Abstract
Is the act of making a decision a process or pulse? Critiques of rational choice theory and models often treat cognitive processes of preference ordering as part of the act of decision that should be incorporated into the models. The failure to account for human psychology, they argue, responds for RCT’s lack of predictability. However, this argument and the models of human mind, such as prospect theory, see decision as a process that begins at the cognitive considerations of preference ordering and extends up to the act of decision. In this paper, I argue that decision is analogous to a pulse rather than a process. I draw this analogy with the Dirac delta function, which in signal theory represents an unitary pulse. In the exact moment of making a decision, all preferences and contextual evaluations must have been already structured in the agent’s mind, otherwise she would not be capable of making the decision. Acknowledging the pulse-like nature of rational choice models allows modellers to eschew the incorporation of complex cognitive processes into their analyses, which has both theoretical and empirical implications to RCT’s representation of real-world phenomena.
Introduction
Rational choice models of decision-making have flourished for the last 50–60 years as a research agenda transversal to various disciplines, such as political science, economics, and International Relations. Decision settings are paramount to understand political and economic phenomena where incentives and constraints affect how agents act. Much of the literature in political science and, more importantly, in rational choice theory (henceforth, RCT) has focused on designing models that structure these settings in ways that facilitate the analysis of post-decision-making scenarios. Game theory, for instance, is one of the primary tools used by rational choice modellers to tailor explanations about the act of choice, highlighting agency and structural aspects that affect decisions (Dowding, 2017, chapter 2).
Nevertheless, the transversality of rational choice models has inescapably attracted criticisms from a variety of disciplines. In political science, the most acclaimed critique of RCT is Green and Shapiro’s (1994) Pathologies of Rational Choice Theory, where they scrutinise rational choice models and their applications in the discipline. Pathologies has influenced much of the debate about the potentials and pitfalls of RCT (Cox, 1999, 2004; Dowding and Hindmoor, 1997; Eriksson, 2011; Fiorina, 1995; Hindmoor and Taylor, 2015; Lohmann, 1995), yet it responds primarily for the empirical indictments against rational choice models. An alternative critique, derived from experiments of cognitive psychology, has also targeted RCT’s approach to decision-making processes and its lack of theoretical foundations about the human mind (Gigerenzer and Selten, 2001; Kahneman and Tversky, 2000; Simon, 1957; Slovic, 2000).
This critique revolves around behavioural and cognitive factors that are absent in mathematical models of rational choice. Herbert A. Simon (1957) was, perhaps, the first one to systematically consider the limits of the rationality assumption entailed in RCT, proposing a model based on bounded rationality. Kahneman and Tversky (2000), as well as Gigerenzer and Selten (2001), further developed studies and methodologies to account for the cognitive aspects of human behaviour. The vast literature of cognitive sciences has provided invaluable insights and criticisms to RCT, despite the limits to their implementation in mathematical models. In addition to these criticisms, simplistic modelling has been on the centre of the discussions, especially through the assumptions of utility and utility maximisation (Hodgson, 2012).
As interesting and relevant these critiques are, they fail to dialogue with rational choice theorists in political science. To be sure, rational choice models are primarily concerned about the settings in which decisions are made, being designed to comprehend the interactions between agents whose preferences have already been structured prior to the moment of choosing a course of action; and the structure of incentives and constraints under which agents operate. More precisely, rational choice models operate at the very last infinitesimal moment before a decision is made, for this is where all incentives, costs and probabilities must have been already incorporated into the agents’ utility functions. Cognitive processes operate prior to the act of choice or, more specifically, to preference ordering.
In this sense, psychologists’ criticisms focus on pre-choice cognitive processes of decision-making, whereas rational choice theorists are primarily interested in the social, collective results of decision-making. In other words, these psychological critiques offer an understanding of the human mind that is beside the point of what matters for RCT’s systemic perspective of social phenomena. Evidently, this does not mean that psychology has not offered any contribution whatsoever that may be of interest to RCT: by saying that psychologists focus on different processes does not mean that their contributions are not useful to reframe RCT research inquiries. Indeed, a great deal of rational choice formal modelling has benefited from these critiques, especially in what matters for preference-ordering and the possibility of mistakes in decision-making (Selten, 1975; Taylor, 2020); how agents assess risks and how risk should be incorporated into models (Congleton, 2019a); how boundedly rational agents make suboptimal or non-optimal decisions given environmental constraints (Alchian, 1950; Ostrom, 1998); how reputation, trust and reciprocity affect agents’ behaviour in decision-making settings (Axelrod, 1984/2006; Rapoport, 1999).
