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

Invariance with respect to positive affine transformations and the Bitcoin value.

Jon Gulson


Detractors of the Bitcoin proposition will speak on price fluctuations which result in risky speculation from wasteful energy expenditure. Alternatively, advocates will point to longer term appreciation against all sovereign nation currency issuances and its representation of the majority decision as insurance against the arbitrary inflation risks of centrally managed money. In the latter case, there has undeniably been an appreciation in the value of a bitcoin measure — but, why?

Explanation and analysis has been offered that the Bitcoin system is an axiomatic design, and that this can be attributed as a genealogy extending and drawing from serious heavyweight players in the combined fields of economics, mathematics, and social science.

Determination and Game Theory

The idea is that a bitcoin allows it user rational anticipation and expectation of the future, maximising the user’s welfare and utility in comparison to other available monetary choices. This in itself isn’t a unique idea because it’s based on already established axiomatic game theory determining non-zero-sum and unique bargaining solutions through unassailable mathematical reasoning.

Of course, the subject known as John Nash isn’t far off the radar when it comes to dialogue of this kind — Nash’s work ranks among the most influential when it comes to axiomatic bargaining and its “idealisations” and the utility of money in facilitating or lubricating this type of beneficial welfare:

“The inner area represents the bargains possible without the use of money. The area between parallel lines represents the possibilities allowing the use of money. Utility and gain measured by money are here equated for small amounts of money. The solution must be formed using a barter-type bargain for which u1 + u2 is at a maximum and using also an exchange of money.” John Nash, The Bargaining Problem, 1950.

The Scale Invariance Axiom and the Bitcoin Value.

In simple terms, affine means a parallel or close relationship; when two people have an affinity, it suggests something like a marriage or similar association. The invariance suggests it doesn’t matter how this relationship is measured or how it passes through time — the affinity will endure and remain.

In respect of the scale invariance possessing agency in a Nash bargain, it means the solution is unchanged by any positive scaling or changes in player’s utilities. In that sense, scale invariance is classical. In the way a Bitcoin advocate will say a bitcoin is a bitcoin and always will be.

Measuring Money

At any immediate moment, a bitcoin has become exchangeable to a sovereign issuance such as an American dollar, or British pound, etc. However, over time the preferences of the bargainer in which these media of their contract is expressed is a matter for them and their logic. Over the longer term, the Bitcoin system appears deterministic by design, presumably because of the axioms it’s “hashing”. However, Nash bargaining is by no means a universally agreed upon idea, but in this context it seems like Bitcoin might be a “short cut” to the application of welfare economics and benefits of coalitions in a conflicting world and the agent makes their choice.