Uncalibrated Models Can Improve Human-AI Collaboration
Abstract
In many practical applications of AI, an AI model is used as a decision aid for human users.
The AI provides advice that a human (sometimes) incorporates into their decision-making process.
The AI advice is often presented with some measure of “conﬁdence” that the human can use to calibrate how much they depend on or trust the advice.
In this paper, we present an initial exploration that suggests showing AI models as more conﬁdent than they actually are, even when the original AI is well-calibrated, can improve human-AI performance (measured as the accuracy and conﬁdence of the human’s ﬁnal prediction after seeing the AI advice).
We ﬁrst train a model to predict human incorporation of AI advice using data from thousands of human-AI interactions.
This enables us to explicitly estimate how to transform the AI’s prediction conﬁdence, making the AI un calibrated, in order to improve the ﬁnal human prediction.
We empirically validate our results across four different tasks—dealing with images, text and tabular data—involving hundreds of human participants.
We further support our ﬁndings with simulation analysis.
Our ﬁndings suggest the importance of jointly optimizing the human-AI system as opposed to the standard paradigm of optimizing the AI model alone.
Introduction
In safety-critical settings like medicine, AI is often integrated as interactive feedback, allowing a human to decide when and to what extent the AI’s “advice” is utilized [29].
For example, in medical diagnosis tasks, this places the AI in a similar category as lab tests or other exams a doctor may order to aid in diagnosis.
This mitigates the often black box nature of AI algorithms that limit a user’s trust and usage of AI [6, 24, 34, 18].
Typically, these algorithms are designed and optimized to minimize loss using standard training objectives independent of the human end user.
In this paper, we question this premise and ask whether a joint optimization of the entire human-AI system is possible.
In particular, we revisit the conventional wisdom that models with calibrated conﬁdence enable more accurate transfer of prediction uncertainty between models and/or humans [8].
We propose calibrating a model for human use, with the intuition that humans are often unable to accurately integrate calibration information correctly and so may beneﬁt from uncalibrated information.
This approach is complementary to existing ideas on how to help humans utilize AI advice (e.g., through some form of explanation accompanying the advice or through feedback on model/human performance)
We draw on prior work modeling human predictions [30] and assume black box access to an AI algorithm which provides advice.
From data collected on thousands of human interactions, we ﬁt a model for how humans incorporate (or ignore) AI advice.
Using this learned human behavior model, we optimize a monotonic transformation that modiﬁes the AI’s reported conﬁdence to maximize human-AI system performance (measured as the human’s ﬁnal accuracy).
We demonstrate ﬁrst in simulations and subsequently with real-world data using crowd-sourced human study participants that our modiﬁed AI advice results in higher overall performance.
These results suggest there is merit to jointly optimizing human-AI systems and that an optimal AI algorithm may not be optimal for human use.
Our contributions We propose and empirically validate a framework for optimizing an AI algorithm with respect to the human end user, resulting in human-calibrated AI that calibrates AI models for human use.
We only require black box access to the AI.
Our framework relies on a model for human behavior, which we create using a large collection of empirical data on human interaction with AI advice.
To the best of our knowledge, these are some of the ﬁrst results using an empirically validated model for human behavior to optimize AI performance in a collaborative human-AI system.
We would like to emphasize, however, that this these methods are not currently suitable for practical use.
Ethical considerations One of the main conclusions of our study is that human users do not consider conﬁdence in the “optimal” way (i.e., how we model conﬁdence in statistics).
In an attempt to correct for this, we present both empirical and simulation-based evidence that modifying the presented AI advice improves overall performance.
One potential question about our approach is: does modifying the AI’s conﬁdence constitute misleading the user?
We agree – in our experiments, the AI can mislead the user.
However, our goal here is not to propose a method that should be used in practical applications, and rather to highlight the importance of modeling human interaction when designing
AI for human use.
Related works The experimental setup we use comes from the advice utilization community.
Here, studies often use the JAS framework for measuring and comparing the effect of different forms of advice [28].
In many of these studies, algorithmic advice is manipulated to elicit various responses from humans.
We take this a step further in our paper by manipulating the advice to improve model calibration with respect to the user and improve performance.
Previous work has studied the effect of peer vs. expert advice [15], automated vs. human advice [22, 5, 15, 21], task difﬁculty [7], and advice conﬁdence [25, 26].
