Ned and Ruth love to play Hide and Seek. It is a simple game, but it continues

Question:

Ned and Ruth love to play €œHide and Seek.€ It is a simple game, but it continues to amuse. It goes like this. Ruth hides upstairs or downstairs. Ned can look upstairs or downstairs but not in both places. If he finds Ruth, Ned gets one scoop of ice cream and Ruth gets none. If he does not find Ruth, Ruth gets one scoop of ice cream and Ned gets none. Fill in the payoffs in the matrix below.

(a) Is this a zero-sum game? __________ What are the Nash equilibria in pure strategies? ___________
(b) Find a Nash equilibrium in mixed strategies for this game.
(c) After years of playing this game, Ned and Ruth think of a way to liven it up a little. Now if Ned finds Ruth upstairs, he gets two scoops of ice cream, but if he finds her downstairs, he gets one scoop. If Ned finds Ruth, she gets no ice cream, but if he doesn€™t find her she gets one scoop. Fill in the payoffs in the graph below.
(d) Are there any Nash equilibria in pure strategies? _________ What mixed strategy equilibrium can you find? ________________ If both use equilibrium strategies, what fraction of the time will Ned find Ruth? ____________