If you think that women don't game the system and have more kids to increase their benefits, you are absolutely kidding yourself.
If they can do math they would be fools to have another kid to pick up less than $100 a month.You're giving them entirely too much benefit of the doubt.Welfare fraud cases are not isolated incidents,.... they are rampant. The state could employ an army of investigators and never keep up. Not to mention many state financial workers don't care about reporting fraud to investigators because "it's all gov't money anyhow".
I think there is a good deal of welfare fraud. Back when I was doing in home visits I saw some but it was not rampant. Today, I don't know. But having more kids, while stupid, is not fraudulent.I should have clarified. Having more children is certainly not fraud. Even if the intended purpose is to collect more benefits, it's almost impossible to prove. Welfare fraud more commonly comes in the form of the father living in the same household while working and contributing to the overall household income. Recipients know full well if that income is included,...... their benefits come to a stop.In your model you're assuming the 4 deadbeat fathers are all paying child support payments on time and in full. They are as much to blame here as any "welfare queen" ever would be. If she isn't receiving any of the support the state is picking up the slack.
Again I was referring to an anecdotal case where purposely having multiple fathers increased the mother’s income. I agree with you such is very rare. Most of these losers marry losers that cannot or will not pay child supportI think you would find you and I agree on things much more than we disagree. It's the way to solve them where we will differ 
They can go after the fathers. Although the average number of cases on a support enforcement officers desk in this state is somewhere in the neighborhood of 150,...... maybe you can shed a little light on that considering your experience.
I have been out of it more than a few years, so I guess I am no expert, but that caseload isn't all that bad. Washington was a national leader in efficient management of cases. The national child support program was modeled after Washington’s. The director at the time, Bob Quarry, had been in the Berlin airlift and he modeled the caseload management after that. Each officer grabs a case does the next logical action, say calling an employer to garnishee wages, then puts that case as idle and does the next logical step in the next case. So he may have one hell of a ,lot of cases "open" but it’s not like he's wading thru one till completion then picking up the next. I worked in a state office as part of my indoctrination. Its damn hard work and often frustrating. If an absent father stays in one job it’s pretty easy to collect support. But if a douche bag is willing to move every three months its damn hard to get anything out of him. But I do agree they could use more officers.
One problem with a meat axe approach to budget balancing is across the board cuts. When I was active in the program each officer was bringing in about ten times his wage. Yet when they had a budget crunch they laid off child support workers. Case in point to the last item. Adding more support enforcement officers in a budgetary crisis isn't currently feesible. Private support enforcement agencies exist that work in conjunction with state enforcement agencies. They are efficient and cost the taxpayer zero.