Some work has also sought to characterize the perceptions people have of an AI when making decisions with AI advice and how these perceptions affect trust and utilization of the AI [13].
Of particular relevance to our work is [25], which provides evidence that uncalibrated, conﬁdent advice is more utilized by humans in certain settings.
We also leverage a proposed model for human behavior from [30].
The conventional approach to human-AI collaboration is to ensure model and human calibration [8, 3] and to make the AI’s predictions explainable [31, 27].
These explanations can affect human understanding of AI conﬁdence [12, 35, 36, 14].
It is also standard practice to optimize the AI in isolation (i.e., maximize the AI performance).
However, recent work suggests that the objectives used to optimize the AI do not adequately consider the human-AI team.
Alternative approaches include learning to defer, where a classiﬁer determines when expert input is required [16, 19]; learning to complement, where the AI is optimized to perform well on only the tasks that humans struggle with [33]; and methods that optimize objectives with costs assigned to requesting AI advice and/or human expert input utilization [1].
Our work is most related to [1], which also emphasizes the importance of conﬁdent AI predictions for human decisions.
Our approach is complementary to these prior works and contains several novel aspects.
We model human behavior using empirical data, demonstrate that this human model allows us to optimize the AI advice, and demonstrate both in simulation and empirically that our modiﬁed advice improves overall system performance.
Furthermore, we only require black-box access to the AI advice.
In relation to [1], we further differentiate our work as the model for human interaction is different – we give advice after the user makes an initial assessment.
Experimental setup
Our goal is to augment the performance of a human-AI system by optimizing the AI for use by a human.
To make this goal tractable, we assume (1) that we have a model for human interaction with
AI advice, and (2) that we have black box access to the AI algorithm producing the advice.
Human-AI interaction
The human-AI interaction involves two stages.
First, a human is shown a task and completes it.
AI advice is then presented, and the human is allowed to modify their response.
As we measure performance before and after receiving advice, this model is conducive to studying the effect of advice and so sees common usage in the literature [22, 30, 17].
Data collection and tasks
We collected data from crowd-sourced participants on a diverse set of tasks that all have the same basic design.
We focused on binary prediction tasks where humans respond on a continuous scale.
The use of the continuous scale allows for a more rich analysis of the user’s prediction than just the simple binary prediction label would permit.
Participants were shown a sliding scale with each end indicating a different binary response.
They were instructed to move a pointer on the scale to the location that indicated their conﬁdence and their response, with the midpoint on the scale indicating an uncertain response.
Participants submitted an initial response; subsequently, the advice was indicated with a visual marker on the scale and participants were allowed to adjust their initial response.
Participants were incentivized so that higher conﬁdence in correct and incorrect answers increased and decreased their monetary reward respectively.
We collected data on a set of four tasks from diverse data modalities.
These tasks were originally proposed in prior work [30].
The tasks were designed to be accessible to the general public to allow use of crowd-sourced study participants for obtaining empirical data.
See below for a brief description of the tasks we used; examples of task instances from each of the four tasks are shown in
Appendix B.2.
The data collected is summarized in Table 1.
Each task consists of 32 binary questions.
For each task, the 32 questions were carefully selected to have a range of difﬁculty.
More details on the distribution of task difﬁculty is presented in Appendix B.3.
All participants received the same set of questions, though the order was randomized for each participant.
All participants also received the same advice, although a small amount of noise was randomly added for each participant.
The advice had an accuracy of ⇠ 80% for each task; participants were informed of this accuracy.
However, no feedback on AI or human accuracy on individual questions was provided during the experiment.
Since the datasets we are using are relatively small and curated primarily for human use rather than
AI training, we opted to use aggregated human responses as a proxy for an AI model.
In particular, the advice was constructed by averaging the responses of a prior group of human participants who responded to the questions without receiving any advice.
This allowed us to ensure the advice given was “reasonable” in the sense that the advice is generally less conﬁdent on difﬁcult tasks and more conﬁdent on easy tasks.
We checked the calibration of this advice by computing the expected calibration error (ECE) [20].
The ECE for AI advice is 0 .
074 , indicating the advice is roughly calibrated ( 0 indicates perfect calibration).
We recruited 49-79 participants for each task, resulting in several thousand datapoints (each question is one datapoint).
Participants are US residents, roughly 50% are female, and the average age is approximately 30 years old.
