The problem is summarized as:
There are two players. Player 1's strategy is
h. Player 2's strategy isw. Both of their strategy sets are within the range [0,500].Player 1's payoff function is:
$ P_h(h, w) = 50h + 2hw-\frac{1}{2}(h)^2 $
Player 2's payoff function is:
$ P_w(h, w) = 50w + 2hw - \frac{1}{2}(w)^2 $
Find a Nash Equilibrium.
I was taught to solve these problems in the following way. Find the first derivative of Player 1's payoff function with respect to h, equate it to 0, then solve for h, and then repeat for Player 2 but with respect to w and solving for w instead. However, I found the first derivatives to be:
$ P_h(h, w)^\prime = 50 + 2w - h $
$ P_w(h, w)^\prime = 50 + 2h - w $
Now after equating these first derivatives to 0 and solving for h and w, we get that h = -50 and w = -50. The issue now is that these strategies aren't within the strategy set [0,500] as mentioned in the problem question. Where am I going wrong?
We see that in the entire region, at least one of the derivatives is always positive. Thus, at least one player gains on increasing w or h.
(We see this because each arrow points either right or up or both).