HELSINKI—Weeks after Chinese investors agreed to buy a controlling interest in the Finnish maker of “Clash of Clans” last summer, chief executive and founder Ilkka Paananen lost his Nordic cool.

While the mobile-gaming transaction led by Chinese internet giant Tencent Holdings Ltd. valued Supercell Oy at more than $10 billion, making it one of Europe’s priciest tech startups, Mr. Paananen was receiving alarming reports: the studio’s signature games no longer ranked among the top-grossing apps in online stores.

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