Re‐Mapation of the Nash Program In Synthesis With The Bitcoin Experiment
The Nash program is a rheomodic experiment-it's meant to evolve by its own implicit introspection. Here we mean to make observations that begin to map the bitcoin experiment and its genesis to the evolution or rounds of the Nash program. This writing then is a proposal as such and an experiment in and of itself. The method of synthesis here is via Bohmian Dialogue and we are in the on going process already of sustain such dialogue with a few other key and early participants.
The following will be updated to collect and narrate this mapation.
Consider the implicit equilibrium solution (NE) as an outcome of the non-cooperative games framework and with respect to the 2 player bargaining problem (as framed by Nash as well).
From here we observe a higher level consideration, not with respect to the bargaining outcomes themselves, but whether or not an agreement is made (agreement on agreement aka move forward) or not made (agreement on no agreement aka walk away) and the bargaining process or flux in between such outcomes.
Then we can consider such bargaining games with THREE players but with a framework in which two of the players, upon coming to agreement, CAN be represented as a single player thus reducing the game back to 2 player negotiation for agreement.
With iteration and the concept of transferable utility we have described the Nash program.
Our observation here is that this also describes the tacit and implicit process that bitcoin's security protocol upholds and guards.
Nash notes the initial considerations for the Nash program were limited by lack of computational power (thus we discover/imply an early evolution of the program):
These computations are found to be “heavy” so that our research could not have been done in the earlier days of game theory, like in the 50’s, because of the inadequacy of the computing resources then.
Nash, in the earlier 50's (of last century), published three papers in Econometrica that were concerned with this area for studies. These specifically consider games of two persons in which cooperative optimization is the concern. First, “The Bargaining Problem” finds an axiomatic approach leading to a definite formula for, effectively, the canonical arbitration of a bargaining problem in which two players (or participants) have the possi-bility of gaining mutual benefits if they can agree on a formula for cooperation. Then “Two-Person Cooperative Games” reviews the bargaining theory in a more general context where the two players have a variety of actions that they can take, before they are cooperating at all, which can variously affect their welfare circumstances (or their “payoffs”). A fresh approach to the bargaining problem side of this total cooperative game problem shows how a “game of demands”, for the two players, has a natural equilibrium which leads to the previously inferred formula for the allocation of payoffs in the simpler “bargaining problem” topic that Nash earlier studied. And the other concept, that of “threats”, links the competitive/non-cooperative side of the general game of two parties with the cooperative side which is modulated through the “demands” of the players.
And also a set of axioms is introduced that, as an alternative, leads to the same found cooperative game solution in a fashion parallel to the derivation of the normative bargaining solution found in “The Bargaining Problem”.
The third early publication in Econometrica was published as a work by three co-authors, Mayberry, Nash, and Shubik. And it was called “A Comparison of Treatments of a Duopoly Problem”. This paper considers a concretely described situation where two producers are producing the same marketable commodity.
Observing this flow chart from Nash's work (UPF.n11.n27.d03.2007.L) we start with the following in the context of grand coalition aka consensus. In such a flow chart there are 3 phases described and with those phases there can be multiple loops of paths etc.
(Hal Finney had talked about not being able to work out the internal algorithm in his early expressions about bitcoin his quote on that belongs here and a comment on how this flowchart could represent that expression Finney was reaching for etc.)
Nash talks about finding 'the ultimate truth' in regard to the study of human nature through 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.
The program started with initial conditions and a direction of understanding cooperation through the comparative game theoretic evolution of it:
This work has in two ways an experimental character: 1) the actual design of a model is like a matter of artistic discretion 2) 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…
A key component of the program is the constant consideration of 3 players and their transformation or reduction to 2 or a consensus (we have mapped this to the Byzantine Generals Problem):
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.
Nash admits from the beginning its understood that there is a limitation to the optimization of the initial choice of conditions (a reason why the evolutionary aspect of the program is implicit and important):
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
From Research Studies Approaching Cooperative Games with New Methods:
I want to say that it is very valuable that the observations derived from one set of experiments, possibly motivated by one theoretical model relating to bargaining or cooperative play and negotiations, would naturally often shed light on other variously differing theoretical models.
In terms of the design of the experiments, the players of a game, as experimental subjects, were not told how they must or should react to the observed behavior of the other players with whom they were interacting repeatedly in the plays of the experimental repeated games. Of course the design idea was that, analogously to a repeated game derived from a stage game of “Prisoners’ Dilemma” form, it would be possible for the experimental player-subjects to interact among themselves, in the repeated play, so that each player would tend to encourage cooperativeness by rewarding behavior (of a reactive sort) that would have comparably cooperative values.
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)).
A specified game may also have intrinsic characteristics that make it plausible that, even though it is a “cooperative game” in that the players are regarded as free to undertake all sorts of cooperative acts of collaboration (outside of the formal structure of the presentation of the game), they might NATURALLY not act in a simple pattern of cooperation (and the sharing, somehow, of wealth and resources) but rather there might be various differing forms of behavior that might possibly emerge as the observable behavior of the players.
