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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.

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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 $h_{-i}$.

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  • $\begingroup$ so is this saying that $h_i^*$ is a constant function of $h_{-i}$? $\endgroup$
    – Kenny Wong
    Apr 6 '20 at 12:25
  • $\begingroup$ @KennyWong: Yes that's right. $\endgroup$
    – Herr K.
    Apr 6 '20 at 18:38
  • $\begingroup$ alright, thanks a lot. i believe i got the answer. $\endgroup$
    – Kenny Wong
    Apr 7 '20 at 8:35

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