Gridworld Value Functions
Value functions organize this information by asking how valuable a situation is when decisions are made optimally.
A State Is More Than a Location
In a gridworld, an agent can be described by the state it occupies, the action it chooses, the resulting transition, and the return that can be expected. Value functions organize this information by asking how valuable a situation is when decisions are made optimally.
The optimal state-value function v*(s) is the maximum expected return from a particular state under an optimal policy.
Reading v* and q*
The key difference is what each function is evaluating. v*(s) evaluates a state by itself: it asks for the maximum expected return starting from that state when behavior is optimal. q*(s,a) evaluates a state-action pair: it asks for the maximum expected return after taking a particular action in that state and then following an optimal policy thereafter.
| Function | Input | Question it answers | Information emphasized |
|---|---|---|---|
| v*(s) | A state | What is the maximum expected return from this state? | The value of the state under optimal behavior |
| q*(s,a) | A state and a particular action | What is the maximum expected return after taking this action and then behaving optimally? | The value of a particular action in a state |
Both functions concern optimal expected return, but q*(s,a) includes a particular action in its input.
Identifying the Function a Question Requires
A problem asks for the maximum expected return from a state without naming a particular action. A second problem asks for the maximum expected return after taking a specified action in that state.
- 1First question
Because the input is a state by itself, interpret the requested quantity as v*(s).
- 2Second question
Because the input includes a state and a particular action, interpret the requested quantity as q*(s,a).
- 3Interpret the continuation
For q*(s,a), remember that the optimal policy is followed after the specified action.
Separate the state-only viewpoint from the state-and-action viewpoint before attempting a calculation.
Following the q* Recursion
The Bellman equation for q* expresses the optimal action-value function recursively. This means that the value of a state-action pair is described using related value information rather than being treated as an isolated quantity. To work with the equation, identify the current state, identify the particular action, and then identify the optimal continuation described by the problem.
- 1Locate the state named in the q* problem.
- 2Locate the particular action paired with that state.
- 3Read the equation supplied by the exercise rather than replacing it with an unstated alternative.
- 4Identify the immediate reward and the related future-value information required by that equation.
- 5Represent the optimal continuation described in the problem.
- 6Evaluate only after the symbolic relationships have been made clear.
Tracing a Gridworld Calculation
A gridworld exercise connects a state, an action, a transition, and a reward. The important skill is not merely identifying a grid cell. It is tracking what happens when an action is selected and then using the provided equation and optimal policy to determine the value associated with the best state.
Planning the Best-State Calculation
A gridworld problem provides an optimal policy and equation (3.2), and asks for the optimal value of the best state.
- 1Read the target
Confirm that the exercise asks for an optimal state value, not for q*(s,a) for one particular action.
- 2Follow the policy
Use the optimal policy to identify the decisions that determine the relevant continuation from the state.
- 3Write symbolically
Use equation (3.2) to express the optimal value before substituting numerical quantities.
- 4Evaluate numerically
Perform the separate numerical evaluation after the symbolic expression is complete. The source exercise states that the value is 24.4 to one decimal place and requests a computation to three decimal places.
The reliable order is policy identification, symbolic expression, and numerical evaluation.
From Policy to Number
Keep symbolic reasoning separate from numerical evaluation. First use the optimal policy and the provided equation to show how the value is assembled. Then substitute the numerical information and calculate the requested precision. This separation makes it easier to detect whether an error came from interpreting the policy, arranging the equation, or performing arithmetic.
Practice Across Three Examples
The golf, recycling robot, and gridworld examples can be approached with the same interpretive discipline: identify the state, identify whether an action is specified, and determine whether the problem asks for a direct value interpretation or a recursive computation. What changes between examples is the description of the states, actions, rewards, and transitions. What stays consistent is the need to keep the three viewpoints separate.
| Problem context | First question to ask | Skill to emphasize |
|---|---|---|
| Golf | What state and action information does the exercise name? | Separate a state-only value from a value tied to a particular action. |
| Recycling robot | Is the exercise asking about a state, a state-action pair, or a continuation? | Track the optimal continuation described by the problem. |
| Gridworld | Does the exercise provide an optimal policy and equation (3.2)? | Build the symbolic expression first, then perform the numerical evaluation. |
The contexts differ, but each problem becomes easier when its requested viewpoint is identified first.
For each prompt, state whether the requested quantity is most naturally interpreted as v*(s), q*(s,a), or a recursive computation involving q*. Then list the information you would need before calculating.
Hints
- Look for whether the prompt names only a state or names a state together with a particular action.
- If the prompt mentions a Bellman equation for q*, identify the current state, action, and optimal continuation.
- If the prompt mentions the gridworld optimal policy and equation (3.2), plan a symbolic step before a numerical step.
Mistakes in Value-Function Reasoning
Common mistake Treating v*(s) and q*(s,a) as interchangeable.
What it looks likeA solution uses q*(s,a) even though the problem asks for the value of a state without specifying an action.
Why it failsThe two functions have different inputs and answer different questions.
Do this insteadCheck whether the problem concerns a state alone or a particular action in that state.
Common mistake Forgetting the optimal continuation in q*(s,a).
What it looks likeA solution considers the named action but ignores what happens after that action.
Why it failsq*(s,a) concerns the return after taking the particular action and following an optimal policy thereafter.
Do this insteadTrace the continuation described by the problem after the specified action.
Common mistake Treating the Bellman equation for q* as a one-step calculation with no recursion.
What it looks likeA solution uses only the immediate reward and leaves out the related future-value information.
Why it failsThe equation expresses q* in terms of related value information.
Do this insteadIdentify both the immediate reward and the optimal continuation represented in the supplied equation.
Common mistake Substituting numbers before writing the symbolic gridworld expression.
What it looks likeA solution jumps directly to a decimal without showing how the policy and equation determine the value.
Why it failsThe gridworld exercise specifically separates symbolic expression from numerical evaluation.
Do this insteadUse the optimal policy and equation (3.2) symbolically first, then calculate.
A problem can mention a gridworld state without asking for v*(s). The decisive clue is the requested quantity: a state-only value, a value after a named action, or a recursive calculation involving q*. The setting alone does not determine which function is required.
Key Takeaways
- 1v*(s) is the maximum expected return from a state under an optimal policy.
- 2q*(s,a) evaluates a particular action in a state and then assumes an optimal policy thereafter.
- 3The Bellman equation for q* is recursive because it defines q* using related value information.
- 4Gridworld symbolic calculations require the optimal policy and the provided equation before numerical evaluation.
- 5In golf, recycling robot, and gridworld exercises, first determine whether the problem concerns a state, an action in a state, or a recursive value computation.
At a glance
- v*(s) describes the maximum expected return available from a particular state when behavior is optimal.
- q*(s,a) adds a particular action to the input and evaluates the return from that action followed by optimal behavior.
- The Bellman equation for q* is recursive, so solving a q* problem requires identifying current and continuation value information.
- For the gridworld exercise, use the optimal policy and equation (3.2) to write a symbolic expression before carrying out the numerical evaluation.
- Classify each exercise by its viewpoint: state value, action value, or recursive computation.