Finding values such that the game is strictly determinable

Question

Please help me to solve these two problems.

Q1)Find the range of values of p and q that will make the entry (2,2) a saddle point of the game.

$$ \begin{pmatrix} &player B\\ 2 & 4 & 5 \\ 10 & 7 & q \\ 4 & p & 6 \\ \end{pmatrix} $$ Q2)For what values of p, the game with following payoff matrix is strictly determinable?

$$ \begin{pmatrix} &player B\\ p & 6 & 2 \\ -1 & p & -7 \\ -2 & 4 & p \\ \end{pmatrix} $$

Please help me to understand how to think when solving this kind of problems.I got the correct answers but that was with a lot of guess and check and it takes a lot of time and I am not sure of the answers.

Answer 1

Q1)

On http://www.zweigmedia.com/RealWorld/Summary3b.html#sp I read that

" A saddle point is a payoff that is simultaneously a row minimum and a column maximum."

So you must ask yourself under which conditions on $p$ and $q$ this is satisfied. Is it clear to you how to do so?

Q2)

On http://www.zweigmedia.com/RealWorld/Summary3b.html#sp again, I read

"A game is strictly determined if it has at least one saddle point."

So from the former definition this is equivalent to the payoff matrix having an entry which is a minimum of its row and a maximum of its column. A systematic way to tackle this problem would be as follows:

• Consider entry $(1,1)$. Under which values of $p$ can entry $(1,1)$ be a saddle point? Define the set of such values as $P_{(1,1)}$ ( if it is an interval of values, you may denote it $(\underline{p}^{(1,1)},\bar{p}^{(1,1)})$ for instance).
• Consider entry $(1,2)$. Under which values of $p$ can entry $(1,2)$ be a saddle point? Define the set of such values as $P_{(1,2)}$.
• ...

Then the set of values of $p$ such that the game is strictly determinable is $P_{(1,1)}\cup P_{(1,2)}\cup \dots \cup P_{(3,2)} \cup P_{(3,3)}$ or shortly $\bigcup_{i=1}^3 \bigcup_{j=1}^3 P_{(i,j)}$.