HMMT 2016 U4

Consider a three-person game involving the following three types of fair six-sided dice.

• Dice of type $A$ have faces labelled $2,2,4,4,9,9$.
• Dice of type $B$ have faces labelled $1,1,6,6,8,8$.
• Dice of type $C$ have faces labelled $3,3,5,5,7,7$.

All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player?