In this context, I argue that RCT has a different approach to decision-making, one that focuses on the infinitesimal moment prior to the act of choice rather than the processes taking place at agents’ minds before the decision is actually made. In terms of modelling, this approach has the advantage of treating the rational decision as a pulse, which takes place in the instantaneous act of choice at a given moment in time. It should cause no surprise to rational choice theorists, for their formal models are cemented on the underlying idea of an instantaneous pulse, meaning that models focus on the preference ordering instead of on the processes that result in that specific preference ordering. Once the decision is made, a scenario unfolds, as predicted by the models. The main implication to this debate is that treating choices as pulses does not demand an overarching explanation of how the human mind works. Although it may be of psychologists’ interest, an account of the human mind is intrinsically complex and perhaps intractable in terms of mathematical modelling and, more importantly, rational choice modelling of decisions. To a large extent, the issues psychologists are interested in differ from those that political scientists are concerned about in their works, and fully incorporating them into formal models may deplete the latter’s explanatory power of social phenomena. Therefore, adopting a pulse-like analogy to rational choice modelling has empirical consequences in terms of which exact slices of the real world should be represented in a model in order for it to provide meaningful insights and explanations.
The paper is divided into three sections. In the one that follows this introduction, I review the literature on cognitive science that has targeted rational choice models. I am particularly interested in Kahneman and Tversky’s works, for they have set the lines of the theoretical critique of RCT from the standpoint of cognitive psychology. In the second section, I discuss the representational character of rational choice models, mobilising arguments developed in the fields of philosophy and political theory. Finally, in the third section, I briefly introduce the Dirac delta function, which is one of the mathematical tools used to model phenomena of an impulsive nature, such as a baseball bat hitting a ball. I then argue that modelling decisions according to the axioms of RCT is analogous to modelling a pulse, meaning that the decision-making process, though intrinsically complex if one wills to account for cognitive processes, is restricted to the act of choice in rational choice models. This should not be seen as a limitation of the theory, but rather as a theoretical and methodological decision about how to represent real world political phenomena.
Psychological models of the human mind
Throughout the development of RCT, social scientists and psychologists alike have criticised the economic and mathematical approach to decision-making entailed in rational choice models. Perhaps the most serious and important indictments against RCT have targeted its theoretical grounds, namely how they conceive of the human mind and its intricate decision-making processes. A traditional criticism that has echoed in many disciplines questions the validity of the rationality assumption and its connection to utility maximisation. According to the critique, the story behind rational behaviour frequently ignores other elements for the sake of modelling, such as norms, cultural references, moral values, altruism and deliberation just to name a few (Hindmoor and Taylor, 2015; Sen, 1994, 1997, 2009). In this context, expected utility theory (EU, henceforth), which is a behaviour model of decision-making under uncertainty, is one of critics’ main target. EU was systematically developed by von Neumann and Morgenstern (1947) as part of their decision theory, and it has soon become widely used in economics and political science. It assumes four axioms (completeness, transitivity, independence of irrelevant alternatives, and continuity), which yield the famous equation represented by equation (1), where ci represents outcome i; pi the corresponding probability; and L a lottery.
The criticisms against EU specifically, and RCT more broadly, are frequently based on the findings and arguments of other disciplines, especially psychology and philosophy respectively. In psychology, the assumption that individuals behave rationally has been contested since Simon’s model of bounded rationality, which was then followed by a myriad experiments, many of them conducted under the Kahneman and Tversky’s framework. In philosophy, especially within the domain of the philosophy of science, the debate has orbited around the explanatory and representational capabilities of models (Cartwright, 2010; Giere, 2004; Morrison and Morgan, 1999).
Simon wrote a series of pieces on the matter of rationality. His approach to the issue was later known as bounded rationality, being often cited in Kahneman’s, Gigerenzer’s and Amartya Sen’s works. Simon’s research is situated on the interfaces of different disciplines, namely economics, political science, cognitive psychology and behavioural sciences. He attempts to provide an alternative interpretation of Homo economicus, favouring a more nuanced and realistic approach to her computational skills. According to his theoretical proposition, a behavioural model must capture the cognitive limitations of human mind and external effects derived from the environment. The basis of Simon’s argument is that agents do not possess the attributes of perfect rationality, but are rather constrained by what he defines as bounded rationality (Simon, 1957). In his words (Simon, 1957: 252): In most global models of rational choice, all alternatives are evaluated before a choice is made. In actual decision-making, alternatives are often examined sequentially. We may, or may not, know the mechanism that determines the order of procedure. When alternatives are examined sequentially, we may regard the first satisfactory alternative that is evaluated as such as the one actually selected.
The essence of Simon’s model lies on the satisfactory alternative, or rather on the concept of satisficing: agents, upon analysing the structure of the environment or context, make decisions that aim to satisfice rather than maximise utility. Simon describes the environment in term of an agent’s needs, desires and goals. Agents possess a variety of goals, but only one matters for a given decision process. As an agent’s perception about the environment is limited, she cannot aim to maximise her utility, but she can satisfice her goal instead. When faced with multiple goals, time becomes a constraint, as the time spent to achieve one goal necessarily reduces the amount of time available to achieve the remainder of them. Once again, Simon affirms that agents facing multiple goals can only think about satisficing limited goals within their needs. In this context, agents use shortcuts or clues to make decisions, meaning that they learn from previous decision contexts and use this information in future decisions.
