arXiv subjects

Anjali Bhagat

Publications and source records attributed to Anjali Bhagat.

2 recordsLinked to original sources

arXiv2026

Additive sink subtraction

Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bound on the period length. Flammenkamp~(1997) conjectures a striking classification for three-move subtraction games: non-additive rulesets exhibit linear period lengths of the form ``the sum of two moves'', where the choice of which two moves displays fractal-like behavior, while additive sets $S=\{a,b,a+b\}$ have purely periodic outcomes with linear or quadratic period lengths. Despite early attention in Winning Ways~(1982), the general additive case remains open. We introduce and analyze a dual winning convention, which we call {\sc sink subtraction}. Unlike the standard {\em wall} convention, where moves to negative positions are forbidden, the sink convention declares a player the winner upon moving to a non-positive position. We show that {\sc additive sink subtraction} admits a complete solution: the nim-sequence is purely periodic with an explicit linear or quadratic period formula, and we conjecture a duality between additive sink subtraction and classical wall subtraction. Keywords: Additive Subtraction Game, Nimber, Periodicity, Sink Convention.

math.CO↗

arXiv2026

Tie-breaking in self interest cumulative subtraction games

Subtraction games have a rich literature as normal-play combinatorial games (e.g., Berlekamp, Conway, and Guy, 1982). Recently, the theory has been extended to zero-sum scoring play (Cohensius et al. 2019). Here, we take the approach of cumulative self-interest games, as introduced in a recent framework preprint by Larsson, Meir, and Zick. By adapting standard Pure Subgame Perfect Equilibria (PSPE) from classical game theory, players must declare and commit to acting either ``friendly'' or ``antagonistic'' in case of indifference. Whenever the subtraction set has size two, we establish a tie-breaking rule monotonicity: a friendly player can never benefit by a deterministic deviation to antagonistic play. This type of terminology is new to both ``economic'' and ``combinatorial'' games, but it becomes essential in the self-interest cumulative setting. The main result is an immediate consequence of the tie-breaking rule's monotonicity; in the case of two-action subtraction sets, two antagonistic players are never better off than two friendly players, i.e., their PSPE utilities are never greater. For larger subtraction sets, we conjecture that the main result continues to hold, while tie-breaking monotonicity may fail, and we provide empirical evidence in support of both statements.

math.CO↗