<p>For the Nash equilibrium of any simultaneous-play game, you are looking for the point where each player is playing a best response to all other players at the same time.</p>

<p>So your steps to solving this game should be:</p>

<ol>
<li>Determine player 1's best response function (<span class="math-container">$g_1$</span> as a function of <span class="math-container">$g_2$</span>)</li>
<li>Determine player 2's best response function (<span class="math-container">$g_2$</span> as a function of <span class="math-container">$g_1$</span>)</li>
<li>Now you have two equations with two unknowns. Solve this system of equations algebraically to get the Nash equilibrium <span class="math-container">$g_1^*$</span> and <span class="math-container">$g_2^*$</span>.</li>
</ol>

<p>So how can you get the best response functions? Take player 1 for example. Player 1 wants to set <span class="math-container">$MB_1 = MC_1$</span>. You have <span class="math-container">$MB_1$</span>, and <span class="math-container">$MC_1 = 4$</span>. So set them equal to each other and solve for <span class="math-container">$g_1$</span> to get player 1's best response function <span class="math-container">$g_1^*$</span> as a function of <span class="math-container">$g_2$</span>. Then similarly for player 2 to get <span class="math-container">$g_2^*$</span> as a function of <span class="math-container">$g_1$</span>.</p>

<p>That's not <em>quite</em> all, though. In this game, the players (presumably) can't contribute negative amounts. So the real best response functions will be of the form</p>

<p><span class="math-container">$$
g_1^*=
\begin{cases}
some.function.of.g_2 \text{ if } some.condition.of.g_2\\
0 \text{ otherwise}
\end{cases}
$$</span></p>

<p>So, first, try to solve your two best response functions together algebraically, assuming that both <span class="math-container">$some.condition.of.g_2$</span> and <span class="math-container">$some.condition.of.g_1$</span> hold so neither is 0. If you don't get a solution, or if that solution makes one of them negative or violates one of the <span class="math-container">$some.condition$</span>s, then instead try setting <span class="math-container">$g_1 = 0$</span>, find <span class="math-container">$g_2^*$</span>, and see if that gives you a Nash equilibrium (i.e. if <span class="math-container">$g_1^*(g_2^*) = 0$</span>). Then try setting <span class="math-container">$g_2 = 0$</span>, find <span class="math-container">$g_1^*$</span>, and see if that gives you a Nash. </p>

CC BY-SA 4.0
2019-10-16T17:13:07.720Z
Thank you! I completed the first 3 steps - I can't find a solution. My response function of Player 1 is g1 = 6 - g2. Of player 2: g2 = 4 - g1. If I set g2 = 0, I'm getting g1 as 6 but I don't think that's correct.
user708015
CC BY-SA 4.0
2019-10-17T18:11:35.907Z
If when you set g_2 = 0, you get g_1 = 6, then you only have one more step to go: if setting g_1 = 6 gets you g_2 = 0 (keep in mind negatives aren't possible), then you have a Nash!
NickCHK
CC BY-SA 4.0
2019-10-16T06:57:55.210Z
2019-10-16T06:57:55.210Z
NickCHK
<p>Two players, 1 and 2, simultaneously choose their consumption of a public good. Given the consumption choices, g1 and g2, player 1 derives a marginal benefit of MB1 = 10 - (g1 + g2), while player 2's marginal benefit is given by MB2 = 8 - (g1 + g2). The unit price of the public good is equal to 4.</p>

<p>The question asks to find a Nash Equilibrium of the game, then calculate the efficient level of public good provision and compare it to the NE.</p>

<p>I can understand the efficient provision of public good which will be nothing but the sum of marginal benefits set equal to the marginal cost. In this case, it turns out to be 7. What will be the Nash equilibrium?  </p>

<p>An update with my attempt: </p>

<p>Player 1's best response function: g1 = 6 - g2
Player 2's best response function: g2 = 4 - g1</p>

<p>Added these two to get the total level of public good - it comes to 5. </p>

<p>Then I equated the two response function to get: g1 = g2 - 2</p>

<p>Substituted it in g1 + g2 = 5 equation with getting g2 = 3.5 and g1 = 1.5 </p>

<p>I don't know if this is correct. </p>

<microeconomics><public-economics><public-goods>
nash equilibrium public good - is this correct?
user708015
# nash equilibrium public good - is this correct?

(Asked by: username_0 on 2019-10-16)

Two players, 1 and 2, simultaneously choose their consumption of a public good. Given the consumption choices, g1 and g2, player 1 derives a marginal benefit of MB1 = 10 - (g1 + g2), while player 2's marginal benefit is given by MB2 = 8 - (g1 + g2). The unit price of the public good is equal to 4.

The question asks to find a Nash Equilibrium of the game, then calculate the efficient level of public good provision and compare it to the NE.

I can understand the efficient provision of public good which will be nothing but the sum of marginal benefits set equal to the marginal cost. In this case, it turns out to be 7. What will be the Nash equilibrium?

An update with my attempt:

Player 1's best response function: g1 = 6 - g2
Player 2's best response function: g2 = 4 - g1

Added these two to get the total level of public good - it comes to 5.

Then I equated the two response function to get: g1 = g2 - 2

Substituted it in g1 + g2 = 5 equation with getting g2 = 3.5 and g1 = 1.5

I don't know if this is correct.

**Tags:** microeconomics, public-economics, public-goods

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## 1 answers

### Answer by username_1 on 2019-10-16. Score: 2

For the Nash equilibrium of any simultaneous-play game, you are looking for the point where each player is playing a best response to all other players at the same time.

So your steps to solving this game should be:

1. Determine player 1's best response function ($g\_1$ as a function of $g\_2$)
2. Determine player 2's best response function ($g\_2$ as a function of $g\_1$)
3. Now you have two equations with two unknowns. Solve this system of equations algebraically to get the Nash equilibrium $g\_1^\*$ and $g\_2^\*$.

So how can you get the best response functions? Take player 1 for example. Player 1 wants to set $MB\_1 = MC\_1$. You have $MB\_1$, and $MC\_1 = 4$. So set them equal to each other and solve for $g\_1$ to get player 1's best response function $g\_1^\*$ as a function of $g\_2$. Then similarly for player 2 to get $g\_2^\*$ as a function of $g\_1$.

That's not *quite* all, though. In this game, the players (presumably) can't contribute negative amounts. So the real best response functions will be of the form

$$
g\_1^\*=
\begin{cases}
some.function.of.g\_2 \text{ if } some.condition.of.g\_2\\
0 \text{ otherwise}
\end{cases}
$$

So, first, try to solve your two best response functions together algebraically, assuming that both $some.condition.of.g\_2$ and $some.condition.of.g\_1$ hold so neither is 0. If you don't get a solution, or if that solution makes one of them negative or violates one of the $some.condition$s, then instead try setting $g\_1 = 0$, find $g\_2^\*$, and see if that gives you a Nash equilibrium (i.e. if $g\_1^\*(g\_2^\*) = 0$). Then try setting $g\_2 = 0$, find $g\_1^\*$, and see if that gives you a Nash.

> **Comments:**
> **username_0**: Thank you! I completed the first 3 steps - I can't find a solution. My response function of Player 1 is g1 = 6 - g2. Of player 2: g2 = 4 - g1. If I set g2 = 0, I'm getting g1 as 6 but I don't think that's correct.  
> **username_1**: If when you set g_2 = 0, you get g_1 = 6, then you only have one more step to go: if setting g_1 = 6 gets you g_2 = 0 (keep in mind negatives aren't possible), then you have a Nash!  

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