Find value of $\beta$ for which there is a strictly dominant strategy The question is as such: $N$ firms are lobbying for subsidies. Let $h_i$ be the number of hours spent by form $i$ for lobbying, with cost $wh_i^2$ where $w$ is a fixed constant. The subsidies granted to each firm will be $\displaystyle\alpha\sum_i h_i+\beta\prod_i h_i$, where $\alpha$ and $\beta$ are constants. We are supposed to find value of $\beta$ such that there exists a strictly dominant strategy for each firm, and with this value of $\beta$, find the Nash equilibrium.

My attempt so far: for firm $i$, the payoff will be $\displaystyle\alpha\sum_i h_i+\beta\prod_i h_i-wh_i^2$; by FOC, we have $\displaystyle\alpha +\beta \left(\prod_{j\neq i} h_{j}\right)-2wh_i=0$. 

My question is: do I attempt to find $\beta$ from here? There will be $N$ such equations, with a product of $N-1$ terms, so solving the system of equations do not seem that feasible. However if I take value of $\beta$ as it is, then there will be many different values.

I feel that the value of $\beta$ should be zero here, but I am not sure how I can show this.
Hint
Let $h_i^*(h_{-i}$) be firm $i$'s best response to the other $N-1$ firms' strategy profile $h_{-i}$. If $h_i^*(\cdot)$ is a dominant strategy, then it must be independent of the other firms' strategies, i.e. $h_i^*(h_{-i})=h_i^*$ for all $h_{-i}$.