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Edited by: Andrew S. Kayser, University of California, San Francisco, United States

Reviewed by: Hansem Sohn, Massachusetts Institute of Technology, United States; John Pearson, Duke University, United States

This article was submitted to Decision Neuroscience, a section of the journal Frontiers in Neuroscience

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In the face of limited computational resources, bounded rational decision theory predicts that information-processing should be concentrated on actions that make a significant contribution in terms of the utility achieved. Accordingly, information-processing can be simplified by choosing stereotypic actions that lead to satisfactory performance over a range of different inputs rather than choosing a specific action for each input. Such a set of similar inputs with similar action responses would then correspond to an abstraction that can be harnessed with possibly negligible loss in utility, but with potentially considerable savings in information-processing effort. Here we test this prediction in an identification task, where human subjects were asked to estimate the roundness of ellipses varying from a straight line to a perfect circle. Crucially, when reporting their estimates, subjects could choose between three different levels of precision corresponding to three levels of abstraction in a decision-making hierarchy. To induce changes in level selection, we manipulated the information-processing resources available at the perceptual and action stages by varying the difficulty of identifying the stimulus and by enforcing different response times in the action stage. In line with theoretical predictions, we find that subjects adapt their abstraction level depending on the available resources. We compare subjects' behavior to the maximum efficiency predicated by the bounded rational decision-making model and investigate possible sources of inefficiency.

Consider the following guessing game where you have to identify different animals drawn from a sample set of photographs. Assume that the sample set includes specimens, such as different kinds of cats (e.g., Persian, Siamese), dogs (e.g., German Shepherd, Rottweiler), snakes (e.g., Ball Python, Corn Snake), lizards (e.g., Chameleon, Leopard Gecko), frogs (e.g., Poison Dart Frog, European Tree Frog), and salamanders (e.g., Axolotl, Fire Salamander). When shown a particular exemplar, you can choose to respond with the precise name of the specimen (e.g., Rottweiler) or you can content yourself with identifying a subset corresponding to an abstract category (e.g., Dog) or even a larger subset corresponding to a more abstract super-category (e.g., Mammal, Reptile, or Amphibian). Let us assume that a precise identification is rewarded with $1, identifying a category with $0.8, and naming a super-category with $0.6. A misclassification results in no payment. Such a payment scheme naturally affords different levels of abstraction, since it allows for generic responses for various subsets of exemplars. The degree of abstraction can be measured by the amount of Shannon information contained in each subset, ultimately counting the effective number of possibilities. For example, in our sample set with 12 possibilities, identifying a specimen requires ~3.6 bits of information, identifying a category ~2.6 bits of information, and selecting a super-category distinguishes between three possibilities, i.e., ~1.6 bits of information. Clearly, in the absence of information-processing limitations—in our example the availability of at least 3.6 bits—the best response is to always identify each exemplar by its exact name.

Limitations in information-processing can have many reasons, such as limited processing time or limited memory capacity (Hick,

The trade-off between utility and information lies at the heart of information-theoretic bounded rationality (Braun et al.,

The bounded rational model of abstraction in the guessing game. In our introductory example there are twelve possible world states

Here, we address the question of how efficiently human subjects can abstract in a hierarchical decision-making task, where subjects (i) select a partition with a given level of abstraction, and (ii) select the correct response inside the partition. As a behavioral assay we use an absolute identification task where subjects are offered multiple levels of precision in which they identify ellipses depending on their degree of roundness. This way we can manipulate the bounds of information-processing both by making the perceptual task more difficult by distorting the visual stimulus and by varying the processing time allowed in the action selection stage. Absolute identification tasks lend themselves for the study of limited information capacity, because of their finite number of states allowing for intuitive information measures. Accordingly, absolute identification tasks have been extensively studied in the literature in the context of information theory (Norwich,

Eleven subjects, six females and five males, participated in this study. The lead author was one of the subjects (S7). All other participants provided written informed consent for participation and were remunerated with a base payment of 8 Euros per hour plus an extra incentive according to performance up to 12 Euros per hour in total. The participants were undergraduate students with normal or corrected to normal vision and no known motor deficits.

The task was run through a graphical user interface based on Psychtoolbox in MATLAB^{TM} R2017a and displayed on a touch screen (Dell 27 Monitor-Touch-P2714T, 27″, 68.6 cm VIS) with maximum refresh rate of 60

The experiment was divided into a training phase to facilitate subjects' adaptation and the subsequent evaluation stage. Both together required around 7 h per subject to be completed, distributed along 2 consecutive days to avoid performance loss due to fatigue.

Subjects were asked to identify the roundness of animated ellipses presented to them in short video clips. They could provide their response by tapping differently sized touch screen buttons on a response panel. The panel represents intervals on a linear roundness scale with hierarchical organization reflecting three different levels of precision decreasing from top to bottom in three stacked rows (see

Experimental setup.

