# Partially Observable Stochastic Game **Domain:** Game Theory / Multi-Agent Systems / Decision Science **Doc Type:** Canonical Concept Node **Maturity:** Developed **Collection:** [[collections/Gamification|Gamification Collection]] ## Definition A **partially observable stochastic game** is a formal model in which multiple agents choose actions over time while the underlying state is not fully visible, actions and chance influence state transitions, and each agent receives observations and rewards according to its position in the system. It extends a partially observable Markov decision process from one decision-maker to strategically interacting agents. A model can be described through: - a set of agents; - possible world states; - actions available to each agent; - observations that reveal only part of the state; - a transition function or probability distribution; - reward functions that may differ across agents; and - policies through which agents choose actions from their histories or beliefs. ## Why it matters here This is the most precise formal bridge between [[wiki/Game Theory|game theory]] and the collection's many-games-at-once thesis. In a networked mystery, platform, security encounter, or political movement, participants may not share a board, objective, or information set. A puzzle solver sees clues; a platform sees engagement telemetry; a recruiter sees prospects; an investigator sees a social graph; an automated system sees features and scores. The same action can therefore update several games simultaneously. The model disciplines analysis without proving hidden coordination. It asks: - What state is actually changing? - What can each participant observe? - Which transitions are controlled and which are stochastic? - Who receives which reward or penalty? - How do beliefs update after ambiguous signals? - Can a participant leave, or does exit itself change status? These questions help distinguish a designed system from an emergent strategic environment. They also reveal why a participant's interpretation can become part of the state: publishing a solution, signaling allegiance, or reacting to a provocation changes what other agents can observe and do. ## Boundary The term is a mathematical abstraction. Applying it to society does not mean people are perfectly rational, rewards are measurable, states are known to an analyst, or a single planner specified the environment. Its value is comparative: it makes partial visibility, sequential action, heterogeneous objectives, and feedback explicit. ## Primary articles [[articles/gamification/The Need for Gamification Literacy|The Need for Gamification Literacy]] uses this family of models to explain unequal roles and information. [[articles/gamification/This Is a Game Planet|This Is a Game Planet]] scales the insight from cybersecurity to digital twins and planetary systems. ## Sources - Lloyd S. Shapley, [Stochastic Games](https://doi.org/10.1073/pnas.39.10.1095), *Proceedings of the National Academy of Sciences*, 1953. - Eric A. Hansen, Daniel S. Bernstein, and Shlomo Zilberstein, [Dynamic Programming for Partially Observable Stochastic Games](https://doi.org/10.1609/aaai.v18i1.20498), 2004. ## Routes [[wiki/Game Theory|Game Theory]] · [[wiki/Gamification|Gamification]] · [[wiki/John Harsanyi|John Harsanyi]] · [[wiki/Tiering|Tiering]] · [[wiki/Information Asymmetry|Information Asymmetry]] · [[wiki/State Estimation|State Estimation]] · [[wiki/Feedback Loops|Feedback Loops]] · [[wiki/Objective Function|Objective Function]]