<p>Usually in (non-coopeative) game theory, one assumes that players take their actions independently.</p>
<p>In this sense, a players' set of feasible actions should be independent of the action taken by the other player. In your setting this is not the case as <span class="math-container">$x_1$</span> can only take values in <span class="math-container">$[0, x_0 - x_2]$</span> and <span class="math-container">$x_2$</span> can only takes values in <span class="math-container">$[0, x_0 - x_1]$</span> (assuming both <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> to be greater or equal to zero).</p>
<p>How does player 1 know that a choice greater than <span class="math-container">$x_0 - x_2$</span> is not allowed if she does not know the action <span class="math-container">$x_2$</span> of player 2?</p>
<ul>
<li><p>One solution could be to define <span class="math-container">$f(x_1, x_2)$</span> and <span class="math-container">$g(x_1, x_2)$</span> on the entire set <span class="math-container">$[0,x_0]\times[0,x_0]$</span> but to define the payoff functions to be very negative, say, <span class="math-container">$-\infty$</span> when <span class="math-container">$x_1 + x_2 &gt; x_0$</span>.
Defining:
<span class="math-container">$$
\tilde f(x_1, x_2) = \begin{cases} f(x_1, x_2) &amp;\text{ if } x_1 + x_2 \le x_0 \\ 
-\infty &amp;\text{ if } x_1 + x_2 &gt; x_0 \end{cases}
$$</span>
<span class="math-container">$$
\tilde g(x_1, x_2) = \begin{cases} f(x_1, x_2) &amp;\text{ if } x_1 + x_2 \le x_0 \\ 
-\infty &amp;\text{ if } x_1 + x_2 &gt; x_0 \end{cases}
$$</span>
The problems of player 1 and player 2 then become:
<span class="math-container">$$
\max_{x_1} \tilde f(x_1, x_2) \text{ s. t. } x_1 \in [0, x_0]\\
\max_{x_2} \tilde g(x_1, x_2) \text{ s.t. } x_2 \in [0, x_0].
$$</span>
Notice that a Nash equilibrium <span class="math-container">$(x_1^\ast, x_2^\ast)$</span> for such game (if it exists) will always have <span class="math-container">$x_1^\ast + x_2^\ast \le x_0$</span> as otherwise every player could improve his or her payoff by satisfying the constraint.</p>
</li>
<li><p>A second option is to simply define the optimization problem of both individuals conditional on the action taken by the other player:
<span class="math-container">$$
\max_{x_1} f(x_1, x_2) \text{ s.t. } x_1 \le x_0 - x_2\\
\max_{x_2} g(x_1, x_2) \text{ s.t. } x_2 \le x_0 - x_1.
$$</span>
The &quot;Nash equilibrium&quot; of this game will be the same as for the previous case.</p>
</li>
<li><p>Anyway, if the payoff functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are continuous and concave in their own strategies one can use Kakutani's fixed point theorem to show that there will always be a Nash equilibrium, i.e. a strategy <span class="math-container">$(x_1^\ast, x_2^\ast)$</span> such that <span class="math-container">$x_1^\ast+ x_2^\ast \le x_0$</span> and:
<span class="math-container">$$
f(x_1^\ast, x_2^\ast) \ge f(x_1, x_2^\ast) \text{ for all } x_1 \le x_0 - x_2^\ast\\
g(x_1^\ast, x_2^\ast) \ge f(x_1^\ast, x_2) \text{ for all } x_2 \le x_0 - x_1^\ast.
$$</span></p>
</li>
</ul>

CC BY-SA 4.0
2022-05-20T18:30:09.267Z
These all make sense. My payoff functions are both continuous and concave in their own strategies. So Kakutani's theorem is applicable here. Thank you very much for the suggestion.
EagleEdge0423
CC BY-SA 4.0
2022-05-19T17:29:23.467Z
2022-05-19T17:29:23.467Z
tdm
<p>I am trying to solve a two player game with constraints on decision variables. The general structure looks something like this:
<span class="math-container">$$\max_{x_1} f(x_1, x_2)$$</span>
<span class="math-container">$$\max_{x_2} g(x_1, x_2)$$</span>
subject to
<span class="math-container">$$x_1 + x_2 \leq x_0$$</span>
What might be a solution approach? or any direction to a resource for solving such problems would be highly appreciated.</p>

CC BY-SA 4.0
2022-05-17T14:19:15.607Z
The kind of object you are looking at is known as a "generalized game" or "abstract economy."
Michael Greinecker
CC BY-SA 4.0
2022-05-17T15:05:17.737Z
Is Arrow K, Debreu G. (1954) a good reference for what you are referring to?
EagleEdge0423
CC BY-SA 4.0
2022-05-17T15:26:28.680Z
They are using them as a tool, but I don't think you'll learn that much from their paper about the general theory. The literature tends to be fairly mathematical. If you are comfortable with that,  chapter 19 of Kim Border's "Fixed Point Theorems with Applications to Economics and Game Theory" discusses the theory in detail.
Michael Greinecker
CC BY-SA 4.0
2022-05-17T15:28:30.727Z
Thanks, I will check that out.
EagleEdge0423
<mathematical-economics><game-theory><optimization><nash-equilibrium>
Nash Equilibrium with Constraints on Decision Variables
EagleEdge0423
# Nash Equilibrium with Constraints on Decision Variables

