What is Markov Decision Process (MDP)?

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A Markov Decision Process (MDP) is a formal mathematical framework used in reinforcement learning and decision-making problems where outcomes are partly random and partly under the control of an agent. It provides a structured way to model environments, actions, rewards, and state transitions.

🔹 Components of an MDP

An MDP is defined by a 4-tuple $(S, A, P, R)$:

1. S (States): The set of all possible states the environment can be in.

2. A (Actions): The set of all actions the agent can take.

3. P (Transition Probability): $P(s'|s, a)$ defines the probability of moving from state $s$ to state $s'$ after taking action $a$.

4. R (Reward Function): $R(s, a, s')$ gives the immediate reward received after transitioning from $s$ to $s'$ via action $a$.

Additionally, MDPs often consider a discount factor $\gamma$ (0 ≤ γ ≤ 1) that balances immediate vs future rewards.

🔹 Key Properties

1. Markov Property:

   • The future state depends only on the current state and action, not on the past history.

   • Formally:

     $$P(s_{t+1} | s_t, a_t, s_{t-1}, a_{t-1}, …) = P(s_{t+1} | s_t, a_t)$$

2. Policy ($\pi$):

   • A mapping from states to actions, guiding the agent’s behavior.

   • The goal is to find an optimal policy $\pi^*$ that maximizes cumulative reward.

3. Value Function (V) and Q-Function (Q):

   • V(s): Expected cumulative reward starting from state $s$.

   • Q(s, a): Expected cumulative reward starting from state $s$ and taking action $a$.

🔹 Example:

• Robot Navigation:

  • States → positions in the maze

  • Actions → move up, down, left, right

  • Transition → moving in a direction may succeed or fail probabilistically

  • Reward → +10 for reaching the goal, -1 per step✅ In short:

An MDP provides a mathematical framework for modeling sequential decision-making under uncertainty, where the agent chooses actions to maximize expected cumulative rewards, obeying the Markov property.

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