Abdallah and Maria have to play arm wrestling. The game goes like this. Maria must decide whether or not to confront Abdallah. If she doesn't fight him, she has to pay him $50. If she decides to confront him, Abdallah must decide whether he accepts the challenge or not. If he refuses, he must pay her $50. If he accepts, Maria and Abdallah arm wrestle and whoever loses has to pay the other $100. There is a 7/13 probability that Maria has trained in an infallible arm-wrestling technique. If so, she will win the challenge. On the other hand, if she does not have this training, Abdallah will win the challenge. Maria knows if she followed this training, on the other hand, Abdallah would not know if she followed the training or not and only knows the probability being 7/13 that she followed it.

a) Represent graphically the extensive form of this game.

b) This game has a hybrid equilibrium in which, if Maria followed the training, she challenges Abdallah with probability 1. If she did not follow the training, she challenges Abdallah with probability . Abdallah accepts the challenge with probability .

Find the equilibrium values of and .

c) What are the expected ex-ante payments of the two players (i.e. before Maria knows if the training will be offered to her)?