Motivation has many definitions (e.g., Deci, 1975, Ryan and Deci, 2000a, Weiner, 1990, White, 1959). Self-determination theory (SDT) puts it concisely as:

In two papers, we propose a mathematical approach to SDT. The present paper represents the first part, referred to as MSDT1. The other part, MSDT2, is the second paper by Ünlü (2023). Throughout, we will write MSDT2 to mean this reference. With MSDT1, we introduce the simpler representation of the theory in real numbers, which will be generalized in MSDT2 to abstract affine spaces.

In this section, we motivate the approach of the two papers. We describe the idea of these papers and review SDT.

In SDT, primarily psychometric statistical analyses are reported, reliability, correlation, factor, and structural equation analyses (e.g., in sport and exercise, Li and Harmer, 1996, Pelletier et al., 2013, Wilson et al., 2012). However, see Ünlü and Dettweiler (2015) and Ünlü, 2016, Ünlü, 2019 for an alternative quantification approach. So far, a mathematical representation and analysis of the fundamental concepts and assumptions of SDT are lacking. Thus, the works MSDT1 and MSDT2 examine the usefulness of a mathematical approach to SDT.

The papers MSDT1 and MSDT2 are not about measurement in the broad sense, including abstract measurement theory or statistical modeling (e.g., Bishop et al., 2007, Narens, 1985, Pfanzagl, 1968). The process of how the measurements can be obtained, which is an important topic on its own, is not the focus of this paper nor of the follow-up work. Rather, presupposing measured values of settled and primitive quantities, in the parametric sense, we ask for the concepts, properties, and interrelations that can be derived from those quantities, independent of the measurement process. This entails mathematical definitions and related mathematical analyses of the fundamental concepts of SDT.

The locus of causality is not considered to be a single continuous dimension as it is assumed in SDT (e.g., Ryan & Connell, 1989). Instead, it is distinguished into three distinct types, namely, the internal, external, and impersonal loci (mixture of internal motivation, external motivation, and amotivation), assumed in coordinates to be shares, relative frequencies, or probabilities, depending on the interpretation, all measured at the absolute scale. They are more settled quantities, something from which you can get the whole theory, if you have a measurement that distinguishes one from the other, the internal, external, and impersonal loci. In particular, there is more information in these primitives, compared to the assumption of a single latent trait.

There can be different ways to come up with a mathematical formalization. The simplest and intuitive approach pursued with the papers is to base the formulation upon the shares representing the loci, thus, multi-dimensional in triples of real numbers (reals) R in MSDT1 or barycentric coordinates of affine space in MSDT2. In particular, the proposed approach delves into a simple yet useful assumption to mathematize the theory. The pertinent assumption is not restrictive from an empirical viewpoint. It is parametric, presupposing that any regulation of motivation is determined by its three coordinates in the internal, external, and impersonal loci, which replace the commonly used single locus of causality trait.

These loci are the primitives of their mathematical representations. The three attributes are known to be continuous quantities, relative frequencies, shares, or probabilities as their levels, measured at the absolute scale in the natural unit interval. Thus, statements formulated with statistics in levels of these attributes, such as differences of differences (Definition 22), are meaningful with respect to this scale type, the scale type the whole theory is built upon. Furthermore, these attributes are assumed to be dependent, that is, related to each other. Their levels for any motivation regulation must add up to one. All other concepts of the proposed mathematical frameworks are mathematical definitions in these primitive quantities.

This approach is simple, but effective. It suffices to develop the concepts, properties, and interrelations of the theory’s mathematization. In fact, the series of two papers serves to delineate generally plausible and necessary properties and interrelations for the concepts of SDT, for any given measurement of the primitive parameters.

However, if measurement is of interest, these results can be considered for alternative formulations of measurement theory structures. As suggested by the reviewers, in a conjoint measurement approach (e.g., Michell, 2014, Suck, 1990), you may reproduce the same or only a subset of the parametric results, presumably under more or abstract assumptions, which need to be tested or may be more easily violated in data. This can be investigated in further research.

