A Stackelberg Game Model of Flocking

TL;DR

This paper analyzes a two-agent Stackelberg flock-formation game where agents arrive at a destination to claim territories, with rewards, travel costs, and predation risk shaping utility. By relaxing the strict flocking requirement to allow arrivals within a window, the authors derive and compare subgame perfect equilibria in continuous and discrete time, revealing a richer set of equilibria and dynamics than prior strict-flocking models. The main contributions are explicit SPE characterizations for both time settings, including a five-type discrete-time taxonomy and a three-type continuous-time taxonomy, plus insights into how stronger-weak differences in territory and agent strength drive strategic timing. The results enhance understanding of cooperative-competition trade-offs in time-dependent group formation and offer groundwork for extending to multi-agent cases and algorithmic SPE computation.

Abstract

We study a Stackelberg game to examine how two agents determine to cooperate while competing with each other. Each selects an arrival time to a destination, the earlier one fetching a higher reward. There is, however, an inherent penalty in arriving too early as well as a risk in traveling alone. This gives rise to the possibility of the agents cooperating by traveling together while competing for the reward. In our prior work [1] we studied this problem as a sequential game among a set of $N$ competing agents in continuous time, and defined the formation of a group traveling together as arriving at exactly the same time. In the present study, we relax this definition to allow arrival times within a small window, and study a 2-agent game in both continuous and discrete time, referred to as the flock formation game. We derive and examine the properties of the subgame perfect equilibrium (SPE) of this game.

Figures (2)

• Figure 1: The values of the four functions determine the type of SPEs in Case (3), based on their relations.
• Figure 2: A simplified illustration of different types of SPEs in continuous-time, discrete-time flock formation game, and SFG. The x-axis is the arrival times of the agents and the y-axis is the difference between the two territories. Depending on the territory differences, there are different cases or games. A solid circle shows a strict flock, while a dashed circle shows a flock but not a strict one.

Theorems & Definitions (11)

• Proposition 1
• Theorem 1
• proof : Proof Sketch
• Corollary 1
• Theorem 2
• proof : Proof Sketch
• Corollary 2
• proof
• Example IV.1
• proof : Proof of Theorem \ref{['thm:SPEcon']}