Encoding Human Behavior in Information Design through Deep Learning
Authors: Guanghui Yu
Wei Tang
Saumik Narayanan
Chien-Ju Ho
Venue: NeurIPS 2023 poster (2023)
Keywords: Information design; Human behavior; Behavioral experiments

Abstract: We initiate the study of $\textit{behavioral information design}$ through deep learning. In information design, a $\textit{sender}$ aims to persuade a $\textit{receiver}$ to take certain actions by strategically revealing information. We address scenarios in which the receiver might exhibit different behavior patterns other than the standard Bayesian rational assumption. We propose HAIDNet, a neural-network-based optimization framework for information design that can adapt to multiple representations of human behavior. Through extensive simulation, we show that HAIDNet can not only recover information policies that are near-optimal compared with known analytical solutions, but also can extend to designing information policies for settings that are computationally challenging (e.g., when there are multiple receivers) or for settings where there are no known solutions in general (e.g., when the receiver behavior does not follow the Bayesian rational assumption). We also conduct real-world human-subject experiments and demonstrate that our framework can capture human behavior from data and lead to more effective information policy for real-world human receivers.

Encoding Human Behavior in Information Design through Deep Learning
Guanghui Yu Washington University in St. Louis
Wei Tang Columbia University wt:3599ccoumbia. edu
guanghuiyu@wustl.cm
Saumik Narayanan
Chien-Ju Ho
Washington University in St. Louis
Washington University in St. Louis chienju. hww.jju..oouust..eee
saumik@vustl.edi
Abstract
We initiate the study of behavioral information design through deep learning. In information design, sender aims persuade receiver to take certain actions by strategically revealing information. We address scenarios in which the receiver might exhibit different behavior patterns other than the standard Bayesian rational assumption. We propose HAIDNet, a neural-network-baae work for information design that can adapt to multiple representations of human behavior. Through extensive simulation, we show that HAIDNet can not only recover information policies that are near-optimal compared with known analytical solutions, but also can extend to designing information policies for settings that are computationally challenging (e.g., when there are multiple receivers) or for settings where there are no known solutions in general (e.g., when the receiver behavior does not follow the Bayesian rational assumption). We also conduct real-world human-subject experiments and demonstrate that our framework can capture human behavior from data and lead to more effective information policy for real-world human receivers.
optimization frame-
Introduction
The problem of information design, where a player with an information advantage (the sender) can strategically reveal information to influence another player (the receiver) to take certain actions, is ubiquitous in everyday applications. For example, online retailers can highlight subset product features influence buyers make purchases [32 46 Recommendation systems might selectively display other users' ratings to persuade users to follow recommendations [75]. Politicians can influence voters' decisions by designing different policy experiments (T] There have been various 44 machine learning and artificial intelligence [79
research efforts from economics [51] 62 29
30
12
33 0 25). and general computer science 23 devoted to the study of information design. Among the growing literature on information design, the model of Bayesian persuasion proposed by Kamenica and Gentzkow (39 is one of the most prominent, and has inspired rich body of studies.
While Bayesian persuasion offers an elegant framework formulating the information design problem, it has two limitations. First, the receiver is assumed to be Bayesian rational. This means that the receiver can form a posterior a Bayesian manner and chooses the action that maximizes his expected utility. However, consistently observed in empirical studies (53) 37], humans often
We use she/he to denote the sender/receiver, respectivell
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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deviate from being Bayesian or rational. Directly applying the techniques from the information design literature that assume Bayesian rational receivers could lead to suboptimal outcomes. this work, we address this limitation by proposing general optimization framework that can integrate a wide range of human behavior, expressed either as traditional analytical dlosedffrmm behavioral models or as data-driven models, and design optimal information policies with respect to the provided human behavior.
