In the previous chapter, we established the fundamental interaction loop in Reinforcement Learning: an agent observes a state, takes an action, receives a reward, and transitions to a new state. This cycle repeats, and the agent's goal is typically to maximize the total reward accumulated over time.

However, many problems we want to solve using RL involve sequences of decisions where the choices made have downstream effects. Think about teaching a robot to navigate a building, training an algorithm to play Go, or optimizing inventory management in a supply chain. These problems share several characteristics:

Sequential Nature: Decisions are not made in isolation. An action taken now directly influences the situations (states) the agent encounters next, potentially opening up or closing off future opportunities. Turning left at an intersection means you won't immediately see what's down the right path.
Delayed Consequences: The impact of an action might not be fully realized immediately. A short-term gain could lead to a long-term loss, and vice versa. Sacrificing a piece in chess might grant a positional advantage that only pays off many moves later. Conversely, maximizing immediate profit might deplete a resource needed for future growth.
Stochasticity: The environment's response to an action might involve randomness. Requesting a web resource might succeed or fail due to network conditions. A robot's command to move forward might result in slightly different actual movements due to wheel slippage. An opponent in a game introduces uncertainty through their actions.

To develop agents that can navigate these complexities effectively, we need more than just the basic agent-environment loop concept. We require a formal way to describe the problem itself, capturing the states the agent can be in, the actions it can take, how the state changes in response to actions (the environment's dynamics), and the rewards received along the way. This formal description allows us to reason rigorously about the problem and develop algorithms capable of finding optimal strategies that balance immediate rewards with long-term objectives.

Consider a simplified representation of navigating between rooms:

An agent needs to reach Room D from Room A. Moving North from A usually leads to B, but sometimes (with probability 0.2) leads to a Trap (Room C). Actions have probabilistic outcomes and lead to different subsequent states and potential rewards.

In this scenario, the agent's location represents the state. Available actions depend on the state (e.g., 'Go North', 'Go East'). The environment's dynamics are captured by the transition probabilities (like the 80%/20% split when moving North from Room A). Reaching the Goal yields a positive reward, while entering the Trap might give a negative reward.

This necessity to formally structure problems involving sequential decision-making under uncertainty leads us directly to Markov Decision Processes (MDPs). MDPs provide the standard mathematical framework used throughout Reinforcement Learning. They offer a precise way to define the components of the environment and the interaction, enabling the development and analysis of learning algorithms. In the sections that follow, we will break down the formal definition of an MDP and its core components: states, actions, transition probabilities, rewards, and the discount factor. Understanding MDPs is fundamental to understanding how RL agents learn optimal behavior in complex, dynamic environments.