The Matthew Effect in Computation Contests: High Difficulty May Lead to 51% Dominance?

We study the computation contests where players compete for searching a solution to a given problem with a winner-take-all reward. The search processes are independent across the players and the search speeds of players are proportional to their computational powers. One concrete application of this abstract model is the mining process of proof-of-work type blockchain systems, such as Bitcoin. Although one's winning probability is believed to be proportional to his computational power in previous studies on Bitcoin, we show that it is not the case in the strict sense. Because of the gaps between the winning probabilities and the proportions of computational powers, the Matthew effect will emerge in the system, where the rich get richer and the poor get poorer. In addition, we show that allowing the players to pool with each other or reducing the number of solutions to the problem may aggravate the Matthew effect and vice versa.