Background
Input matrix:
$$ \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 4 & 3 & 2 \\ \end{bmatrix}$$
We have a game with 2 players. The game is antagonistic e.g value functions P(x,y) for player 1 and Q(x,y) for player 2: $P(x,y) = -Q(x,y)$
graph
We can see that all lines merge in one point -3: https://www.graphsketch.com/?eqn1_color=1&eqn1_eqn=1%2B4x&eqn2_color=2&eqn2_eqn=2%2B2x&eqn3_color=3&eqn3_eqn=3&eqn4_color=4&eqn4_eqn=4-2x&eqn5_color=5&eqn5_eqn=&eqn6_color=6&eqn6_eqn=&x_min=0&x_max=1&y_min=-3&y_max=5&x_tick=1&y_tick=1&x_label_freq=5&y_label_freq=5&do_grid=0&do_grid=1&bold_labeled_lines=0&bold_labeled_lines=1&line_width=4&image_w=850&image_h=525
We know that the price $v=3$, and so:
$$ \begin{array}{lcl} y_1 + 2y_2 + 3y_3 + 4y_4 & = & 3 \\ 5y_1 + 4y_2 + 3y_3 + 2y_4 & = & 3 \\ \sum y_i = 1 \end{array} $$
Solution:
The following is given as a hint but I don't understand, what is the idea behind the following?:
$$P(\overline{x}, 1) = v$$ $$P(\overline{x}, 3) = v$$ $$P(\overline{x}, 2) > v$$ $$P(\overline{x}, 4) > v$$ $$\overline{y_2}=0$$
Question:
- How can I get the Nash equilibrium?
- What is the idea of the hint?