<p>This game need not have any equilibrium as defined here with the lexicographic structure. Consider a situation where the payoff depends only on the action <span class="math-container">$x$</span> of player <span class="math-container">$1$</span> with <span class="math-container">$u_1(x)=\sin x$</span> and <span class="math-container">$u_2(x)=x$</span>. There are infinitely many maxima of <span class="math-container">$u_1$</span>, but no largest maximizer. So one cannot select a best choice for player <span class="math-container">$2$</span> among those that are optimal for player <span class="math-container">$1$</span>.</p>

CC BY-SA 4.0
2021-04-26T19:06:19.533Z
I added the unique interior maximum which was missing. Thx for catching it.
Paul
CC BY-SA 4.0
2021-04-26T17:22:11.287Z
2021-04-26T17:22:11.287Z
Michael Greinecker
<p>Let me first try to answer Statement 2:</p>
<blockquote>
<p>The agent's utility profiles are continuous in x</p>
</blockquote>
<p>Let's try to flesh out the problem. Let <span class="math-container">$S_1$</span> and <span class="math-container">$S_2$</span> be compact, convex, strategy sets of the players (say subsets of <span class="math-container">$\mathbb{R}^n$</span>) (for simplicity) . Let <span class="math-container">$u_1(s_1, s_2, x)$</span> and <span class="math-container">$u_2(s_1, s_2,x)$</span> be the continuous utility functions of the two players.</p>
<p>Let's solve for the SPNE. At stage 2, player 2 maximizes her utility given the action <span class="math-container">$s_1$</span> of player 1. This gives:
<span class="math-container">$$
v_2(s_1, x) = \max_{s_2 \in S_2} u_2(s_1, s_2, x)\\
s_2(s_1, x) = \arg\max_{s_2 \in S_2} u_2(s_1, s_2, x).
$$</span></p>
<ul>
<li>The problem is well defined due to the theorem of the maximum.</li>
<li>The function <span class="math-container">$v_2(s_1, x)$</span> is continuous (Berge's maximisation theorem).</li>
<li>The best response <span class="math-container">$s_2(s_1, x)$</span> is only guaranteed to be an upper hemi-continuous, non-empty correspondence.</li>
<li>If <span class="math-container">$s_2(s_1, x)$</span> is single valued, then it is continuous. However single valuedness will depend on the functional form of <span class="math-container">$u_2(s_1, s_2, x)$</span> (e.g. strict quasi-concavity) and therefore does not follow from the assumptions so far.</li>
</ul>
<p>Let <span class="math-container">$\tilde s_2(s_1, x)$</span> be a selection of <span class="math-container">$s_2(s_1, x)$</span>. In stage 1, Player 1's optimisation problem is:
<span class="math-container">$$
v_1^\ast(x) = \max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).\\
s_1^\ast(x) = \arg\max_{s_1 \in S_1} u_1(s_1, \tilde s_2(s_1, x), x).
$$</span></p>
<ul>
<li>If the selection <span class="math-container">$\tilde s_2(s_1, x)$</span> of Player 2 is the optimal choice from <span class="math-container">$s_2(s_1, x)$</span> for Player 1, then the objective function appears to be upper semi-continuous, (but not, in general, continuous). Due to upper semi-continuity of the objective function, the maximisation problem is well defined.</li>
<li>Although <span class="math-container">$v_1^\ast$</span> will be (I think) upper semi-continuous, it is not necessarily continuous. Anyway, the best response correspondence <span class="math-container">$s_1^\ast(x)$</span> does not need to be continuous.</li>
<li>If <span class="math-container">$s_2(s_1, x)$</span> was single valued, then it is continuous, which means that <span class="math-container">$v_1^\ast(x)$</span> is also continuous. If, in addition <span class="math-container">$s_1^\ast(x)$</span> is also single valued, then it will also be continuous.</li>
</ul>
<p>Let <span class="math-container">$\tilde s_1(x)$</span> be a selection from the best response <span class="math-container">$s_1^\ast(x)$</span> from player 1. Then, we can define:
<span class="math-container">$$
v_2^\ast(x) = v_2(\tilde s_1(x), x),\\
s_2^\ast(x) = \tilde s_2(\tilde s_1(x), x).
$$</span></p>
<ul>
<li>In general, neither <span class="math-container">$v_2^\ast(x)$</span> nor <span class="math-container">$s_2^\ast(x)$</span>  need to be continuous.</li>
<li>If both <span class="math-container">$s_1^\ast(x)$</span> and <span class="math-container">$s_2(s_1, x)$</span> are single valued, then <span class="math-container">$v_2^\ast(x)$</span> and <span class="math-container">$s_2^\ast(x)$</span> will both be continuous.</li>
</ul>
<p>It thus appears that Statement 2 holds if both <span class="math-container">$s_2(s_1, x)$</span> and <span class="math-container">$s_1^\ast(x)$</span> are single valued.</p>
<ul>
<li>Single valuedness of <span class="math-container">$s_2(s_1, x)$</span> holds if (for example) <span class="math-container">$u_2(s_1, s_2, x)$</span> is strictly quasi-concave in <span class="math-container">$s_2$</span></li>
<li>In turn, single valuedness of <span class="math-container">$s_1^\ast(x)$</span> holds if <span class="math-container">$u_1(s_1, \tilde s_2(s_1, x), s_1)$</span> is strictly quasi-concave in <span class="math-container">$s_1$</span>. Notice that strict quasi-concavity of <span class="math-container">$u_1(s_1, s_2(s_1, x), x)$</span> is a strong assumption as it puts shape restrictions on the best response <span class="math-container">$s_2(s_1, x)$</span> (and not only on the function <span class="math-container">$u_1$</span> itself).</li>
</ul>
<p>I hope this answers Statement 2.</p>
<p>For statement 1:</p>
<blockquote>
<p>This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.</p>
</blockquote>
<ul>
<li>As shown above, the existence of a SPNE can be shown using only continuity of <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span> and compactness of the strategy sets, and under the assumption that <span class="math-container">$\tilde s_2(s_1, x)$</span> selects the best option for Player 1.</li>
<li>Different SPNE will in general not give the same utility. Assume that at stage 2, Player 2 is indifferent between more than one optimal strategy, so <span class="math-container">$s_2(s_1, x)$</span> is multivalued (for example, assume that <span class="math-container">$u_2(s_1, s_2, x)$</span> is the constant function, so Player 2 is indifferent between every strategy in <span class="math-container">$S_2$</span>). If different selections <span class="math-container">$\tilde s_2(s_1, x)$</span> from <span class="math-container">$s_2(s_1, x)$</span>, provide different utilities for player 1, then it is possible that <span class="math-container">$v_1^\ast$</span> takes different values depending on the choice of player 2 in stage 2 of the game. So different SPNE may give different utilities.</li>
</ul>

