But I wonder about the value in the end. If we spend X to prevent Y dollars in fraud, then Y - X can go to people who need it more. As long as Y > X, that's good for the deserving beneficiaries.

However, from a society-level view, how much value do we get from the X spent in prevention? We're funneling money to a specific goal, so some verification is needed. But the more we verify, the more man-hours we spend on filling out and checking the paperwork. None of that work benefits society. Those people could've been doing other jobs that perform useful services. And it's not like the money going to fraudsters vanishes. At least it goes back into the economy. We should require only enough to make sure we achieve the program's goals within budget.

Figuring that out is the hard part. Hopefully smart minds have been tackling this.

The fraud is entirely due to the structural problems (like hard cut offs) of these programs.

(I mean entirely, because if there are people living comfortably on the program without no motivation to do anything else in their life, then a) maybe that's okay, b) maybe that's a mental health issue [and fixing it might cost a lot more], c) at some point the fraud becomes so encompassing, that it's not the program's fault, but it's someone literally robbing taxpayers via whatever means they can. Furthermore any social program that only addresses and takes into account some narrow slice of people's life problems will be easily gameable, so let's have comprehensive integrated programs. Yes, they are hard, but it's not like putting people on the moon is that much easier.)