If Google (specifically GMail) is rent-seeking, the Republican lawsuit isn't an example of it.
The study Republicans cite only displayed a bias in the default training of a new email account that had never trained the spam filter:
> "Shahzad said while the spam filters demonstrated political biases in their 'default behavior' with newly created accounts, the trend shifted dramatically once they simulated having users put in their preferences by marking some messages as spam and others as not," the Post article said. [1]
In other words, Republican email messages are more likely to use spammy phrases. Once a user starts marking any messages as "mark as spam" or "move to inbox", the bias dissipates.
This is not anywhere close to a good example of rent seeking.
The study Republicans cite only displayed a bias in the default training of a new email account that had never trained the spam filter:
> "Shahzad said while the spam filters demonstrated political biases in their 'default behavior' with newly created accounts, the trend shifted dramatically once they simulated having users put in their preferences by marking some messages as spam and others as not," the Post article said. [1]
In other words, Republican email messages are more likely to use spammy phrases. Once a user starts marking any messages as "mark as spam" or "move to inbox", the bias dissipates.
This is not anywhere close to a good example of rent seeking.
[1] https://arstechnica.com/tech-policy/2022/10/republicans-sue-... (money shot in last 4 paragraphs)