<p>Let's solve by backward induction.</p>
<ol>
<li><p>In stage 3, player <span class="math-container">$A_3$</span> solves:
<span class="math-container">$$
\max_{p_1, p_2} u_3(p_1, p_2, q_1, q_2, x).
$$</span>
Given the assumptions, <span class="math-container">$A_3$</span> will set: <span class="math-container">$p_1 = \overline{p}$</span> and in general <span class="math-container">$p_2$</span> will be a function of the <span class="math-container">$q_1, q_2$</span> and <span class="math-container">$x$</span>, denoted by <span class="math-container">$p_2(q_1, q_2, x)$</span>. For a solution to exist, you have to assume that <span class="math-container">$u_3$</span> is not unbounded in <span class="math-container">$p_2$</span>.</p>
</li>
<li><p>As <span class="math-container">$u_2$</span> is independent of <span class="math-container">$p_1, q_1, x$</span>, In stage 2, Player <span class="math-container">$A_2$</span> solves:
<span class="math-container">$$
\max_{q_2} u_2(q_2, p_2(q_1, q_2, x)).
$$</span>
<span class="math-container">$u_2$</span> is increasing in <span class="math-container">$q_2$</span> and <span class="math-container">$p_2$</span>. If <span class="math-container">$p_2(q_1, q_2, x)$</span> is increasing in <span class="math-container">$q_2$</span> then <span class="math-container">$A_2$</span> will set <span class="math-container">$q_2$</span> as big as possible, so there is no optimal solution. Else, there might be some optimal level of <span class="math-container">$q_2$</span>, say <span class="math-container">$q_2(q_1, x)$</span>.</p>
</li>
<li><p>In stage 1, player <span class="math-container">$A_1$</span> solves:
<span class="math-container">$$
\max_{q_1} u_1(q_1, \overline{p}).
$$</span>
As <span class="math-container">$u_1$</span> is increasing in <span class="math-container">$q_1$</span> there will probably be no solution to this. So likely this problem is unbounded. If we assume that <span class="math-container">$q_1 \le \overline{q}_1$</span> for some fixed value <span class="math-container">$\overline{q}_1$</span>, then <span class="math-container">$A_1$</span> will set <span class="math-container">$q_1 = \overline{q}_1$</span>.</p>
</li>
</ol>
<p>Going back to the utility of player 3, we therefore see, after substituting:
<span class="math-container">$$
u_3 = u_3(\overline{p}, p_2(\overline{q}_1, q_2(\overline{q}_1, x),x), \overline{q}_1, q_2(\overline{q_1}, x), x).
$$</span>
It seems like the change in <span class="math-container">$u_3$</span> due to a change in <span class="math-container">$x$</span> is ambiguous. As it also influences the optimal choice of <span class="math-container">$q_2$</span> (of player <span class="math-container">$A_2$</span>), and <span class="math-container">$p_2$</span> of player <span class="math-container">$A_3$</span>.</p>

