  We analyze the dynamics of the Sisyphus random walk model, a discrete Markov
chain in which the walkers may randomly return to their initial position $x_0$.
In particular, we present a remarkably compact derivation of the time-dependent
survival probability function $S(t;x_0)$ which characterizes the random walkers
in the presence of an absorbing trap at the origin. The survival probabilities
are expressed in a compact mathematical form in terms of the $x_0$-generalized
Fibonacci-like numbers $G^{(x_0)}_t$. Interestingly, it is proved that, as
opposed to the standard random walk model in which the survival probabilities
depend linearly on the initial distance $x_0$ of the walkers from the trap and
decay asymptotically as an inverse power of the time, in the Sisyphus random
walk model the asymptotic survival probabilities decay exponentially in time
and are characterized by a non-trivial (non-linear) dependence on the initial
gap $x_0$ from the absorbing trap. We use the analytically derived results in
order to analyze the underlying dynamics of the `survival-game', a highly risky
investment strategy in which non-absorbed agents receive a reward $P(t)$ which
gradually increases in time.
