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Nash equilibrium and pro-cooperative games in the bitcoin context

A seemingly light hearted remark about aliens being accounted for (when designing bitcoin) by Satoshi Nakamoto, alludes to considered subject matter of cooperative games in the determinative context of how to measure the outcome or rationality of engagement when playing any kind of game, whether non-cooperative or cooperative.

This is because the idea of cooperation is based not only on the understanding of rules or context of a game, but also on how such rules are enforceable to be taken seriously, and is why the idea of prisoner dilemma games evolved from John F Nash Jr.’s Nobel winning thesis on non-cooperative games (1950).

It is therefore interesting to consider how non-cooperative games (where players have limited or no opportunity to communicate with each other) transform into pro-cooperative games (where players can not only communicate with each other and commit to a strategy, but encourage and induce honesty in doing so) have evolved in parallel with the evolution of modern computing.

Uniqueness in Games

“A non-cooperative game does not always have a solution, but when it does the solution is unique. Strong solutions are solutions with special properties. Sub-solutions always exist and have many of the properties of solutions, but lack uniqueness” John F Nash Jr., Non-Cooperative Games, 1950

John Nash’s later works in game theory, not only include the Ideal Money (money intrinsically free of inflation), but also an agencies based method in the study of coalition forming games of a repeated and recursive character resulting in enhanced cooperation between players.

It is noted on the possibilities of such agency being made possible by technology:

“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.” John F Nash Jr., The Agencies Method for Modeling Coalitions and Cooperation in Games, 2008

In typical Nash style, these works aren’t prescriptive or prognostic (which can make Nash difficult to understand), but contain elements of commonality, such as uniqueness:

“I must cautiously remark here that I DO NOT have a PRECISE concept of a difference between pro-cooperative games and those not of that type. But if we have any specific modeling of the games in terms of a repeated games equilibrium then there could be values of b1, b2, and b3 for which the equilibrium would be unique and other values for which it would be non-unique and this phenomenon could be viewed as corresponding to the concept or issue of pro-cooperative games.” John F Nash Jr., Pro-cooperative Games, 2005


When Nash entered Princeton University as an undergraduate in 1948, he described — according to his biographer Sylvia Nassar — his religion on his entrance papers as Shinto, implying superior lineage to fellow students.

Shinto isn’t an official creed or codified system of theology, but emphasizes Makoto — sincerity in the heart. Makoto, “Kannagara-no-Michi”, is the basis of Shinto ethics. It is a distinct sensibility that underlies an entire approach to life and the world.

Interestingly, Makoto Shinto is an anagram of Satoshi Nakamoto, with obvious connections to decentralisation.

Contracts and Agreements

In Two Person Cooperative Games (1953), Nash introduces the idea of an umpire who can enforce agreements (such as contracts). In the Ideal Money, Nash repeatedly speaks to the importance of contracts in commerce and their trustworthiness, and how the money unit contained in the contract is causative to complete contractual performance.

Nash’s axiomatic approach to game theory is apparent throughout, with mathematics regarded almost genealogical in its problem solving method:

“The Sumerian scribes wrote down records for the quantities of grain received or sent from their central granaries. So mathematical logic itself looks like a language that is naturally capable of evolution like also mathematics as a language and as an encyclopedia.” John F Nash Jr., Hierarchical Introspective Logics, 1998

There is a close parallel here between Nash and Satoshi, where Satoshi expresses his doubt that a country could become totally airtight from the internet, such that it couldn’t communicate with the rest of the world, to Nash’s view on “translatability”:

“But we feel that a “translatability” property should hold true here. Thus, for example, a relatively modern proof in geometry by Pascal should be translatable into a form, written in Greek, that Euclid would have found acceptable.” John F Nash Jr., Hierarchical Introspective Logics, 1998

The idea therefore of a sense of anticipation and expectation from a probability based system such as bitcoin becomes apparent in the idea of a universally open state of play, such that if bitcoin was designed with aliens in mind, then an alien can’t really be an alien if proper communication is possible.

Ideas in games, language, and trust.