Despite limitations in the age distribution and geographic location of participants, a survey question we asked indicated a varied range of AI familiarity in our participants.
We followed standard practice in collecting data.
Our study protocol was approved by our institution’s
IRB board.
The data collection process was low risk, and we ensured samples contained diverse participants across age, sex, and ethnicity.
We compensated participants at  $10 .
per hour (depending on bonus pay).
The total cost was around $600 (Section 2) + $750 (Section 5.2).
We have four types of data with corresponding user tasks.
Art dataset (Image data) Contains images of paintings from 4 art periods: Renaissance, Baroque, Romanticism, and Modern Art.
Participants were asked to determine the art period a painting is from given a binary choice.
Cities dataset (Image data)
This dataset contains images from 4 major US cities: San Francisco, Los
Angeles, Chicago, and New York City.
The task is to identify which city an image is from given a binary choice.
Sarcasm dataset (Text data)
This dataset is a subset of the Reddit sarcasm dataset [10], which includes text snippets from the discussion forum website, Reddit.
Participants were asked to detect whether sarcasm was present in a given text snippet.
Census dataset (Tabular data)
This dataset comes from US census data [32].
The task is to identify an individual’s income level given some of their demographic information—e.g. state of residence, age, education level.
[Table]
Human behavior model
Table 1 provides a summary of the data collected for developing our human behavior model.
We ﬁnd that humans actually changed their response after receiving advice roughly 50% of the time, with the exact number shown in the “Activation Rate” column of Table 1.
When a person does change their response, we call this person “activated.”
Speciﬁcally, a person is activated if their ﬁnal response changes by at least a small amount.
This observation motivates a simple, two-stage model for human behavior: a human ﬁrst decides whether to modify their response (“activation stage”), and subsequently, if they were activated, they modify their response (“integration stage”).
A diagram of this model is shown in Figure 1.
This two-stage model is consistent with previously proposed models from the psychology and HCI literature [9, 30].
We ﬁt functions for each of the two stages: an activation model that predicts how likely it is that a person is activated, and an integration model that predicts how the person modiﬁes their advice, assuming they are activated.
Taken together, we can model human behavior in the two-stage humanAI system we use to run our experiments.
We deﬁne this model’s behavior more rigorously in
Section 4.
Our ﬁtted models will only be used to optimize our advice-modifying function and will not be used a test-time.
As such, we can utilize demographic information about the participants to ﬁt the model, though such information may not be available during a real-world deployment.
We opt to ﬁt a small neural network for each of these stages, as we require a differentiable model (for easy optimization) with moderately more complexity than a linear model.
3.1 Activation model
Our activation model takes 12 input features extracted from a single person-task instance interaction.
It outputs a probability for whether the person will be activated on the given task instance.
The input features are based on the person’s initial response, the AI advice, and several demographic features including age and sex.
We do not encode the initial response or AI advice directly as the label (the
[Figure]
left or right slider location of the correct answer) is randomized for each participant.
Instead, we use the response and advice conﬁdence, as well as a binary feature that encodes whether the response and advice agreed on the label.
This implies a symmetry in the model’s predictions.
Please see
Appendix D.1 for a description of all the features used.
We use a 3 layer neural network with ReLU non-linearities for the activation model.
We also attempted to use other simple algorithms for modeling the activation behavior of humans, like linear models, but found the neural networks to perform better.
We trained the network using binary cross-entropy loss with an Adam optimizer [11] and stopped training when validation loss increased, as is standard.
The activation model achieves a ROC-AUC > 0 .
81 on held-out test participants, suggesting reasonable performance despite the high variability in our dataset.
In Figure 2a, we show a partial dependence plot for the learned activation model.
This plot shows the average behavior (across our entire test dataset) of the activation model when we ﬁx the AI advice to be a certain value (the x-value) for all data points.
The AI advice can be positive (correct advice) or negative (incorrect advice), while the magnitude corresponds to the advice conﬁdence.
We note that (1) there is a base level of activation around 50% and (2) our activation model predicts people will be most activated when the advice is conﬁdently incorrect.
This occurs because (a) most people in our dataset get answers correct, on average, and seeing conﬁdent advice of the opposite label tends to make people change their initial response (e.g. by decreasing conﬁdence even if the predicted label does not change).
