Codes and demo are available. This article explores what are states, actions and rewards in reinforcement learning, and how agent can learn through simulation to determine the best actions to take in any given state.

## Intuition

After a long day at work, you are deciding between 2 choices: to head home and write an article or hang out with friends at a bar. If you choose to hang out with friends, your friends will make you feel happy; whereas heading home to write an article, you’ll end up feeling tired after a long day at work. In this example, enjoying yourself is a reward and feeling tired is viewed as a negative reward, so why write articles?

Because in life, we don’t just think about immediate rewards; we plan a course of actions to determine the possible future rewards that may follow. Perhaps writing an article may brush up your understanding of a particular topic really well, get recognised and ultimately lands you that dream job you’ve always wanted. In this scenario, *getting your dream job* is a delayed reward from a list of actions you took, then we want to assign some *value* for being at those states (for example “going home and write an article”). In order to determine the *value* of a state, we call this the “value function”.

So how do we learn from our past? Let’s say you made some great decisions and are in the best state of your life. Now look back at the various decisions you’ve made to reach this stage: what do you attribute your success to? What are the previous states that led you to this success? What are the actions you did in the past that led you to this state of receiving this reward? How is the action you are doing now related to the potential reward you may receive in the future?

## Reward vs Value Function

A *reward* is immediate. It can be scoring points in a game for collecting coins, winning a match of tic-tac-toe or securing your dream job. This *reward* is what you (or the agent) wants to acquire.

In order to acquire the *reward*, the value function is an efficient way to determine the *value* of being in a state. Denoted by *V(s)*, this value function measures potential future *rewards* we may get from being in this state *s*.

## Define the Value Function

In figure 1, how do we determine the *value* of state A? There is a 50–50 chance to end up in the next 2 possible states, either state B or C. The value of state A is simply the sum of all next states’ *probability* multiplied by the *reward* for reaching that state. The *value* of state A is 0.5.

In figure 2, you find yourself in state D with only 1 possible route to state E. Since state E gives a *reward* of 1, state D’s *value* is also 1 since the only outcome is to receive the *reward*.

If you are in state F (in figure 2), which can only lead to state G, followed by state H. Since state H has a negative *reward* of -1, state G’s *value* will also be -1, likewise for state F.

*X*s in a row (state J in figure 3) does not win the game, hence there is no

*reward*. But being at state J places you one step closer to reaching state K, completing the row of

*X*to win the game, thus being in state J yields a good

*value*.

In figure 4, you’ll find yourself in state L contemplating where to place your next *X*. You can place it at the top thus bringing you to state M with 2 *X*s in the same row. The other choice would be to place it at the bottom row. State M should have a higher significance and *value* as compared to state N because it results in a higher possibility of victory.

Therefore, at any given state, we can perform the *action* that brings us (or the agent) closer to receiving a *reward*, by picking the state that yields us the highest *value*.

## Tic Tac Toe — Initialise the Value Function

The Value function *V(s)* for a tic-tac-toe game is the probability of winning for achieving state *s*. This initialisation is done to define the winning and losing state. We initialise the states as the following:

*– V(s)* = 1 — if the agent won the game in state *s*, it is a terminal state

– *V(s)* = 0 — if the agent lost or tie the game in state *s*, it is a terminal state

– *V(s)* = 0.5 — otherwise 0.5 for non-terminal states, which will be finetuned during training

## Tic Tac Toe — Update the Value Function

Updating the value function is how the agent learns from past experiences, by updating the *value* of those states that have been through in the training process.

State *s’* is the next state of the current state *s*. We can update the *value* of the current state *s* by adding the differences in *value* between state *s* and *s’*. α is the *learning rate*.

*rewards*. With a good balance between exploring and exploiting, and by playing infinitely many games, the

*value*for every state will approach its true probability. This good balance between exploring and exploit is determined by the epsilon greedy parameter.

We can only update the *value* of each state that has been played in that particular game by the agent when the game has ended, after knowing if the agent has won (reward = 1) or lost/tie (reward = 0). A terminal state can only be 0 or 1, and we know exactly which are the terminal states as defined in during the initialisation.

The goal of the agent is to update the value function after a game is played to learn the list of actions that were executed. As every state’s *value* is updated using the next state’s *value*, during the end of each game, the update process read the state history of that particular game backwards and finetunes the *value* for each state.

## Tic Tac Toe — Exploit the Value Function

*value*(or probability of winning) of any given state. So, when we play a game against our trained agent, the agent uses the exploit strategy to maximise winning rate. See if you can win against the agent.

At each state of the game, the agent loop through every possibility, picking the next state with the highest *value*, therefore selecting the best course of action. In figure 6, the agent would pick the bottom-right corner to win the game.

## Conclusion

At any progression state except the terminal stage (where a win, loss or draw is recorded), the agent takes an action which leads to the next state, which may not yield any *reward* but would result in the agent a move closer to receiving a *reward*.

The value function is the algorithm to determine the *value* of being in a state, the probability of receiving a future reward.

*value*of each state is updated reversed chronologically through the state history of a game, with enough training using both explore and exploit strategy, the agent will be able to determine the true

*value*of each state in the game.

There are many ways to define a value function, this is just one that is suitable for a tic-tac-toe game.