On the unimportance of distant players in sparse stochastic differential network games

TL;DR

This paper demonstrates that in large sparse stochastic differential network games, a player's optimal strategy can be effectively determined by considering only nearby players, ignoring distant ones.

Contribution

It provides explicit non-asymptotic bounds showing the unimportance of distant players in large sparse network games, applicable to both open-loop and distributed strategies.

Findings

Optimal trajectories depend only on local neighborhood in the graph.

Explicit bounds are provided independent of the time horizon.

Results hold under convexity and monotonicity assumptions.

Abstract

We study stochastic differential games with $N$ players, where interactions are determined by sequences of graphs in which the number of neighbours of each node remains bounded as $N$ grows, such as chain graphs or lattices. Our main goal is to quantify the phenomenon of the "unimportance of distant players" in such a large population, sparse regime: we show that, in order to determine the optimal trajectory in open-loop strategies of a given player with an arbitrarily small error, it suffices to consider a reduced game involving only the players at a certain distance in the graph, assigning arbitrary trajectories to the farther ones. Our main result provides an explicit non-asymptotic estimate in terms of the graph distance, valid independently of the time horizon $T$, under suitable convexity and monotonicity assumptions on the costs. Similar results are obtained for games in distributed strategies.