On the Foundations of Dynamic Games and Probability: Decision Making in Stochastic Extensive Form

TL;DR

This paper develops a comprehensive abstract framework for dynamic games under probabilistic uncertainty, integrating decision trees, exogenous information, and stochastic processes to address complex decision-making scenarios over time.

Contribution

It introduces stochastic decision forests and extensive forms, generalizing previous models and providing a foundation for continuous-time stochastic game analysis.

Findings

Characterization of well-posedness via order-theoretic properties

Development of a stochastic process form for extensive games

Application to stochastic differential and timing games

Abstract

In this work, an abstract and general language for the fundamental objects underlying dynamic games under probabilistic uncertainty is developed. Combining the theory of decision trees by Al\'os-Ferrer--Ritzberger (2005) and a Harsanyian notion of exogenous uncertainty, the concept of stochastic decision forests is introduced. Exogenous information is modelled via filtration-like objects providing dynamic updates on the "realised tree", and an abstract decision-theoretic model of adapted choice is formulated. Based on this, a consistent model of "rules" is introduced, leading to the notion of stochastic extensive forms, generalising the works of Al\'os-Ferrer--Ritzberger (2008, 2011). Well-posedness is completely characterised in terms of order-theoretic properties of the underlying forest. Moreover, the language of stochastic extensive forms addresses a vast class of dynamic decision problems formulated in terms of time-indexed paths of action -- a first step towards an approximation theory of continuous-time games based on stochastic processes. In this formulation, a well-posed theory obtains if and only if the time half-axis is essentially well-ordered. Therefore, a relaxed game-theoretic model of "extensive form characteristics" is introduced: the stochastic process form. Its action processes arise from well-posed action path stochastic extensive forms under tilting convergence, which is introduced in order to faithfully describe accumulating reaction behaviour. The problem of instantaneous reaction and information about it is tackled by introducing vertically extended continuous time, for which a suitable stochastic analysis is developed. Stochastic process forms admit a natural notion of information sets, subgames, and equilibrium. The theory applies to stochastic differential and timing games, e.g., addressing open issues in Fudenberg--Tirole (1985) and Riedel--Steg (2017).