A mathematical theory of the state of Flow

A knowledge-centric approach


There are many studies of the Skill-Challenge Balance Model based on games. The issues with that is games cannot offer an infinite imbalance between skills and challenges. That's why such studies are not able to properly show an inverted-U relationship. We use knowledge workers like software developers. In software development it is much easier to get into anxiety e.g. if one needs to master a new technology in a very short time.

What is Flow?

Flow is a state of optimal experience that generates feelings of happiness and enjoyment. Flow states are commonly referred to in many ways in society, such as "wired in," "in the groove," "in the moment," and "the zone."

A state of "flow" is a key indicator of optimal experience and happiness in work.

Research indicates that flow experiences represent a distinct state that can be identified not only through self-report data but also through physiological measures[13]. The key characteristics of a person experiencing flow include:

  • A sense of loss of self-consciousness.
  • A perception of time passing faster.
  • A sense of control over one's actions, as increased capabilities reduce the margin of error.
  • A state of security and relaxation with the complete absence of worry.
It is the experience of complete, yet effortless, attention that is specifically associated with being in the enjoyable state of flow. The effortlessness of this experience does not stem from expending less attention, but rather from the feeling that investing more attention requires less effort[9].

Individuals who have experienced flow often wish to replicate the activity, regardless of whether it provides material rewards or not. This suggests that flow is an optimal experience that facilitates the fulfillment of individual potential.

The Skill-Challenge Balance and Flow

According to Csikszentmihaly, flow appears at the boundary between boredom and anxiety, when the challenges are balanced with the individual's capabilities[1][2].

The relation between the skill-challenge balance and optimal flow is classically understood by means of the “channel model” depicted in the image below.

  • Graph Axes: X = personal capability; Y = challenge (task)
  • FLOW: a balance between an individual's perceived skills and the perceived challenges
  • Boredom: Low challenge, high capability — If challenges are too low, one gets back to Flow by increasing them.
  • Anxiety: High challenge, Low capability — If challenges are too great, one can return to the Flow state by reducing the challenge or developing new capabilities.

The model posits entering a flow state depends on finding a balance between an individual's perceived skills and the perceived challenges of a task. In other words, the individual's capability should match the complexity of the work. Numerous studies have shown that when skill and challenge are approximately equal, flow is experienced[15][16][17][18][19][20][21][22]. In the channel model, it does not matter at what level of skill and challenge players are at. Even for beginners with little skill, so long as they are playing an easy game, flow should be experienced since their minimal skill is matched by the minimal challenge of the game. Controlled experiments have shown that optimal challenges lead to flow[10]. This is why flow is considered to be intrinsically motivating, rather than extrinsically motivating: the motivation comes from within the activity[5]. Moreover if flow is a highly motivating state[19][20], then even relatively novice players should be motivated to keep playing right from the introduction to a new game.

An alternative to the classic model of flow is what can be described as a quadrant model of flow as presented in the image below.

  • Graph Axes: X = personal capability; Y = challenge (task)
  • FLOW: High challenge, high capability — "the goal".
  • Apathy: Low challenge, low capability. — A general lack of interest in the task at hand.
  • Boredom: Low challenge, high capability — If challenges are too low, one gets back to Flow by increasing them.
  • Anxiety: High challenge, Low capability — If challenges are too great, one can return to the Flow state by reducing the challenge or developing new capabilities.

In this model, flow is experienced only when players feel that they have reached a high level of skill (offset by a high level of challenge). The quadrant model posits that the balance between challenge and skill does not always lead to optimal flow[23][24][25][26]. For players who feel they have minimal skill and are playing what they feel is a minimally challenging game, apathy rather than flow should ensue. In the quadrant model it does matter at what level of skill and challenge players are at. Players who have played more frequently to develop their skill are at a point where they can take on higher game demands confidently. These players would fall in the “optimal flow” quadrant of the model. By contrast, novice players may not enter flow as easily as they are focusing on building fundamental skills and becoming familiar with a game environment by playing minimally challenging (yet balanced) levels. Novices would thus land in the “apathy” quadrant.

Another alternative is the skill-challenge balance model. Empirical support for the notion that flow within video games is engendered by the balance of challenge and skill comes from studies which manipulate how challenging the game is by increasing or decreasing the speed at which players must play. When speed of play is manipulated, an inverted-U relation between the perceived skill-challenge balance and flow is produced[15][16][17][18][19][20][21][22][27], Specifically, optimal flow is produced when the speed-based demands matches the player's skill, and less flow ensues when skills far exceed demands (e.g., slow-paced and too easy). Similarly, less flow also ensues if the demands far exceed skills (e.g., far too fast-paced and hence too difficult). These relations are depicted in the image below.

