<p>First verify that $\overline{g}_1 = \theta$ and $\overline{g}_2 = \beta$. We'll now find Subgame perfect equilibrium for all possible values of $(\theta, \beta)$ satisfying $\theta &gt; \beta&gt; 1$.</p>

<p>To do so, we first maximize player 2's payoff with respect to his contribution taking as given player 1's contribution:
\begin{eqnarray*} \max_{g_2 \geq 0} &amp; \ \beta\ln (g_1+g_2)- g_2 \end{eqnarray*}</p>

<p>and we get the best response strategy of player 2 as a function of player 1's contribution:</p>

<p>\begin{eqnarray*} g_2 = \max(\beta - g_1, 0) \end{eqnarray*}</p>

<p>Next, solve player 1's payoff maximization problem taking as given player 2's strategy</p>

<p>\begin{eqnarray*} \max_{g_1 \geq 0} &amp; \ \theta\ln (g_1+g_2)- g_1 \\  \text{s.t.} &amp; \ g_2 = \max(\beta - g_1, 0)\end{eqnarray*}</p>

<p>and we get
\begin{eqnarray*} g_1^* = \begin{cases} 0 &amp; \text{if } \theta &lt; e\beta \\ \theta &amp; \text{if } \theta \geq e\beta\end{cases} \end{eqnarray*}</p>

<p>Consequently, the contribution of player 2 in a subgame perfect outcome is
\begin{eqnarray*} g_2^* = \begin{cases} \beta &amp; \text{if } \theta &lt; e\beta \\ 0 &amp; \text{if } \theta \geq e\beta\end{cases} \end{eqnarray*}</p>

<p>For the next one, the cost of player 2 is 
\begin{eqnarray*} c_2(g_2) = \begin{cases} g_2 &amp; \text{if } g_1 \geq \overline{g}_1 = \theta \\ \lambda g_2 &amp; \text{if } g_1 &lt; \overline{g}_1 = \theta\end{cases} \end{eqnarray*}</p>

<p>First verify that in this case $\overline{g}_1 = \theta$ and $\overline{g}_2 = \dfrac{\beta}{\lambda}$. We'll now find Subgame perfect equilibrium for all possible values of $(\theta, \beta, \lambda)$ satisfying $1 &lt;\theta \leq  \dfrac{\beta}{\lambda} &lt; \beta$.</p>

<p>To do so, we first maximize player 2's payoff with respect to his contribution taking as given player 1's contribution:
\begin{eqnarray*} \max_{g_2 \geq 0} &amp; \ \beta\ln (g_1+g_2)- c_2(g_2) \end{eqnarray*}</p>

<p>and we get the best response strategy of player 2 as a function of player 1's contribution:</p>

<p>\begin{eqnarray*} g_2 = \begin{cases} \dfrac{\beta}{\lambda} - g_1 &amp; \text{if } g_1 &lt; \overline{g}_1 = \theta \\ \max\left({\beta} - g_1, 0\right) &amp; \text{if } g_1 \geq \overline{g}_1 = \theta \end{cases} \end{eqnarray*}</p>

<p>Next, solve player 1's payoff maximization problem taking as given player 2's strategy 
\begin{eqnarray*} \max_{g_1 \geq 0} &amp; \ \theta\ln (g_1+g_2)- g_1 \\  \text{s.t.} &amp; \ g_2 = \begin{cases} \dfrac{\beta}{\lambda} - g_1 &amp; \text{if } g_1 &lt; \overline{g}_1 = \theta \\ \max\left({\beta} - g_1, 0\right) &amp; \text{if } g_1 \geq \overline{g}_1 = \theta \end{cases}\end{eqnarray*}</p>

<p>and we get
\begin{eqnarray*} g_1^* = \begin{cases} 0 &amp; \text{if } \lambda &lt; e \\ \theta &amp; \text{if } \lambda \geq e\end{cases} \end{eqnarray*}</p>

<p>Consequently, the contribution of player 2 in a subgame perfect outcome is
\begin{eqnarray*} g_2^* = \begin{cases} \dfrac{\beta}{\lambda} &amp; \text{if } \lambda &lt; e \\ \beta - \theta &amp; \text{if } \lambda \geq e\end{cases} \end{eqnarray*}</p>