And because (b) conﬁdent advice that agrees with a person’s initial response (whether correct or incorrect) tends to increase their conﬁdence.
We also observe that (3) the activation rate increase when advice is conﬁdent and correct for the same reason previously noted in (2b).
The activation rate is lower for correct advice than for incorrect advice as most people in our dataset are, on average correct, which tends to invoke a smaller response.
3.2 Integration model
Our integration model is similar to our activation model, but optimized for a different function.
In particular, we use the same input features, model architecture, and training procedure.
The integration model outputs a prediction for sign ( r 1 ) ( r 2  r 1 ) , where r 1 , r 2 {+
+} [  1 , 1] are the initial and ﬁnal responses respectively.
Note that r 1 , r 2 are probabilities normalized to the [  1 , 1] scale.
The training minimizes the RMSE between the predicted change and the actual change.
The optimization is performed on only the subset of the dataset that was actually activated (roughly 50% of the data).
Our integration model has a reasonable performance, with an RMSE of 0 .
25 (the maximum absolute error possible is 2 ) and R 2 of 0 .
73 on test data.
In Figure 2b, we plot a heatmap of the ﬁtted integration model’s behavior.
We plot the average output of our integration model across our test data split when the person’s initial response (x-axis) and
AI advice (y-axis) are ﬁxed to a certain value for all datapoints.
The sign and magnitude of the initial response and AI advice correspond to correctness (negative is incorrect) and conﬁdence.
The integration model prediction can be positive (increasing conﬁdence in the initial predicted label) or negative (decreasing conﬁdence in the initial predicted label / change in the predicted label).
We observe three features in the heatmap: (1) The heat map is symmetric across quadrants 1, 3 and 2, 4.
This is by design.
As the model does not use the initial response and AI advice values directly – it uses their magnitudes (conﬁdence) as well as a third, binary term indicating whether they agree on label – the integration model can only predict a delta change relative to the sign of the initial response.
The output has largest magnitude when the advice is conﬁdent and opposite the person’s initial response.
If the advice and the person’s initial response share the same label, the integration model output is largest when the person’s initial response has low conﬁdence and the model has high conﬁdence.
These observations conﬁrm our integration model has reasonable operation behavior.
Modifying the AI advice
[Figure]
Here we discuss how we modify the raw AI advice output.
First let’s establish some notation.
The notation described below is in reference to a single datapoint; later, we will add the subscript i to index across datapoints.
Let A 2 R denote the inverse sigmoid of the advice presented to participants (i.e., we present the
AI advice as  ( A ) ).
We will work with A rather than  ( A ) here, as we will aim to replace the  transform with an optimized function that improves human-AI performance.
Let g : R ! [0 , 1] be a function (which we will optimize) that maps the AI advice to a probability that the user is shown.
Note that for the baseline, “unmodiﬁed advice,” we set g to be the sigmoid function,  ( x ) = 1 {+
 ++} e  x .
Let r 1 , r 2 be the initial and ﬁnal responses from a participant ( r 2 is a function of r 1 , A , and other features), and let y 2 { 0 , 1 } denote the true label.
Let u represent demographic features of a person, and let x = ( r 1 , A, u ) be the input feature vector to f activation and f integration , our activation and integration models for human behavior models described in Section 3.
To simplify notation, we let f integration denote the function that outputs the predicted r 2 for an activated person (rather than the delta from r 1 that we optimize for).
Finally, let f HB represent the entire model for human behavior.
We assume all feature preprocessing is hidden inside f activation and f integration for convenience of notation.
4.1 Optimizing for human-AI performance
We consider g of the form
[Equation]
where ↵ ,  2 R  0 .
Note that setting ( ↵ ,  ) =
(1 , 0) results in g 1 , {+
+} =  , the sigmoid function used to produce the unmodiﬁed advice.
Note that we do not exactly perform a linear transformation on A : the | A | and sign ( A ) terms are included to ensure that g does not change the label recommended as it makes the function symmetric around A = 0 .
↵ modulates the rate at which the presented advice increases with AI conﬁdence, while  adjusts the minimum conﬁdence level of the presented advice.
For example, if  = 1 , the presented conﬁdence is > e 1+ e > 0 .
73 (relative to the chosen label).
In general, we can use other differentiable functions for g ; we chose the form in Equation 1 because it is easy to optimize, simple to understand, and is still ﬂexible enough to ﬁt our data.
