# 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.

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 $$h_{-i}$$.
• so is this saying that $h_i^*$ is a constant function of $h_{-i}$? Apr 6 '20 at 12:25