This notion of boundedly rational individuals contrasts with the widely known conception of rationality that became popular in the 1950s and 1960s in Economics (Gigerenzer and Selten, 2001; Selten, 1975). Rationality was linked to optimisation, as if individuals were capable of acquiring and processing information in a calculative fashion. Critics considered this model to be implausible, not fitting the reality of human mind. In this sense, they favour what they call the theory of bounded rationality. 1 In their view, ‘[b]ounded rationality means rethinking the norms as well as studying the actual behaviour of minds and institutions’ (Gigerenzer and Selten, 2001: 6).
The bounded rationality model laid the groundwork for the psychological analyses that followed. Perhaps, the most consistent research agenda on the cognitive aspects of human rationality was established by Daniel Kahneman and Amos Tversky, two renowned psychologists who formulated a framework of judgment and decision-making process known as prospect theory. They published a myriad results of psychological experiments on bounded rationality. Many of their findings reverberate across disciplines (for the study of rationality is transversal to many sciences), and constitute the basis of the cognitive critique of RCT. They mainly focus on individuals instead of roles to build their account of human behaviour.
Kahneman and Tversky’s (2000) research generated invaluable findings that contribute to our understanding of decision-making. Perhaps, the most striking result obtained in their experiments and which eventually became part of their theory of bounded rationality, is that individuals are subject to framing effects, no matter how sophisticated players/decision-makers they might be. In a straightforward definition, framing is a cognitive bias in decision-making generated by how alternatives are presented to the individual, whether she perceives them with positive or negative connotations.
Most of Kahneman and Tversky’s arguments revolve around the concept of framing. Their experiments are basically tailored to unravel the nature of this phenomenon. A typical experiment poses individuals with pairs of probabilistic scenarios, whose statements are written with subtle changes in how probabilities are assigned to events, affecting how individuals perceive them in terms of positive and negative connotations. A cautious evaluation of the statements would make it clear that they correspond to the same scenario, but framing the sentence in different ways leads to different results in individuals’ choices. This is what they call framing effects, and the cognitive bias they entail are in disaccord with the tenets of perfect rationality: if individuals were perfectly rational agents, they would not fall prey to framing effects.
Kahneman and Tversky’s challenge the ideal of rationality professed by rational choice scholars and their attempts to predict social phenomena. According to them, failing to incorporate the cognitive bias generated by how settings are framed renders rational choice modellers’ conclusions unrealistic. However, Kahneman and Tversky also recognise that it is impossible to predict ex ante the effects of framing. Experiments would have to be carried out in order to understand the precise effect and the underlying cognitive bias operating in the decision-making process. Without doing so, one can only expect vague conclusions about an individual’s behaviour, such as: (1) individuals value stability over change because of their fear for losses; (2) the instability of preferences favours the preference for stability; (3) individuals value the real experience of outcomes; (4) individuals value the decision itself; (5) individuals value outcomes that are certain relative to outcomes that are uncertain; (6) individuals, when ranking their preferences and alternatives, focus on their distinctive elements, which may lead to inconsistent orderings.
Despite the limitations on predictability, these findings were essential to build the two-phase framework known as prospect theory, which attempts to account for the probabilistic reasoning of individuals instead of just the probabilities of final states (as EU does). The first phase, editing, consists in a preliminary analysis of prospects, resulting in a simple representation of them. In the second phase, evaluation, prospects are evaluated and the highest in value is chosen. Editing consists of three operations: coding (gains and losses are defined in relation to the reference status), combination (prospects with similar probabilities and associated to the same outcome are combined) and segregation (separation of any risk components from riskless components of a given prospect). This theory is based on changes of wealth and welfare because humans are cognitively better equipped to deal with gains and losses rather than with absolute magnitudes. In addition, in a second version of their theory – cumulative prospect theory –, Kahneman and Tversky claim that the process of making a decision is essentially constructive and contingent, but not necessarily rational as a result of framing effects. Applications of prospect theory in the real world can be found in stock markets (Barberis et al., 2006; Benartzi and Thaler, 1995) and auctions (Rosenkranz and Schmitz, 2007).
Following Kahneman and Tversky’s works and teaming with them in further research, Slovic (2000) developed an integrative decision theory that could account for the effects of preference reversals. Preference reversals poses a serious epistemological challenge to RCT for it demonstrates that human action is devoid of any principle of optimisation, which is the fundamental assumption of rational choice analysis (Grether and Plott, 1979). According to Slovic, this phenomenon of preference reversals violates the principle of invariance of rational choice analysis: ‘[t]he principle of procedure invariance is violated by preference reversals that are induced by changing from one mode of eliciting a preference to another, formally equivalent, mode of response’ (Slovic, 2000: 491). In his understanding of preference formation, rational choice theorists have underestimated the complexity of this process and the subtleties of preference ordering. In his account, ‘[c]onstruction strategies[for preference orderings] include anchoring and adjustment, relying on the prominent dimension, eliminating common elements, discarding nonessential differences, adding new attributes into the decision frame in order to bolster one alternative, or otherwise restructuring the decision problem to create dominance and thus reduce conflict and indecision’ (Slovic, 2000: 500). In other words, a psychological account of cognitive processes is missing in rational choice analyses, resulting not only in an incomplete picture of the human mind and how it processes decisions; but also in a fundamentally unrealistic understanding of the decision-making setting.