As a shorthand for when we modify the features input to f activate and f integrate , we denote g ( x ) =
( g ( A ) , r 1 , u ) .
We will optimize for ↵ ,  to maximize the expected ﬁnal accuracy using our human behavior model, f HB .
As discussed in Section 3, f HB has the following representation:
[Equation]
We then solve the following optimization problem to obtain ↵ ,  :
[Equation]
where L is the binary cross entropy loss function.
The expectation is taken with respect to the randomness in f HB and the data distribution.
Given training dataset D , we optimize the following expression:
[Equation]
X ( x i ,y i ) 2 D
L ( r 1 i , y i ) · (1  f activate ( g ( x i )))
+ L ( f integrate ( g ( x i )) , y i ) ·
f activate ( g ( x i ))
In practice, we carry out this optimization using stochastic gradient descent, which is possible as we choose f activate and f integrate to be differentiable functions (see Appendix D.2).
For data, we used the empirical data described in Section 2.
The full training procedure is visualized in Figure 4.
[Figure]
Once we have optimized for ↵ ⇤ ,  ⇤ , we can apply it to new data without needing the user demographics u (i.e., we do not personalize the conﬁdence transformation to individual users).
Note that the choice of loss function is arbitrary here (we used logistic loss as our task is binary classiﬁcation).
In particular, for high risk settings where a mistake is very costly, the ↵ ⇤ ,  ⇤ could be selected accordingly using an appropriate loss function.
[Table]
Here we describe simple theoretical situations where uncalibrated advice improves performance.
We consider a binary classiﬁcation task and assume we want to minimize the logistic loss of the human-AI classiﬁer.
We also assume (1) a constrained version of the activation-integration model perfectly models human behavior; in particular, humans either use their initial response or follow the given advice exactly.
And (2) we have an oracle for human behavior and know exactly the calibration of the human and AI advice.
We will now construct a function f ( a, r 1 ) that outputs a ⇤ , the optimal advice to minimize logistic loss of the human-AI system.
To select a ⇤ , we need to consider two quantities:
[Equation]
[Equation]
Here, p a and p r 1 are the probabilities the advice and initial response are correct respectively.
Then L rrelates the log-loss for non-activated humans and L a ⇤ relates the log-loss of following the modiﬁed advice, a ⇤ .
Now we simply select a ⇤ to minimize the expected loss incurred by the human-AI system
[Equation]
(5) for p activation = f activation ( a ⇤ , r 1 , u ) (which we assumed we know exactly).
For example, consider a setting where
[Equation]
where ¯ x = max { x, 1  x } , ✏ 2 [  0 . 5 , 0 . 5] , and for a generic function f : [0 , 1] !
Here a human under/overvalues their response relative to the advice (depending on ✏ ), and the human is not calibrated (depending on f ).
In this setting, it is indeed optimal for the advice, a ⇤ , to be under/overconﬁdent to correct for human misutilization of advice (depending on the advice and initial response).
An example where ✏ = 0 .
1 and f ( r 1 )
=
r 21 is shown in Figure 3b.
Further discussion is included in Appendix C.
Results 5.1 Simulation results
Optimizing Equation 2 results in an optimal ↵ ⇤ ,  ⇤ .
Using f HB , we can then analyze the expected beneﬁt of g ↵ ⇤ ,  ⇤ over the baseline, g 1 , 0 .
In Figure 3a, we show g ↵ ⇤ ,  ⇤ .
We plot three separate curves.
The curve labeled “Advice given” is optimized using the previously described dataset, and this is the function we use in our empirical results.
We also synthetically modify the dataset by biasing the advice towards the correct label and adding noise to the advice to respectively increase and decrease the average advice accuracy in the dataset to 75% and 85% .
We then ﬁt a new ( ↵ ,  ) to the modiﬁed dataset and plot these curves.
Note that f HB is ﬁt to the unmodiﬁed dataset and is ﬁxed for optimizing all three of these curves.
[Table]
For all three curves, we notice there is a large discontinuity at A = 0 .
This is a result of a large  ⇤ , and suggests that advice is not useful to humans unless it is at a certain level of conﬁdence (e.g., to make the human activated).
That all three curves follow relatively similar paths suggests the performance of g ↵ ⇤ ,  ⇤ may be robust to small changes in advice accuracy.
