Self interest cumulative subtraction games

TL;DR

This work introduces self-interest cumulative subtraction games, where two players accumulate their own removals from a shared heap under deterministic tie-breaking. The authors prove a key monotonicity result: for two-action subtraction sets $S={s_2,s_1}$ with $s_2<s_1$, if both players use friendly tie-breaking in indifference, each player's PSPE payoff is never worse than under antagonistic tie-breaking. They develop a recursive framework for four outcome regimes and establish the main two-action theorem via lemmas about unilateral deviation and greedy/sacrificial strategies, while also showing that discrepancies arise in larger sets and offering empirical evidence and conjectures. The paper surveys broad connections to tie-breaking across domains—voting, fair division, tournaments, auctions, mechanism design, and more—emphasizing that seemingly local choices to resolve indifference can substantially impact global strategic outcomes. The results provide both a foundational monotonicity property in two-action subtraction games and a roadmap for exploring richer behavior in multi-action settings and related scoring-play models.

Abstract

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.

Figures (3)

• Figure 1: Points $(s_2, s_1)$ where there exists a heap with a friendly/antagonistic discrepancy for $S = \{s_2, s_1\}$, $s_2 < s_1$. Colors indicate similarly-sized heaps where the outcomes differ. An online version can be found https://www.desmos.com/calculator/ynbxlvaatq.
• Figure 2: Points $(s_3, s_2, s_1)$ where there exists a heap with a ${\rm FvF}$-${\rm AvA}$ discrepancy for $S = \{s_3, s_2, s_1\}$, with $s_3<s_2<s_1$. Color indicates the value of $s_3$ for readability. An online rotatable version can be found https://www.desmos.com/3d/pqirkkmvjq.
• Figure 3: Points $(s_3, s_2, s_1)$ where there exists a heap that causes a discrepancy between the ${\rm AvA}$ and zero-sum games for $S = \{s_3, s_2, s_1\}$. Color indicates the value of $s_3$ for readability. All cases where $s_3 \not< s_2 \not< s_1$ are ignored. An online rotatable version can be found https://www.desmos.com/3d/xbazasvgxs.

Theorems & Definitions (23)

• Definition 1: Outcomes
• Theorem 2: Main Theorem
• Conjecture 3
• Lemma 4: First Player Advantage
• proof
• Lemma 5: Dominance and Greedy
• proof
• Lemma 6: Last Move and Sacrifice
• proof
• Lemma 7: Tie-breaking Monotonicity, FvF