<h2>Hint</h2>
<p>Let <span class="math-container">$h_i^*(h_{-i}$</span>) be firm <span class="math-container">$i$</span>'s best response to the other <span class="math-container">$N-1$</span> firms' strategy profile <span class="math-container">$h_{-i}$</span>. If <span class="math-container">$h_i^*(\cdot)$</span> is a dominant strategy, then it must be independent of the other firms' strategies, i.e. <span class="math-container">$h_i^*(h_{-i})=h_i^*$</span> for all <span class="math-container">$h_{-i}$</span>.</p>

CC BY-SA 4.0
2020-04-06T12:25:15.077Z
so is this saying that $h_i^*$ is a constant function of $h_{-i}$?
Kenny Wong
CC BY-SA 4.0
2020-04-06T18:38:37.387Z
@KennyWong: Yes that's right.
Herr K.
CC BY-SA 4.0
2020-04-07T08:35:31.020Z
alright, thanks a lot. i believe i got the answer.
Kenny Wong
CC BY-SA 4.0
2020-04-06T05:32:10.367Z
2020-04-06T05:32:10.367Z
2020-06-17T09:40:54.643Z
Herr K.
<p>The question is as such: <span class="math-container">$N$</span> firms are lobbying for subsidies. Let <span class="math-container">$h_i$</span> be the number of hours spent by form <span class="math-container">$i$</span> for lobbying, with cost <span class="math-container">$wh_i^2$</span> where <span class="math-container">$w$</span> is a fixed constant. The subsidies granted to each firm will be <span class="math-container">$\displaystyle\alpha\sum_i h_i+\beta\prod_i h_i$</span>, where <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> are constants. We are supposed to find value of <span class="math-container">$\beta$</span> such that there exists a strictly dominant strategy for each firm, and with this value of <span class="math-container">$\beta$</span>, find the Nash equilibrium.</p>

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

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

<p>I feel that the value of <span class="math-container">$\beta$</span> should be zero here, but I am not sure how I can show this.</p>

<game-theory><nash-equilibrium><dominant-strategy>
Find value of $\beta$ for which there is a strictly dominant strategy
Kenny Wong
# Find value of $\beta$ for which there is a strictly dominant strategy

(Asked by: username_0 on 2020-04-06)

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.

**Tags:** game-theory, nash-equilibrium, dominant-strategy

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## 1 answers

### Answer by username_1 on 2020-04-06. Score: 1 | Accepted answer

## 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 $h\_{-i}$.

> **Comments:**
> **username_0**: so is this saying that $h_i^*$ is a constant function of $h_{-i}$?  
> **username_1**: @KennyWong: Yes that's right.  
> **username_0**: alright, thanks a lot. i believe i got the answer.  

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