A Markov Decision Process (MDP) is a framework for modeling sequential decision-making under uncertainty. Its key components are states, actions, transition probabilities, reward functions, and a discount factor. These elements work together to define how an agent interacts with an environment, learns from outcomes, and optimizes decisions over time. Understanding each component is essential for implementing reinforcement learning algorithms or dynamic programming solutions.
The first two components are states and actions. States represent the distinct configurations of the environment the agent can be in, such as a robot’s position on a grid or a game’s current board state. Actions are the choices available to the agent in each state, like moving north/south in a grid or placing a mark in a game. For example, in a navigation task, states could be coordinates on a map, and actions might include “move forward” or “turn left.” States and actions define the problem’s structure, ensuring the agent can perceive and interact with its environment meaningfully.
Next, transition probabilities and reward functions determine how the environment responds to actions. Transition probabilities describe the likelihood of moving from one state to another after taking an action. For instance, a robot attempting to move forward might have an 80% chance of succeeding and a 20% chance of sliding sideways due to slippery terrain. The reward function assigns a numerical value to each state-action pair, reflecting immediate outcomes (e.g., +10 for reaching a goal, -1 for each step taken). These components model uncertainty and guide the agent’s learning by quantifying trade-offs between different actions.
Finally, the discount factor (gamma, γ) balances immediate and future rewards. A value between 0 and 1, it reduces the weight of rewards further in the future, encouraging the agent to prioritize near-term gains without ignoring long-term outcomes. For example, a discount factor of 0.9 means a reward received after two steps is valued at 0.81 times its original value. This ensures the agent’s strategy remains practical and avoids infinite reward loops. Together, these components create a mathematically rigorous model for optimizing decisions in dynamic, uncertain environments.