CC BY-SA 4.0
2018-05-24T00:29:08.507Z
Thank you for your great answer. Your solution was really helpful to understand. Also can you look at my another solution. We have tried to solve it but really stack. And we need your help. Thank you so much. https://economics.stackexchange.com/questions/22121/question-on-uncertainity
b11bb
CC BY-SA 4.0
2018-05-19T09:08:16.543Z
2018-05-19T09:14:01.197Z
2018-05-19T09:14:01.197Z
Amit
<p>Note, there is a mistake in FOC---it should be 
$$\beta/(g_1+g_2) - 1 =0,$$
so the response function of firm $2$ is
$$g_2 = \beta - g_1.$$</p>

<p>This means that a unit increase in a public good contributed by firm $1$ causes firm $2$ to decrease its contribution by the same unit. Then, notice that firm $1$, who correctly anticipates such a (optimal) response by firm $2$, is better off by not contributing anything (hint: $g_1$ only affects $C(g_1)$ up to $g_1 \in [0,\beta]$).</p>

<p>I guess you can take from here to solve the second part</p>

CC BY-SA 4.0
2018-05-18T14:31:45.410Z
2018-05-18T14:31:45.410Z
Green.H
<p>The main part of the question is as follows </p>

<p><a href="https://i.sstatic.net/puDLr.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/puDLr.jpg" alt="enter image description here"></a></p>

<p>(if you cannot read this, I can immediately write it)</p>

<p>My question is how to find the subgame perfect Nash equilibrium for the both cases $\bar{g_2}\ge \bar{g_1}$ and $\bar{g_2}&lt; \bar{g_1}$</p>

<p>My attempt is </p>

<p>For firm 2 </p>

<p>$$max [f_2(G)-c_2(g_2)]$$
$$max [\beta ln(g_1+g_2)-g_2]$$</p>

<p>By FOCs</p>

<p>$$\beta / (g_1+g_2)- 1=0$$</p>

<p>$$g_2=\beta -g_1$$</p>

<p>For firm 1</p>

<p>$$max [f_2(G)-c_2(g_2)]$$
$$max [\theta ln(g_1+\beta -g_1)-g_1]$$</p>

<p>I cannot get FOC with respect to $g_1$ </p>

<p>At this point I’m stuck. How can I proceed this solution?</p>

<p>And I have another question </p>

<p>For the same question, I assume the following part</p>

<p><a href="https://i.sstatic.net/PFcIW.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PFcIW.jpg" alt="enter image description here"></a></p>

<p>Again I need to find subgame perfect Nash equilibrium of this game where player 1 contributes a positive amount of public good.? </p>

<p>Sorry but I could not this part completely. Any helps will be appreciated. </p>

<p>Since I could not understand such type of questions for the SPE, I asked lots of time. Thanks.  </p>

<microeconomics><mathematical-economics><game-theory><self-study>
Finding subgame perfect equilibrium
b11bb
# Finding subgame perfect equilibrium

(Asked by: username_0 on 2018-05-17)

The main part of the question is as follows

[![enter image description here](https://i.sstatic.net/puDLr.jpg)](https://i.sstatic.net/puDLr.jpg)

(if you cannot read this, I can immediately write it)

My question is how to find the subgame perfect Nash equilibrium for the both cases $\bar{g\_2}\ge \bar{g\_1}$ and $\bar{g\_2}< \bar{g\_1}$

My attempt is

For firm 2

$$max [f\_2(G)-c\_2(g\_2)]$$
$$max [\beta ln(g\_1+g\_2)-g\_2]$$

By FOCs

$$\beta / (g\_1+g\_2)- 1=0$$

$$g\_2=\beta -g\_1$$

For firm 1

$$max [f\_2(G)-c\_2(g\_2)]$$
$$max [\theta ln(g\_1+\beta -g\_1)-g\_1]$$

I cannot get FOC with respect to $g\_1$

At this point I’m stuck. How can I proceed this solution?

And I have another question

For the same question, I assume the following part

[![enter image description here](https://i.sstatic.net/PFcIW.jpg)](https://i.sstatic.net/PFcIW.jpg)

Again I need to find subgame perfect Nash equilibrium of this game where player 1 contributes a positive amount of public good.?

Sorry but I could not this part completely. Any helps will be appreciated.

Since I could not understand such type of questions for the SPE, I asked lots of time. Thanks.

**Tags:** microeconomics, mathematical-economics, game-theory, self-study

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## 2 answers

### Answer by username_1 on 2018-05-19. Score: 1 | Accepted answer

First verify that $\overline{g}\_1 = \theta$ and $\overline{g}\_2 = \beta$. We'll now find Subgame perfect equilibrium for all possible values of $(\theta, \beta)$ satisfying $\theta > \beta> 1$.