The set of world states in our task consists of twelve possible ellipses whose major axis is vertically oriented with a constant unit length that is later scaled up to ~10

with

The stimuli are generated from the twelve world states by creating video clips showing a (~12 × 12^{2}) square filled with black moving dots on a gray background. In order to manipulate subjects' perceptual information-processing capacity, the majority of dots are

Stimulus parameters.

Easy | 100 | 250 | 35 | 500 |

Medium | 75 | 400 | 40 | 500 |

Hard | 50 | 450 | 40 | 500 |

Importantly, the manipulation of the perceptual difficulty does not render the stimuli ambiguous, but only makes them more difficult to process for human subjects. In all three perceptual conditions the stimuli can, in principle, be identified perfectly, as the identity of the stimulus is preserved. To demonstrate this fact, we designed an automatic recognition algorithm for ellipses with _{w} of the minor semi-axis for every point with coordinates (

where for points that are part of the ellipse _{w} will also be random. To distinguish better between random and non-random points, the recognition algorithm compares two consecutive video frames with the idea that for pairs of non-random points the estimate â_{w} should be consistent, whereas for pairs of random points the estimates for â_{w} will differ. Pairs of points across two frames are determined simply by minimum distance. We throw away all pairs of points whose estimates for â_{w} are larger than a threshold value of 0.1. With the remaining points we create a histogram of all values of â_{w} over all frames. The stimulus _{w}. As shown in

In the top left corner of the screen subjects could activate a start/pause button to commence each batch of trials. Each trial involves two parts: a perception stage during which the stimulus is displayed, and a subsequent action stage where subjects indicate their response on the touch screen. The perception stage starts with the presentation of a video clip according to one of the above-mentioned stimulus conditions. After 500 ms the video disappears and the response panel shown in

The design of three perceptual processing conditions and two response time conditions leads to a total of six conditions:

First, subjects are allowed to experience a linear roundness scale where they can observe how their horizontal finger position is mapped continuously into the roundness of an ellipse with 0 ≤ λ ≤ 1. Second, subjects are exposed to stimuli of the easy perceptual condition with a duration of 2 × 500

Subjects are presented with a stimulus

where _{0}(

and (ii) by choosing the optimal prior _{0}(

where _{p(w)}[_{KL}(

with partition sum

By traversing β from zero to infinity in Equation (4) we generate a family of bounded rational solutions

between stimuli

Both quantities can be determined by estimating subjects' response probabilities _{exp}(^{exp}, ^{exp}} in the utility-information-plane, as shown in

where ^{min} is the highest feasible theoretical utility in the absence of computational resources, and ^{max} indicates the maximal theoretical utility of a channel whose information processing rate is equal to ^{exp}.

Subjects' response distributions. Estimated response distributions _{exp}(

The optimization problem in Equation (3) penalizes computational complexity in terms of the mutual information between actions and stimuli. An action that is exclusively selected for a particular stimulus, and that is not chosen under other circumstances, is expensive in terms of mutual information. One way to reduce informational costs while optimizing the expected utility consists in selecting an action that yields a “good enough” expected utility for many different inputs. In other words, different world states end up being treated as if they were the same. This is the essence of abstraction (Genewein and Braun, _{l}). The high-level decision with distribution _{l}|_{1} ∈ {Mammal, Reptile, Amphibian}. At the intermediate level _{2} ∈ {Dog, Cat, Snake, Lizard, Frog, Salamander}. And at the highest resolution _{3} = Rottweiler. Since the level is part of the decision, it is treated as a random variable _{l} = _{l}) =

Mathematically, the decision-making problem (3) can be equivalently reformulated as

which trades off expected utility against computational costs of the abstraction level selection

where

The marginal _{l} ∈ {0.6, 0.8, 1.0}, _{l} ∈ {3, 6, 12} and ^{*}(

and accordingly the choice of the abstraction level

to characterize the level transitions, as indicated by the shaded region in

When we measure subjects' efficiency based on Equation (7) and find a substantial deviation from the efficiency frontier, we can ask what causes may underlie this inefficiency. In general, one could argue that there might be more specific constraints that we have not (yet) considered in the basic form of the bounded rational model—for example, the generic distributions _{Ω} we are searching could be further constrained to be of Gaussian shape _{θ}(_{w} = (μ_{w}, σ_{w}). Formally, such constraints are represented by restricting the search space in the optimization problems (1)–(3) to a permissible subset Γ, i.e.,

where

By solving (11) in the constraint set Γ for different values of the Lagrange multiplier β and determining the corresponding expected utility 𝔼[

Non-adaptive priors _{0}(

Subjective utility functions

– utility distortion of the actual payoffs (e.g., risk attitude), or

– utility that allows for neighborhood relationships (e.g., Shepard's similarity).

