for playing chess or Go by simply letting the programs compete in playing the game.)
We developed a series of Mathematica programs for improving the quality of an initially given approximate solution of a system of simultaneous equations. (Versions of these programs and of associated computational data developed in the work on this research project will be made available in a reference web site that we are preparing to be available to readers of this publication.) These programs work by modifying the variables (e.g.: y1,...,y21) of an approximate solution