Prior to a major battle on the Veracruz coast, there are rumors of soldiers deserting Cortez by seizing ships and sailing home rather than fighting. Cortez has two options -- burn the ships to prevent deserters from leaving; or not burn the ships. With all ships left, all 10,000 soldiers can desert before the battle begins; but with no ships left, only 1,000 soldiers can desert using makeshift vessels before the battle begins (thus, with no ships left, if 1,000 soldiers decide to leave, there are 9,000 soldiers left before the battle starts). The soldiers in aggregate care about the number of soldiers alive and whether they win, while Cortez only cares about winning and having ships to sail back with. The payoff function for the two separate players is: Usoldier = N +500 * W U Cortez = 10,000 W 500 * B Where N is the number of soldiers alive after battle (note, N is max when all soldiers leave); W is 1 for winning the battle and 0 for losing; and B is 1 for burning ships and 0 for not burning ships. Cortez first decides whether to burn the ships; and then soldiers decide whether to leave before the battle starts. Cortez always goes to battle if there are some soldiers present, and he always loses 5,000 soldiers in battle. The army only wins when all 10,000 soldiers stay to fight. A) Fill in the game tree where Cortez first decides whether to burn the ships, and then soldiers decide whether to leave. You need to determine which of the "?" branches represent Cortez's decision to burn or not burn ships; and you need to fill in the payoffs "(#, #)": Cortez Soldiers Stay (#, #) ? Leave Stay (0, (9,500, 10.000) 5,500) ? Leave (#, #) B) Recursively solve the game. (That is, use backward induction to find the subgame perfect Nash equilibrium). C) Suppose Cortez knows that the soldiers highly value winning (this particular Aztec city holds a lot of gold) such that the aggregate soldier utility is now: Usoldier = N +5,500 * W Find the new payoffs and create a new tree for this game (note the structure is exactly the same as above, but the payoffs are now different). D) Recursively solve the new game. (That is, use backward induction to find the subgame perfect Nash equilibrium). E) Can you explain why Cortez decides to do something different in the second game?