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.

翻译：减法博弈作为正常玩法组合博弈已有丰富文献记载（例如 Berlekamp、Conway 和 Guy，1982）。最近，该理论已扩展到零和计分玩法（Cohensius 等人，2019）。本文采用 Larsson、Meir 和 Zick 在近期框架预印本中提出的累积自利博弈方法。通过改编经典博弈论中的标准纯子博弈完美均衡（PSPE），玩家在无差异情况下必须声明并承诺采取“友好”或“对抗”行为。当减法集大小为二时，我们建立了平局决胜规则的单调性：友好玩家绝不可能通过确定性偏离到对抗玩法而获益。这类术语对“经济”博弈和“组合”博弈而言都是新的，但在自利累积设定中变得至关重要。主要结果是平局决胜规则单调性的直接推论；在双动作减法集情形下，两个对抗玩家绝不会比两个友好玩家获得更好结果，即其 PSPE 效用值绝不会更高。对于更大规模的减法集，我们推测主要结论仍然成立，而平局决胜单调性可能失效，并为此提供了支持这两项陈述的经验证据。