  • Graph Axes: X = ratio between an individual's perceived skills and the perceived challenges
  • FLOW: a balance between an individual's perceived skills and the perceived challenges
  • Boredom: Low challenge, high capability — If challenges are too low, one gets back to Flow by increasing them.
  • Anxiety: High challenge, Low capability — If challenges are too great, one can return to the Flow state by reducing the challenge or developing new capabilities.

A heightened desire to re-engage in the activity appears to only be experienced at the apex of the inverted-U flow function[20]. That is the urge to keep playing appears to be maximized by flow.

Entering Flow

To enter and maintain a flow state, the following conditions should be met[3]:

  1. The opportunity for action, with a clear and attainable goal. That doesn't meant having an overall goal for the activity, but knowing what to do next from moment to moment[9]
  2. An action with a balance between skill level and challenge, meaning that the activity is neither too easy nor too difficult.
  3. Clear and immediate feedback, so that successes and failures are apparent and behavior can be adjusted as needed.

The balance should be established for each task the individual will face. Once the conditions for action are in place, the individual is able to engage in a series of challenging tasks that are neither too difficult (not overwhelming) nor too easy (not boring). By balancing each task, we can create the conditions for entering flow for each set of tasks, each feature, and ultimately, for the scope of the project.

To achieve flow, one must find a balance between individual capability and work complexity, with a slight inclination towards challenges.

Sustaining flow is dependent on the fact that neither capabilities nor challenges remain constant. When the challenges presented exceed an individual's capabilities, they may experience states of perplexity, worry, and anxiety. On the other hand, if there are no challenges in a task that match the individual's capabilities, they may become bored. Prolonged boredom can eventually lead to a state of anxiety, as there may be a perception that current capabilities may be lost[4].

When the conditions for flow are met, they create a feedback loop between action and feedback that allows for continuous, effortless tuning of performance while taking action. This feedback loop makes an activity worth doing for its own sake[3].

In order to have an opportunity to act, an individual needs to have a clear proximal goal[3]. This is not the overall goal of the activity, but rather a clear goal for the next step in the action sequence, and then the next, and so on, until the final goal is reached. The relationship between clear, attainable proximal goals and flow may be self-evident. If an individual does not know what to do next, they are less likely to enter a flow state. In order to have a clear and attainable proximal goal, an individual needs to:

  • Know what to do
  • Know how to do it
  • Know where to go (where navigation is involved)
These three conditions ensure that the individual has a clear proximal goal.

After the goal is established, the individual needs to subjectively evaluate if their capabilities are in balance with the work complexity. If that is the case, then there will be a balance between:

  • High perceived challenges
  • High perceived skills
And this balance will facilitate the individual to be in the flow state.

To maintain the flow state, it's also important to provide the individual with an environment free from distractions. Additionally, we must ensure that the individual receives immediate feedback that allows them to continuously adjust their performance as they tackle these challenges. This kind of feedback communicates how well they are performing and how they can improve their performance.

A mathematical model of The Skill-Challenge Balance and Flow

Existing models

There are proposals for quantifying the state of flow. One study used the channel model of flow. They examine the role of a single variable, the difficulty of training, on the rate of learning, to a broad class of learning algorithms in the context of binary classification tasks[28]. Identifying the precision, β, with the level of skill and the level challenge with the inverse of true decision variable, 1/Δ, they show that when challenge equals skill, flow is associated with a high learning rate and accuracy, anxiety with low learning rate and accuracy and boredom with high accuracy but low learning rate

Another study proposed a formal theoretical structure — the informational theory of flow[29]. They propose that the mutual information I(M:E) between desired end states (E) and means of attaining them (M) gives rise to flow. They show that increasing I(M:E) increases flow and has important downstream benefits, including enhanced attention and enjoyment. However, this approach is not able to explain the effects of I(M:E) on flow in terms of expected value or skill-challenge balance.

Our model

Our proposal is to use the Skill-Challenge Balance model. In order to determine whether a person was in a state of flow, it's essential to assess the balance between their perceived individual capabilities and the complexity of each task they worked on. This assessment demands precise and dependable information regarding both factors.

From a knowledge-centric perspective, we define the person's perception of the complexity of the task as the "required knowledge." Similarly, we refer to the person's perception of their individual capabilities as the "prior knowledge". Prior knowledge is known to facilitate the learning of new information. Estimates suggest that between 30% and 60% of the variance in learning outcomes can be explained by prior knowledge[11]. Additionally, prior knowledge of different domains can jointly support the recall of arguments[12].

The difference between the prior knowledge a person possesses and the required knowledge needed to complete a task is what we term as the knowledge to be discovered, which we measure in bits of information.

Imagine a scale: on one side is the required knowledge and on the other is the prior knowledge. They will be balanced if they are equal, implying that the knowledge to be discovered equals zero bits. To quantify this balance, we need to assess how a person felt while working on a task. This is difficult, if not impossible, to achieve accurately as it involves tapping into the individual's cognitive process.

