Trigger Warning

Even the King's favourite mistress can wield a lot of influence, especially if they're from a powerful family. It's royal politicking bread and butter, even if he cannot get Robert to put Cersei aside it still benefits him, spreading Tyrell influence at court still strengthens him and diminishes the Lannisters. Robert is going to make Jaime the Warden of the East, with his favoured mistress whispering in his ear perhaps such an appointment could go to Garlan Tyrell etc and so on. Being favoured by the king is a direct line to power. 

4 hours ago, Craving Peaches said:

From what I recall Jon Arryn and Stannis were going to inform Robert together, but then Jon Arryn died, Stannis felt Robert wouldn't listen to him alone and left because he thought the Lannisters would come after him next. So he was motivated by self preservation. But regardless of the threat to his life or whether Robert would listen or not, it was still Stannis' duty to inform him. And I don't think Stannis would view self-preservation as acceptable grounds for any of his vassals to shirk their duties to him.

If Stannis left out of self preservation he wouldn't have been crying to Cat about Ned being made Hand instead of him.

13 hours ago, Thedog said:

A Queen he prestige through her husband. Tywin having men at court would seem to indicate that he thought that there would be situations where he thought Cersei might need swords who are not beholden to Robert (at least not any more beholden than any random commoner in Westeros). I guess that knowing Cersei her getting into a situation needing men did seem like a good precaution. 
 

nevertheless, i still Don’t get why Robert would let a force of men not loyal to him be permanently lodged under his roof.
 

Obviously there is precedent for a Queen having a sworn shield (in this case the Hound). But a force of armed men? That seems to undercut his power. 

I think you're thinking of this in too rigid terms. She might be the queen but she's still an individual with her own retinue, as long as Tywin's footing the bill I'm sure Robert doesn't give a shit if she wants to have a household guard. Ned brought his household guard to King's Landing too because they're his retinue. Basically every noble staying in the Red Keep would likely have some armed retainers accompanying them as I'm sure Robert did as well the difference is Tywin can casually afford to give his daughter a large one so he does. 

I'm not sure what your point is about prestige, you can have many sources of prestige and a distinctly Lannister source in the capital benefits Tywin, paying for the Queen to have a load of men at arms in her personal service decked out in his livery is a potent reminder of who she represents and how much wealth he commands. 

Robert's accompanied by a retinue of knights and retainers when he heads North, it's likely that they're with him in King's Landing too but not worth mentioning. Tywin likely sent Cersei her guard to add to the prestige of the queen and to serve as an ever present reminder to the court of who she represents, besides it never hurts to ensure your daughter has the protection of men personally loyal to you when paying for them wouldn't even make you bat an eyelid.