Amy and Ben are playing a 99-round match of Rock-Paper-Scissors. Find Amy’s best

Question

Amy and Ben are playing a 99-round match of Rock-Paper-Scissors. Find Amy’s best response to each of the following Ben strategies. If she has more than one, describe all best responses.

a) Ben: 100% Rock.

(b) Ben: First time Rock, then copy Amy’s last move.

(c) Ben: First time Rock, then play whatever would have beaten Amy’s last move.

(d) Ben: Mixture of 50% Scissors, 50% Paper.

e) Your friend says, “No, Amy’s best response to Ben’s mixture of 50% Scissors, 50% Paper is obviously a mixture of 50% Rock, 50% Scissors.” Tell him why he’s wrong.

(f) Ben: Mixture of 2/3 Scissors, 1/3 Paper.

(g) Ben: Mixture of 1/3 Rock, 1/3 Scissors, 1/3 Paper

Explanation / Answer

a.) Given Ben's strategy, Amy should go for 100% paper

b.) When Ben's strategy is, First time Rock, then copy Amy’s last move.

Amy should use this strategy: 1st-time paper, 2nd-time scissor since ben would be using paper 2nd time, 3rd-time rock since ben would be using paper.

In the 4th turn, Ben would be going for rock, just like the 1st turn Amy should go for paper, and in 5th turn, Amy should go for scissors.

And thus we see a pattern emerging here.

And the best strategy for Amy would be cycling through the ordered set

(Paper, Scissors, Rock) i.e. 1st go for paper, then scissors, then rock, then again paper, then scissors and so on.

c.) When Ben uses the strategy "First time Rock, then play whatever would have beaten Amy’s last move".

Amy should use this strategy: 1st-time paper, 2nd-time rock since ben would be using scissors has that would have beaten Amy's previous move paper, 3rd-time scissors since ben would be using paper.

In the 4th turn, Ben would be going for rock since it would have beaten Amy's previous choice scissors, just like the 1st turn Amy should go for paper, and in 5th turn, Amy should go for rock just like the 2nd turn.

And thus we see a pattern emerging here.

And the best strategy for Amy would be cycling through the ordered set

(Paper, Rock, Scissors) i.e. 1st go for paper, then rock, then Scissors, then again paper, then rock and so on.

d.) When Ben: Mixture of 50% Scissors, 50% Paper. In this case, Amy should go for 100% scissors as that would lead to her winning 50% of the time and rest of the times it would be a draw.

e.)The friend is wrong because if we use the strategy proposed by him, in every turn there is a 50% chance of winning in that turn and 25% chance losing in that turn. Whereas the strategy we proposed we have a 50% chance of winning every turn and 0% chance of losing

f.) when uses Ben: Mixture of 2/3 Scissors, 1/3 Paper as his strategy, in this case, the best strategy would be 100% rock as using it we would be winning 66 of the 99 turns whereas we would lose the rest

e.) When Ben uses Mixture of 1/3 Rock, 1/3 Scissors, 1/3 Paper as his strategy, Amy would be better of using the same strategy as Ben, as this strategy is the one corresponding to mixed strategy Nash equilibrium ( a concept in game theory).

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