CC BY-SA 4.0
2021-04-27T13:13:48.427Z
The question states that "each agent's utility function has a unique interior maximum in its own decision $s_i$". So assuming "that at stage 2, Player 2 is indifferent between more than one optimal strategy" seems to be ruled out by assumption.
VARulle
CC BY-SA 4.0
2021-04-27T16:41:11.273Z
@tdm: Thank you soo much for that extensive and great answer.
Paul
CC BY-SA 4.0
2021-04-27T06:25:22.987Z
2021-04-27T06:25:22.987Z
tdm
<p>Consider 2 agents <span class="math-container">$A_i$</span>. <span class="math-container">$A1$</span> moves before agent <span class="math-container">$A2$</span>. Each of their utility functions is continuous in each agents' decision <span class="math-container">$0&lt;s_i\in \mathbb{R}$</span> and a parameter <span class="math-container">$x$</span>. Additionally, each agent's utility function has a unique interior maximum in its own decision <span class="math-container">$s_i$</span>.</p>
<p>The agents' have lexicographic prefeences (primarily needed so that the other agents can anticipate strategies): <span class="math-container">$A_1$</span>'s maximize its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for <span class="math-container">$A_2$</span>. <span class="math-container">$A_2$</span>'s maximizes its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for <span class="math-container">$A_1$</span>.</p>
<p>Statement one: This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.</p>
<p>Statement two: The agent's utility profiles are continuous in <span class="math-container">$x$</span>
(Remember, each of their utilities was continuous in <span class="math-container">$x$</span>.) My reasoning for statement two is that an incremental change in x implies that the ne NE will be very close to one of the NE before the change in <span class="math-container">$x$</span>. And as these NE had the same utility profile, the change in <span class="math-container">$x$</span> will only have an incremental effect on the utility profiles.</p>
<p>Do you agree with the two statements and do you perhaps have a more solid argumentation (partcularly for the second).</p>

