Two players each chooses a number from the set $\{1,2,4\}$ and correspondingly bets an amount of \$$1$, \$$2$, or \$$4$. There is no collaboration between players.
Rules:
$1.$ If the two chosen numbers are identical, then the amounts are returned to each player.
$2.$ If the two numbers differ by $1$, the first player receives the whole amount.
$3.$ If the numbers have difference greater than $1$, then the second player receives the whole amount.
Problem:
Show that the equilibria can be found by the following LP: \begin{equation} \text{ minimize } \quad x_1 +x_2+x_3 \end{equation} \begin{equation} \text{subject to } \end{equation} \begin{eqnarray} 5x_1 + 6x_2 + x_3 & \geq & 1 \\ 7x_1 + 5x_2 + x_3 & \geq & 1 \\ 4x_1 + 3x_2 + 5 x_3 & \geq & 1\\ x_1, x_2, x_3 & \geq & 0. \end{eqnarray}
Remark:
I know how to transform from the game problem to an LP, however the payoff matrix I obtained for the row player (first player) is \begin{equation} P= \left( \begin{array}{ccc} 1 & 3 & 0 \\ 3 & 2 & 0 \\ 0 & 0 & 4 \end{array} \right), \end{equation} which is totally different from the coefficients in the LP. Any ideas??? Any help will be appreciated.
The coefficients in the LP are shifted by $+5$ from the coefficients of the correct payoff matrix that is given in the comments.
If you express the constraints as $$A^ty\ge 1$$ then observe that $$A=P+5\cdot\mathbb 1_{3\times 3}$$ where the correct payoff matrix $P$ is given in the comments and $1_{3\times3}$ denotes the matrix with all it's elements equal to 1. Now use that if $A$ is such that $a_{ij}=p_{ij}+c$ then $$\operatorname{val}(A)=\operatorname{val}(P)+c$$ and that a strategy is optimal in the game with payoff matrix $A$ iff this strategy is optimal in the game with payoff matrix $P$ to conclude that the given LP indeed gives you the optimal solution.