A Stochastic Game Framework for Analyzing Computational Investment Strategies in Distributed Computing with Application to Blockchain Mining
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We study a stochastic game framework with varying number of players, for modeling and analyzing their computational investment strategies in distributed computing, for solving a problem such as in blockchain mining. In particular, we propose a continuous time Markov chain model, where players arrive and depart according to a stochastic process, and determine their investment strategies based on the number of other players in the system. The players obtain a certain reward for being the first to solve the problem, while incur a certain cost based on the time and computational power invested in the attempt to solve it. In this paper, we consider that the players are Markovian, that is, they determine their strategies which maximize their expected utility, while ignoring past payoffs. We first study a scenario where the rate of problem getting solved is proportional to the total computational power invested by the players. We show that, in statewise Nash equilibrium, players with costs exceeding a particular threshold do not invest, while players with costs less than this threshold invest maximum power. Further, we show that Markov perfect equilibrium follows a similar threshold policy. We then consider a scenario where the rate of problem getting solved is independent of the computational power invested by players. Here, we show that, in statewise Nash equilibrium, only the players with cost parameters in a relatively low range, invest. We also show that, in Markov perfect equilibrium, players invest proportionally to the reward-cost ratio. Using simulations, we quantify the effects of arrival and departure rates on players' expected utilities and provide insights.
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