The Tight Bound for Pure Price of Anarchy in an Extended Miner's Dilemma Game

Pool block withholding attack, which reduces the effective mining power in the system and leads to potential systemic instability in the blockchain, can be modeled as a non-cooperative game called "the miner's dilemma". However, existing literature on the gametheoretic properties of this attack only gives a preliminary analysis. In this paper, we establish the existence and uniqueness of pure Nash equilibrium for the two-player miner's dilemma. Then we give a tight upper bound 2 for PPoA, which measures how much mining power is wasted in the game. Moreover, we show the uniqueness and the tight bound holds in a more general setting with betrayal assumption. Inspired by the experiments on the games among three mining pools, we conjecture that similar results should hold for the N-player miner's dilemma game (𝑁 ≥ 2).