(c) Explain i U5. Two players, Amy and Beth, play the following game with a jar
ID: 1135942 • Letter: #
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(c) Explain i U5. Two players, Amy and Beth, play the following game with a jar containing 100 pennies. The players take turns; Amy goes first. Each time it is a players turn, she takes between one and 10 pennies out of the jar. The pläyer whose move empties the jar wins. (a) If both players play optimally, who will win the game? Does this game have a first-mover advantage? Explain your reasoning. (b) What are the optimal strategies (complete plans of action) for each player? (e) Now suppose we change the rules so that the player whose move emp- ties the jar loses. Does this game have a first-mover advantage? Explain your reasoning (d) In this second variant, what are the optimal strategies for each player U6. Now Amy and Beth play a game with two jars, each containing 100 pennles The players take turnsExplanation / Answer
a) When both players play optimally, then Amy will be a winner and yes,this game does have a first mover advantage.Beth can pick upto 10 pennies but if she picks leeser then Amy will take as many as she wants to be able to pick up the 88th penny and by doing so,she will ensure that 11 pennies are left and since one can only take 1-10,Amy will be the one to pick up the last penny and win.
b) Amy would leave Beth to pick up from 11 pennies which is the optimal strategy.Amy will remove pennies in multiples of 11 and if there are 10 or less than 10 pennies in the jar,Amy will pick up all to empty the jar.
c) The game will now have a second mover advantage as the person who removes the 11th penny from the jar can control the game and first move will be at disadvantage,
d) The optimal strategy will be leave the opponent with 12 pennies in their next to last move.
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