In Aug. 2017, Bitcoin was split into the original Bitcoin (BTC) and Bitcoin Cash (BCH). Since then, miners have had a choice between BTC and BCH mining because they have compatible proof-of-work algorithms. Therefore, they can freely choose which coin to mine for higher profit, where the profitability depends on both the coin price and mining difficulty. Some miners can immediately switch the coin to mine only when mining difficulty changes because the difficulty changes are more predictable than that for the coin price, and we call this behavior fickle mining. In this paper, we study the effects of fickle mining by modeling a game between two coins. To do this, we consider both fickle miners and some factions (e.g., BITMAIN for BCH mining) that stick to mining one coin to maintain that chain. In this model, we show that fickle mining leads to a Nash equilibrium in which only a faction sticking to its coin mining remains as a loyal miner to the less valued coin (e.g., BCH), where loyal miners refer to those who conduct mining even after coin mining difficulty increases. This situation would cause severe centralization, weakening the security of the coin system. To determine which equilibrium the competing coin systems (e.g., BTC vs. BCH) are moving toward, we traced the historical changes of mining power for BTC and BCH. In addition, we analyze the recent "hash war" between Bitcoin ABC and SV, which confirms our theoretical analysis. Finally, we note that our results can be applied to any competing cryptocurrency systems in which the same hardware (e.g., ASICs or GPUs) can be used for mining. Therefore, our study brings new and important angles in competitive coin markets: a coin can intentionally weaken the security and decentralization level of the other rival coin when mining hardware is shared between them, allowing for automatic mining.