Constraints Γ ⊂ ℙ_{Ω} on the shape of permissible distributions, modeling

– irreducible perceptual or motor execution noise, or

– parameterized decision strategies with fixed noise structure.

Note that in contrast to Equations (1)–(3), Equation (11) does in general not allow for analytical solutions.

_{0}(

The prior that is uniform across levels would be

Instead of uniform priors, subjects could of course also have arbitrary prior beliefs across levels

where the probabilities of the levels are related according to 3_{1} + 6_{2} + 12_{3} = 1. When assuming a fixed prior, we can find the best-fit values of _{i} for each subject and all conditions. Necessarily, all these priors will induce inefficiency compared to the utility-information efficiency frontier under optimal priors.

^{α}. For concave utility functions, decision-makers are risk-averse, for convex utility functions they are risk-seeking. Since we only have four utility values {0, 0.6, 0.8, 1} in our experiment, we can also explore the space of all possible local distortions by replacing 0.6 with _{0.6} ∈ (0, 1) and 0.8 with _{0.8} ∈ (_{0.6}, 1). This way, the subjects can express locally different risk attitudes, while the order of the utilities is preserved and only their absolute and relative values to each other change.

More radically, the subject could have a completely different utility function than the one stipulated by the experimenter in the task. In particular, we consider blurring the utility function as a direct way of introducing neighborhood relationships between world states. This presumes that in subjects' minds it is better to have a close miss than a distant miss. A simple way to obtain a blurred utility function

where

Applying the same kernel parameter θ across the three levels implies that a near-miss of a button leads to the same relative reduction in utility in all levels.

with θ = {(μ_{w}, σ_{w})}_{w}. Since decision noise cannot be optimally exploited in this case, such decision-makers will be inefficient. Note that, just like in (16), parameterized decision strategies may introduce neighborhood relationships.

More sophisticated parameterized models can be obtained by assuming decision-makers with internal states. We consider a decision-maker that is not fully able to identify the state of the world _{θ}(_{θ}(_{θ}(_{θ}(_{pθ(w|x)}[

As a possible transducer model we consider a truncated Gaussian model,

where

Thus, _{T}(_{0}(

In the case of additional execution noise, we may assume that the observed action _{p(x|a)}[

Unlike in our previous study (Schach et al.,

Our experiment is a hierarchically organized absolute identification task involving two information-processing stages, stimulus perception and action planning, where action planning again consists of two stages, choosing a level of abstraction and identifying the stimulus given that level. This way we can manipulate the information processing capacity both in the perception and action channel to determine the relationship between overall information resources and abstraction through level selection. During stimulus presentation subjects are faced with a video clip representing the world state

Experimental conditions.

easySlow | Easy | 5,000 |

easyFast | Easy | 450 |

mediumSlow | Medium | 5,000 |

mediumFast | Medium | 450 |

hardSlow | Hard | 5,000 |

hardFast | Hard | 450 |

We evaluate subjects' performance by their average utility 𝔼[_{exp}(_{exp}(

In

Subjects' performance depending on abstraction level across conditions.

In the first part of the study we compare subjects' performance against the normative performance of a bounded rational decision-maker with information constraints. To this end, we determine subjects' average utility 𝔼[_{exp}(

Changes in mutual information and expected utility. ^{2} = 0.78. ^{2} = 0.93.

The main result comparing the absolute values of mutual information and expected utility to the bounded rational optimum are presented in

Subjects' choice efficiency.

When comparing the theoretical curves and experimental data points, it appears that there is a systematic performance gap in

_{0}(_{0}(

Non-optimal priors.

Subjective utility distortion.

A slightly more radical reason why subjects' behavior might seem inefficient could always be that they optimize a completely different utility function than proposed by the experimenter. One example, is the valuation of a near-miss. The basic bounded rationality model with discrete stimuli and actions does not care whether we miss by a small or large margin, however subjects may feel that it is better to have a near-miss than a far-miss. We therefore consider blurred utility functions generated by the kernels in Equation (16), thereby creating neighborhood relationships by assigning positive utility to near misses. The blurred utility with exponentially decaying utility is depicted in

Subjective utility with neighborhood relationships. _{1} in Equation (16) with a decay constant of θ = 2.05 leads to non-zero off-diagonal entries.

Process-dependent noise: Gaussian responses.

Other search space constraints like perceptual noise with a fixed structure or neighborhood relationships in perceptual maps could further compromise performance, leading to a decrement both in expected utility and in mutual information. A standard model for perceptual noise is the Thurstonian model (19) that assumes internal states

We can see subjects' behavior quantified with respect to this constrained performance curve in

Process-dependent perceptual noise: Thurstonian model.