(Asked by: username_0 on 2022-05-17)

I am trying to solve a two player game with constraints on decision variables. The general structure looks something like this:
$$\max\_{x\_1} f(x\_1, x\_2)$$
$$\max\_{x\_2} g(x\_1, x\_2)$$
subject to
$$x\_1 + x\_2 \leq x\_0$$
What might be a solution approach? or any direction to a resource for solving such problems would be highly appreciated.

**Tags:** mathematical-economics, game-theory, optimization, nash-equilibrium
> **Comments:**
> **username_1**: The kind of object you are looking at is known as a "generalized game" or "abstract economy."  
> **username_0**: Is Arrow K, Debreu G. (1954) a good reference for what you are referring to?  
> **username_1**: They are using them as a tool, but I don't think you'll learn that much from their paper about the general theory. The literature tends to be fairly mathematical. If you are comfortable with that,  chapter 19 of Kim Border's "Fixed Point Theorems with Applications to Economics and Game Theory" discusses the theory in detail.  
> **username_0**: Thanks, I will check that out.  

---

## 1 answers

### Answer by username_2 on 2022-05-19. Score: 4 | Accepted answer

Usually in (non-coopeative) game theory, one assumes that players take their actions independently.

In this sense, a players' set of feasible actions should be independent of the action taken by the other player. In your setting this is not the case as $x\_1$ can only take values in $[0, x\_0 - x\_2]$ and $x\_2$ can only takes values in $[0, x\_0 - x\_1]$ (assuming both $x\_1$ and $x\_2$ to be greater or equal to zero).

How does player 1 know that a choice greater than $x\_0 - x\_2$ is not allowed if she does not know the action $x\_2$ of player 2?

* One solution could be to define $f(x\_1, x\_2)$ and $g(x\_1, x\_2)$ on the entire set $[0,x\_0]\times[0,x\_0]$ but to define the payoff functions to be very negative, say, $-\infty$ when $x\_1 + x\_2 > x\_0$.
  Defining:
  $$
  \tilde f(x\_1, x\_2) = \begin{cases} f(x\_1, x\_2) &\text{ if } x\_1 + x\_2 \le x\_0 \\
  -\infty &\text{ if } x\_1 + x\_2 > x\_0 \end{cases}
  $$
  $$
  \tilde g(x\_1, x\_2) = \begin{cases} f(x\_1, x\_2) &\text{ if } x\_1 + x\_2 \le x\_0 \\
  -\infty &\text{ if } x\_1 + x\_2 > x\_0 \end{cases}
  $$
  The problems of player 1 and player 2 then become:
  $$
  \max\_{x\_1} \tilde f(x\_1, x\_2) \text{ s. t. } x\_1 \in [0, x\_0]\\
  \max\_{x\_2} \tilde g(x\_1, x\_2) \text{ s.t. } x\_2 \in [0, x\_0].
  $$
  Notice that a Nash equilibrium $(x\_1^\ast, x\_2^\ast)$ for such game (if it exists) will always have $x\_1^\ast + x\_2^\ast \le x\_0$ as otherwise every player could improve his or her payoff by satisfying the constraint.
* A second option is to simply define the optimization problem of both individuals conditional on the action taken by the other player:
  $$
  \max\_{x\_1} f(x\_1, x\_2) \text{ s.t. } x\_1 \le x\_0 - x\_2\\
  \max\_{x\_2} g(x\_1, x\_2) \text{ s.t. } x\_2 \le x\_0 - x\_1.
  $$
  The "Nash equilibrium" of this game will be the same as for the previous case.
* Anyway, if the payoff functions $f$ and $g$ are continuous and concave in their own strategies one can use Kakutani's fixed point theorem to show that there will always be a Nash equilibrium, i.e. a strategy $(x\_1^\ast, x\_2^\ast)$ such that $x\_1^\ast+ x\_2^\ast \le x\_0$ and:
  $$
  f(x\_1^\ast, x\_2^\ast) \ge f(x\_1, x\_2^\ast) \text{ for all } x\_1 \le x\_0 - x\_2^\ast\\
  g(x\_1^\ast, x\_2^\ast) \ge f(x\_1^\ast, x\_2) \text{ for all } x\_2 \le x\_0 - x\_1^\ast.
  $$

> **Comments:**
> **username_0**: These all make sense. My payoff functions are both continuous and concave in their own strategies. So Kakutani's theorem is applicable here. Thank you very much for the suggestion.  

---

