On the waiting time in a janken game

Consider a janken game (scissors-paper-rock game) started by n players such that (1) the first round is played by n players, (2) the losers of each ro und (if any) retire from the rest of the game, and (3) the game ends when o nly one player (winner) is left. Let W-n be the number of rounds played thr ough the game. Among other things, it is proved that (2/3)W-n(n) is asympto tically (as n --> infinity) distributed according to the exponential distri bution with mean 1/3, provided that each player chooses one of the three st rategies (scissors, paper, rock) with equal probability and independently f rom other players in any round.