Sports Direct was forced to make a full takeover offer for GAME Digital after taking its stake in the business to 38.49% from 29.9%.
Shareholders that own more than 30% of a company are required to make an offer for the rest of the shares under UK takeover rules.
Sports Direct offered 30p per share for GAME, representing a 27.4% premium to the closing price of 23.55p on Tuesday and valuing the business at £51.88mln.
The sports retailer said the offer is final and the price will be not be increased.
"The retail and gaming sectors are fast moving and currently subject to challenging conditions,” Sports Direct said.
"Sports Direct does not believe that, as a standalone business, GAME is able to weather the pressures that it is facing."
In February 2018, GAME agreed to sell half its Belong esports business to Sports Direct for £3.2mln.
Sports Direct, which bought a 25.8% stake in Game Digital in 2017, plans to introduce concessions of the chain in its own stores along with Belong-branded gaming “arenas” where people can play computer games.
Sports Direct said on Wednesday: “Sports Direct believes that the offer, in providing GAME with the wider benefit of Sports Direct's operating and other experience and increased support, will secure GAME's future and allow it to navigate these pressures whilst contributing to GAME's long-term sustainability by escalating the evolution of the Belong business and accelerating the next stage of GAME's development.”