Abstract
We study a variation of Nim-type subtraction games, called Cumulative Subtraction (CS). Two players alternate in removing pebbles out of a joint pile, and their actions add or remove points to a common score. We prove that the zero-sum outcome in optimal play of a CS with a finite number of possible actions is eventually periodic, with period $2s$, where $s$ is the size of the largest available action. This settles a conjecture by Stewart in his Ph.D. thesis (2011). Specifically, we find a quadratic bound, in the size of $s$, on when the outcome function must have become periodic. In case of exactly two possible actions, we give an explicit description of optimal play.
Highlights
Two players, Alice and Bob, stand next to a single pile of 7 pebbles, alternately taking pebbles from it
In this paper we study generalizations of this game, called cumulative subtraction (CS)
Note that increasing the starting score by p points will increase the outcome by p, but it will not change the optimal sequence of actions
Summary
Alice and Bob, stand next to a single pile of 7 pebbles, alternately taking pebbles from it. Uniqueness is a convenient tool in proofs of optimal play, as is further explained via Lemma 11).. Uniqueness is a convenient tool in proofs of optimal play, as is further explained via Lemma 11).1 To this purpose we define the opt-function. Given a game S, the optimal action, opt : Z min S → S, is a mapping from the set of non-terminal positions to the maximum action s, such that o(x) = s − o(x − s). Note that increasing the starting score by p points will increase the outcome by p, but it will not change the optimal sequence of actions. Is the greedy action optimal in every position of CS game? Since Positive plays at least the same number of actions as Negative plays, by playing greedily she guarantees a result of at least 0. Since Positive plays at most one more action than Negative, if Negative plays greedily he guarantees a result of at most max S
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