So it looks like Robinhood did not prohibit customers from buying GameStop or AMC as part of a conspiracy to protect short sellers. It did it because it literally couldn't afford to let them keep trading. https://www.nytimes.com/2021/01/29/technology/robinhood-fundraising.html?smid=tw-share

Short explanation of why having lots of customers buying GameStop put Robinhood into a cash crunch: Because stock trades don't settle for 2 days, brokerages are required to post capital to cover the risk that a clearinghouse will fail in that period.

The more volatile the underlying stocks are, the more capital you have to post. GME's (and AMC's) stock prices have become incredibly volatile, which meant Robinhood had to put up more of its own money (and it has to use its own money, not client money).

The paradox here is that you'd normally assume that more trading is good for a stock-trading platform. But as a brokerage, Robinhood has to be able to meet those capital requirements. And it clearly got worried that if people just kept buying GME, it wouldn't be able to.

The culprit behind the GameStop buying ban, in other words, seems not to have been a sinister hedge-fund cabal. Instead, disappointing as it is from a dramatic perspective, the culprit seems to have been government- and industry-imposed capital requirements.

Here is a longer, more detailed thread explaining exactly how the capital requirements are set, and why Robinhood, et. al., found itself facing a liquidity crunch. Well worth reading if you're interested. https://twitter.com/KralcTrebor/status/1354952689478754309