Optimal strategies for a generalized “scissors, paper, and stone” game

In the game of Scissors, Paper, and Stone, two players together chant "one, two, three" (see Figure 1). On the count of three, they independently select either "Scissors" (shown by a "V" formed with the index and middle fingers), "Paper" (shown by extending all fingers) or "Stone" (shown by a clenched fist). If both players pick the same object, the game is tied. Otherwise, a player picking Scissors beats a player picking Paper (Scissors "cut" Paper), but loses to a player picking Stone (Stone "smashes" Scissors). A player picking Paper beats a player picking Stone (Paper "smothers" Stone). Williams [5] gives two similar games "for older children" (see Figure 2) which use five objects instead of three.'