R2.1. The concept of sensibleness

Some pro-optimality commentators (e.g., Stocker) saw the alternative to optimal behavior to be “deliberately random” behavior. However, neither we nor a single commentator endorsed a view that behavior is random or makes no sense. In fact, impassioned arguments against optimality were followed by statements about how animals “adaptively cope with the environment” (Withagen et al., para. 6) or that “it is not difficult to imagine why” a particular case of suboptimality occurs (thus indicating that the suboptimality is not random [Salinas et al., para. 6]).

We think it is fair to say that every single commentator agreed that behavior “makes sense” on some level. To highlight the substantive agreement, we find it useful to invent a term unburdened by previous inconsistent usage in order to communicate the concept that behavior “makes sense.” We propose to use the term sensibleness (see Glossary).

Sensibleness is a weak concept. We define sensible behavior as behavior that is motivated by a combination of our goals and constraints, driven by the environment and evolutionary history. It is generally accepted that natural selection produces sensible behavior in that behavior is adaptive and functional (Rosenberg & McShea Reference Rosenberg and McShea2008) – though it could be maladaptive in particular circumstances or environments. Sensible behavior may be suboptimal according to some definitions while remaining a “feature” and not a “bug” (Summerfield & Li).

R2.2. The concept of global optimality

Many commentators embraced sensibleness but argued against optimality. Rejecting optimality for these commentators does not mean that behavior is random, but that behavior is not always as optimized as it possibly could be. Another way to put this is that behavior cannot be predicted by optimality considerations alone. However, statements like “people are optimal” or “perception is optimal” are not uncommon in the literature. These statements give the impression that optimality can in fact be used to predict all of our behavior.

Just as with the concept of sensibleness, we find it useful to introduce an unburdened term for this interpretation of optimality. We propose the term global optimality (see Glossary). Global optimality is the view that behavior can be fully predicted by optimality considerations alone after taking into account our environment and evolutionary history, which together determine our goals and the constraints on our behavior. Concretely specifying globally optimal behavior would require full knowledge of these goals and constraints. At an abstract level, global optimality entails that (1) given enough information, we can unambiguously specify the globally optimal behavior, and (2) people's behavior perfectly matches the globally optimal behavior.

Let us illustrate this position with two examples. In the famous marshmallow task, people are allowed to have one marshmallow immediately or two after a certain delay. The global optimality assumption holds that in this task one can identify the globally optimal course of action for every individual and that everybody takes this course of action (though what is globally optimal may be different for different people). Both behaviors are sensible because they can easily be justified, but only one of them can be globally optimal for a given individual. A second example, from perception, is the following: Should different images presented to the two eyes be merged, should they alternate, or should one image dominate in conscious perception? Again, each possibility is sensible (can be reasonably justified), but for every pair of images only one of these possibilities can be globally optimal. The global optimality assumption holds that it is theoretically achievable to identify the best possible percept and that people actually form that percept.

Global optimality is a strong concept. It implies that each organism is constantly at its most optimized state. In other words, global optimality holds that complex organisms are able to optimize jointly the thousands of behaviors in which they engage.

R2.3. The case against global optimality

There are both theoretical and practical arguments against global optimality. Such arguments are featured in the target article and in several of the commentaries.

Theoretically, it is unlikely that any optimization algorithm could find the global optimum in an optimization task as complex as jointly optimizing all behaviors, especially considering that the algorithm would also need to optimize resource allocation to different behaviors (Mastrogiorgio & Petracca). Given the trial-and-error process behind natural selection, genetic drift, and random genetic variation across individuals, it is doubtful that all behaviors are as optimized as they possibly could be (Bachmann; Withagen et al.; Zhao & Warren).

More practically, our target article illustrates that a global optimalist view is not appealing given existing data. Our approach was to define optimality in the most “standard” way and then survey hundreds of instances of suboptimality. As might be expected, several commentators took issue with the assumptions we used to define optimality. For example, Shimansky & Dounskaia argued that “inclusion of neural effort in cost function can explain perceptual decision suboptimality” (similar sentiments were expressed by Howes & Lewis; Stocker), whereas Moran & Tsetsos showed that assuming late noise in the system leads to qualitatively different predictions for optimal behavior. We anticipated such criticisms and in the second part of our target article argued that it is impossible to specify what is optimal in a global sense.