This is conﬁrmed (in simulation) in Table 2, where we show the simulated performance difference between modiﬁed and unmodiﬁed advice.
We see here that the ﬁnal accuracy, conﬁdence in correct result, and activation rate all increase with the modiﬁed advice.
5.2 Human experiments
[Figure]
We verify our simulation results across the four previously described tasks (with the exception that we do not add noise to the advice, so all participants see the exact same advice now).
Adult,
US-based participants were recruited through the Proliﬁc crowdworker platform [23].
We recruited 50 participants for each task, and randomly assigned either the unmodiﬁed (baseline) or modiﬁed advice to each person.
In Figure 5, we show the average change in (a) accuracy and (b) conﬁdence of the correct label (the average conﬁdence, with correct and incorrect labels receiving positive and negative weight respectively), partitioning by the advice received by participants.
Accuracy and conﬁdence increase across all tasks when using advice ( p = 3 .
84 ⇥ 10  6 ).
Moreover, we can largely conﬁrm our simulation results.
Accuracy and conﬁdence increased due to the modiﬁed advice across nearly all tasks.
Additionally, activation rate (not plotted) increased with the modiﬁed advice, with an average increase of 3 . 3% across tasks.
Another interesting ﬁnding is that tasks with higher advice accuracy (Art and Cities) exhibit larger increases in accuracy and conﬁdence.
The AI advice is more accurate in Art and Cities (accuracy: 90% and 87%) compared to Sarcasm and Census (accuracy: 78% for both).
This intuitively makes sense – we are making the advice more conﬁdent in its predictions, and so while this beneﬁts all tasks, it disproportionately beneﬁts tasks where the advice is more accurate.
This effect was predicted by our simulation results.
Our primary human experiments involved US-based participants.
To show our ﬁndings generalize, we took the advice modiﬁcation learned on the US data and applied it to new participants based in the UK for the Art and Cities tasks.
The ﬁndings are highly consistent: modiﬁed advice increases activation and improves ﬁnal human accuracy and conﬁdence (Table 3).
Combining US and UK data, performance improvement is signiﬁcant for both accuracy ( p = 8 . 86 ⇥ 10  3 ) and conﬁdence ( p = 4 .
43 ⇥ 10  11 ).
Discussion, Limitations, and Future Work
In this paper, we proposed optimizing a simple function to better calibrate AI advice for human use in a collaborative human-AI system.
To optimize the reported conﬁdence, we trained a human behavior model from experimental data and used it simulate the full human-AI system.
The modiﬁed AI advice are explicitly uncalibrated, but in both simulated and empirical results, it resulted in substantially higher ﬁnal human performance.
We further validated our model using new participants from a different country.
These results highlight the importance of recognizing that AI, when deployed in collaborative systems, should not be optimized in isolation.
Our results also suggest that we can view modifying AI advice as a nudge for the human user.
If the AI says its conﬁdence is 0.7, the user may not fully absorb the import of 0.7 since humans are not great at intuitive probability.
Modifying the reported conﬁdence could be a useful nudge to help users to better absorb the information.
This view is supported by prior work suggesting humans prefer clean decision boundaries [2].
Our work has several limitations that would be interesting to address in future studies.
In particular, it is unclear how reliable such a method would be in practice.
We argue that our method for creating human-calibrated AI can be deployed in practice, but requires an accurate model for human behavior.
In particular, if we have access to an oracle for an individual’s utilization of AI advice, we can exactly optimize the output of a calibrated AI model to correct for the individual’s biases.
Note that this may lead to under- or overcalibrated advice depending on the user and the advice conﬁdence.
However, as we cannot perfectly model any given individual, the question to ask is whether we can create a robust model for human behavior (e.g., with conﬁdence intervals etc.) and then optimize with respect to that model while accounting for error.
This extension was out-of-scope for this paper, and is an important direction of future research.
Other limitations include: we considered tasks where advice accuracy generally exceeded individual human performance and did not evaluate expert tasks (e g., involving clinicians making clinical decisions).
We also did not consider how human interaction with an AI evolves over time – that is, we assumed each individual to be static over the course of the study, while in general, humans may naturally adapt to the AI and modifying the AI may affect this process.
These are also important directions for future research.
Finally, we recognize that the human behavior model we used in this work does not capture all real-world settings.
It may be important to consider different models for different applications, and our results will need to be validated in these new settings.