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?