Finding Nash Equilibria with Calculus

Question

The problem is summarized as:

There are two players. Player 1's strategy is h. Player 2's strategy is w. 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?

Answer 1

(I have not studied Nash equilibria before, but I'll take a stab at this anyway).

Ok, so say h is on the x axis, and w is on the y axis. Both players are trying to maximize the profit, i.e. they change their strategies according to the derivatives of the payout functions.

Plotting this stream plot, we get the graph below:

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).

From this, we can see that the point (h,w)=(500,500) is an equilibrium, and you can verify this by seeing that both the derivatives in this point are positive, and even more, $P_h(h,500)>0$ for all $h\in[0,500]$, and similarly $P_w(500,w)>0$ for all $w \in [0,500].$ Thus, no player would gain on changing the strategy if they are in (500,500).