To do so, we first maximize player 2's payoff with respect to his contribution taking as given player 1's contribution:
\begin{eqnarray\*} \max\_{g\_2 \geq 0} & \ \beta\ln (g\_1+g\_2)- g\_2 \end{eqnarray\*}

and we get the best response strategy of player 2 as a function of player 1's contribution:

\begin{eqnarray\*} g\_2 = \max(\beta - g\_1, 0) \end{eqnarray\*}

Next, solve player 1's payoff maximization problem taking as given player 2's strategy

\begin{eqnarray\*} \max\_{g\_1 \geq 0} & \ \theta\ln (g\_1+g\_2)- g\_1 \\ \text{s.t.} & \ g\_2 = \max(\beta - g\_1, 0)\end{eqnarray\*}

and we get
\begin{eqnarray\*} g\_1^\* = \begin{cases} 0 & \text{if } \theta < e\beta \\ \theta & \text{if } \theta \geq e\beta\end{cases} \end{eqnarray\*}

Consequently, the contribution of player 2 in a subgame perfect outcome is
\begin{eqnarray\*} g\_2^\* = \begin{cases} \beta & \text{if } \theta < e\beta \\ 0 & \text{if } \theta \geq e\beta\end{cases} \end{eqnarray\*}

For the next one, the cost of player 2 is
\begin{eqnarray\*} c\_2(g\_2) = \begin{cases} g\_2 & \text{if } g\_1 \geq \overline{g}\_1 = \theta \\ \lambda g\_2 & \text{if } g\_1 < \overline{g}\_1 = \theta\end{cases} \end{eqnarray\*}

First verify that in this case $\overline{g}\_1 = \theta$ and $\overline{g}\_2 = \dfrac{\beta}{\lambda}$. We'll now find Subgame perfect equilibrium for all possible values of $(\theta, \beta, \lambda)$ satisfying $1 <\theta \leq \dfrac{\beta}{\lambda} < \beta$.

To do so, we first maximize player 2's payoff with respect to his contribution taking as given player 1's contribution:
\begin{eqnarray\*} \max\_{g\_2 \geq 0} & \ \beta\ln (g\_1+g\_2)- c\_2(g\_2) \end{eqnarray\*}

and we get the best response strategy of player 2 as a function of player 1's contribution:

\begin{eqnarray\*} g\_2 = \begin{cases} \dfrac{\beta}{\lambda} - g\_1 & \text{if } g\_1 < \overline{g}\_1 = \theta \\ \max\left({\beta} - g\_1, 0\right) & \text{if } g\_1 \geq \overline{g}\_1 = \theta \end{cases} \end{eqnarray\*}

Next, solve player 1's payoff maximization problem taking as given player 2's strategy
\begin{eqnarray\*} \max\_{g\_1 \geq 0} & \ \theta\ln (g\_1+g\_2)- g\_1 \\ \text{s.t.} & \ g\_2 = \begin{cases} \dfrac{\beta}{\lambda} - g\_1 & \text{if } g\_1 < \overline{g}\_1 = \theta \\ \max\left({\beta} - g\_1, 0\right) & \text{if } g\_1 \geq \overline{g}\_1 = \theta \end{cases}\end{eqnarray\*}

and we get
\begin{eqnarray\*} g\_1^\* = \begin{cases} 0 & \text{if } \lambda < e \\ \theta & \text{if } \lambda \geq e\end{cases} \end{eqnarray\*}

Consequently, the contribution of player 2 in a subgame perfect outcome is
\begin{eqnarray\*} g\_2^\* = \begin{cases} \dfrac{\beta}{\lambda} & \text{if } \lambda < e \\ \beta - \theta & \text{if } \lambda \geq e\end{cases} \end{eqnarray\*}

> **Comments:**
> **username_0**: Thank you for your great answer. Your solution was really helpful to understand. Also can you look at my another solution. We have tried to solve it but really stack. And we need your help. Thank you so much. https://economics.stackexchange.com/questions/22121/question-on-uncertainity  

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### Answer by username_2 on 2018-05-18. Score: 1

Note, there is a mistake in FOC---it should be
$$\beta/(g\_1+g\_2) - 1 =0,$$
so the response function of firm $2$ is
$$g\_2 = \beta - g\_1.$$

This means that a unit increase in a public good contributed by firm $1$ causes firm $2$ to decrease its contribution by the same unit. Then, notice that firm $1$, who correctly anticipates such a (optimal) response by firm $2$, is better off by not contributing anything (hint: $g\_1$ only affects $C(g\_1)$ up to $g\_1 \in [0,\beta]$).

I guess you can take from here to solve the second part


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