Even when eligible, not everyone chooses to receive payments from welfare. This may be by ignorance or by choice. In the latter case, welfare payments may obliterate other incomes, or vice-versa. A particularly interesting problem is that of single mothers reporting (or not) paternity.
Indeed, reporting who the father of the child induces most of the time procedures by the state to obtain child support from the father. If the mother is a welfare recipient, most US states capture most of this child support. It becomes thus a strategic decision for the mother: report the father and see little in child support, or seek child support informally (and possibly not strain a relationship with the father).
Jennifer Roff shows that this can be formulated as a Stackelberg game where the mother is the leader. She shows in particular that the outcome of this game can have perverse consequences when states capture more of the child support payments: mothers report fewer fathers, and the total captured amounts are lower. Also, awarding high child support payments decreases expected payments as mothers expect fewer fathers to comply, mostly because because of low paternal incomes. In other words, it does not necessarily pay to play tough.