On a stochastic game with one-chance recovery

Summary

Bernoulli trials with success ratep are considered. Peter, who is a gambler of success ratep, gets 1 unit if the first trial results in success and loses the same unit otherwise. For thekth trial (k≧2), he gets or loses 1 according as success or failure unless his previous gainS _k−1 is negative. WhenS _k−1 is minus, Peter gets or loses −S _k−1 . Then Peter's gainS _n inn trials is the sum of “dependent” random variables. Therefore, Peter has always the chancep of recovering his minus gain instantaneously.

The probability function ofS _n is given and the expected gain is compared with the ordinary (non-symmetric) random walk situation. It will be concluded that Peter should not play the game with one-chance recovery because whenp is less than 1/2, he must be afraid of suffering a bigger risk than the usual case.