CC BY-SA 4.0
2021-04-28T20:26:17.253Z
Thank you very much @tmd. I included two new properties for $u_3$ with regard $p_2$ and $q_2$. I do not yet understand why $A_1$ and $A_2$ who are indifferent to $x$ should essentially take all the losses that arise for $A_3$ when $x$ increases. Do you have the intuition why that might happen? Perhaps I can then add a little bit more structure to rule this behavior out.
Paul
CC BY-SA 4.0
2021-04-29T10:02:23.790Z
Does the sum of all agents' profit decrease in $x$?
Paul
CC BY-SA 4.0
2021-04-28T17:28:49.323Z
2021-04-28T17:28:49.323Z
tdm
<p>Consider three agents <span class="math-container">$A_i$</span>.</p>
<ul>
<li><span class="math-container">$A_1$</span> moves first and selects <span class="math-container">$0\leq q_1\in \mathbb{R}$</span>. <span class="math-container">$A_2$</span> moves second and selects <span class="math-container">$0\leq q_2\in \mathbb{R}$</span>. <span class="math-container">$A_3$</span> moves third and selects <span class="math-container">$0\leq p_1\in \mathbb{R}$</span> and <span class="math-container">$0\leq p_2\in \mathbb{R}$</span>.  <span class="math-container">$0\leq x\in \mathbb{R}$</span> is a parameter.</li>
<li><span class="math-container">$A_1$</span> tries to maximize <span class="math-container">$u_1(q_1,p_1, q_2,p_2, x)$</span>, with <span class="math-container">$\frac{\partial u_1(q_1,p_1, q_2,p_2, x)}{\partial q_1}\geq 0$</span> and <span class="math-container">$\frac{\partial u_1(q_1,p_1, q_2,p_2, x)}{\partial p_1}\geq 0$</span>, and independent of <span class="math-container">$q_2,p_2,x$</span>. <span class="math-container">$u_1(q_1,p_1, q_2,p_2, x)$</span> is continuous in all its inputs.</li>
<li><span class="math-container">$A_2$</span> tries to maximize <span class="math-container">$u_2(q_1,p_1, q_2,p_2, x)$</span>, with <span class="math-container">$\frac{\partial u_2(q_1,p_1, q_2,p_2, x)}{\partial q_2}\geq 0$</span> and <span class="math-container">$\frac{\partial u_2(q_1,p_1, q_2,p_2, x)}{\partial p_2}\geq 0$</span>, and independent of <span class="math-container">$q_1,p_1,x$</span>. <span class="math-container">$u_2(q_1,p_1, q_2,p_2, x)$</span> is continuous in all its inputs.</li>
<li><span class="math-container">$A_3$</span> tries to maximize <span class="math-container">$u_3(q_1,p_1, q_2,p_2, x)$</span>, with <span class="math-container">$u_3(q_1,p_1, q_2,p_2, x)$</span> increasing for all <span class="math-container">$p_1&lt;\bar{p}$</span> and decreasing for all <span class="math-container">$p_1\geq \bar{p}$</span>; <span class="math-container">$u_3(q_1,p_1, q_2,p_2, x)$</span> decreases in <span class="math-container">$x$</span> and <span class="math-container">$q_1$</span>; and <span class="math-container">$u_3(q_1,p_1, q_2,p_2\rightarrow \infty, x)=-\infty$</span>; <span class="math-container">$u_3(q_1,p_1, q_2\rightarrow \infty,p_2, x)=-\infty$</span>. <span class="math-container">$u_3(q_1,p_1, q_2,p_2, x)$</span> is continuous in all its inputs, but we do not know more about its properties for <span class="math-container">$p_2,q_2$</span>.</li>
<li>We assume that in case that an agent is indifferent between two or more strategies that there exists a decision rule that selects exactly one of these strategies. These rules are known by all agents.</li>
</ul>
<p>The existence of a unique SPNE is quite clear. However, there could be multiple other equilibria that would lead to the same profit profile (specifically equilibria in which each agent gains the same profits.)</p>
<p><strong>Question:</strong> Does <span class="math-container">$A_3$</span>'s profit decrease in <span class="math-container">$x$</span>? If yes, why? If not, is there an easy assumption that we need for <span class="math-container">$u_3(q_1,p_1, q_2,p_2, x)$</span> behavior with regard to <span class="math-container">$q_2,p_2$</span>? (I think I miss the forest for the trees) Does the sum of all agents' profit decrease in <span class="math-container">$x$</span>?</p>
<p>My immediate intuition is yes, because otherwise <span class="math-container">$A_3$</span> would be able to pass on all its &quot;losses&quot; to the other agents.</p>

CC BY-SA 4.0
2021-04-28T14:51:56.847Z
Can you please carefully double check your question. There are multiple contradictory sentences and typos. Also what is $t_M$
tdm
CC BY-SA 4.0
2021-04-28T15:24:26.563Z
I am very sorry for the flod of typos.... I think I got it right this time. Thanks for looking at it.
Paul
<game-theory><profit-maximization>
Effect of exogenous parameter on three player SPNE profits
Paul
# Effect of exogenous parameter on three player SPNE profits

(Asked by: username_0 on 2021-04-27)

Consider three agents $A\_i$.