Indeed, although we define global optimality in the abstract, we doubt it can be specified in a way that would make it empirically testable. The commentators who defended optimality unfortunately did not seriously grapple with the issue of whether it is possible to define global optimality unambiguously – though this issue was addressed by many others such as Bachmann; Brette; Love; Meyer; Noel; and Wyart. It is easy to argue that a conclusion of suboptimality in some task is invalidated because it is based on wrong assumptions. However, that argument also invalidates all of the conclusions of optimality that are based on the same assumptions.

If we follow this line of reasoning, then we must conclude that there is no firm evidence that people are either globally optimal or suboptimal. This is in fact the conclusion we reached in our target article and is precisely why we argued that the optimal/suboptimal distinction is not useful. We were especially surprised to read Stocker’s admission that this distinction is “ill-defined,” as it was followed by a fervent defense of optimality. Later on, Stocker claimed that optimality allows us to build “a quantitative model hypothesis before actually running the experiment” (para. 3). We see a contradiction here, which is at the heart of the issue at hand: If we do not have enough information to specify what is globally optimal, how can we generate any firm predictions based on the concept of optimality? All predictions can only be based on the assumptions built into a specific model and are therefore tentative and dependent on the validity of the model's assumptions.

We do not think that at present there is evidence for a global optimalist view and, given the definitional difficulties, are doubtful that there ever could be. We therefore consider the argument between sensibilists and global optimalists to be philosophical rather than scientific.

R2.4. Pinning down the commentators’ philosophical commitments

The majority of our commentators (e.g., Booth; Chambers & Kording; Cicchini & Burr; Love; Ma; Meyer; Salinas et al.; Simen & Balcı; Summerfield & Li; Wyart; Zednik & Jäkel) appeared to be sensibilists but not global optimalists. Although it is impossible to prove that all behavior is sensible, both we and all commentators consider behavior to be generally sensible.

However, it was more difficult to discern whether any commentator is a global optimalist. In particular, the commentaries of Geurts et al.; Howes & Lewis; Moran & Tsetsos; Shimansky & Dounskaia; and Stocker could be interpreted as supporting a global optimalist position. Another interpretation, however, is that these commentators only argued that optimality is a fruitful empirical strategy, without committing themselves to a global optimalist view. We explore this position in section R3. We hope researchers can use the sensibilist/global optimalist distinction to clarify their positions regarding the nature of human behavior.

Even if most (or all) researchers turn out to agree that human behavior is sensible but not globally optimal, it is still likely that there are remaining differences in opinion about the substantive question of how close behavior is to global optimality. There is potentially a vast space between a sensibilist and a global optimalist. The disagreement is perhaps best captured by the simple question: “Assuming perfect knowledge of an organism, what percent of its behavior can be predicted using optimality considerations alone?” This question can be seen as defining a “global optimality scale.” To be a global optimalist is to answer 100%. It is likely that researchers who are moved to defend optimality are high on the global optimality scale, whereas researchers who are moved to attack optimality are low on the global optimality scale. It should be stressed that where one places oneself on the global optimality scale is based on a philosophical perspective rather than empirical findings; in fact, it is unclear that we can ever find the correct answer or that such an answer even exists. We think of the global optimality scale as a useful shorthand for highlighting differences in opinion rather than as a topic for empirical investigation.

Hence, appreciating the distinction between sensibleness and global optimality does not necessarily resolve all our differences. However, it (1) forces us to clarify our positions, which may otherwise slide back and forth between implying global optimality or only sensibleness; (2) highlights agreements, which we think are more extensive than they appear on the surface; and (3) discourages arguments that misrepresent other researchers’ views as belonging to 0% or 100% on the global optimality scale. We hope this distinction provides the language to express more precisely one's position and escape false either/or dichotomies. We are all optimalists when that means sensibleness; few (if any) of us are optimalists when that means 100% global optimality.

R2.5. Relating sensibleness and global optimality to other optimality-related terminology

We defined the terms sensibleness and global optimality to clarify areas of agreement and disagreement in the optimality debate. We think these terms are useful, because careful consideration of the commentaries convinced us that common optimality-related terms are used inconsistently. Here we briefly review how our commentators used such terms and relate their meaning to our sensibleness/global optimality distinction.

