Three-player Differential Game Logic

2026-07-24

AI Summary

This work addresses the verification challenge of non-zero-sum, coalition-forming differential games among three players with distinct objectives in hybrid dynamical systems by introducing a novel formalism, three-player differential Game Logic (dGL3). dGL3 establishes the first formal framework capable of modeling non-zero-sum, coalition-enabled hybrid games involving three players, thereby overcoming the restrictive assumptions of traditional zero-sum formulations. The logic features a rigorously defined syntax and semantics, accompanied by a sound and relatively complete proof calculus, enabling precise representation of cooperative and competitive behaviors under shared safety constraints. Case studies demonstrate that dGL3 effectively guarantees safety in multi-agent hybrid systems amidst complex strategic interactions.

Abstract

This paper introduces the three-player differential game logic dGL3, which enables the verification of hybrid games of discrete and differential equation dynamics with three players who may or may not form coalitions. Each player has an individual goal they try to fulfill, so the game becomes non-zero-sum when the goals of the players overlap. This enables dGL3 to verify complex situations involving multiple players, taking into account their coalitional power. dGL3 excels at verifying games where players share safety objectives but otherwise pursue different goals, so that they may or may not collaborate. In this case, zero-sum assumptions lead to overly conservative results by neglecting the potential of coordination amongst the players. In this paper, the syntax and semantics of dGL3 are presented and crucial properties of dGL3 are proved. A sound and relatively complete proof calculus for dGL3 is introduced and the use of dGL3 is illustrated in a canonical example.

Problem

• three-player differential games
• non-zero-sum games
• coalition formation
• hybrid systems verification
• safety objectives

Innovation

• three-player differential games
• non-zero-sum logic
• coalitional power
• hybrid game verification
• proof calculus