This is analogous to the patterns observed in international politics and warfare, where shifting alliances and patterns of opposition have emerged regularly, for example, in European history.
The approach of the Nash program seems to be conducive with our re-mapation experiment (that natural observation are approximately conducive to a re-mapation of the Nash Program and the Bitcoin experiment):
Methods Used for Improving Approximations to Solutions Working either with 42 equations for x1, x2,...,x42 or with 21 equations for y1, y2,...,y21, we used Mathematica (on Linux) as the general framework for calculations. After initially obtaining one or a few good numerical solutions then others could be obtained by the use of what are called, mathematically, “homotopy methods”, and this was essentially a matter of always simply finding a new solution that would be numerically in close approximation to a known solution that had been previously found."
In the early iterations of the program there was no official funding grant:
Currently I am very much into some computational work for a model which would progress from the previously studied model which led to the publication in the IGTR journal. I have no assistants right now and no NSF project grant is applicable at this time (late 2010); so I am limited to doing the calculations myself and attempting to check the formal accuracy of my work myself.
Nash eventually noted:
And with the general move towards a more elaborated model to be studied (with the use of modern resources and technology for actual computations!) we have also the idea of setting up a game model with properly transferable utility.
And in Ideal Money he notes the formal benefits of TU with respect to cooperative games:
When one studies what are called “cooperative games”, which in economic terms include mergers and acquisitions or cartel formation, it is found to be appropriate and is standard to form two basic classifications:
(1): Games with transferable utility.
(and)
(2): Games without transferable utility (or “NTU” games).
In the world of practical realities it is money which typically causes the existence of a game of type (1) rather than of type (2); money is the “lubrication” which enables the efficient “transfer of utility”. And generally if games can be transformed from type (2) to type (1) there is a gain, on average, to all the players in terms of whatever might be expected to be the outcome.
Here we map this with the integration of Bitcoin as funding mechanism embedded in the program which doubles as a TU between players (however the Nash program defined or defines players etc). For this the components of Bitcoin are re-levant and necessary:
Unfortunately, proof of work is the only solution I've found to make p2p e-cash work without a trusted third party. Even if I wasn't using it secondarily as a way to allocate the initial distribution of currency, PoW is fundamental to coordinating the network and preventing double-spending. ~ Satoshi Nakamoto
Depending on the evolution, observations, and design there can be symmetrical reductions as noted in regard to previous iterations:
When the game has symmetries the equation set can be much reduced.
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.
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.
Perhaps there can be determined some of the evolving definitions with regard to the evolution framework or scope of the games, subprogram, and program etc. There is some description of the concept of iterations of rounds in the early Nash program modeling which we will collect and gather here to explain and extrapolate from.
For now there is the obvious consideration that the new 'bell' or 'gong' or signal for rounds or iterations might be best considered to be each Bitcoin halvening. This would then extend Bitcoin's consensus and coordination mechanisms out as a very general and broadened consensus field.
One possible approach for the early Nash program model involved "random proposers":
I have been myself, in recent years, one of the game theorists who have sought to (somehow) reduce 3-person cooperative games to non-cooperative games so that equilibrium methods could be applied to these games. And the ultimate objective could be merely to estimate “values” for the players or to also, conceivably, obtain predictions as to which of the coalitions might tend to form in intermediate negotiations of the players. A group of these approaches depend on the device of “random proposers” to achieve the descent from the level of the difficulties suggested by three-party games in general to the level of non-cooperative games of three players.
Via our studies of Ideal Poker as a generalization and specification of Ideal Money this gives us the idea to map or use a poker application for bitcoin. The synthetic narrative here is that early in the Nash program poker failed to provide a random proposer solution that was re-elevant or superior later solutions or evolutions etc.
This could be a folklore type explanation in which the truth of it becomes as irrelevant as both the poker software in Satoshi's implementation and the random proposers methods themselves etc.
In "THE AGENCIES METHOD FOR MODELING COALITIONS AND COOPERATION IN GAMES 2008" Nash describes a phenomenon that naturally maps to Bitcoin:
This suggests the economic concept of a “market price” which is associated with the “market clearing” concept.
This is reminiscent perhaps of the limited supply and block limitations that imply a fee market for bitcoin transactions (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
newer.perspective.304.2005.txt
The concept of the next major research project, in terms of mathematical efforts of calculation, is that of studying again the same sort of three-person games (of "bargaining and negotiation") but with a model of agencies in which the "agents" will be like attorneys of a robotic variety. The agents will themselves play to maximize their income from fees, but this will be as if their cost (in fees) is truly infinitesimal and paid by "the state".
This video might serve as a good primer for those new to game theory etc. and perhaps the Axlerod tournaments referenced can be considered proto-Nash Program iterations.