# QUESTION 1 (The Centipede Game). Consider another two-player centipede game described asfollows: Eac

QUESTION 1 (The Centipede Game). Consider another two-player centipede game described asfollows: Each player starts with one dollar. Starting with player 1, they alternately say Stop orContinue. When a player says Continue, one dollar is taken from her pile, and two dollars areadded to her opponents pile. As soon as either player says Stop, the game terminates, and eachreceives the dollars in her own pile. Alternately, the game stops if each players pile reaches fivedollars.(a) Draw the complete extensive form of this game.(b) What is the backward induction outcome of this game?QUESTION 2 (Choosing Pennies Sequentially). Two players, Amy and Beth, playing the followinggame with a jar containing 100 pennies. The players take turns; Amy goes first. Each time it is aplayers turn, she takes between one and 10 pennies out of the jar. The player whose move emptiesthe jar wins.(a) If both players play optimally, who will win the game? Does this game have a first-moveradvantage? Explain your reasoning.(b) What are the optimal strategies (complete plans of action) for each player?(c) Now suppose we change the rules so that the player whose move empties the jar loses. Does thisgame have a first-mover advantage? Explain your reasoning.(d) In the second variant, what are the optimal strategies for each player?QUESTION 3. The members of a hierarchical group of hungry lions face a piece of prey. If lion 1does not eat the prey, the prey escapes and the game ends. If it eats the prey, it becomes fat and slowand lion 2 can eat it. If lion 2 does not eat lion 1, the game ends. If it eats lion 1, then it may beeaten by lion 3, and so on. Each lion prefers to eat than be hungry, but prefers to be hungry than beeaten. Find the backward induction outcomes for any number n of lions(Do not just write down the equilibrium outcome). Explain your answer in detail and clearly.Remarks:(1) Notice that there is a strict hierarchy among the lions: if lion k becomes fat and slow, it can onlybe eaten by lion (k + 1).)(2) You can, for example, specify that eating the prey or another lion gives a payoff 1, the payoff ofstaying hungry is 0, while the payoff of being eaten is -1. You can start with the case where thereare only 3 or 4 lions to gain some intuition. With these two cases, you can then get an idea of thebackward induction outcome for the general case.