Markov Decision Process

What is a Markov Decision Process? Markov Decision Process Explained

A Markov Decision Process (MDP) is a mathematical framework used to model decision-making problems in a stochastic environment. It is an extension of the Markov chain concept and incorporates decision-making and rewards to determine optimal strategies for sequential decision problems. MDPs are widely used in reinforcement learning and operations research. Here are the key components of an MDP:

States: An MDP consists of a set of states that represent the possible configurations or conditions of the system. The system transitions from one state to another based on actions taken by the decision-maker.

Actions: At each state, the decision-maker has a set of possible actions to choose from. Actions represent the decisions or choices made by the decision-maker that influence the state transitions.

Transition probabilities: For each state-action pair, there are associated transition probabilities that indicate the likelihood of moving from the current state to the next state. These transition probabilities capture the stochastic nature of the environment.

Rewards: At each state, the decision-maker receives a reward or a penalty based on the action taken and the resulting state. The rewards can be positive or negative values and reflect the desirability or undesirability of being in a particular state.

Discount factor: MDPs often include a discount factor, denoted by γ (gamma), which determines the importance of future rewards compared to immediate rewards. The discount factor balances the trade-off between short-term rewards and long-term rewards in decision-making.

Policy: A policy in an MDP specifies the decision-maker’s strategy, which determines the action to take at each state. It maps states to actions and can be deterministic or stochastic.

Value functions: Value functions are used to evaluate the quality of different states or state-action pairs in an MDP. The state value function (V(s)) represents the expected total cumulative rewards starting from a particular state and following a specific policy. The action value function (Q(s, a)) represents the expected total cumulative rewards starting from a state, taking a specific action, and following a specific policy.

Optimal policy: The goal in an MDP is to find an optimal policy that maximizes the expected cumulative rewards over time. The optimal policy maximizes the value function and determines the best action to take at each state to achieve the highest rewards.

MDPs provide a formal framework for modeling decision-making problems in uncertain and stochastic environments. They enable the development of optimal strategies for sequential decision problems and are extensively used in reinforcement learning algorithms, such as Q-learning and policy gradient methods. By incorporating rewards and stochastic state transitions, MDPs allow for the analysis and optimization of decision-making processes in a wide range of applications, including robotics, game theory, operations research, and autonomous systems.