More recent studies have advanced these critiques by offering new readings on the theory of bounded rationality (Gigerenzer and Selten, 2001; Selten, 2001). New models now scrutinise the decision-making process whilst relaxing the highly technical language of cognitive processes. Gigerenzer and Selten’s (2001) adaptive toolbox model, for example, adopts the language of heuristics to argue that agents make decisions based on a set of rules applied to specific contexts and environments, reflecting the idiosyncratic assessments agents make about their motivations and goals. Selten’s (2001) adaptive aspirations model, although more sophisticated in terms of describing certain cognitive processes, also rests on the assumption that individuals make decisions under the constraints of bounded rationality, which leads them to an evaluative process of their goals vis-à-vis aspirations. These models have been tested by Scheibehenne et al. (2013) using a Bayesian approach in an attempt to check for the validity of the principle of bounded rationality. Other models based on similar assumptions are listed in Table 1.
Models of bounded rationality.
Summing up, the aforementioned models display a similar concern when it comes to decision-making: their primary focus rests on the cognitive processes taking place inside the human mind that tailor a decision. There is no surprise that their critique of RCT is directed to the allegedly incapability of rational choice models to account for neurological processes of reasoning, preferences edition and evaluation, cognitive heuristics etc. Yet, two questions should be raised in this context. First, does RCT claim to be (or should be) a theory of the mind, attempting to map the cognitive processes that psychologists are interested in? The answer is no. In analysing agents’ interactions, rational choice theorists are interested in solving for their formal models, attempting to explain the outcomes of specific social settings. Hardly ever they are concerned about the psychological/cognitive microfoundations of human behaviour, or what Dowding (2005: 453) calls ‘deeper reasons’. As Dickson (2006: 455) declares: The epistemology of rational choice in positive political theory involves learning of a very different kind. Typically, positive rational choice models seek to explain, or at least to provide a mechanism for or an account of, macro-level phenomena. What might be considered the microfoundations of political science – the cognitive pathways through which individual members of society form political judgments, learn about political questions, or come to make political choices – are generally not the objects of interest for rational choice political theorists. Instead, these aspects of human nature are stipulated by assumption, almost always in the form of standard decision theoretic axioms. Investigation of these microfoundational questions is generally left as an exercise for another field – psychology, perhaps, or the behavioral branch of political science – to the extent that rational choice theorists conceptualize it as a task at all.
Secondly, how do cognitive processes enhance rational choice models in terms of the latter’s representation of decision settings? RCT is primarily preoccupied with the collective outcomes yielded by the interplay of agents’ preferences and decisions, on the one hand; and structural constraints imposed by the environment, on the other. This structural element seems to be missing in the aforementioned psychological models of the mind. Yet structures are fundamental to drawing a complete picture of the decision setting. As Dowding states (2017: 55): ‘Not only do structures create incentives for agents to behave differently given some underlying set of preferences, but structures can be thought to delve into agents themselves to create those underlying sets of preferences’. In this sense, a typical rational choice model represents real world phenomena through rather different lenses, incorporating in its assumptions not only the necessary elements of agents’ preference orderings, but also the environmental constraints on these orderings. What cognitive elements, if any, are incorporated into models is a matter of what the model aims to represent.
Rational choice modelling and representation
Rational choice models have a long tradition in the discipline and are quintessential to RCT as a whole. Although not all RCT is formalised via mathematical constructa (Snidal, 2006), a great deal of it relies upon formal models. Thus, what a model is and what functions models perform are fundamental questions to be raised if one is willing to understand how psychological concepts can be incorporated into them.
As mentioned previously, philosophers and social scientists alike debate the contentious nature of models: Morrison and Morgan (1999: 10) define them as ‘autonomous agents’ that ‘function as instruments of investigation’; Giere (2004: 747) declares that models are ‘abstract objects constructed in conformity with appropriate general principles and specific conditions’; and Cartwright (2010: 19) understands models as ‘experiments in thought about what would happen in a real experiment’. Each of these ‘definitions’ of models are imbued not only with certain characteristics, but also advance certain ideas about what models do regarding phenomena either natural or social. Nonetheless, they share an underlying understanding that models are not exactly a portrait of the real world, but rather a representation of that world. Dowding (2016: 80) summarises this idea: ‘Models are usually simplified versions of the things they represent, eliminating aspects that are not important for the use to which the model is being put’.