R3.1. What is optimality as an empirical strategy?

Cicchini & Burr; Geurts et al.; Howes & Lewis; Moran & Tsetsos; Shimansky & Dounskaia; Simen & Balcı; Stocker; and Zednik & Jäkel argued that, at least in some cases, optimality is useful as an a priori assumption for studying behavior. In this empirical strategy, it is assumed that the correct model of a task will have a locally optimal decision rule. The idea is that assuming local optimality (often along with various other standard assumptions, as described in the target article) constrains the problem of generating a model of a task. Critically, a finding of suboptimality leads to proposals about new constraints that will preserve the presumed optimality, and these new constraints are then independently validated.

For example, in a Bayesian framework, one specifies the relevant likelihood function, prior, cost function, and decision rule – what we call the LPCD components (see Glossary). Optimality as an empirical strategy (see Glossary) consists of assuming that the decision rule (D) should be optimal and adjusting the LPC components to make it so. These LPC components are new hypotheses that are then independently tested and either supported – leading to new discoveries – or disconfirmed – prompting new LPC hypotheses to be generated. Several commentaries (e.g., Cicchini & Burr; Geurts et al.; Stocker) provided compelling examples of this strategy at its best.

R3.2. Pitfalls of optimality as an empirical strategy

Although optimality as an empirical strategy is fruitful when used carefully, it invites a number of pitfalls that require consideration by researchers who use this strategy.

R3.3. Is optimality as an empirical strategy a fruitful strategy?

In the commentaries, it seemed that researchers who thought optimality as an empirical strategy was useful were high on the global optimality scale, and vice versa. However, views on tools and substance need not be this tightly linked. We believe that as long as behavior is sensible, optimality as an empirical strategy is likely to be fruitful.

At the same time, other research strategies can also be fruitful. A great many discoveries have been made by explicitly rejecting optimality (e.g., Kahneman & Tversky Reference Kahneman and Tversky1979) or not considering optimality (e.g., Gibson & Radner Reference Gibson and Radner1937; Hubel & Wiesel Reference Hubel and Wiesel1970; Müller-Lyer Reference Müller-Lyer1889). Hence, we agree with Barth et al.; Danks; and Zednik & Jäkel on the importance of maintaining a diversity of strategies for understanding human behavior. What is critical is to uncover phenomena that are replicable and generalizable, not the strategies that we use to uncover them.

R4.1. Bayesian tools

Multiple commentators argued for the usefulness of Bayesian tools in research on perceptual decision making (Chambers & Kording; Cicchini & Burr; Geurts et al.; Howes & Lewis; Nardini & Dekker; Schultz & Hurlemann; Stocker). Commentators noted that Bayesian tools provide a compact, transparent, and explicit formulation of how to integrate different pieces of information. The Bayesian formulation can also provide a benchmark for human performance (but see Noel for an argument against this claim). Commentators who seemed to adopt optimality as an empirical strategy also noted that Bayesian tools are a natural analytic framework for this strategy (Geurts et al.; Howes & Lewis; Stocker); optimality is naturally expressed in Bayesian terms. We agree with all of these arguments for the usefulness of Bayesian tools, and we used these tools extensively in our target article.

At the same time, it is important to acknowledge the limitations of Bayesian tools. Several commentators pointed out that Bayesian approaches do not provide insight into some of the hardest problems of perceptual decision science. Summerfield & Li noted that the LPCD framework does not automatically provide meaningful explanations of behavior and pointed to the idea of efficient coding as a more satisfying kind of explanation. Brette observed that a major challenge in developing a standard observer model under the LPCD framework is specifying the generative model. Perhaps another way to say this is that the hardest part of understanding perceptual decision making is understanding perception itself. In fact, according to Brette, an overemphasis on using Bayesian approaches can obscure the difficulties that perception poses and can lead researchers to ignore or underemphasize these difficulties. Finally, Bachmann argued that Bayesian models may not map well onto the phenomenology of conscious perception. We agree with these perspectives, too.

