A Classic Mixed Strategy Game: Rock Paper Scissors
Two paleontologists, Al O. Sorus and Pei Kingman, are whiling away some time by playing the classic childrens' game Rock-Paper-Scissors. The players hold one hand behind their backs. On the count of three they display one of three hand signals: A fist for Rock, an open hand for Paper, and first two fingers extended for Scissors. The winner is determined by the rule - - Rock breaks Scissors, Scissors cut Paper, and Paper covers Rock. If both players play the same hand signal then it is a tie. To make the game interesting the loser pays the winner one dollar. The payoffs are shown in the table.
Neither player has a dominant strategy, nor is any strategy dominated. There are no Nash equilibria in this symmetric game, yet we see kids the world over playing this game.
Suppose Al plays Rock with probability q_{1}, Paper with probability q_{2}, and Scissors with probability 1-q_{1}-q_{2}. What is Al's expected payoff when Pei plays Rock? 0xq1+1xq_{2}+(-1)(1-q_{1}-q_{2}) = 2q_{2}+q_{1}-1. Can you figure out the rest?
Al O. Sorus |
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Rock | Paper | Scissors | q-mix | ||
Pei Kingman | Rock | 0, 0 | -1, 1 | 1, -1 | 2q_{2}+q_{1}-1 |
Paper | 1, -1 | 0, 0 | -1, 1 | 1-2q_{1} - q_{2} | |
Scissors | -1, 1 | 1, -1 | 0, 0 | q_{1} - q_{2} | |
p-mix | 2p_{2}+p_{1}-1 | 1-2p_{1} - p_{2} | p_{1} - p_{2} |
Pei plays his strategies with probabilities p_{1}, p_{2} and 1-p_{1}-p_{2}. His expected payoffs against any of Al's pure strategies are shown in the last row.
If Al plays his mixed strategy then he should choose the probabilities so that Pei is indifferent to the pure strategy he (Pei) plays. Use this fact to determine the probability with which Al plays each strategy. Now use the same reasoning to find the probabilities that Pei will use. What are Al's expected winnings on any play of the game when he uses his mixed strategy?