Toward Equilibria and Solvability of Blockchain Pooling Strategies: A Topological Approach

Abstract: In 2015, Eyal proposed the first game-theoretical model for analyzing the equilibrium of blockchain pooling: when the blockchain pools are abstracted as a non-cooperative game, two pools can reach a Nash equilibrium with a closed-form formula; Moreover, an arbitrary number of pools still exhibit an equilibrium as long as the pools have an equal number of miners. Nevertheless, whether an equilibrium exists for three or more pools of distinct sizes remains an open problem. To this end, this paper studies the equilibrium in a blockchain of arbitrary pools. First, we show that the equilibrium among $q$ identical pools, coinciding the result demonstrated by Eyal through game theory, can be constructed using a topological approach. Second, if the pools are of different size, we show that (i) if the blockchain's pools exhibit two distinct sizes, an equilibrium can be reached, and (ii) if the blockchain has at least three distinct pool sizes, there does not exist an equilibrium.