In the end, this is almost always the nature of a single tool (e.g., a hammer): It is useful in certain contexts (e.g., putting a nail in a wall) but is virtually never sufficient by itself to solve a larger problem (e.g., building a house). Overly high expectations of Bayesian tools can lead to both excessive reliance on them and excessive criticism of them.

R4.2. Bayesian theories

In the field at large, there is a substantive debate about Bayesian theories (i.e., theories that hold that the brain explicitly represents LPCD components and explicitly computes using Bayes’ theorem). However, no commentator defended Bayesian theories outright, so here we focus on their limitations. Simen & Balcı reminded us of the computational demands of exact Bayesian inference, which they argued make Bayesian theories unlikely to be true. “Surely then,” they remarked, “heuristics are the best we can do, in general” (para. 8). Turner et al. argued that sometimes behavior is not “best understood in terms of the elements of LPCD” (para. 3) and suggested that process models (see Glossary) could better capture the true mechanisms underlying behavior. We agree that these are both serious difficulties for Bayesian theories. Withagen et al. gave an evolutionary argument for rejecting a Bayesian approach and questioned a foundation of Bayesian theories, the notion of inference, arguing that the primary challenge for human perception does not stem from uncertainty about the environment. We agree that more attention should be given to the interaction between perception and action, but we consider dealing with uncertainty to be a central issue in perceptual decision making.

R4.3. Conflating Bayesian tools and Bayesian theories

In section R3, we distinguished between optimality as a substantive hypothesis and optimality as an empirical strategy. In a similar fashion, it is important to distinguish between Bayesian theories (which are substantive claims about the nature of the internal representation) versus Bayesian tools (which simply refer to an analytical framework). Barth et al. discussed this issue extensively and pointed out that successful use of Bayesian tools does not imply that Bayesian theories are true. Danks emphasized a similar distinction. He referred to Bayesian theories as a “realist” interpretation and to Bayesian tools as an “as-if” interpretation of the Bayesian formalism. Danks explained the difference between Bayesian theories and Bayesian tools and the importance of not conflating them.

In fact, Danks criticized our target article for starting out by using Bayesian approaches as tools and then switching to using them as substantive theories. Let us clarify that we intended to use the LPCD framework purely as a tool. The positive proposal in section 5 of our target article was intended to be completely approach-agnostic. This is why we stated that the standard observer model may eventually take the form of “a Bayesian model … a ‘bag of tricks’ … a neural network … and so forth” (sect. 5.2, para. 1). The hypotheses that we thought might generalize across tasks, such as “placing a premium on accuracy” (sect. 5.2, para. 5) were not intimately linked to the Bayesian approach (we classified them under different LPCD components purely for convenience). We see the question of whether such hypotheses generalize across tasks as independent from whether one supports Bayesian theories. Nevertheless, Danks's criticism of our target article shows the practical difficulties involved in keeping Bayesian tools clearly separated from Bayesian theories and indicates the need for enhanced conceptual clarity on this issue.

R4.4. Bayesian approaches and optimality

Continuing the theme of conflating concepts that should be kept separate, the term Bayesian was sometimes equated with optimal in the commentaries, a common issue we addressed in the target article. For example, Stocker argued that “Bayesian decision theory and the optimality assumption are in many ways synonymous; without the latter, the former is not meaningful” (para. 3). This view was also endorsed by Zhao & Warren. We agree with these commentators that Bayesian tools are a natural choice if one adopts the use of optimality as an empirical strategy. However, optimality as an empirical strategy can be pursued independently of Bayesian tools (e.g., maximizing a non-Bayesian utility function), and Bayesian tools can be used without assuming optimality (e.g., an LPCD model with a suboptimal decision rule) – so the two should not be equated.

Bayesian tools and theories should also be clearly separated from claims about optimality (Ma Reference Ma2012). As just noted, the Bayesian formalism can be used to model decision behavior, whether that behavior is optimal or not, as with a suboptimal decision rule (Chambers & Kording). Similarly, Bayesian theories could be substantively correct in that the brain explicitly represents LPCD components and performs Bayesian computations, but if the brain represents the wrong components, then behavior could still be suboptimal. Hence, claims about both local and global optimality should be kept separate from Bayesian tools and theories.

