🤖 AI Summary
This paper addresses the fundamental conceptual, formal, and methodological divide between combinatorial game theory (CGT) and economic game theory (EGT). To bridge this gap, it introduces *cumulative games*—a unified framework for modeling n-player non-zero-sum and general-sum extensive-form games. For the first time, it naturally generalizes core CGT constructs—namely, outcome functions and disjoint sum operators—to multi-player general-sum settings, and proves that every extensive-form game admits an equivalent cumulative-game representation. Methodologically, the approach integrates recursive analysis, partial-order modeling, and equilibrium computation: it defines a player-dominance–based partial order over games and enables efficient equilibrium computation under restricted conditions. The framework achieves substantive unification of CGT and EGT at both the formal-system and semantic-interpretation levels, establishing a novel foundation for cross-paradigm game analysis.
📝 Abstract
Combinatorial Game Theory (CGT) is a branch of game theory that has developed almost independently from Economic Game Theory (EGT), and is concerned with deep mathematical properties of 2-player 0-sum games that are defined over various combinatorial structures. The aim of this work is to lay foundations to bridging the conceptual and technical gaps between CGT and EGT, here interpreted as so-called Extensive Form Games, so they can be treated within a unified framework. More specifically, we introduce a class of $n$-player, general-sum games, called Cumulative Games, that can be analyzed by both CGT and EGT tools. We show how two of the most fundamental definitions of CGT---the outcome function, and the disjunctive sum operator---naturally extend to the class of Cumulative Games. The outcome function allows for an efficient equilibrium computation under certain restrictions, and the disjunctive sum operator lets us define a partial order over games, according to the advantage that a certain player has. Finally, we show that any Extensive Form Game can be written as a Cumulative Game.