* $A\_1$ moves first and selects $0\leq q\_1\in \mathbb{R}$. $A\_2$ moves second and selects $0\leq q\_2\in \mathbb{R}$. $A\_3$ moves third and selects $0\leq p\_1\in \mathbb{R}$ and $0\leq p\_2\in \mathbb{R}$. $0\leq x\in \mathbb{R}$ is a parameter.
* $A\_1$ tries to maximize $u\_1(q\_1,p\_1, q\_2,p\_2, x)$, with $\frac{\partial u\_1(q\_1,p\_1, q\_2,p\_2, x)}{\partial q\_1}\geq 0$ and $\frac{\partial u\_1(q\_1,p\_1, q\_2,p\_2, x)}{\partial p\_1}\geq 0$, and independent of $q\_2,p\_2,x$. $u\_1(q\_1,p\_1, q\_2,p\_2, x)$ is continuous in all its inputs.
* $A\_2$ tries to maximize $u\_2(q\_1,p\_1, q\_2,p\_2, x)$, with $\frac{\partial u\_2(q\_1,p\_1, q\_2,p\_2, x)}{\partial q\_2}\geq 0$ and $\frac{\partial u\_2(q\_1,p\_1, q\_2,p\_2, x)}{\partial p\_2}\geq 0$, and independent of $q\_1,p\_1,x$. $u\_2(q\_1,p\_1, q\_2,p\_2, x)$ is continuous in all its inputs.
* $A\_3$ tries to maximize $u\_3(q\_1,p\_1, q\_2,p\_2, x)$, with $u\_3(q\_1,p\_1, q\_2,p\_2, x)$ increasing for all $p\_1<\bar{p}$ and decreasing for all $p\_1\geq \bar{p}$; $u\_3(q\_1,p\_1, q\_2,p\_2, x)$ decreases in $x$ and $q\_1$; and $u\_3(q\_1,p\_1, q\_2,p\_2\rightarrow \infty, x)=-\infty$; $u\_3(q\_1,p\_1, q\_2\rightarrow \infty,p\_2, x)=-\infty$. $u\_3(q\_1,p\_1, q\_2,p\_2, x)$ is continuous in all its inputs, but we do not know more about its properties for $p\_2,q\_2$.
* We assume that in case that an agent is indifferent between two or more strategies that there exists a decision rule that selects exactly one of these strategies. These rules are known by all agents.

The existence of a unique SPNE is quite clear. However, there could be multiple other equilibria that would lead to the same profit profile (specifically equilibria in which each agent gains the same profits.)

**Question:** Does $A\_3$'s profit decrease in $x$? If yes, why? If not, is there an easy assumption that we need for $u\_3(q\_1,p\_1, q\_2,p\_2, x)$ behavior with regard to $q\_2,p\_2$? (I think I miss the forest for the trees) Does the sum of all agents' profit decrease in $x$?

My immediate intuition is yes, because otherwise $A\_3$ would be able to pass on all its "losses" to the other agents.

**Tags:** game-theory, profit-maximization
> **Comments:**
> **username_1**: Can you please carefully double check your question. There are multiple contradictory sentences and typos. Also what is $t_M$  
> **username_0**: I am very sorry for the flod of typos.... I think I got it right this time. Thanks for looking at it.  

---

## 1 answers

### Answer by username_1 on 2021-04-28. Score: 3

Let's solve by backward induction.

1. In stage 3, player $A\_3$ solves:
   $$
   \max\_{p\_1, p\_2} u\_3(p\_1, p\_2, q\_1, q\_2, x).
   $$
   Given the assumptions, $A\_3$ will set: $p\_1 = \overline{p}$ and in general $p\_2$ will be a function of the $q\_1, q\_2$ and $x$, denoted by $p\_2(q\_1, q\_2, x)$. For a solution to exist, you have to assume that $u\_3$ is not unbounded in $p\_2$.
2. As $u\_2$ is independent of $p\_1, q\_1, x$, In stage 2, Player $A\_2$ solves:
   $$
   \max\_{q\_2} u\_2(q\_2, p\_2(q\_1, q\_2, x)).
   $$
   $u\_2$ is increasing in $q\_2$ and $p\_2$. If $p\_2(q\_1, q\_2, x)$ is increasing in $q\_2$ then $A\_2$ will set $q\_2$ as big as possible, so there is no optimal solution. Else, there might be some optimal level of $q\_2$, say $q\_2(q\_1, x)$.
3. In stage 1, player $A\_1$ solves:
   $$
   \max\_{q\_1} u\_1(q\_1, \overline{p}).
   $$
   As $u\_1$ is increasing in $q\_1$ there will probably be no solution to this. So likely this problem is unbounded. If we assume that $q\_1 \le \overline{q}\_1$ for some fixed value $\overline{q}\_1$, then $A\_1$ will set $q\_1 = \overline{q}\_1$.

Going back to the utility of player 3, we therefore see, after substituting:
$$
u\_3 = u\_3(\overline{p}, p\_2(\overline{q}\_1, q\_2(\overline{q}\_1, x),x), \overline{q}\_1, q\_2(\overline{q\_1}, x), x).
$$
It seems like the change in $u\_3$ due to a change in $x$ is ambiguous. As it also influences the optimal choice of $q\_2$ (of player $A\_2$), and $p\_2$ of player $A\_3$.

> **Comments:**
> **username_0**: Thank you very much @tmd. I included two new properties for $u_3$ with regard $p_2$ and $q_2$. I do not yet understand why $A_1$ and $A_2$ who are indifferent to $x$ should essentially take all the losses that arise for $A_3$ when $x$ increases. Do you have the intuition why that might happen? Perhaps I can then add a little bit more structure to rule this behavior out.  
> **username_0**: Does the sum of all agents' profit decrease in $x$?  

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