# Problem 481: Chef Showdown
A group of chefs (numbered \#1, \#2, etc) participate in a turn-based
strategic cooking competition. On each chef's turn, he/she cooks up a
dish to the best of his/her ability and gives it to a separate panel of
judges for taste-testing. Let S(k) represent chef \#k's skill level
(which is publicly known). More specifically, S(k) is the probability
that chef \#k's dish will be assessed favorably by the judges (on
any/all turns). If the dish receives a favorable rating, then the chef
must choose one other chef to be eliminated from the competition. The
last chef remaining in the competition is the winner. The game always
begins with chef \#1, with the turn order iterating sequentially over
the rest of the chefs still in play. Then the cycle repeats from the
lowest-numbered chef. All chefs aim to optimize their chances of winning
within the rules as stated, assuming that the other chefs behave in the
same manner. In the event that a chef has more than one equally-optimal
elimination choice, assume that the chosen chef is always the one with
the next-closest turn. Define Wn(k) as the probability that chef \#k
wins in a competition with n chefs. If we have S(1) = 0.25, S(2) = 0.5,
and S(3) = 1, then W3(1) = 0.29375. Going forward, we assign S(k) =
Fk/Fn+1 over all 1 ≤ k ≤ n, where Fk is a Fibonacci number: Fk = Fk-1 +
Fk-2 with base cases F1 = F2 = 1. Then, for example, when considering a
competition with n = 7 chefs, we have W7(1) = 0.08965042, W7(2) =
0.20775702, W7(3) = 0.15291406, W7(4) = 0.14554098, W7(5) = 0.15905291,
W7(6) = 0.10261412, and W7(7) = 0.14247050, rounded to 8 decimal places
each. Let E(n) represent the expected number of dishes cooked in a
competition with n chefs. For instance, E(7) = 42.28176050. Find E(14)
rounded to 8 decimal places.