Tie-breaking in self interest cumulative subtraction games

TL;DR

This paper introduces a novel tie-breaking rule for self-interest cumulative subtraction games, demonstrating that with two actions, friendly players cannot be worse off than antagonistic ones, and explores conjectures for larger sets.

Contribution

It adapts classical game theory equilibrium concepts to cumulative subtraction games and establishes a new monotonic tie-breaking rule for two-action sets.

Findings

Tie-breaking rule is monotonic for two-action sets.

Two antagonistic players are never better off than friendly players.

Empirical evidence supports conjectures for larger subtraction sets.

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.