Decision making in stochastic extensive form II: Stochastic extensive forms and games

TL;DR

This paper develops a comprehensive theory of stochastic extensive forms that integrates decision trees, information flow, and probability, enabling modeling of continuous-time stochastic processes and differential games.

Contribution

It introduces a unified framework for stochastic extensive forms, addressing limitations in modeling continuous-time processes and providing a foundation for differential game approximation.

Findings

Framework bridges decision trees and filtrations.

Models continuous-time stochastic processes like Brownian motion.

Lays groundwork for stochastic differential game approximation.

Abstract

A general theory of stochastic extensive forms is developed to bridge two concepts of information flow: decision trees and refined partitions on the one side, filtrations from probability theory on the other. Instead of the traditional "nature" agent, this framework uses a single lottery draw to select a tree of a given decision forest. Each "personal" agent receives dynamic updates from an own oracle on the lottery outcome and makes partition-refining choices adapted to this information. This theory addresses a key limitation of existing approaches in extensive form theory, which struggle to model continuous-time stochastic processes, such as Brownian motion, as outcomes of "nature" decision making. Additionally, a class of stochastic extensive forms based on time-indexed action paths is constructed, encompassing a wide range of models from the literature and laying the groundwork for an approximation theory for stochastic differential games in extensive form.