SDT (Deci and Ryan, 1985, Deci and Ryan, 2000, Ryan and Deci, 2000b, Ryan and Deci, 2002, Ryan and Deci, 2017) is probably the most researched theory of motivation. Practices of SDT are numerous, such as in education (e.g., Deci et al., 1991, Reeve and Cheon, 2021, Ryan and Deci, 2020) and physical activity (e.g., Ntoumanis, 2012, Standage et al., 2019). A plethora of other publications and applications in areas like health, technology, psychotherapy, or workplace can be found on the website of the Center for Self-Determination Theory at https://selfdeterminationtheory.org.

The general idea of the SDT approach can be described as follows. The theory distinguishes between external and internal factors for motivational behavior. Whereas the external factors such as reward, grade, punishment, evaluation, or reputation are outside the person, the internal factors such as fun, enjoyment, curiosity, interest, or excitement are within the person. A major aim of SDT is to investigate the interrelationships between external and internal motivation factors. (Amotivation as the presumably extreme case of lack of any motivational factor, be it external or internal, is also addressed in SDT.)

By SDT researchers, and probably in motivation research in general, internal factors are preferred over external factors. Behaviors or outcomes driven by internal factors are generally better, as they entail, for instance, deeper learning processes or higher performance (e.g., Chiu, 2021, Goodman et al., 2021, Sheldon and Niemiec, 2006, Vallerand et al., 2008). This is intuitive. External factors are imposed extrinsic forces acting on people, whereas internal factors are natural intrinsic motives and needs inherent in humans. Intrinsic motivation is desired in applications and central to SDT. SDT claims that intrinsically motivated people, implied by three basic psychological needs (described below), generally behave in effective and healthy ways, for instance, grow and feel well mentally. This seems to hold irrespective of culture (e.g., Chen et al., 2015). The basic psychological needs can be cross-culturally validated, and there are studies that empirically corroborated the fact that satisfaction of these needs remained essential to well-being regardless of cultural contexts. In particular, if autonomy was not satisfied, this had a negative impact on well-being, independent of people’s residence.

We look at the different motivation types posited within the theory. For orientation, SDT is recapped in compact form in Table 1, a tabular summary of the theory of sorts commonly reported in SDT. Subsequently, we specify all of its components textually.

SDT was initiated with research about intrinsic motivation (Deci, 1975). Intrinsic motivation or intrinsic regulation, the terms are synonymously used, subsumes behaviors or activities that are characterized by genuine interest, inherent satisfaction, enjoyment, and enthusiasm. The unforced motives for the behaviors lie within the person and do not depend on external contingencies. If I do math for fun, for its own sake, I am intrinsically motivated.

SDT argues that intrinsic motivation (or internalization of extrinsic motivation) derives on the satisfaction of three, innate and universal, basic psychological needs (e.g., Ryan et al., 2019). When persons’ needs for competence (to be good at something), autonomy (choices and control over decisions and actions), and relatedness (to be connected to others through positive relationships) are satisfied, they become intrinsically (or internally) motivated. This is a fundamental empirical postulate of SDT, with far-reaching applications (Fig. 1).

Intrinsic regulation is fully internal, the most self-determined or autonomous form of motivation. We call it uncontrolled, in the sense that no external and internal controls, always defined to be forced factors, are moving people to act. External controls are triggered by external factors that constrain motivational behavior, such as bad grades or reprimand by parents. Internal controls relate to internal factors constraining behavior, which could be pride, shame, or ego involvement.

SDT is composed of subtheories, each of which studies a specific aspect of motivation. Intrinsic motivation is the topic of cognitive evaluation theory, extrinsic motivation of organismic integration theory (Deci & Ryan, 1985).

