Image taken from “The Bargaining Problem” by John Nash.

Further Analysis on “Independence of Irrelevant Alternatives” as a Bitcoin Design Axiom.

Jon Gulson
5 min readSep 3, 2023

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This analysis extends previous rudimentary work on the absence of a third party in the core Bitcoin system representing Independence of Irrelevant Alternatives (IIA) in the proof-of-work Bitcoin uses to determine the majority decision of its chain.

In addressing transaction settlement, this is how the Bitcoin system design is described:

“A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution. Digital signatures provide part of the solution, but the main benefits are lost if a trusted third party is still required to prevent double-spending.” Bitcoin: A Peer-to-Peer Electronic Cash System, 2008.

The obvious connection between IIA and Bitcoin proof-of-work is the irrelevance in both ideas of a third party in a two player or peer-to-peer game (as represented by Jack and Bill, in the header illustration). This goes to the heart of the value proposition behind a bitcoin: there isn’t a formally or legally recognised “principal” (third party) and so the core Bitcoin system avoids all the associated mediation, insurance, legal, representation, and dispute resolution costs incurred where a principal (third party) is involved:

“Commerce on the Internet has come to rely almost exclusively on financial institutions serving as trusted third parties to process electronic payments. While the system works well enough for most transactions, it still suffers from the inherent weaknesses of the trust based model. Completely non-reversible transactions are not really possible, since financial institutions cannot avoid mediating disputes. The cost of mediation increases transaction costs, limiting the minimum practical transaction size and cutting off the possibility for small casual transactions, and there is a broader cost in the loss of ability to make non-reversible payments for non-reversible services.” Bitcoin: A Peer-to-Peer Electronic Cash System, 2008.

IIA Agency in Nash Bargaining

John Nash offered a new solution in his first game theory paper (The Bargaining Problem, 1950) to a classical economic problem:

“A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. It may also be regarded as a nonzero-sum two-person game. In this treatment a few general assumptions are made concerning the behavior of a single individual and of a group of two individuals in certain economic environments. From these, the solution (in the sense of this paper) of the classical problem may be obtained. In the terms of game theory, values are found for the game.” John F Nash Jr., The Bargaining Problem, 1950.

The “assumptions” Nash speaks about in the opening to The Bargaining Problem are also referred to as “idealizations” in the paper, and show a realisation that Nash doesn’t know or understand enough about the preferences of the bargainers to determine or maximise their utility so he creates them in a set of axioms to be satisfied. IIA is the most controversial of these and continues to generate dispute — Nash himself continued to research and question IIA well into his advanced years.

The other assumptions Nash uses for his bargaining solution are Pareto efficiency, symmetry, and scale invariance. Significantly, Nash recognised the utility of money in satisfying his “idealizations” and therefore lays foundations for his later works on Ideal Money and Asymptotically Ideal Money:

“When the bargainers have a common medium of exchange the problem may take on an especially simple form. In many cases the money equivalent of a good will serve as a satisfactory approximate utility function.” John F Nash Jr., The Bargaining Problem, 1950.

A Bitcoin Granularity Philosophy

Nash’s bargaining solution resolved a problem economists had been considering regarding the determination of a bargain which produced maximum monetary gain according to each player’s utility. This was a problem first posed by Francis Edgeworth in 1881, who wanted to get away from the political economic traditions of Adam Smith and David Ricardo, and alternatively adopted a mathematical approach to understand the nature of human bargaining which wasn’t just limited to large nation players.

As Nash develops his ideas on money, he makes this observation:

“In Game Theory there is generally the concept of ‘pay-offs’, if the game is not simply a game of win or lose (or win, lose, or draw). The game may be concerned with actions all to be taken like at the same time so that the utility measure for defining the payoffs could be taken to be any practical currency with good divisibility and measurability properties at the relevant instant of time.” John F Nash Jr., Ideal Money and the Motivation of Savings and Thrift, 2011

This leads into a question of IIA and whether the granularity — or divisibility — of a bitcoin is causal to IIA? This can be demonstrated against supply inflation of a bitcoin and how this appears determinative to the value of a US$ against a bitcoin:

Source: Nakamoto Institute
Source: Google currency converter

The conclusion being the annualised money supply growth rate of bitcoin suggests determinism to the value of a bitcoin as measured by the value of a United States dollar — to four decimal points, a United States dollar can’t purchase any bitcoin as of 20 August 2023. In that respect, the asymptotic money supply targeting of the Bitcoin system is demonstrating independence from alternative macroeconomic traditions or influences as well as its future value laying in the “long tail” part of the distribution.

And while IIA will continue to generate controversy, disagreement, and potential breakthroughs, it does at least court the question that if Bitcoin is axiomatic by design, then it raises possibilities for currency coalitions to arise from similar principles or values rather than discretionary central bank policy.

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