State sufficient conditions for $p$ and $q$ to be optimal mixed strategies for the row and column players in a zero-sum game with payoff matrix $A$ and value $v$.

Rowena and Colin play a hide-and-seek game. Rowena hides in one of 3 locations, and then Colin searches them in some order. If he searches in order $i, j, k$ then his search cost is $c_{i}, c_{i}+c_{j}$ or $c_{i}+c_{j}+c_{k}$, depending upon whether Rowena hides in $i, j$ or $k$, respectively, and where $c_{1}, c_{2}, c_{3}$ are all positive. Rowena (Colin) wishes to maximize (minimize) the expected search cost.

Formulate the payoff matrix for this game.

Let $c=c_{1}+c_{2}+c_{3}$. Suppose that Colin starts his search in location $i$ with probability $c_{i} / c$, and then, if he does not find Rowena, he searches the remaining two locations in random order. What bound does this strategy place on the value of the game?

Guess Rowena's optimal hiding strategy, show that it is optimal and find the value of the game.