The representational character of models is paramount to debates on what exactly social scientists expect from the formalisation of their theories and claims. As objects designed to represent the world, models must identify features in reality in order to generate the explanations they are designed for in the first place. That is to say that models should to some extent represent reality, sharing similarities with the real world (Dowding, 2016; Giere, 2004).
However, what exactly is this extent of representation? There is no uncontentious answer to this question, especially in terms of how realistic the assumptions entailed in a model are. Models are often constructed upon assumptions that are not observable directly in the real world, at least in part. Frequently, idealisations such as Homo economicus or the perfectly rational voter are targeted as unrealistic constructions of human behaviour – hence the indictments of psychologists. Radical critiques along those lines tend to dismiss the explanatory power of models, because their design is flawed from its inception, which in turn leads to false conclusions about real-world phenomena (Alexandrovna and Northcott, 2013; Cartwright, 2010; Reiss, 2013).
As powerful this critique may be, it fails to acknowledge the representational character of models as the defining factor of at what extent reality has to be represented in order to explain natural or social phenomena. Models are designed to generate explanations of a certain target system (Dowding, 2016), which comprises the set of phenomena (and the underlying assumptions) of the modeller’s interest. Thus, a model is ‘partially isomorphic to the real world’ to the extent that ‘some assumptions that define the model match some of the assumptions met in the real world, the target system. What we need in terms of truth values to make this happen is a claim indicating precisely which aspects of the model identify which aspects of the target system’ (Rol, 2013: 246). This process of matching what the model represents and what in the real world is represented matters for the modeller’s decision of what she wants to explain: it is not given a priori. Different modellers may make different modelling decisions, and in doing so they take into account the quality of their assumptions and the tractability of the model. They also understand that representing the totality of reality is impractical, and even if it were possible, there is no guarantee that the model would generate a better understanding of the phenomena it aims to explain (Morton, 1999: 53).
Rational choice models are designed having these concerns in mind. The assumptions entailed in such models derive from the features of a real world where collective interaction generates effects in the social realm. As a theory in political science, RCT is concerned about the slice of reality that explains phenomena at the collective level. To be sure, individual agency matters, as long as it is associated with collective outcomes. Therefore, a certain degree of understanding of how the individual conceives of her preferences is paramount to the tailoring of explanations at the aggregate level. Nevertheless, how deep one has to delve into the human mind is a matter of which elements of the real world the modeller is willing to explain in order to represent them in her model. As Cox (2004: 172–173) suggests: Another way to think about rational choice in social science (. . .) is that it focuses on the system of human interaction and black boxes the constituent parts of the system (humans). (. . .) The argument is about how much internal structure, how much human nature, we need to bring in to our models of social interaction. However much you decide to bring in, you can presumably always be criticized for not appreciating even richer conceptions that tap into levels even lower in the architecture of complexity (. . .). Moreover, however much you decide to bring in, the resulting social science is not comparable to Newtonian mechanics in precision.
Political scientists have not ignored the psychological critiques of RCT’s allegedly flawed picture of the human mind and responses to critiques have been varied. Many rational choice theorists incorporated altruistic behaviour into their models (Fehr and Fischbacher, 2002; Quackenbush, 2004); others preferred to relax the ideal of universalism, accepting partial universalism in RCT (Satz and Ferejohn, 1994); and there are still those who favour a view of models as stories, fables or credible worlds that represent some aspects of the real world to convey a narrative and generate predictions (Johnson, 2017; Rubinstein, 1991; Sugden, 2011). In her presidential address to the American Political Science Association, Ostrom (1998) acknowledged that individuals’ preferences might change throughout their lives as part of their learning about social norms from past interactions with other individuals. According to her, second-generation models of rationality attempt to introduce–via different mathematical approaches–psychological processes of learning to formal models, which to some extent addresses the concerns of cognitive psychologists. Similarly, Axelrod’s (1984/2006) and Rapoport’s (1966/1999) approaches to cooperation have enhanced our understanding of the roles reputation and reciprocity play in agents interactions, which has been utterly important to the incorporation of a more nuanced picture of human behaviour into formal models. Behavioural economists, such as Herbert Gintis, have sought similar approaches to deal with human behaviour, incorporating cooperative behaviour (Gintis, 2000) and social norms (Gintis and Helbing, 2015) into their models.
Nevertheless, in specific contexts,
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RCT’s assumptions hold and offer a fairly representation of the decision setting, both for the environmental and contextual structure, and for the agent’s preferences and strategies. Cox (2004) and Ostrom (1998) agree that the core assumptions entailed in thin models of decision-making where environmental pressures are in play approximate individuals’ behaviour, generating not only the outcomes expected from the theory, but also those observed in the real world. This systemic approach to decision-making aims to represent the social setting within which the individual is immersed, providing a picture of what matters not only in respect to her cognitive processes, but also to the environmental circumstances under which she chooses a course of action. Even if uncertainty plays a role at the individual level, the collective outcome does not necessarily reflect the choices (and the cognitive processes leading to them) of one single individual. As Alchian (1950: 216) suggests: ‘Where there is uncertainty, people’s judgments and opinions, even when based on the best available evidence, will differ; no one of them may be making his [sic] choice by tossing coins; yet the aggregate set of actions of the entire group of participants may be indistinguishable from a set of individual actions, each selected at random. (. . .) With a knowledge of the economy’s realized requisites for survival and by a comparison of alternative conditions, he [sic] can state what types of firms or behavior relative to other possible types will be more viable, even though the firms themselves may not know the conditions or even try to achieve them by readjusting to the changed situation if they do know the conditions.’
