Credit for this problem goes to The Riddler. Thanks!

Rock, Paper, Scissors, Double Scissors

From “The Riddler”:

From Patrick Coate, a new and improved version of the game rock-paper-scissors:

Besides the usual three options that players have — rock, paper or scissors — let’s add a fourth option, double scissors, which is played by making a scissors with two fingers on each side (like a Vulcan salute). Double scissors, being larger and tougher, defeat regular scissors, and just like regular scissors, they cut paper and are smashed by rock. The three traditional options interact just as they do in the standard game.

A rock-paper-scissors-double scissors match is always played best two out of three (or, more precisely, first to win two throws, since there can be an unlimited number of ties). There is just one exception: If your opponent throws paper and you throw regular scissors, you immediately win the match regardless of the score.

What is the optimal strategy at each possible score (0-0, 1-0, 0-1, 1-1)? (You can ignore any ties.) What is the probability of winning the match given a 1-0 lead?