Second, despite a decade of effort, characterizing the optimal information policy remains notoriously difficult. Dughmi and Xu [8] have shown that it is #P-hard to compute the optimal expected sender utility, and in multi-receiver settings where each receiver only has binary actions, it is #P-hard to even approximate the optimal sender utility within any constant multiplicative factor [78]. Moreover, most previous works have assumed that the receiver follows the Bayesian rational assumption. When this assumption is relaxed [14 70), there are generally no known analytical solutions for finding the
optimal information policy yet.
In this work, we initiate the study of automated information design that encodes human behavior into the design process, Inspired by the recent effort in utilizing deep learning for auction design 20, 58). we propose HAIDNet, an optimization framework that leverages neural-network architectures for information design. Unlike existing works that assume rational human behavior, our optimization framework can adjust to multiple representations of human behavior pattems, including standard behavioral models represented in analytic forms, and data-driven models trained using machine learn- ing upproaches. More specifically, We encode receiver behavior as function and represent the loss in our optimization framework as function of the receiver's 's responses to the disclosed information. This approach enables our optimization framework to accommodate different representations of human behavior and can lead to corresponding optimal information policies. Our contributions can be summarized as follows:
We initiate the study of automated information design that encodes human behavior in the design process. The proposed end-to-end Human behavior encoded neural network mechanism for Automated Information Design, namely HAIDNet, enables to optimize the sender's 's information policy based on a given model of human behavior. We incorporate a neural network architecture for information design problems. Moreover, we extend the literature on deep learning for mechanism design to encode realistic human behavior in the design process.
We evaluate our approach via extensive simulations. In simpler settings with known analytical solutions, we show that HAIDNet can recover the optimal information policies. We also show that HAIDNet can extend to design information policies for settings that are computationally challenging (e.g., multiple receivers involved), or for settings with no known solutions in general (e.g., when the receiver's ' behavior does not follow the standard Bayesian rationality assumption).
Through real-world human-subject experiments, we demonstrate that our framework can adapt to scenarios where we do not have access to human models a priori. We demonstrate that our approach can accurately learn human descriptor from behavioral data, incorporate it in our optimization framework, and result in more effective information design policy in the real world.
Related Work. Our work joins a growing line of research that leverages computational tools for
automated mechanism design [II 65
10, 66 More recently, deep neural networks have been utilized
42 [1 45 [2]. Our work differs from this line of work in two ways. First, we extend the approach beyond auction design to address the automated information design problem. Second and more importantly, we have incorporated human behavior in our design, while prior works mostly require standard rationality assumptions.
for optimizing auction design 200 27
Our information design formulation builds on the seminal work of Bayesian persuasion (39 which inspired a rich line of research in information design [e.g., see the recent surveys by 38 4]. In particular, given the practical relevance of information design, there an increasing number of information design studies in the research community in machine learning and artificial intelli-
gence 79 12
economics [5
62
29
30 34 and operations research 44
76 Our
work differs from most of the existing works in that we integrate human behavior into the design of information policy. There have been few works addressing non-Baaesi belief updating [4] and non-rational receiver behavior 770 26 with stylized models in information design. Our work extends previous research by designing an framework that can accommodate both the analytical form and the data-driven form of human behavior. More discussions on related work can be found in Appendix A
2

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2 Preliminary- - Bayesiin Persuasion Basics
In Bayesian persuasion, there are two players: a sender and receiver. The sender's goal is to design an information disclosure policy that persuades the receiver to take certain actions maximizing the sender's objective. The state of nature 0 is drawn from finite set Q _ {l....,m, m) according to prior distribution À (A()))age E A()). The prior is common knowledge to both the sender and the receiver. The receiver's utility R (a. ()) depends on the receiver action a E AA from an action set A and the state 0. The sender's 's utility s 0) also depends on the receiver's s action and the state.