The task for the rational choice modeller, then, consists in assessing which courses of action–based on certain preference-orderings–are viable and more likely to be chosen. This can be achieved by analysing the pressures the environment exert on individuals and how they affect the probability of each choice and resulting scenario. Moreover, by focusing on the set of individuals immersed in a given environment, rational choice theorists do not have to offer a complete picture of single agents’ cognitive processes to generate the kind of explanations of social phenomena that they are interested in. Indeed, the more complexity is added, the less tractable a model may become and, as a consequence, less explanatory (Cox, 2004: 177).
Modelling choice as pulses
At this point, I shall turn to the central argument of this paper, which favours a pulse-like approach to the decision settings familiar to RCT. The Dirac delta analogy that I develop here aims to represent the nature of decision-making processes as rational choice modellers frequently conceive of them. As a representational analogy, it shares similarities with those slices of the real world for which rational choice theorists design their models. In doing so, they consider certain assumptions about agents’ behaviour in a given environmental setting, for their focus rests on the social outcomes generated by the decision.
Dirac delta function
In 1958, physicist Paul Dirac proposed a function to model an idealised point mass or point charge, whose main characteristic consists in approaching infinity at a single point in the domain; and zero elsewhere. Dirac was working on quantum mechanics and his model was later implemented in other fields of mathematics and physics, namely those that model impulses.
Impulses in the context of natural phenomena are strong forces, momenta or stresses acting for a short period of time. In more technical terms, impulses have a great numerical value and are instantaneous or quasi instantaneous (Baowan et al., 2017, chapter 2; Boyce and DiPrima, 2009, chapter 6; Rapp, 2017, chapter 3). For example, a billiard ball moving across the table and colliding with another ball at rest exchanges momentum via an impulse. The force acting on the stationary ball also displays an impulsive nature. The time scales in these phenomena are usually smaller than milliseconds, and in some contexts, they approach zero. Dirac was particularly interested in describing phenomena where a single, unitary peak approaches infinity at t = 0. This is what has become known as the Dirac delta function – δ(x) function –, a generalised function whose main characteristic is that it makes a discontinuous function behave more or less like continuous ones.
The Dirac delta function is represented in Figure 1. Across the x axis, all values equal to zero, except for an instantaneous point at t = 0 where the function yields a peak. In particular applications of signal theory, namely the Fourier integral, the Dirac delta function may represent a chain of pulses. This is of uttermost importance to convert signals from the domain of time to the domain of frequency (and vice-versa), a very common procedure in signal modelling.
Typical representation of the Dirac delta function.
Mathematically speaking, the Dirac delta behaves like a distribution rather than a function. This is why it is usually converted into a Fourier series, for the mathematical treatment is more palatable (Rapp, 2017). Nonetheless, the intuition of the Dirac delta remains unchanged, for the phenomena it models are characterised by an impulsive nature, namely the displaying of an infinite value at a single moment in time.
Scientists interpret the concept of infinity as a great value beyond the ordinary idea associated with the term ‘great’. In our present context, it is analogous to a phenomenon that has a clearly distinguishable signature from the rest of the domain (or phenomena). The act of choice possesses such characteristic, for it is distinguishable from the cognitive process of preference formation; and from the scenario that unfolds once the decision is made. Should we plot the decision-making, the act itself would be represented as the Dirac delta depicted in Figure 1.
But why the Dirac delta? In rational choice models, the act of choice takes the form of a distinguishable pulse in the whole domain of what we call decision-making. To be sure, the making of a decision is equivalent to the choice itself, for the decision can only be made once the cognitive processes have already structured the preference-ordering. In this sense, rational choice models represent the specific slice of the world that contains the pulse-like form of the act of decision with the factors (namely, a structured set of preference ordering vis-à-vis environmental constraints) that lead to the unfolding of a certain scenario. This is they key to understand the representational character of rational choice models.
The case for pulse-like decision settings
Models in political science represent the political world, which is comprised of agents and structure (manifested in institutions, contexts and other environmental factors). Modelling decisions in political settings means pairing agents and structural incentives as well as constraints to human action. In doing so, models ‘capture some important features of a real world process in order to highlight some interesting causal relationship’ (Taylor, 2020: 73) and frequently that means eschewing a full description of the functioning of the human mind.