R5.1. The benefits and limitations of a standard observer model

In our target article, we communicated a vision of a standard observer model that will predict observers’ behavior on a wide variety of perceptual tasks. We then urged researchers to actively work toward building such a model. A number of commentators reflected on the benefits and limitations of this type of effort.

The benefits of building a standard observer model were recognized by a number of commentators (e.g., Nardini & Dekker; Meyer; Simen & Balcı; Schultz & Hurlemann; Turner et al.; Wyart). They noted the utility of having a model that applies across a variety of tasks and quantitatively predicts novel behavior. Further, the ability to make quantitative predictions would also make the standard observer model readily falsifiable.

An observer model is a prerequisite for making any new predictions. Stocker appeared to disagree, contending that the principle of optimality could generate predictions from first principles, because optimality “allows us to formulate with relative ease an observer model for any specific task” (para. 3). However, neither global nor local optimality lead to specific models. Global optimality is not specified in sufficient detail to model specific tasks, and local optimality can only determine the decision rule after the rest of the model is formulated (e.g., as Moran & Tsetsos showed, local optimality makes different predictions depending on whether observer models feature late noise or not). At the same time, we agree with Simen & Balcı that whatever form the future standard observer model takes, some “utility function will be found at [its] core” (para. 2). In other words, the model would imply that behavior is sensible.

Some commentators were concerned that a focus on building a standard observer model eschews the question of why behavior is the way it is (Summerfield & Li; Zednik & Jäkel) and that such a model is too descriptive (Howes & Lewis; Mastrogiorgio & Petracca; Simen & Balcı). We are sympathetic to these concerns, but at the same time, we doubt one could satisfactorily answer why behavior is the way it is before understanding in much more detail what people actually do and how they do it. Nevertheless, we think that various components of the standard observer model can and will be inspired by normative (i.e., sensibilist) considerations like the ones Summerfield & Li discussed. In the end, we support any approach that yields predictive, generalizable models of human behavior.

Finally, Howes & Lewis criticized observer models for being overly flexible because they are not constrained by a locally optimal decision rule. However, it is important to point out that a locally optimal decision rule is no solution for excessive flexibility either (Bowers & Davis Reference Bowers and Davis2012a). Avoiding excessive flexibility is why the components of any model need to be independently verified, and this is exactly the principle that we advocated for building the standard observer model.

R5.2. What to do about individual differences?

Perception science has traditionally focused on what is common among all observers rather than what is different. However, as several commentators pointed out, there are meaningful individual differences in how people perform perceptual decision-making tasks (Bachmann; Booth; Love; Mastrogiorgio & Petracca; Schultz & Hurlemann; Withagen et al.; Zhao & Warren). Such differences create a big additional hurdle for the global optimalist view (Bachmann; Booth; Withagen et al.), which must explain why such differences should be globally optimal in the first place. Individual differences create challenges for observer models, too. Such differences are typically accommodated by free parameters that can vary across individuals (such as the level of internal noise), but it is possible that in some cases different model components may be needed for different people (e.g., different observers may use qualitatively different decision rules). In our view, individual differences ultimately bring opportunities to develop and test observer models. The predictive validity of a model can be tested not only by generalizing to a new stimulus or task but also to the individual differences that could be expected for the stimulus or task.

R5.3. Finding the best approach for building a standard observer model

One of the most pressing questions for a researcher who wishes to contribute to building a standard observer model is what form this model should take. Stocker criticized our vision as “fuzzy,” which is not wrong. We certainly do not claim to know what the final model will be. As some commentators thought we were advocating for bag of tricks or neural network models (Geurts et al.; Stocker) and others thought we were advocating for LPCD models (Danks), we realized we did not make clear that our position is one of agnosticism, not advocacy. We support a diversity of modeling approaches in our field (Barth et al.; Danks; Zednik & Jäkel), with the common theme that all model components should be tested in multiple ways. There are many types of modeling efforts in the field right now, and this seems to us like a good thing.

