bibkey: bhagatkulkarnilarssonmurali2026tiebreaking
authors: Anjali Bhagat; Tanmay Kulkarni; Urban Larsson; Divya Murali
year: 2026
title: Tie-breaking in self interest cumulative subtraction games
doi: null
url: https://arxiv.org/abs/2510.24280v2
claim: Conjecture 4 predicts that friendly tie-breaking for both players is never worse for either player than antagonistic tie-breaking for both.
strata_touched:
  - D5/S0/Certificates/Games/BhagatKulkarniLarssonMuraliConjectureFourRefutation
license: citation-only
triage: anchor

Tie-breaking in self interest cumulative subtraction games

Conjecture 4 (printed page 5) states:

Consider any subtraction set S, and suppose both players act friendly in case of indifference. Then each player’s PSPE utility is never worse than if both players have antagonistic tie-breaking rules.

The paper defines order on pairs coordinatewise. Definition 2 gives the terminal zero outcome and, at every nonterminal heap, maximization of the mover’s total with friendly or antagonistic tie-breaking on the opponent’s total. The dual convention exchanges player roles after a move. Theorem 3 proves the prediction when the subtraction set has two elements; the paragraph after it says no method is known to pass directly from FvF to AvA for arbitrary sets. Section 6 reports no positive FvF-to-AvA discrepancy in random sets with maximum step at most 25. This note cites the conjecture, not its truth.

Verified locator

• https://arxiv.org/abs/2510.24280v2
• Definition 2: printed pages 4-5; Conjecture 4 and pair order: printed page 5; numerical-support paragraph: section 6, printed page 12.
• Version 2 posted 20 January 2026; no journal reference or DOI is listed.

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