Cognitive psychologists depart from a different approach, representing decisions as a process encompassing various mental mechanisms that cannot be ignored, for they affect the way an agent interprets alternatives and makes a choice. This is certainly true, but the crucial point in this kind of approach – which may be modelled following the lines of prospect theory or adaptive toolbox theory – is that it focuses on the processes that leads to a certain preference ordering which will then fundament the act of choice. As I have stated previously, an agent cannot make the actual decision if her preference ordering is not yet defined. You cannot choose between alternatives if your preferences about them are not clear, especially if you are still mentally editing their ordering and assessing the risks they entail or the scenarios that may unfold. 3
In this sense, transitivity, which is a pillar of rational choice models, may not offer a real picture of the human mind, but it still allows for the sidestepping of the psychological aspects of decision whilst generating relevant insights about social outcomes. 4 On average, rational choice models of first and second generation offer fairly approximate results of decision-making processes even if they do not incorporate into their assumptions a complete picture of cognitive processes (Congleton, 2019a; Levy, 1987; Ostrom, 1998; Taylor, 2020). Furthermore, the structure where agents are immersed provide the set of incentives and constraints that affects their preferences in ways that make the latter less unstable than the psychological critique frequently assumes (Alchian, 1950; Congleton, 2019a; Dowding, 2017; Shepsle and Weingast, 1981). RCT is primarily preoccupied with this kind of decision-making setting, and its models target specifically those phenomena that are on the verge of the act of choice. Thus, the conceptual key to rational choice models is the combination of agent’s context-oriented preference orderings and their evaluation of the scenarios that may unfold once a decision is made. In other words, rational choice models operate in the domain of a pulse-like setting such as that of the Dirac delta.
This approach is evident in various rational choice models across social sciences. The literature on collective action clearly models the decision settings as pulses where agents make decisions in face of environmental constraints: departing from certain assumptions about agents’ preference orderings and the set of incentives and constraints imposed by the institutional architecture, modellers focus on the verge of the act of decision, depicting the likely scenarios according to the combination of agents’ preference orderings and strategic courses. In doing so, modellers are capable of explaining a variety of phenomena, such as the free rider, cooperation, and the tragedy of commons (Axelrod,1984/2006; Olson, 1971; Ostrom, 1990, 1998). By eschewing the incorporation of sophisticated cognitive processes into their models, not only modellers simplify the intricate complexity of the human mind, but also offer understandings about social phenomena where the structure matters for the agent’s decision-making process.
Similarly, the literature on party competition frequently departs from certain assumptions about voters’ preferences in order to design spatial models. Originally, Downs’ (1957) model, which identifies the centripetal forces at work in party competition, assumes that voters’ preferences are fixed in order to solve for the model and advance the median voter theorem. In more recent developments in this field (as well as in the fields of coalition theory and electoral behaviour), learning processes have been incorporated into models, in order to explain how agents make decisions based on their past experience in similar decision settings (Axelrod, 1984/2006; Congleton, 2019b; Ostrom, 1998; Rapoport, 1966/1999), as well as to address phenomena such as ignorance (Congleton, 2019a; Taylor, 2020). In either case, the mathematical model concedes to representing the cognitive process of learning, yet it is still preoccupied with depicting the decision setting as a whole, which eventually leads to treating the latter in a pulse-like fashion. This means that the learning process only makes sense as long as it is paired with the political context where the decision has to be made. In doing so, the setting comprises the agents’ preference orderings as structured by her past experiences, which, in its turn, has been informed by her learning process; and the environmental constraints and incentives that affect her courses of action.
It is also important to emphasise that rational choice models are generally specified in terms of mathematical expressions and are designed to solve for particular puzzles, or, more precisely, target systems (Dowding, 2016, chapter 4; Rol, 2013). Therefore, certain assumptions have to be mathematically worked out in order for the model to be tractable and operative; and it must clearly specify the target system for which it is designed. Different target systems require different models and, as a consequence, different assumptions. Rational choice’s target systems are situated around the act of choice, namely the elements immediately before a choice is made (namely, preference ordering, sets of incentives and constraints, strategies) and immediately after it (the scenario that unfolds or the predicted scenarios that may unfold). 5
Thus, the decision settings rational choice models are designed for are best seen as pulses rather than a long-phased process that culminates in the act of choice. This is especially true of decision settings where the environment exerts selective pressures on agents. In such contexts, rational choice principles–such as utility maximisation–approximate agents’ behaviour without having to resort to sophisticated psychological assumptions about the real functioning of the mind (Cox, 2004; Ostrom, 1998). Within these settings, agents are more likely to resort to simplifying heuristics that eschews the profound examination of preferences and strategies (Congleton, 2019b; Taylor, 2020). Furthermore, environmental constraints structure preference orderings such that they are well-behaved, rendering a fairly reasonable approximation of the decision-making process. Hence, this decision-making scenario resembles a discrete, single-peaked pulse, where cognitive processes lead to the final infinitesimal moment prior to t = 0, when the decision is finally made.
