Consider two players
Player 1 choose $x_1$
Player 2 choose $x_2$
$x_1, x_2 \in [0,b]$ and $b>0$
The payoffs for players 1 and 2 are identical and equal
$$Z=u(x_1+x_2)+v(2b-x_1-x_2)$$
$u$ and $v$ are strictly concave functions.
I want to find best response functions for both players and unique Nash equilibrium. And what can I say about the slope of the best response functions?
Solution
For player 1
$\partial Z /\partial x_1=0$
$$Z_1=u_1(x_1+x_2)*(1)+v_1(2b-x_1-x_2)(-1)=0$$
I obtain best response of player 1 to player 2. I.e. $R_1(x_2)$
For player 2
$\partial Z /\partial x_2=0$
$$Z_2=u_2(x_1+x_2)*(1)+v_2(2b-x_1-x_2)(-1)=0$$
I obtain best response of player 2 to player 1. I.e. $R_2(x_1)$
So far I obtain best response functions.
Now, if I solve these above two equations simultaneously,
That’s, $R_2(x_1)=x_2$ and $R_1(x_2)=x_1$
Then I will get results $x_1^*$ and $x_2^*$ which are unique Nash equilibrium.
As for the slope of the vest response function which is equal to $-{z_1\over z_2}$
I can only say these about this question. I cannot write specific answer with using this payoff function z. What is your ideas? How can I solve this question expect for what I did. I am stuck at this point. I am waiting for your helps please. Thanks.
Best Answer
You derivation of the first order condition is correct and this is indeed a necessary and sufficient condition for optimal since the objective function is concave. One detail is that $u_1(x_1+x_2)=u_2(x_1+x_2)=u'(x_1+x_2)$ and the same for $v$.
To obtain the slope of the best response function for player $1$, for instance, you can, then, use the implicit function theorem. Let
$$ F(x_1,x_2)=u'(x_1+x_2)-v'(2b-x_1-x_2)=0, $$
Then,
$$\frac{dx_1}{dx_2}=-\frac{F_{x_2}(x_1,x_2)}{F_{x_1}(x_1,x_2)}=-\frac{u''(x_1+x_2)+v''(2b-x_1-x_2)}{u''(x_1+x_2)+v''(2b-x_1-x_2)}=-1.$$