Alice and Bob.

“Independence of Irrelevant Alternatives” in Bitcoin Proof of Work

Jon Gulson


John Forbes Nash Jr.’s “The Bargaining Problem” (1950) sets a reasonable desiderata stated as axioms to be satisfied for an important and unexpected conclusion, derived by clear and unassailable mathematical reasoning (according to Kuhn and Nasar, in their biography, “The Essential John Nash”).

The most controversial of these axioms is the Independence of Irrelevant Alternatives, which generated dispute for years after publication (of The Bargaining Problem): Independence of Irrelevant Alternatives (IIA) is relevant to voting systems, in that adding irrelevant (non-winning) candidates should not be able to change the election results.

For example, if Alice and Bob are in an election, and Alice wins, then adding a third candidate should not change this outcome.

The Mystique of Values; Can There Be Independence of Irrelevant Alternatives?

The objective of The Bargaining Problem was to predict how players of a two person game would settle their differences, or in another interpretation, how an impartial arbitrator would identify a fair compromise to recommend to them.

Nash, in further papers, then elaborates and extends on the two-person nonzero-sum bargaining problem into an n-person case, and then into analysis as a non-cooperative game. In Two Person Cooperative Games (1953), Nash introduces a threat approach, whereby there is an umpire who can enforce contracts.

Fifty years later, Nash is still considering the IIA axiom, this time in context of coalition formation where possible alternative bargaining compromises or arrangements can become “irrelevant” in relation to the determination of the axiomatically preferred bargaining solution.

Bitcoin Proof of Work and a “World Empire Context”.

In relation to Nash’s bargaining solution, there appears a connection between IIA in voting systems and the bitcoin proof of work consensus: the proof of work is said to solve the problem of the majority decision by the majority being defined as one-CPU-one-vote, rather than one-IP-one-vote (as IP allocation could be subverted by anyone able to allocate many IP addresses).

Nash’s idea in Two-Person Cooperative Games of an umpire, then also seems to appear in bitcoin, in that the majority decision is represented by the longest chain with the greatest proof of work effort invested in it, so the honest chain grows fastest, outpacing competing chains.

This means irrespective of varying interest in running nodes over time and increasing hardware speed, the core rules of the game should remain self-enforcing with the majority decision unchanged (thanks also to the proof of work difficulty adjustment mechanism averaging a target number of block generation per hour) — and through this repetitive and recursive election procedure, cooperation can be expected to arise or derive, where the latent possibility exists for a “grand coalition” in a “world empire context” between all nations of the world which would potentially render the decentralization of bitcoin mute.