Principals and Agents — John F Nash Jr. (centre left).

Bitcoin and Repeated Principal-Agency Games

Jon Gulson


Principal and Agent games are well established in game theory — this writing explains how they relate to Bitcoin, in respect to contracts at large, and how decentralisation has long been applied by “non-cooperative” game methodology.

Game theoretically expressed, the principal founds the expectations or the bounds of an agreement, contract, or game; where the agent then carries out their responsibilities as so defined to the rules or conditions set out.

In everyday life, the principal might issue an employment contract, and the agent carries out those obligations because the principal — for whatever reason — can’t or doesn’t want to themselves. For example, at a wedding, the vows in which the marriage is expressed could be thought as principal — with the subjects becoming agents in that circumstance. In poker too, the casino is the principal, the player the agent.

The agent therefore gets to choose on behalf of the principal, in a random environment, the outcome which can’t be exactly predicted but is made to a pre-announced reward function (contractual remuneration or otherwise), which depends only on the directly observed consequence.

“This last restriction expresses the fact that the principal cannot directly observe the agent’s action, nor can the principal observe the information on which the agent bases his action. This situation is one of the simplest examples of decentralized decision-making in which the interests of the decision-makers do not coincide.” [1]

Nash Bargaining and the Cost of Trust

When corporations or firms cooperate in economic bargaining, it’s generally believed to work better on smaller scale — it means the communication is more limited and the mechanisms by which players make binding commitments more trustworthy [2], with less room for “verbal complication” or for things to go wrong. [3]

Modelling this cooperation has long since concerned the field of game theory, in respect of outcomes and welfare for those involved.

Cooperative bargaining ideas developed in 1944 with von Neumann’s and Morgenstern’s Theory of Economic Games and Human Behavior. These ideas then extended into John F Nash Jr.’s The Bargaining Problem (1950), where a set of idealisations, axioms, values, or principles, were to be satisfied for a non-zero sum two person game to be achieved. [4]

Bitcoin as a Set of Satisfied Idealisations for a Non-Zero Sum Two Person Game

Bitcoin in itself doesn’t produce an “equal outcome” or equality for all, but the pseudonymity (in theory at least) does provide a kind of symmetry (or equality of bargaining skill). [5]

The symmetry provides for a characteristic of Bitcoin being open source — i.e. its participants are voluntary — and it’s why Nash’s Agencies Method is highly relevant, and why the decentralisation creates agency in the bargaining environment [1].

The paradox here is a principal will usually set the conditions for a game, by which the agents then “play”, but usually in a setting of contract offer or self-enforcement. In Bitcoin, players can just “play” — it’s permissionless by that dichotomy of non-cooperation and cooperation in the game theory accepted definitions.

So where one of the conditions of Bitcoin is pseudonymity—as if Bitcoin is working on a set of automated and pre-conceived assumptions — is the decentralisation creating the symmetry and equality of bargaining?

We can’t be certain on the actual principal’s identity in this respect — but this doesn’t seem to stop the recursive and repeatable Bitcoin block discovery continuing, in the fashion of a coalitional game occurring over time without verbally expressed or formally enforceable rules by the traditional legal definitions or assumptions on the first person or so-called “principal” involved.

Witness how Bitcoin’s value proposition is to mitigate on mediation costs in that regard.

[1] Radner, R; Repeated Principal-Agent Games with Discounting; Econometrica, 1985; Available online

[2] Radner, R; Monitoring Cooperative Agreements in a Repeated Principal-Agent Relationship; Econometrica, 1981; Available online

[3] Nash Jr., JF; The Agencies Method for Modeling Coalitions and Cooperation in Games; International Game Theory Review, 2008; Available online

[4] Gulson, J; Bitcoin as an Implementation of John F Nash Jr.’s Bargaining “Idealizations”, 2022; Available online

[5] Nash Jr., JF; The Bargaining Problem; Econometrica, 1950; Available online