This approach matches the archetypical design of rational choice models. Game-theoretical analyses, for example, specify utility functions of incentives, constraints and probabilities. Solving for these functions produce a prediction – a scenario – that will follow once the decision is made. In modelling these functions, rational choice theorists do not depart from a model of the mind to build the mathematical model – and they should not, for their goal consists in explaining the act of choice with the behavioural and structural elements it entails (Taylor, 2020). Furthermore, cognitive processes such as editing and evaluation and framing effects do not offer a clear mathematical design that could be incorporated into the rational choice model. To be sure, the specification of framing effects is only made possible by conducting social experiments with real human beings. 6 Most rational choice modellers do not subscribe to this empirical commitment: they rather model political situations where decisions are expected to be made under a set of incentives and constraints. Understanding decisions as pulses is also fundamental in settings where a chain of decisions is made. Subgames are structured into a sequence of decisions that are interdependent: what happens in one branch conditions the results in the next branch. Solving for this model often requires the modeller to use backward induction, which is only possible due to the fact that she ignores the cognitive dimension underlying preference formation. In fact, in subgames, either the agent evaluates the whole chain of decisions and the corresponding scenarios for each branch of the decision tree; or she treats the various branches as a single decision scenario. In both cases, nonetheless, the act of choice preserves its pulse-like nature: she can only make a choice once the preference ordering is complete (and well-behaved) in her mind. If this preference ordering is structured for a specific branch or for the whole tree, this is a matter that may interest the modeller to the extent that it does not require an in-depth immersion into cognitive processes. For what rational choice modellers are interested in are the predictions generated by the setting, not the mental process behind that setting.
Summing up, the Dirac pulse approach allows us to assess the theoretical and empirical value of rational choice formal models on the terms set by the models themselves. Instead of dwelling about the complex nature of processes taking place in the human mind and diverting rational choice theorists’ attention from the social implications of decisions, acknowledging the pulse-like nature of rational choice models draws our attention to the scenarios unfolding from the interplay of ordered preferences and strategic decision-making based on this very set of ordered preferences. Furthermore, by treating decisions in pulse-like scenarios, modellers can mathematically simplify their models, for accounting for psychological processes may incorporate a great deal of nonlinearities into the model. Nonlinear phenomena, which are pervasive in nature and social life, are substantially more complicated, and add extra difficulties to formal modelling (Signorino, 2003; Signorino and Yilmaz, 2003) – some might even be mathematically intractable. 7 Thus, requiring rational choice models to incorporate cognitive processes may jeopardise their capability of explaining political phenomena. The challenge for rational choice theorists, more than providing an all-embracing account of the human mind, consists in designing models that are capable of representing the nature of the decision setting in a mathematically tractable way, yet true to the reality they aim to represent.
Conclusion
The analogy of rational choice modelling with Dirac pulses draws attention to the nature of phenomena political science aims to explain. Cognitive psychologists and critics in political science have indicted RCT for its lack of an overarching explanation of the human mind, more precisely of the processes of preference formation, and framing effects. As the summary of critiques has demonstrated, the literature on cognitive sciences has developed various understandings about the functioning of the human mind prior to making an actual decision.
I do not ignore nor reject the contributions of psychology: they have shed light on various issues that remained unclear, especially when confronting theoretical predictions with empirical evidence. Framing effects, perhaps, best illustrate the contribution of this research agenda, having illuminated our understanding of certain results that deviate from theoretical predictions. Nevertheless, the focus of RCT is not to offer an explanation of the mechanisms operating in the human mind. Its goal is rather modest in comparison to cognitive psychology: RCT aims to represent the interaction of agents making decisions under certain structural frameworks that offer incentives and constraints to their courses of action. As limited this goal may sound to psychologists, it serves the research interests of RCT and political science.
Therefore, treating decisions as pulses allows us to devote our attention to what matters for RCT, id est, the act of choice and the scenarios that unfold once a decision is made. Rational choice modellers should not be concerned about mental processes that cannot be precisely incorporated into the model, for this might render their models intractable and purposeless. Nor they should be expected to represent this aspect of the world into their models, for their interest is on how agents and structure operating together within a decision setting generate predictions. For a modeller, preferences must be defined prior to the act of choice, even if this definition comes in the very last infinitesimal moment before the decision is actually made. Furthermore, even if modellers were to incorporate mental processes of preference formation, risk assessment and the more controversial framing effects, this extra complexity would probably render the formal model mathematically intractable, which would, in turn, make all these efforts of full representation of the human mind counterproductive. The empirical reality models attempt to explain would be lost in the process of incorporating complex cognitive processes.
Modelling decisions is paramount to various fields of scientific inquiry in political science. Yet, as a discipline whose concerns are directed to phenomena in aggregate levels, the theoretical and empirical questions raised by cognitive psychologists are considerably misplaced, especially for RCT. Acknowledging the raison d’être of formal models in political science is fundamental to advance the debate on the issues that actually matter for modelling.
