Bitcoin as an Implementation of John Forbes Nash Jr.’s Axiomatic Bargaining “Idealizations”.
Any serious analysis of modern game theory can’t look beyond John Von Neumann and Oskar Morgenstern’s ground-breaking Theory of Games and Economic Behavior (1944), for a definitive mathematical explanation of human economic interaction.
Von Neumann provided the mathematical proofs, Morgenstern the economic theory. It’s noteworthy in this regard Morgenstern attended the University of Vienna in the 1920’s, and so bringing with him an “Austrian” influence — Morgenstern believed governments were not capable of coordinating markets “because the whole economic process cannot be statistically portrayed”.
Von Neumann and Morgenstern establish a zero sum game with winners and losers. Subsequently, John Forbes Nash Jr.; then a young and ambitious mathematician, makes the first significant extension to this thesis by establishing the conditions for a two person non-zero sum game, which works towards “certain idealizations”.
The Bargaining Problem (1950)
Just as Morgenstern brought an “Austrian” influence to Theory of Games and Economic Behavior, Nash — prior to joining Von Neumann and Morgenstern at Princeton University in 1948 — had taken just one elective course in “International Economics” taught — which Nash later acknowledges as a good influence — by an Austrian (Bert F. Hoselitz) who, like Morgenstern, was a former student of Mises.
Nash’s breakthrough in “The Bargaining Problem” is to satisfy the preferences of parties to an economic or commercial situation by meeting assumptions, conditions, or axioms, which result in a determined bargaining outcome.
These axioms, or “idealizations” (as Nash termed them) were Pareto efficiency, scale invariance, symmetry, and independence of irrelevant alternatives. Each of these axioms appear present in bitcoin.
Asymptotically Ideal Money and Realizing Pareto
Toward the end of the last century, Nash starts to lecture on Ideal Money, which Nash defines as “intrinsically free of inflation”. Nash criticises central banks for adopting an arbitrary approach to inflation targeting with an insufficient axiom set — that if they are to target inflation, that rate should be zero — redolent to Satoshi’s claim bitcoin escapes the arbitrary inflation risk of centrally managed currencies.
However Nash realises early in his work on Ideal Money, that a money can’t be so free of inflation that it won’t circulate (Nash calls this a safe-deposit box singularity), in that a problem for the issuer of a coinage (whether in electronic form, or other) is such a money being “too good” becomes exploitable by those not party to its issuance.
Nash therefore introduces a “steady and constant” rate of inflation so this problem is avoided — Nash thereon in refers to Asymptotically Ideal Money. Here we can make obvious comparison with the bitcoin asymptotic money supply targeting:
This means the projected long term supply issuance of bitcoin is generally believed to be 21 million — but more accurately, and due to a side-effect of the data structure of the blockchain, the exact value is 20,999,999.9769 coins — so that the total number only approaches 21 million without becoming tangential to the “ideal” (thus asymptotic).
The asymptote demonstrably provides the characteristic function of Pareto cumulative distribution in total bitcoin supply, and leads into a further Nash bargaining axiom.
Difficulty Adjustment and “Determination” in Bitcoin
The scale invariance Nash uses in The Bargaining Problem, and as demonstrated in the graph above by the orange line representing the annualised money supply growth rate, occurs in bitcoin through the difficulty adjustment mechanism. This means, no matter how popular or unpopular bitcoin becomes to mine, the scale of its inflation supply density shouldn’t vary over time (remaining “steady and constant”).
By unpegging bitcoin supply from its demand, rational expectations as to the performance of bitcoin over time need not be fettered by the threat of debasement of the issuer. Nash’s belief that inflation can indeed be controlled by the supply of money — perhaps considered an “Austrian virtue” — can then be demonstrated by a “long tail distribution” in relation to the US dollar’s purchasing power against bitcoin over all time:
Symmetry in Bitcoin and Nash Bargaining
Satoshi implies on the importance of symmetry in his design when he speaks on a dislike for forks and how they would sound to the network:
“It could tell if it’s not hearing the hum of the world anymore…In practice, splits are likely to be very asymmetrical.” Satoshi Nakamoto, 3 August, 2010
The symmetry axiom Nash uses in The Bargaining Problem can be summarised as the players being indistinguishable so any agreement should not discriminate against them — Nash phrases this as “equality of bargaining skill” — and probably the most relevant characteristic to this in bitcoin is the network pseudonymity/anonymity and the absence for the necessity for first person identity in participation (this might even explain why Satoshi used a pseudonym himself, i.e. out of a desire to represent symmetry in his design).
One may also speculate it’s the symmetry in bitcoin which creates the decentralisation. In a futuristic paper Nash wrote in 1954 called Parallel Control, a design is presented for “electronic brains of the future”, and where the idea is to “decentralize control”. Nash closes with a line alluding to symmetry, which reads as “…the human brain is a highly parallel setup. It has to be.” — and as Nash moves his bargaining idea into a non-cooperative context, he continues with symmetrical solutions:
“The main mathematical result is the proof of the existence in any game of at least one equilibrium point. Other results concern the geometrical structure of the set of equilibrium points of a game with a solution, the geometry of sub-solutions, and the existence of a symmetrical equilibrium point in a symmetrical game”. John Forbes Nash Jr., Non-Cooperative Games, 1950
The Bitcoin Proof of Work Voting Consensus
This could have later provided motivation for Nash, when resuming his game theory ideas at a similar time to writing Ideal Money, to devise an Agencies Method for modelling cooperation from a non-cooperative context using “attorneys of a robotic variety” in games of bargaining and negotiation. Agencies therefore becomes an election procedure where agents vote on the agencies of others, even if they exhibit reluctant acceptance behavior, but do so in order to gain benefits.
The parallel to draw with bitcoin is the proof of work consensus, where voting is assigned on a one-CPU-one-vote basis, rather than a one-IP-one-vote basis (as IP allocation could be subverted by anyone able to allocate many IP addresses) is the rules of the majority decision become self-enforcing through the longest chain and therefore the benefit of this chain (BTC) in comparison to the irrelevance of those it’s competing against is that it is outpacing them by growing faster through node validation and accepting the most trustworthy version of events (solving the Byzantine Generals’ Problem in the process).
It may not be fashionable or popular to link the advent of bitcoin and the machinations of blockchain technology back to the development of game theory in the 1940’s and 1950’s which took hold at Princeton University at that time as a new branch of mathematics.
However, there seem to be so many coincidences in the bitcoin design to Nash bargaining “idealizations” that the connection should be taken seriously. Consider the William Feller reference in bitcoin — the obvious question is why did Satoshi Nakamoto feel the need to make this? Why is Satoshi referencing a Princeton University mathematician — who Nash would have known — from the 1950’s in the bitcoin white paper?
These questions can be considered in light of understanding where bitcoin came from and why — and to ultimately garner a better insight into its actual purpose and end game.