Alice and Bob are both well-known perfect logicians who have the ability to internally randomize and often pass their time by playing rock-paper-scissors. One day, tired of the usual RPS game, Alice comes to Bob with a variation: Rock, Paper, Scissors, Anti-Rock. It works similarly to RPS, on the count of three each player shows one of the possible symbols, except this time there are 4 options:
Rock, paper and scissors have the usual relationship: Rock beats scissors, scissors beats paper, paper beats rock, anything against itself is a tie.
Anti-Rock, on the other hand, ties against rock (and itself), beats paper, and loses to scissors.
Upon being presented with this variation, Bob is skeptical: "It seems kind of dumb, and it feels like there'll be way too many ties, but I guess I'll give it a shot."
If they both play logically and they both play each of the four throws at least some of the time, how frequently will Bob and Alice tie in the new version of the game?