Why do Claudius and Gert go along with Polonius's spy schemes?

Polonius's theory presented to Claudius and Gert of how Hamlet went mad is easily seen by the audience as an idiotic Rube Goldberg process and a reflection, in reality, of his own obsessions. Claudius at a minimum should be shrewd enough to see that his adviser is off his rocker, and that throwing Ham and Oph together while others watch behind an arras is a preposterous act. It's true Claudius wants to glean whatever intel he can about Hamlet, but this arras business is so crazy it's downright risky.

Is it possible that Claudius goes along with Polonius's schemes with the private intention to get rid of the old fool as his next order of business (after he takes care of the Hamlet problem)? Does he refrain from dismissing Polonius right away because Polonius knows too much from his long tenure in Elsinore?

I just can't for the life of me imagine someone with a modicum of savvy acceding to Polonius's nutty proposals. I'd look at that guy and say "You're the mad one, not Hamlet."

If I have not said so, allow me to say that I find your attentiveness to this play admirable, and your questions have been a pleasure to ponder. If I seem to disagree with your points it is merely because I have given a lot of time to them.

As for this one. Dealing with Gertrude first.

I am not convinced she is as scheming as you make her out to be. She knows that her son is somehow "afflicted" and she admits to Claudius in 2.2 that is it most likely because of his father's death and the king and queen's hasty marriage. She is genuinely concerned about her son and finding out the cause of his madness. Regarding Polonius, she clearly thinks he is a mumbling fool ("more matter with less art" she said to Polonius in 2.2), but when he presents the letter that Hamlet gave to Ophelia, she seems to be convinced that Polonius may have a point worth testing.

As for Claudius - the way I see him is that Claudius is a very different character pre-Mousetrap and post-Mousetrap. You are looking at Claudius post-Mousetrap: the one who schemes and does what he can to gain the upper hand on Hamlet. However there is really no indication pre-Mousetrap that this is so: Claudius' care for Hamlet's only extends as far as Gertrude. It is not until post-Mousetrap that Claudius mentions Hamlet not in Gertrude's presence (except when they are actually spying on him in 3.1). Pre-Mousetrap he cares for Hamlet as far as any stepfather would who doesn't like his stepson. He must keep Gertrude happy and he must keep his people happy, and Gertrude and his people like Hamlet and so he must like Hamlet. So he may or may not believe Polonius' plan, but going along with it makes it seem like he cares and that he is trying to better the situation.

Claudius, unlike pretty much all real Shakespeare villains (Richard III, Iago, Edmund, L. Macbeth) is not governed by ambition. He wanted something, he got it, he doesn't need any more. Richard III would not have rested until Hamlet was dead, neither would Iago or the rest - but Claudius was perfectly content to let Hamlet live on and probably take the throne when he was done with it (had not legitimate child been produced). He had no reason to take any action against Hamlet until the Mousetrap when he first realized that his position was in danger.

Why thank you, Charles Darnay. I was afraid my questions were getting tedious.

Hmm, yes, Claudius does in fact defer the question to Gertrude ("Do you think ’tis this?"). I had an idea, however, that Claudius is nervous about Hamlet even pre-Mousetrap. I thought he hired R&G because he'd like Hamlet to cheer the hell up and quit making a mockery of his court and fretting Gertrude, but also because he's nervous about what's inside Hamlet's head and he'd like R&G to do a little digging. Could the moody runt possibly know?

Hmmm, a fair point. This warrants more thought. More later

Gertrude's and Claudius's intentions seem a little different in the scene where we first meet R&G. Gertrude wants to cheer Hamlet up.

GUILDENSTERN: I hope to God we can make him happy and do him some good!

GERTRUDE: Ay, amen!

While Claudius wants to find out what's eating him. He tells them to:

CLAUDIUS
draw him on to pleasures and to gather,
So much as from occasion you may glean,
Whether aught, to us unknown, afflicts him thus
That, opened, lies within our remedy.


Later, when Gert & Claudius question R&G, their concerns still seem a little different:

CLAUDIUS
And can you by no drift of conference
Get from him why he puts on this confusion,
Grating so harshly all his days of quiet
With turbulent and dangerous lunacy?

While Gertrude still seems more concerned about his mood and behavior:

GERTRUDE
Did he receive you well? Did you assay him? To any pastime?

This is an important distinction, but suggests rather differing approaches rather than intent.

Gertrude cares about Hamlet as her son, cares about his happiness and well-being - I think this case can be clearly made. However much she `goes along with Claudius`she does not wish harm to come to Hamlet. So she sends R&G to cheer him up

As for Claudius, going back to what I wrote earlier, I think he is more concerned for the stability of his state than Hamlet as a person. So he doesn't care if Hamlet is happy, he just needs to make sure that he is not a threat to the peace of his state. Whether he believes that Hamlet is a threat to himself - I still don't accept that at this point he is - but in the end, Hamlet's madness is a problem, particularly in the unstable times and Claudius cannot afford to look weak in the eyes of Norway et al.

That works.