Stochastic Extensive Forms in Dynamic Games

• Stochastic extensive forms are a framework that generalizes decision trees by integrating stochastic events and order theory to model dynamic games under uncertainty.
• The structure incorporates filtration mechanisms and adapted choice models to dynamically update information and capture decision-theoretic contingencies.
• By linking discrete extensive forms with continuous-time process forms via vertically extended time and tilting convergence, the approach rigorously analyzes timing and preemption in games.

Stochastic extensive forms provide a rigorous, order-theoretic language for modeling dynamic games and decision problems under probabilistic uncertainty. This framework generalizes classical extensive-form representations (based on decision trees and refined partitions) by introducing stochastic decision forests indexed by exogenous scenarios and equipping the structure with filtration-like objects that dynamically represent the evolution of information. The approach establishes necessary and sufficient order-theoretic conditions for well-posedness, proposes a model of adapted choice to capture decision-theoretic contingencies under stochastic information, and defines a relaxed game-theoretic process form suitable for continuous-time analysis and approximation of stochastic differential games, enabling the precise treatment of timing and instantaneous reaction phenomena.

1. Stochastic Decision Forests: Generalization of Extensive Forms

A stochastic decision forest is defined as a triple (F, π, 𝒳), where F is a decision forest on a set of outcomes W, π: F → Ω is a surjection onto an exogenous scenario space (Ω, 𝔼), and 𝒳 is a family of random moves (sections) mapping measurable sets D_ξ⊆Ω to nonterminal nodes X of F such that for each ω∈D_ξ, ξ(ω) is a move in the tree T_ω = π⁻¹({ω}) (Rapsch, 26 Nov 2024). Maximal chains in (F, ⊇) are in bijection with outcomes W and encode histories, while the duality f(v)={x ∈ F : v ∈ x} ensures an outcome v is associated with its unique decision path.

This construction subsumes the deterministic extensive form (single decision tree) and generalizes it via realized stochastic events: a lottery on Ω selects the tree T_ω, thereby determining which possible future unfolds. This representation allows a decision-theoretic account of stochastic dynamic games, decision problems, and timing games under general forms of uncertainty, including continuous-time evolutions of noise.

2. Modeling Exogenous Uncertainty: Filtration Structures

Exogenous uncertainty is represented by the scenario space (Ω, 𝔼), where 𝔼 is a σ-algebra of measurable events affecting the outcome. Each random move ξ: D_ξ → X carries an attached sub–σ-algebra 𝔽ξ⊆𝔼, governing the information revealed at that move. Thus, endogenous structure (order of F) and exogenous information (π, 𝔽ξ) are woven together, and as the realized tree is traversed, the agents receive dynamically updated information sets, reflecting stochastic evolution (Rapsch, 26 Nov 2024).

This language captures the continuous revelation of information in economic games (e.g., type revelation in Bayesian games) and stochastic control problems, paving the way for a process-oriented perspective where random moves interface with filtration updates as in Harsanyi-type models.

3. Adapted Choice Model and Rules

Choices are understood not as primitive moves but as contingent plans c⊆W, characterized order-theoretically via immediate predecessor sets P(c) = {x ∈ F : ∃y s.t. x is the immediate predecessor of y and y⊆c}, indicating where c is available (Rapsch, 26 Nov 2024). The adaptedness of choices demands that measurability of their availability be preserved under the filtration structure at each random move: for reference choices C and information structures 𝓕 = (𝔽ξ)ξ∈𝒳, a choice c is F–C–adapted if for all ξ and c'∈C, the set ξ⁻¹(P(c∩c'))⊆D_ξ is measurable in 𝔽_ξ.

This abstraction ensures that plans adjust to both endogenous histories and exogenous signals, allowing for consistent modeling of agent behavior in games with imperfect or dynamically evolving information.

4. Order-Theoretic Conditions and Well-Posedness

Well-posedness of a stochastic extensive form—the property that every strategy profile induces a unique outcome—is completely characterized by the forest's order-theoretic properties [(Rapsch, 26 Nov 2024); os-Ferrer--Ritzberger (2008, 2011)]:

• Up-discreteness: Every nonempty upward-directed chain has a least element, ensuring that moves refine the outcome space in well-ordered fashion.
• Coherence: Every history without a minimum has at least one continuation with a maximum, ensuring extension of partial histories.
• Regularity: The infimum (limit) of any history is unique.

The main theorem asserts that a stochastic extensive form is well-posed exactly when each scenario-wise classical extensive form is well-posed, i.e., these properties are satisfied. This result clarifies that order-theoretic structure—rather than probability per se—determines the clarity of strategic contingency in stochastic games (Rapsch, 26 Nov 2024).

5. Stochastic Process Forms and Continuous-Time Analysis

The stochastic process form (SPF) is introduced to treat dynamic games whose outcomes are stochastic processes indexed by continuous or vertically extended time (Rapsch, 26 Nov 2024). Vertically extended time Ṫ = [0,∞)×α, where α is a well-order, permits separation of instantaneous reactions; lexicographic order on Ṫ ensures that “reaction order” is faithfully encoded even when multiple actions coincide in real time.

This extension allows the development of tilting convergence: discrete-time grids with lags converge order-theoretically to outcome processes on Ṫ, preserving dynamic information about instantaneous reactions. SPF thus bridges discrete extensive forms and continuous-time dynamic games (e.g., stochastic differentials, timing/preemption games), admitting natural notions of strategies, subgames, information sets, and (subgame-perfect or Bayesian) equilibrium as limits of discrete approximations [(Rapsch, 26 Nov 2024); Fudenberg–Tirole (1985); Riedel–Steg (2017)].

6. Applications: Differential and Timing Games, Bayesian Analysis

Stochastic extensive forms and process forms are suited to a wide class of applications:

• Stochastic differential games: State processes such as χₜ = V(ξₜ, χₜ)ηₜ are interpreted as outcomes of action path extensive forms, allowing game-theoretic analysis of equilibria under stochastic evolution.
• Timing and preemption games: Problems involving stopping times (e.g., Dynkin games) are rigorously modeled via vertically extended time so that preemption equilibria arise as limits of well–posed stochastic extensive forms [Fudenberg–Tirole (1985), Riedel–Steg (2017)].
• Bayesian games and type spaces: The scenario space Ω represents types; the approach generalizes Harsanyi’s model, allowing clean separation of exogenous and endogenous uncertainties.

7. Continuous-Time Games: Vertically Extended Time and Tilting Convergence

Addressing the issue of instantaneous reaction in continuous-time games—where standard models “collapse” concurrent actions—the vertically extended time axis T̃=[0,∞)×α is proposed (Rapsch, 26 Nov 2024). The tilting convergence framework formalizes the limit of discrete approximations, distinguishing actions not only by time but by reaction order via the vertical coordinate. Stochastic analysis is then developed on T̃: measurable functions, random times, and feedback controls conform to this extended order, ensuring well-posedness and accurate modeling of strategic situations where timing is decisive.

Summary

Stochastic extensive forms unify refined partitions, decision trees, and filtrations, providing an abstract, general language for dynamic games under probabilistic uncertainty. Stochastic decision forests indexed by exogenous scenarios, with adapted choices and precise order-theoretic characterizations guarantee well-posedness and unique outcome induction. Vertically extended time and tilting convergence allow continuous-time games to be approximated and analyzed rigorously, facilitating transparent definitions of subgames, information sets, and equilibrium in a wide spectrum of stochastic process-driven strategic contexts. This framework reconciles classical extensive-form reasoning with stochastic process theory, resolving longstanding issues in timing, information, and adaptation in dynamic games.