A row $r$ of the payoff matrix is said to dominate a row $s$ if $a_{rj}\geq a_{sj}$ for all $j$ = 1,2,......,$n$. Similarly, a column $r$ of the payoff matrix is said to dominate a column $s$ if $a_{ir}\geq a_{is}$ for all $i$ = 1,2,......,$m$.
Prove:
$(i)$ If a row $r$ is dominated by another row, then the row player has at least one optimal strategy $x^{*}$ in which $x_r^{*}=0$. In particular, if row $r$ is deleted from the payoff matrix, then the value of the game does not change.
$(ii)$ If a column $s$ is dominated by another column, then the column player has at least one optimal strategy $y^{*}$ in which $y_s^{*}=0$. In particular, if column $s$ is deleted from the payoff matrix, then the value of the game does not change.
I do not know how to start on this proof, can anyone give me some hints?
As a hint, I think the best way to proceed is to think about each player's maximization problem. Player A maximizes his/her payoff given player B's strategy. For the row player, given that row $r$ always gives equal or higher payoffs, her/she can do just as well by playing $r$ instead of $s$. Thus, if you deleted the row, the payoff in the equilibrium strategy could not be less.