# Decentralized partially observable Markov decision process

The decentralized partially observable Markov decision process (Dec-POMDP)  is a model for coordination and decision-making among multiple agents. It is a probabilistic model that can consider uncertainty in outcomes, sensors and communication (i.e., costly, delayed, noisy or nonexistent communication). it is a generalization of a Markov decision process (MDP) and a partially observable Markov decision process (POMDP) to consider multiple decentralized agents.

## Definition

### Formal definition

A Dec-POMDP is a 7-tuple $(S,\{A_{i}\},T,R,\{\Omega _{i}\},O,\gamma )$ , where

• $S$ is a set of states,
• $A_{i}$ is a set of actions for agent i, with $A=\times _{i}A_{i}$ is the set of joint actions,
• $T$ is a set of conditional transition probabilities between states, $T(s,a,s')=P(s'\mid s,a)$ ,
• $R:S\times A\to \mathbb {R}$ is the reward function.
• $\Omega _{i}$ is a set of observations for agent i, with $\Omega =\times _{i}\Omega _{i}$ is the set of joint observations,
• $O$ is a set of conditional observation probabilities $O(s',a,o)=P(o\mid s',a)$ , and
• $\gamma \in [0,1]$ is the discount factor.

At each time step, each agent takes an action $a_{i}\in A_{i}$ , the state updates based on the transition function $T(s,a,s')$ (using the current state and the joint action), each agent observes an observation based on the observation function $O(s',a,o)$ (using the next state and the joint action) and a reward is generated for the whole team based on the reward function $R(s,a)$ . The goal is to maximize expected cumulative reward over a finite or infinite number of steps. These time steps repeat until some given horizon (called finite horizon) or forever (called infinite horizon). The discount factor $\gamma$ maintains a finite sum in the infinite-horizon case ($\gamma \in [0,1)$ ).