In organismic integration theory, the three manifestations of external regulation, introjected regulation, and identified regulation of extrinsic motivation are specified, shown in Table 1. In SDT literature (e.g., Niemiec & Ryan, 2009), the regulations of extrinsic motivation are assumed to form the internalization continuum. Identified regulation is more internalized (thus more autonomous) than introjected and external regulation, and introjected regulation is more internalized than external regulation. We may think of internalization as the degree to which a person values the activity. However, based on this paper, we suggest to include into this continuum also non-regulation of amotivation and intrinsic regulation of intrinsic motivation, to allow for all potential cases. Behaviors may traverse any compositions of amotivation and extrinsic and intrinsic motivation. For example, activities may move from being completely or mainly amotivated to more extrinsically motivated to more intrinsically motivated, and vice versa. This gives an extended internalization continuum. It can be defined a continuum based on the underlying continuous dimensions of self-determination (or relative autonomy) and locus of causality.

A possible situation encountered in school class is that the activity of a student may start with being externally regulated, not liking or enjoying doing arithmetic algebra, but learning only to avoid bad grades. However, while working on one or more tasks about arithmetic algebra in a motivating learning environment, for example in group work assisted by the teacher, that student may value the activity more and more, and it may become part of one’s self, thereby being gradually internalized.

Extrinsic motivation encompasses behaviors that depend on external factors. The motives (e.g., money) for an extrinsically motivated activity lie outside the person. In that sense, extrinsic motivation describes instrumental behavior. The extent of instrumentality, or dependence on external factors, may vary and defines the distinct regulation types of extrinsic motivation. The more internalized extrinsic motivation, along the internalization continuum, the more autonomous, or self-determined, volitional, or unforced, the person is in her behavior, the terms being synonymously used.

External regulation is the fully external, thus least autonomous, type of extrinsic motivation. Colloquially, it is what is often referred to as extrinsic motivation. The behaviors are dictated by external constraints, by and large, to avoid a form of punishment (e.g., bad grades, reprimand by parents) or to obtain a reward. She goes to work in order to earn money. Because of enforced external controls, extrinsic regulation is classified as non-autonomous and controlled.

Introjected regulation refers to behavior enacted on the basis of external societal expectations, only partially internalized and “somewhat external” to the self. For example, a student is concerned about what opinion other classmates and the teacher may have of her, she wants to be recognized as a hardworking student, and self-imposes the forced internal controls of pride or shame for success or failure, respectively. Introjected regulation also pertains to partially internalized behavior determined by self-esteem, ego involvement, or guilt (Nicholls, 1984, Ryan, 1982). The student works hard for the exam because, complying with internal contingencies, she wants to feel worthy, to avoid shame, or she does not want to blame herself for not having studied enough. Constrained by particularly internal controls, introjected regulation is also categorized as non-autonomous and controlled form of motivation.

Identified regulation occurs if a person recognizes and consciously identifies with the value of an activity. The goals of the activity are personally endorsed, valuable, and important. In contrast to intrinsic regulation, the activity is not associated with enjoyment or fun. Although she does not enjoy doing math, she learns math because knowledge of math is important for her studies in physics. Behaviors triggered by identified regulation are “somewhat internal” but still instrumental extrinsic motivation. Identified regulation is a relatively highly, both, volitional and externally and internally uncontrolled form of motivation. We classify identified regulation as quasi-autonomous and quasi-uncontrolled.

Amotivation (Gravel et al., 2016, Legault et al., 2006, Pelletier et al., 1999, Pelletier et al., 2013, Vlachopoulos et al., 2010) is the absence of extrinsic and intrinsic motivation of any form. We can define it to be complementary to extrinsic and intrinsic motivation. Amotivation describes behavior that is characterized by lack of value, interest, or perceived competence. The activity is of no relevance to the person who does not exhibit any intention to act. In contrast, all other motivation types involve intention in one way or the other. For example, a student must follow math instructions in class, she does not see any value in learning math, nor does she feel competent enough to perform, and she shows no intention at all to work on the subject.

As might be expected, amotivation is strongly detrimental to outcomes such as performance and psychological well-being. Amotivation lacks intentionality, is impersonal. No unforced or self-determined internal motives drive behavior, it is the strongest non-autonomous form of the continuum. Due to also independence of forced external and internal controls, amotivation may be viewed as uncontrolled.

In this paper, we interpret, redefine, and unify those motivations based on their decomposability into internal and external motivation and amotivation, and thereby, we also embed the above common motivations into a broader continuous spectrum of graded motivations.