Alternatively, to ascertain balance between individual capability and work complexity can be quantified by measuring how much of the required knowledge for a successful work completion is covered by the prior knowledge. We can employ Knowledge Discovery Efficiency (KEDE) for this measurement, presented in details here..

By implementing the Knowledge Discovery Efficiency (KEDE), we can assess whether a balance was struck between a person's abilities and the complexity of the work.

KEDE is a ratio between the knowledge discovered and the maximum knowledge that could be discovered for a time period, as explained in details here:


where Q is the total number of questions asked in a time interval, S is the total number of answers acquired for the same time interval.

KEDE is continuous in the closed interval of (0,1]. KEDE is inversely proportional on the questions asked i.e. on the difference between knowledge required by a task and the prior knowledge of a person, and proportional to the answers acquired

We assume that the number of questions Q reflects the complexity of the work, and the number of symbols produced S reflects the individual capabilities of a person. When they are in balance the person is in a state of flow.

The output of a knowledge discovery process has only two possible outcomes: symbols S and questions Q), with probabilities KEDE and (1-KEDE), respectively. For calculating the balance between questions and symbols we use Shannon's formula[1]

Balancep1, p2=-i=12pilog2pi


In this case, p1 = KEDE and p2 = (1-KEDE) and the Balance function of one variable is:



Figure below shows the function Balance(KEDE).

Balance as a function of KEDE

The balance function is always positive, concave (or concave downward), and has a maximum value at KEDE = 1/2. It is zero at both KEDE = 0 and KEDE = 1.

This property is consistent with what we intuitively expect from a quantity that measures the balance between questions and answers. If KEDE = 1, then we know for certain that now questions were asked. If KEDE = 0, then we know for certain that no answers were acquired. In both cases, there was no balance between questions and answers.

When KEDE is equal to 0, the person may be in a state of anxiety, as the challenges are too great. On the other hand, when KEDE is equal to 1, the person may be in a state of boredom, as the challenges are too low.

The optimal state is when KEDE is equal to 1/2, as this indicates a balance between questions and answers. For this case we get:


The numerical value of balance in this case is one. This state of KEDE = 1/2 is referred to as flow, and it is characterized by a balance between the challenges of software development and the individual's capabilities.

It is clear that for any value of 0 < KEDE < 1/2, we have less balance than in the case KEDE = 1/2. For the case of KEDE=0.1 we get:


This is the case of individual experience tending to anxiety, as its limiting case from the left.

It is also clear that for any value of 1/2 < KEDE < 1, we again have less balance than in the case KEDE = 1/2. For the case of KEDE=0.9 we get:


This is the case of individual experience tending to boredom, as its limiting case from the right.

In general, values of KEDE less than 1/2 indicate a lack of balance and a tendency towards anxiety, while values greater than 1/2 indicate a lack of balance and a tendency towards boredom. In both cases, the level of balance is less than in the case of KEDE=1/2.




We modify the equation

H(X|Y) = H(X)-I(X:Y)

by dividing each term by H(X), as follows:

KnowledgE Discovery Game

To practice measuring person happiness, you can try the Knowledge Discovery Game. This game simulates the knowledge discovery process, enabling you to evaluate the balance between person abilities and task complexity.

Works Cited

1. Csikszentmihalyi, M. 1990. Flow: the psychology of optimal experience. United States of America: Harper & Row.

2. Csikszentmihalyi, M 1975. Beyond Boredom and Anxiety: The Experience of Play in Work and Games. San Francisco: Jossey-Bass

3. Schaffer, O. 2013. Crafting Fun User Experiences: A Method to Facilitate Flow. Human Factors International Whitepaper. Retrieved from: https://scholar.google.com/scholar?cluster=9760324565241258858

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13. Keller J., Bless H., Blomann F., Kleinböhl D. Physiological aspects of flow experiences: Skills-demand-compatibility effects on heart rate variability and salivary cortisol. J. Exp. Soc. Psychol. 2011;47:849–852. doi: 10.1016/j.jesp.2011.02.004.

14. Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal. 1948;27(3):379-423. doi:10.1002/j.1538-7305.1948.tb01338.x

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    28. Wilson, R.C., Shenhav, A., Straccia, M. et al. The Eighty Five Percent Rule for optimal learning. Nat Commun 10, 4646 (2019). https://doi.org/10.1038/s41467-019-12552-4

    29. Melnikoff, D.E., Carlson, R.W. & Stillman, P.E. A computational theory of the subjective experience of flow. Nat Commun 13, 2252 (2022). https://doi.org/10.1038/s41467-022-29742-2

    How to cite:

    Bakardzhiev D.V. (2023) A mathematical theory of the state of Flow : A knowledge-centric approach https://docs.kedehub.io/kede/kede-flow.html

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