In the Gaussian model (19) we need to artificially truncate the hypothesis space which introduces asymmetry, as roundness is only defined on the unit interval [0, 1]. We therefore also consider an inference model (20) that is slightly more abstract, but naturally constrained on the unit interval. In particular, we investigate a Binomial model, where the latent variable θ ∈ [0, 1] can be thought to reflect the unobserved roundness of an ellipse, and where the observation is given by binary strings, i.e., sequences of 0 and 1 outcomes, reflecting round and non-round cues present in the stimulus. Naturally, this is not a mechanistic model of perceptual processing, but simply phenomenological. Accordingly, the variance of the distribution is elevated for medium roundness values, and smaller for extreme roundness values, which makes it easier to identify extreme stimuli. The Bayesian belief

Process-dependent inference noise: Binomial model.

In order to assess the adequacy of our models in a quantitative manner, we have performed a 10-fold cross-validation over the empirical posteriors

Model cross validation.

Instead of selecting between 21 actions (_{1} and β_{2}, respectively), thereby optimizing

Crucially, such a hierarchical model allows modeling the precision of choosing a level specific for each world state, that could for example reflect the decision-maker's ability to estimate the effective utility Δ_{1} across all three levels (see _{1}, as they rarely spread across all three levels, and if so, then only with high stochasticity (compare

Hierarchical decision models.

In this study we have argued that the theory of bounded rationality with information constraints provides a conceptual and normative framework to reason about abstraction and hierarchical decision-making. We have demonstrated the application of this theory to an absolute identification task where subjects could not only identify a given stimulus on a scale, but also choose the precision of that scale, which corresponds to different levels of abstraction. In order to encourage subjects to change their chosen abstraction levels, we manipulated information resources in two ways, by corrupting the stimuli and thereby aggravating stimulus identifiability and by constraining reaction time. This allowed for testing subjects' efficiency of identifying different stimuli and selecting the appropriate level of abstraction across a broad range of information conditions. We found a systematic efficiency gap in subjects' behavior across that range (~70% efficiency), which implies that with the measured amount of choice uncertainty it would in principle be possible to achieve considerably better performance (see

The assumption of bounded rationality with fixed priors is amongst the worst explanations of subjects' choice behavior and the efficiency gap—see cross validation results in

Finally, we analyze process-dependent search space constraints, where the nature of the constraint depends on the particular framework under consideration, such as Markov Chain Monte Carlo planning or particular reinforcement learning models (Wang and Sandholm,

From our models, the two perceptual models are the best fitting models explaining subjects' responses in a cross-validation. Prima facie, it seems natural to classify these inference models as perfectly rational decision-making models under uncertainty. However, the perceptual uncertainty in our experiment was not a result of ambiguity inherent in the stimuli, but a result of imperfect perceptual processing. Instead of conceptualizing the decision-maker as a composite system of noisy perception and perfect action selection, where the action selection stage is trying to undo the noise induced by perception by doing inference, we may thus regard the inference process as an instance of bounded rational decision-making under information constraints, where the noise parameter effectively plays the role of a rationality parameter. This can be seen as follows. Given a parameterized distribution _{θ}(_{Ω} in the optimization problem in Equation (11)—of a system are responsible for not reaching the optimum. In our analysis we found that subjects' level selection was close to optimal in terms of utility, but subjects also showed bias in the information produced at each level because of neighborhood relationships that are neglected in the basic model.

It has been previously criticized (Luce, _{2}21 ≈ 4.4bits. One might argue, however, that this overstates the number of possibilities in some sense because these 21 actions are not independent and, in fact, represent only 12 possible actions with varying degree of granularity. In this interpretation, pressing a large button in the bottom is equal to pressing 4 of the 12 actions with uniform distribution, and pressing a medium sized button to pressing 2 out of 12 actions with equal probability. In this case, a random action selection would generate log_{2}12bits for a small button and log_{2}3bits in case of a large button. We could then pose a new optimization problem, where the decision-maker maximizes expected utility over the space of probability distributions over 21 actions, but where the information constraint is determined as if there were only 12 actions. However, the efficiency gap widens under this assumption which results in a decrement in the cross validation performance compared to the original information cost, both in action and level selection (see

Another possible criticism is that our experimental design conflates bounded rationality with the study of confidence in decision-making (Fleming and Daw,

Our study is part of a large body of research that has investigated absolute identification with information-theoretic means in tasks involving judgements on figures (Lacouture and Marley,

The formation of partitions, categories and abstractions has a long history in the psychological sciences (Reznikova,

The datasets generated for this study are available on request to the corresponding author.

The studies involving human participants were reviewed and approved by Ethics committee of Ulm University. The patients/participants provided their written informed consent to participate in this study.

CL-L, SG, and DB designed the experiment. CL-L performed the experiments and analyzed the data. CL-L and SG generated the predictions from computer simulations. SG and DB supervised the project. CL-L and DB wrote the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at: