Group specific public goods, orchestration of interest groups with free riding

We consider a situation in which two groups contest group-specific public goods. Each member of the group may have a different valuation of the prize. Our model can be interpreted as the contestants' cost of lobbying efforts being non-linear, or the returns to their efforts being decreasing. We show that the level of free-riding depends on the return of the investment. We consider the situation under which one group initiates a contest to which different individuals and/or groups can be added. The question we pose is what is the optimal structure of additional groups?.