The basis is set out in the Final Report of the Microsoft / Activision Blizzard Merger Inquiry, in particular Chapters 1 (p24) and 3 (p35-36). The role and powers of the Competition and Markets Authority are set out in the Enterprise Act 2002, as amended in 2013 when it replaced the Competition Commission (with predecessor bodies going back to 1949).
The overall procedure is pretty complex, but the main point is that a CMA investigation will happen whenever there is a "relevant merger situation", and the result of that may be an order telling the companies what to do. It could stop them from merging, or impose various conditions about the conduct of business - see Schedule 8 for a long list of possibilities. Their jurisdiction over this anticipated "relevant merger situation" is on the basis that
- The two enterprises, namely Microsoft and Activision Blizzard, would cease to be distinct enterprises (s.23(1)(a)) because they would be brought under common ownership or control (s.26).
- The acquired company, Activision Blizzard, has a UK turnover of more than £70m (s.23(1)(b)), in fact £716m in FY2021.
The report dedicates some words in Chapter 3 to proving that Microsoft and Activision are both businesses, just in case anybody was in doubt about that.
In essence, the CMA has these powers because a lot of people in the UK play World of Warcraft (etc.) and there could be a "substantial lessening of competition" in the console gaming market as a result of the acquisition. If Activision had fewer UK customers then the £70m threshold would not be reached.
While the investigation overall is addressed to "Microsoft" and "Activision Blizzard" without analyzing the precise structure of the enterprises, any final order from the CMA would be addressed to more specific entities - perhaps Microsoft Ltd. in the UK as well as its American parent, among others. In other cases, orders have been similarly directed to various non-UK companies as well as domestic ones. But the CMA does not require a company to be UK-registered in order to exercise power over it. The relevant statutory condition is about how the enterprise (of whatever kind, wherever incorporated, however structured) does business in the UK.
Other countries' equivalent bodies do basically the same thing, and can similarly impose conditions on foreign companies.
The structure and safeguards in the law mean that the CMA cannot act totally capriciously, or impose conditions that go beyond the scope of its remit (competition in the UK market). So even though Schedule 8 seems very broad on its face, it actually is more limited.