Decentralized Collaborative Action: Cryptoeconomics in Space

Blockchains and peer-to-peer systems are part of a trend toward computer systems based on decentralized collaborative action, by which we mean that they (1) run across many participants, (2) without central control, and (3) are such that qualities (1) and (2) are essential to the system’s intended use cases.

We propose a notion of topological space, which we call a semitopology, to help us mathematically model such systems. We treat participants as points in a space, which are organized into actionable coalitions. An actionable coalition is any set of participants who collectively have the resources to collaborate (if they choose) to progress according to the system’s rules, independently of the rest of the system.

Mathematicians will recognize semitopology as a generalization of the notion of point-set topology, where actionable coalitions correspond to open sets.

It turns out that much useful information about the system can be obtained just by viewing it as a semitopology and studying its actionable coalitions. For example, we will prove a mathematical sense in which if every actionable coalition of some point p has nonempty intersection with every actionable coalition of another point q—note that this is the negation of the famous Hausdorff separation property from topology—then p and q must remain in agreement. Remarkably, since this observation depends only on the semitopological structure, it holds for any possible concrete algorithm.

This matters because remaining in agreement is a key correctness property in many distributed systems. For example, in blockchain, participant disagreement is called forking, and blockchain designers try hard to avoid it.

We provide an accessible introduction to the following: the technical context of decentralized systems; why we build them and find them useful; and how they motivate the theory of semitopological spaces. We also sketch some basic theorems and applications of the resulting mathematics.