CC BY-SA 4.0
2021-04-26T13:52:03.553Z
This is confusing. Utility represents preferences, so preferences cannot depend on utilities. Do you mean something like lexicographic preferences on own and other's, say, monetary payoffs?
VARulle
CC BY-SA 4.0
2021-04-26T14:06:08.100Z
Also, is $s_i$ a real number?
Giskard
CC BY-SA 4.0
2021-04-26T16:42:16.843Z
@Giskard I made the requested changes. Thank you for highlighting the shortcomings in my post.
Paul
CC BY-SA 4.0
2021-04-26T17:03:25.873Z
Are you looking at NE or subgame perfect NE (as you have sequential moves). Concerning statement two: even if utlities are continuous in $x$, best responses don't need to be continuous. Usually they will be upper hemi-continuous though.In such case, however, I don't think there is any particular reason for the NE to be continuous in $x$.
tdm
CC BY-SA 4.0
2021-04-26T19:12:08.707Z
@tdm: I am looking at SPNE. I do not need that the best responses are continuous $x$, just that the resulting utilities in the SPNE are continuous in $x$.  Thx for your input and perhaps you can follow up on it.
Paul
<game-theory><nash-equilibrium>
Game Theory: Continuity in equilibrium profits?
Paul
# Game Theory: Continuity in equilibrium profits?

(Asked by: username_0 on 2021-04-26)

Consider 2 agents $A\_i$. $A1$ moves before agent $A2$. Each of their utility functions is continuous in each agents' decision $0<s\_i\in \mathbb{R}$ and a parameter $x$. Additionally, each agent's utility function has a unique interior maximum in its own decision $s\_i$.

The agents' have lexicographic prefeences (primarily needed so that the other agents can anticipate strategies): $A\_1$'s maximize its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for $A\_2$. $A\_2$'s maximizes its own utility, if two or more strategies maximize its own strategy it chooses among these the one that is best for $A\_1$.

Statement one: This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.

Statement two: The agent's utility profiles are continuous in $x$
(Remember, each of their utilities was continuous in $x$.) My reasoning for statement two is that an incremental change in x implies that the ne NE will be very close to one of the NE before the change in $x$. And as these NE had the same utility profile, the change in $x$ will only have an incremental effect on the utility profiles.

Do you agree with the two statements and do you perhaps have a more solid argumentation (partcularly for the second).

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

> **Comments:**
> **username_1**: This is confusing. Utility represents preferences, so preferences cannot depend on utilities. Do you mean something like lexicographic preferences on own and other's, say, monetary payoffs?  
> **username_2**: Also, is $s_i$ a real number?  
> **username_0**: @username_2 I made the requested changes. Thank you for highlighting the shortcomings in my post.  
> **username_3**: Are you looking at NE or subgame perfect NE (as you have sequential moves). Concerning statement two: even if utlities are continuous in $x$, best responses don't need to be continuous. Usually they will be upper hemi-continuous though.In such case, however, I don't think there is any particular reason for the NE to be continuous in $x$.  
> **username_0**: @username_3: I am looking at SPNE. I do not need that the best responses are continuous $x$, just that the resulting utilities in the SPNE are continuous in $x$.  Thx for your input and perhaps you can follow up on it.  

---

## 2 answers

### Answer by username_3 on 2021-04-27. Score: 3 | Accepted answer

Let me first try to answer Statement 2:

> The agent's utility profiles are continuous in x

Let's try to flesh out the problem. Let $S\_1$ and $S\_2$ be compact, convex, strategy sets of the players (say subsets of $\mathbb{R}^n$) (for simplicity) . Let $u\_1(s\_1, s\_2, x)$ and $u\_2(s\_1, s\_2,x)$ be the continuous utility functions of the two players.

Let's solve for the SPNE. At stage 2, player 2 maximizes her utility given the action $s\_1$ of player 1. This gives:
$$
v\_2(s\_1, x) = \max\_{s\_2 \in S\_2} u\_2(s\_1, s\_2, x)\\
s\_2(s\_1, x) = \arg\max\_{s\_2 \in S\_2} u\_2(s\_1, s\_2, x).
$$

* The problem is well defined due to the theorem of the maximum.
* The function $v\_2(s\_1, x)$ is continuous (Berge's maximisation theorem).
* The best response $s\_2(s\_1, x)$ is only guaranteed to be an upper hemi-continuous, non-empty correspondence.
* If $s\_2(s\_1, x)$ is single valued, then it is continuous. However single valuedness will depend on the functional form of $u\_2(s\_1, s\_2, x)$ (e.g. strict quasi-concavity) and therefore does not follow from the assumptions so far.

