<p>Solution:</p>
<ul>
<li>Set of players: <span class="math-container">$\{1,2\}$</span></li>
<li>Action sets: <span class="math-container">$A_1 = \mathbb{R^+}$</span>, <span class="math-container">$A_2 = \mathbb{R^+}$</span></li>
<li>The payoffs are,</li>
</ul>
<p><span class="math-container">$u_1(c_1,c_2)= c_{1} (c_{2}-c_{1})$</span></p>
<p>and, <span class="math-container">$u_2(c_1,c_2)= c_{2} (1 - c_{1}-c_{2})$</span></p>
<p>We can figure out the Best Response Set of player 1 and 2 and then look the intersection of the Graphs of Best responses to figure out the Nash Equilibria,</p>
<p><span class="math-container">$BR_1(c_2)= \{c_1\in A_1 | u_1(c_1,c_2) \ge u_1(c_1^1,c_2) 
\hspace{2mm}, \forall c_1^1 \in A_1\}$</span>
and similarly <span class="math-container">$BR_2(c_1)$</span></p>
<p><strong>Figuring out the Best Response of Player 1:</strong></p>
<p><span class="math-container">$\max_{c_1} \hspace{2 mm} u_1(c_1,c_2)= \max_{c_1} \hspace{2mm} (c_1c_2-c_1^2)$</span> (Notice that function <span class="math-container">$u_1$</span> is concave in <span class="math-container">$c_1$</span>, therefore the First order condition is both necessary and sufficient)</p>
<p>The First order condition is,</p>
<p><span class="math-container">$$c_2-2c_1=0 \\
\Rightarrow c_2=2c_1$$</span>
The Best Response of player 1 as a function of <span class="math-container">$c_2$</span> is
<span class="math-container">$BR_1(c_2)=\{c_1 \in A_1 | c_2=2c_1\}$</span></p>
<p><strong>Figuring out the Best Response of Player 2:</strong></p>
<p><span class="math-container">$\max_{c_2} \hspace{2 mm} u_2(c_1,c_2)= \max_{c_2} \hspace{2mm} (c_2-c_1c_2-c_2^2)$</span> (Notice that function <span class="math-container">$u_2$</span> is concave in <span class="math-container">$c_2$</span>, therefore the First order condition is both necessary and sufficient)</p>
<p>The First order condition is,</p>
<p><span class="math-container">$$1-c_1-2c_2=0 \\
\Rightarrow c_1=-2c_2+1$$</span>
The Best Response of player 2 as a function of <span class="math-container">$c_1$</span> is
<span class="math-container">$BR_2(c_1)=\{c_2\in A_2 | c_1=-2c_2+1\}$</span></p>
<p>Now we can plot the <span class="math-container">$GR(BR_1(c_2))=\{(c_1,c_2)\in A_1 \times A_2 |c_1 \in BR_1(c_2)\}$</span> and <span class="math-container">$GR(BR_2(c_1))=\{(c_1,c_2)\in A_1 \times A_2 |c_2 \in BR_2(c_1)\}$</span> and look at their intersection to figure out the equilibrium, which will be equivalent to solving,
<span class="math-container">$$ c_2=2c_1 \\ 1-c_1-2c_2=0$$</span>
<strong>which will give us the Nash-Equilibrium as, <span class="math-container">$(c_1^\ast,c_2^\ast)=(\frac{1}{5},\frac{2}{5})$</span></strong></p>

CC BY-SA 4.0
2024-03-18T05:20:34.123Z
2024-03-24T07:17:50.987Z
2024-03-24T07:17:50.987Z
SGP
<p>Find the non-randomized Nash Equilibria of this two-player strategic game, in which each player's set of actions is the non-negative real numbers, and the players' payoff functions are:</p>
<p><span class="math-container">$$u_{1} (c_{1},c_{2}) = c_{1} (c_{2}-c_{1})$$</span></p>
<p><span class="math-container">$$u_{2} (c_{1},c_{2}) = c_{2} (1 - c_{1}-c_{2})$$</span></p>
<p>How does one approach this problem without a payoff matrix?</p>

