Cumulative Games: Who is the current player?

TL;DR

The paper introduces Cumulative Games to bridge CGT and EGT, providing an n-player general-sum framework that unifies core CGT tools with extensive-form analysis. It defines a three-layer Cumulative Game Form, a layered disjunctive sum, and a generalized outcome function capable of capturing PSPE-like equilibria under self-interest utilities, enabling efficient computation under a Heap Dynamic restriction. The main contributions are two theorems: a generalized outcome framework linking PSPE utilities to CGT-style outcomes, and an efficient dynamic-programming approach for Heap Dynamic games, along with a demonstration that any Extensive Form Game can be represented as a Cumulative Game. The work lays a foundational bridge between CGT and EGT, offering a unified language for comparing and composing games and suggesting future directions on memory, cycles, and more complex multi-player economic settings.

Abstract

Combinatorial Game Theory(CGT)is a branch of Game Theory that has developed largely independently of Economic Game Theory (EGT), and is concerned with deep mathematical properties of two-player zero-sum games recursively defined over various combinatorial structures. The aim of this work is to lay the foundations for bridging the conceptual and technical gaps between CGT and EGT, here interpreted as multiplayer Extensive Form Games, so that they can be treated within a unified framework. More specifically, we introduce a class of $n$-player, general-sum games, called {\sc Cumulative Games}, which can be analyzed using tools from both CGT and EGT. 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 {\sc Cumulative Games}. The outcome function allows for efficient equilibrium computation under certain restrictions, while the disjunctive sum operator lets us define a partial order over games according to the advantage that a given player has. Finally, we show that any Extensive Form Game can be written as a {\sc Cumulative Game}.

Figures (5)

• Figure 1: A game of Nim.
• Figure 2: The picture is an Extensive Form Game tree, and it illustrates the self-interest game Squirrel Pebbles where the squirrels remove 2 or 3 nuts, and where the initial heap size is 7; Alice (A) starts, and the players take turns. Each node shows the heap size, the current cumulation for each player, and the current player. In Figure \ref{['fig:ext_23']} we compute the PSPE of this grounded position.
• Figure 3: The picture is a CGT-type game tree, and it illustrates a heap position of the self-interest game Squirrel Pebbles where the squirrels remove 2 or 3 nuts, and where the initial heap size is 4; each node shows the heap size, and the current cumulation for each player. The left slanting edges represent Alice's options, whereas the right slanting edges represent Bob's options.
• Figure 4: This is a reiteration of Figure \ref{['fig:ext_23-squirrel']}, to exemplify a PSPE analysis: the Extensive Form Game induced by the self-interest symmetric fixed subtraction set $\mathcal{S}=\{2,3\}$Squirrel Pebbles and the grounded position $(7,(0,0),2)$ (where Player 1 starts). In each node we show $x;(C_1,C_2)$, while the current player is shown to the left. The cumulations at the leaf levels are also the utilities of the players. The thick edges mark the PSPE of this game.
• Figure 5: To the left: symmetric Cumulative Subtraction with $S=\{2,3\}$ represented as a DAG played from a heap of size 7. The middle picture displays the zero-sum outcomes, and to the right, the general-sum outcomes. The double edges indicate which child determines the parent's value.

Theorems & Definitions (48)

• Remark 1
• Remark 2: Greedy Play
• Definition 1: Two-player Cumulative Subtraction
• Definition 2: Outcome Symmetric Zero-sum CLMW
• Proposition 1: Symmetric, Zero-sum, CLMW
• proof
• Definition 3: Outcome Symmetric Self-interest Antagonistic
• Example 1
• Definition 4: Outcome Zero-sum
• Definition 5: Outcome Self-interest Antagonistic