Let $\tilde s\_2(s\_1, x)$ be a selection of $s\_2(s\_1, x)$. In stage 1, Player 1's optimisation problem is:
$$
v\_1^\ast(x) = \max\_{s\_1 \in S\_1} u\_1(s\_1, \tilde s\_2(s\_1, x), x).\\
s\_1^\ast(x) = \arg\max\_{s\_1 \in S\_1} u\_1(s\_1, \tilde s\_2(s\_1, x), x).
$$

* If the selection $\tilde s\_2(s\_1, x)$ of Player 2 is the optimal choice from $s\_2(s\_1, x)$ for Player 1, then the objective function appears to be upper semi-continuous, (but not, in general, continuous). Due to upper semi-continuity of the objective function, the maximisation problem is well defined.
* Although $v\_1^\ast$ will be (I think) upper semi-continuous, it is not necessarily continuous. Anyway, the best response correspondence $s\_1^\ast(x)$ does not need to be continuous.
* If $s\_2(s\_1, x)$ was single valued, then it is continuous, which means that $v\_1^\ast(x)$ is also continuous. If, in addition $s\_1^\ast(x)$ is also single valued, then it will also be continuous.

Let $\tilde s\_1(x)$ be a selection from the best response $s\_1^\ast(x)$ from player 1. Then, we can define:
$$
v\_2^\ast(x) = v\_2(\tilde s\_1(x), x),\\
s\_2^\ast(x) = \tilde s\_2(\tilde s\_1(x), x).
$$

* In general, neither $v\_2^\ast(x)$ nor $s\_2^\ast(x)$ need to be continuous.
* If both $s\_1^\ast(x)$ and $s\_2(s\_1, x)$ are single valued, then $v\_2^\ast(x)$ and $s\_2^\ast(x)$ will both be continuous.

It thus appears that Statement 2 holds if both $s\_2(s\_1, x)$ and $s\_1^\ast(x)$ are single valued.

* Single valuedness of $s\_2(s\_1, x)$ holds if (for example) $u\_2(s\_1, s\_2, x)$ is strictly quasi-concave in $s\_2$
* In turn, single valuedness of $s\_1^\ast(x)$ holds if $u\_1(s\_1, \tilde s\_2(s\_1, x), s\_1)$ is strictly quasi-concave in $s\_1$. Notice that strict quasi-concavity of $u\_1(s\_1, s\_2(s\_1, x), x)$ is a strong assumption as it puts shape restrictions on the best response $s\_2(s\_1, x)$ (and not only on the function $u\_1$ itself).

I hope this answers Statement 2.

For statement 1:

> This game has at least one SPNE, but in general multiple. In case of multiple SPNE these will have the same utility profiles, that is, yield each agent the same utility.

* As shown above, the existence of a SPNE can be shown using only continuity of $u\_1$ and $u\_2$ and compactness of the strategy sets, and under the assumption that $\tilde s\_2(s\_1, x)$ selects the best option for Player 1.
* Different SPNE will in general not give the same utility. Assume that at stage 2, Player 2 is indifferent between more than one optimal strategy, so $s\_2(s\_1, x)$ is multivalued (for example, assume that $u\_2(s\_1, s\_2, x)$ is the constant function, so Player 2 is indifferent between every strategy in $S\_2$). If different selections $\tilde s\_2(s\_1, x)$ from $s\_2(s\_1, x)$, provide different utilities for player 1, then it is possible that $v\_1^\ast$ takes different values depending on the choice of player 2 in stage 2 of the game. So different SPNE may give different utilities.

> **Comments:**
> **username_1**: The question states that "each agent's utility function has a unique interior maximum in its own decision $s_i$". So assuming "that at stage 2, Player 2 is indifferent between more than one optimal strategy" seems to be ruled out by assumption.  
> **username_0**: @username_3: Thank you soo much for that extensive and great answer.  

---

### Answer by username_4 on 2021-04-26. Score: 2

This game need not have any equilibrium as defined here with the lexicographic structure. Consider a situation where the payoff depends only on the action $x$ of player $1$ with $u\_1(x)=\sin x$ and $u\_2(x)=x$. There are infinitely many maxima of $u\_1$, but no largest maximizer. So one cannot select a best choice for player $2$ among those that are optimal for player $1$.

> **Comments:**
> **username_0**: I added the unique interior maximum which was missing. Thx for catching it.  

---

