Random Strategies with Historical Memory for the Robin Hood Game

The Robin Hood game is played as follows: On day i, the Sheriff puts s(i) bags of gold in the cave. On night i, Robin removes r(i) bags from the cave. The game is played for each i ∈ N. Robin wins if each bag which was put in the cave is eventually removed from it; otherwise the Sheriff wins. Gasarch, Golub, and Srinivasan studied the Robin Hood game in the case of random strategies where Robin has no historical memory. We extend their main result to the case of bounded historical memory, and obtain a hierarchy of provably distinct games. 1. The Robin Hood game The Robin Hood game RH(r, s, A) is defined for functions r, s : N → N such that 1 ≤ r(i) < s(i) for each i, and for a set A, as follows: (1) On day i, the Sheriff (of Nottingham) puts s(i) bags of gold in the cave, each labelled by an element of A. No label is used twice (over the course of the entire game). (2) On night i, Robin (Hood) removes r(i) bags from the cave. The game is played for each i ∈ N. Robin wins if each bag which was put in the cave is eventually removed from it; otherwise the Sheriff wins. It is easy to see that if Robin has an unlimited historical memory (knowing at each night i which of the bags in the cave appeared first), then he has a winning strategy: On night i pick r(i) bags out of those which arrived first. Deterministic strategies for this game were studied, from the settheoretic point of view, in [2, 3]. Gasarch, Golub and Srinivasan [1] consider the case where Robin has no historical memory, that is, he cannot distinguish between the days where the bags were put in the cave. They suggest the following probabilistic strategy for Robin: On night i, remove random r(i) bags out of the cave (with uniform distribution). They say that Robin wins almost surely if for each bag put in the cave, its probability of being eventually removed is 1. The probability is taken over Robin’s coin tosses. More precisely, the probability 1