Many of you probably played the game “Rock, Paper, Scissors” as a child. Consider the following variation of that game. Instead of two players, suppose three players play this game, and let us call these players A, B, and C. Each player selects one of these three items—Rock, Paper, or Scissors—independent of each other. Player A will win the game if all three players select the same item, for example, rock. Player B will win the game if exactly two of the three players select the same item and the third player selects a different item. Player C will win the game if every player selects a different item. If Player B wins the game, he or she will be paid $1. If Player C wins the game, he or she will be paid $3. Assuming that the expected winnings should be the same for each player to make this a fair game, how much should Player A be paid if he or she wins the game?
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