Inform friends, do not inform enemies

A random walk that carries information starts at 0 and moves inside the set of integers ℤ. When the random walk reaches a new point this point obtains the information. A given set Vf containing 0 is considered to be the set of friendly stations and its complement Vh=ℤ\Vf is considered to be the set of hostile stations. An observer who knows the partition of ℤ into Vf and Vh obtains a signal only if a new station is informed. There are two kinds of signal: from friendly stations and from hostile ones. This is the whole knowledge about the random walk the observer possesses. Thus, the observer's time is measured by the number of informed stations. The observer can stop the walk at a newly informed point and wins n if there have been n friendly stations informed and no hostile one or wins 0 if at least one hostile station has been informed. We consider two models one with Vf={−m+1,−m+2,…, m−2,m−1} and one with Vf={0,1,2,…} for which we find the observer's optimal stopping time maximizing the expected value of the win.