  Motivated by earlier results on universal randomized guessing, we consider an
individual-sequence approach to the guessing problem: in this setting, the goal
is to guess a secret, individual (deterministic) vector $x^n=(x_1,\ldots,x_n)$,
by using a finite-state machine that sequentially generates randomized guesses
from a stream of purely random bits. We define the finite-state guessing
exponent as the asymptotic normalized logarithm of the minimum achievable
moment of the number of randomized guesses, generated by any finite-state
machine, until $x^n$ is guessed successfully. We show that the finite-state
guessing exponent of any sequence is intimately related to its finite-state
compressibility (due to Lempel and Ziv), and it is asymptotically achieved by
the decoder of (a certain modified version of) the 1978 Lempel-Ziv data
compression algorithm (a.k.a. the LZ78 algorithm), fed by purely random bits.
The results are also extended to the case where the guessing machine has access
to a side information sequence, $y^n=(y_1,\ldots,y_n)$, which is also an
individual sequence.
