Random Strategies with Memory for the Robin Hood Game

Gasarch, Golub, and Srinivasan studied the Robin Hood game in the case of random, memoryless strategies. In this brief note we extend their main result to the case of bounded memory, and clarify several questions raised by them. 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. (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 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 any r(i) bags out of those which arrived first. Deterministic strategies for this game were studied, e.g., in [2, 3]. Gasarch, Golub and Srinivasan [1] consider the case where Robin is memoryless, and cannot distinguish between the days where the bags were put in the cave. They suggest the following natural 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; see [1] for a formalization of this. Let L(i) = ∑i j=1 s(j)− r(j) denote the number of bags in the cave after night i. The main result in [1] is that Robin wins almost surely if, and only if, the series