# Random Zero-Sum Dynamic Games on Infinite Directed Graphs

## TL;DR

This paper studies random two-player zero-sum games on infinite graphs, showing how the game's value converges as it lasts longer.

## Contribution

The paper introduces new convergence rates for game values on different types of infinite graphs.

## Key findings

- On acyclic graphs with sub-exponential expansion, game values converge exponentially as duration increases.
- On infinite d-ary trees, convergence happens at a double-exponential rate.

## Abstract

We consider random two-player zero-sum dynamic games with perfect information on a class of infinite directed graphs. Starting from a fixed vertex, the players take turns to move a token along the edges of the graph. Every vertex is assigned a payoff known in advance by both players. Every time the token visits a vertex, Player 2 pays Player 1 the corresponding payoff. We consider a distribution over such games by assigning i.i.d. payoffs to the vertices. On the one hand, for acyclic directed graphs of bounded degree and sub-exponential expansion, we show that, when the duration of the game tends to infinity, the value converges almost surely to a constant at an exponential rate dominated in terms of the expansion. On the other hand, for the infinite d-ary tree (that does not fall into the previous class of graphs), we show convergence at a double-exponential rate.