Self interest cumulative subtraction games

2025-10-28

AI Summary

This paper investigates how players’ behavioral choices—“friendly” (cooperative) versus “adversarial” (competitive)—in tie-breaking scenarios affect equilibrium utility in selfish cumulative subtraction games. We propose and formalize two behavioral commitment mechanisms, and within a pure-strategy subgame-perfect equilibrium framework, integrate combinatorial and economic game-theoretic modeling to conduct a rigorous analysis on two-element subtraction sets. We prove that adversarial behavior never improves a player’s equilibrium utility: friendly strategies are always weakly dominant. Moreover, we establish the monotonicity of tie-breaking rule perturbations. This work provides the first systematic characterization of how behavioral predispositions fundamentally shape the equilibrium structure of cumulative games—challenging the conventional intuition that adversarial behavior is advantageous—and furnishes both theoretical grounding and empirical evidence supporting the monotonicity conjecture for larger subtraction sets.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Adversarial Agents
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Extends subtraction games to cumulative self-interest play with PSPE adaptation
• Analyzes tie-breaking monotonicity in two-action subtraction sets for friendly players
• Compares PSPE utilities between antagonistic and friendly players in subtraction games

Innovation

• Adapted PSPE for self-interest cumulative games
• Introduced friendly vs antagonistic tie-breaking rules
• Proved tie-breaking monotonicity for two-action sets