I found the article confirming your stat of the gov pays more to enforce than it loses. Apparently, it is per $1 stolen it costs the Gov $3.72 in investigation. Also 20% of the fraud is benefit trafficking, 31% involve identity theft.
Given that in 2016 alone there was $592 million in SNAP fraud. If that ratio holds $1.5 billion in investigation costs. Sound like the US gov to me. Lol
Also, I didn't realize how much fraud there was. This makes me think, it's not the guy selling his food stamps outside the Kroger. It has organized crime involved.
Edit: after thought, it probably works same way as the IRS scam, just have a script submit requests for benefits for random people in data dumps. Even if only a fraction of requests make it through. Do it for both SNAP and state benefits. That's easily millions in defrauded benefits.