In a distant galaxy, a war pits the forces of the Interplanetary Federation against those of the rebels of planet X₂₂₃₄₁. Suppose these rebel forces have a probability of 0.10 that they have military logistics to strike the capital of the Interplanetary Federation. The Secretary General of the rebel forces knows full well if he has this ability, but the President of the Interplanetary Federation only knows the probability distribution (0.10 that the rebels have this ability and 0.90 that they do not). The head of the rebel armed forces presents the Secretary General with a plan to attack

Federation forces stationed on the neighboring planet. The Secretary General must decide to either approve the attack or decide not to do anything. If he decides to do nothing, the Secretary General of the rebel forces and the President of the Interplanetary Federation will both have a payment of 0. If the Secretary General decides to strike the forces stationed on the neighboring planet, the President must decide whether 'he does not order a military response on X₂₂₃₄₁. If the President orders a military response on X₂₂₃₄₁ and the rebels have the logistical capacity to strike the capital of the Interplanetary Federation, the Secretary General's payment is 0 and the President's payment is -200. If the President orders a military response on X₂₂₃₄₁and the rebels do not have this capacity, the Secretary General's payment is -100 and that of the President is 100. If the President refuses to order a military response on X22341, the payment of the Secretary General is 200 and that of the President -100, and this regardless of whether the rebels have or do not have the capacity to strike the capital of the Interplanetary Federation.

a) Graphically represent the extensive form of this game.

b) This game has a hybrid equilibrium in which, if the rebels have the logistical capacity to strike the capital of the Interplanetary Federation, the Secretary General orders an attack on the troops stationed on the neighboring planet with probability of 1 and if they do not does not have this ability, he proceeds with this attack with probability p1. The President after an attack on the troops of the Interplanetary Federation orders a

military response on X₂₂₃₄₁ with probability p2. Find the equilibrium values of p₁ and p₂.

c) What is the probability of having an interplanetary war (i.e. that the rebels have the logistical capacity to attack the capital of the Interplanetary Federation and that the President orders a military response on X₂₂₃₄₁)?