CC BY-SA 4.0
2024-03-18T04:14:07.737Z
Here is a hint: Every player needs to maximize their utility. So, try to determine the maximum $u_i$ with respect to $c_i$.
Max
<game-theory>
Find the Nash Equilibria in this Two-Player Strategic Game
user9487
# Find the Nash Equilibria in this Two-Player Strategic Game

(Asked by: username_0 on 2024-03-18)

Find the non-randomized Nash Equilibria of this two-player strategic game, in which each player's set of actions is the non-negative real numbers, and the players' payoff functions are:

$$u\_{1} (c\_{1},c\_{2}) = c\_{1} (c\_{2}-c\_{1})$$

$$u\_{2} (c\_{1},c\_{2}) = c\_{2} (1 - c\_{1}-c\_{2})$$

How does one approach this problem without a payoff matrix?

**Tags:** game-theory

> **Comments:**
> **username_1**: Here is a hint: Every player needs to maximize their utility. So, try to determine the maximum $u_i$ with respect to $c_i$.  

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## 1 answers

### Answer by username_2 on 2024-03-18. Score: 2 | Accepted answer

Solution:

* Set of players: $\{1,2\}$
* Action sets: $A\_1 = \mathbb{R^+}$, $A\_2 = \mathbb{R^+}$
* The payoffs are,

$u\_1(c\_1,c\_2)= c\_{1} (c\_{2}-c\_{1})$

and, $u\_2(c\_1,c\_2)= c\_{2} (1 - c\_{1}-c\_{2})$

We can figure out the Best Response Set of player 1 and 2 and then look the intersection of the Graphs of Best responses to figure out the Nash Equilibria,

$BR\_1(c\_2)= \{c\_1\in A\_1 | u\_1(c\_1,c\_2) \ge u\_1(c\_1^1,c\_2)
\hspace{2mm}, \forall c\_1^1 \in A\_1\}$
and similarly $BR\_2(c\_1)$

**Figuring out the Best Response of Player 1:**

$\max\_{c\_1} \hspace{2 mm} u\_1(c\_1,c\_2)= \max\_{c\_1} \hspace{2mm} (c\_1c\_2-c\_1^2)$ (Notice that function $u\_1$ is concave in $c\_1$, therefore the First order condition is both necessary and sufficient)

The First order condition is,

$$c\_2-2c\_1=0 \\
\Rightarrow c\_2=2c\_1$$
The Best Response of player 1 as a function of $c\_2$ is
$BR\_1(c\_2)=\{c\_1 \in A\_1 | c\_2=2c\_1\}$

**Figuring out the Best Response of Player 2:**

$\max\_{c\_2} \hspace{2 mm} u\_2(c\_1,c\_2)= \max\_{c\_2} \hspace{2mm} (c\_2-c\_1c\_2-c\_2^2)$ (Notice that function $u\_2$ is concave in $c\_2$, therefore the First order condition is both necessary and sufficient)

The First order condition is,

$$1-c\_1-2c\_2=0 \\
\Rightarrow c\_1=-2c\_2+1$$
The Best Response of player 2 as a function of $c\_1$ is
$BR\_2(c\_1)=\{c\_2\in A\_2 | c\_1=-2c\_2+1\}$

Now we can plot the $GR(BR\_1(c\_2))=\{(c\_1,c\_2)\in A\_1 \times A\_2 |c\_1 \in BR\_1(c\_2)\}$ and $GR(BR\_2(c\_1))=\{(c\_1,c\_2)\in A\_1 \times A\_2 |c\_2 \in BR\_2(c\_1)\}$ and look at their intersection to figure out the equilibrium, which will be equivalent to solving,
$$ c\_2=2c\_1 \\ 1-c\_1-2c\_2=0$$
**which will give us the Nash-Equilibrium as, $(c\_1^\ast,c\_2^\ast)=(\frac{1}{5},\frac{2}{5})$**


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