Several broad views of how to proceed in studying perceptual decision making were brought up in the commentaries. None of them are specific observer models – they do not generate predictions for behavior – but they provide overarching frameworks and philosophies that guide research and can lead to the development of observer models. We have already discussed optimality as an empirical strategy (Geurts et al.; Howes & Lewis; Moran & Tsetsos; Shimansky & Dounskaia; Stocker) and the LPCD framework as potential approaches. Howes & Lewis argued for “computational rationality” (or “bounded optimality”), which is a particular take on optimality as an empirical strategy. Mastrogiorgio & Petracca advocated for “bounded rationality,” in which decision rules are selected to reach a good enough level of performance, rather than the best possible performance. Such “satisficing” is a clear example of sensibleness without global optimality. (In the target article, we mischaracterized bounded rationality as emphasizing “the optimality of the decision rule”; bounded rationality does not assert that decision rules should be globally or locally optimal.) Turner et al. and Salinas et al. discussed the option of implementing process models at Marr's algorithmic and implementational levels, which need not be formulated using Bayesian tools. We see value in all of these approaches and perspectives.

R6.1. Modeling suggestions

A few commentators made specific methodological suggestions related to modeling human behavior, which may prove useful in the development of a standard observer model. Ma and Love both discussed the importance of model comparison, a point with which we wholeheartedly agree. Ma introduced the notion of factorial model comparison, which allows researchers to systematically explore a large number of factors that could jointly affect behavior. Turner et al. advocated for the use of Bayesian statistics to compute posterior distributions for each model parameter rather than a single point estimate of the parameter's most likely value. Shimansky & Dounskaia proposed that to validate models, a useful strategy is to use a subset of the data to determine unknown parameters and test the resulting model using the rest of the data. Such cross-validation is already common in many fields and should indeed be adopted more often in perceptual decision making. Finally, Wyart identified a key issue in evaluating observer models: Given that behavioral data are always noisy, how will we know when a model predicts the data well enough? Wyart proposed a strategy for determining this “good enough” level, also called a “noise ceiling,” for an individual observer by measuring the observer's decision consistency across repeated trials.

R6.2. Specific topics

Many commentators gave examples of suboptimality not included in our target article: Salinas et al. on reward processing; Noel; Turner et al.; and Zhao & Warren on cue combination; Chambers & Kording on sensorimotor behavior; and Turner et al. on category learning. We appreciated these additional examples of the inability of “standard” optimality considerations to predict behavior. Though we strove to be comprehensive in our survey, there are likely many more such examples in the published literature.

Some commentators argued that a specific effect we cited as locally suboptimal is in fact locally optimal if further factors are taken into account. Cicchini & Burr contended that serial dependence is optimal if the previous perceptual information can be used to reduce the uncertainty about the current stimulus. Summerfield & Li suggested that many of the surveyed suboptimalities may arise from the principles of efficient coding. We find both of these to be promising hypotheses (efficient coding considerations were discussed in the target article). Finally, Simen & Balcı pointed out that our list of suboptimal findings in speed-accuracy tradeoff (SAT) tasks included two articles (Balcı et al. Reference Balcı, Simen, Niyogi, Saxe, Hughes, Holmes and Cohen2011b; Simen et al. Reference Simen, Contreras, Buck, Hu, Holmes and Cohen2009) that in fact primarily showed optimal behavior. Their commentary provided a balanced view of findings of optimality and suboptimality in SAT tasks.

Other commentators showed how our target article can be extended to development (Nardini & Dekker), computational psychiatry (Schultz & Hurlemann), detection of abnormalities in medical images (Meyer), and other fields beyond perceptual decision making (Barth et al.). Although we focused our target article on perceptual decision making in healthy adults, we strongly support such extensions.

Finally, Chambers & Kording criticized our claim that there is a “current narrow focus on optimality” (target article, sect. 1, para. 5). They surveyed articles from the last 23 years and found a similar proportion of optimality and suboptimality claims. However, their data, which they graciously shared with us, suggested a potential difference in the visibility of optimality versus suboptimality claims as measured by the impact factors (IFs) of the journals in which they appeared. Indeed, the numbers of optimality/“near optimality”/suboptimality claims were: 4/0/0 for IF > 10, 7/2/1 for IF > 8, and 9/2/1 for IF > 6. We note that it is difficult to know what the unbiased proportions of optimality and suboptimality claims should be.