The sender can observe the realized state while the receiver cannot, and the sender utilize this information advantage to persuade the receiver to take the desired action. In particular, before observing the realized state, the sender can commit an information policy т,, specifying what signal to present to the receiver conditional on the realized state. More formally, an information policy x consists of a signal space Σ and a set of conditional probabilities (x(1)))pee where (-100) = (T(a(()))es) E A(E) and (( Ø) E [0, 1] denotes the probability to send signal a E Σ given the realized state 0. This information disclosure policy is known to the receiver and specifies how the sender discloses information to the receiver. When state A E θ is realized, the sender sends a signal a π (-16) according to the policy.
In Bayesian persuasion, the receiver is assumed to be Bayesian rational in the sense that upon seeing the signal a, the receiver forms his posterior belief about the state in Bayesian manner and takes an action that maximizes his expected utility. Formally, upon seeing the signal realization o, the receiver updates his posterior belief over the state of nature, denoted by u(o) - (p(0lo))oee E A())) by
applying Bayes' rule: ()((0r) - Ee'ce *(a (')A(("'
=(alo)A(0)
Given the posterior induced from the observed signal σ E Σ, the receiver takes an action a (R)( E A
that maximizes his expected utility? F7 namely, (BR (or) _ arg max.e.A Ence H(() l / (a, 0). The
(
sender's information design problem is to find the optimal information policy that maximizes her expected payoff induced by the receiver's action, follows:
max Σx(0) Σ T(al0)u S (a
(PBR)
In this work, our goal is to design an automated framework to solve the above bi-level optimization problem while encoding realistic human behavior in the design process (i.e., replacing the Bayesian rational human model (a) with general human behavior).
Example. Consider the scenario in which an online retailer (the sender) aims to persuade a buyer (the receiver) to make a purchase. The retailer's s products are directly coming from the factory. and the product quality (represented by the binary state 0)) drawn from a prior distribution X. The buyer's utility u "a 0) depends on both his binary purchase decision and the binary product quality 0, while the retailer's utility u (a, f) = u (a). V is mate-independen and only depends on the buyer's purchase decision. For example, the goal of the retailer is to persuade the buyer to make a purchase, for a = 1 and S (a) = 0 for a = O. The buyer only wants to purchase when the product is good, i.e., (a, = 1 if 0- = and u (a, 0) 0 otherwise.
ie., u$ (a)
In order to persuade the buyer to purchase, the retailer can commit to performing (noisy) product inspections (a 0) to reveal information about the product quality. For example, the inspection might satisfactory with 80% chance if the quality of the product indeed satisfactory (i.e., <(a- 1 = 1) = 0.8) and signal the product quality is unsatisfactory with 90% chance if the quality is indeed unsatisfactory (i.e., aa 0/0 0) = 0.9). The information design problem for the retailer is to identify an inspection policy that maximizes the probability on selling the product to the buyer.
signal the product quality
3 HAIDNet: Encoding Human Behavior in Automated Information Design
In this section, we introduce HAIDNet, an optimization framework based on neural network architecture that can adjust to various forms of human behavior. In the following discussion, we first describe how we modularize human behavior in information design. We then explain the neural
2 BR here stands for Bayesian rational.
3

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Sender's Optimization
Das
vlol 01
Human Descriptor
[(..a) Ha 6)
Information Policy
Receiver
Hic.a]
Hidden layax
L (T,,
Sende: Unliy
Beotivar
Figure 1: The overall HAIDNet framework. The human descriptor module given to the optimization module before training. The optimization is performed through back propagation which evaluates the gradient of the loss to update the weights in the neural network structure.
network architecture of our proposed HAIDNet that can adapt to different forms of human behavior. Finally, we outline the procedures for optimizing the information policy in HAIDNet.
3.1 Encoding Human Behavior
Bayesian persuasion assumes that the receiver is Bayesian rational. However, in practice, this assumption often does not hold. The receiver may exhibit systematic biases both in belief updating and in decision making. In the following discussion, formulate the sender's problem finding the optimal information policy when taking more general human behavior into account.
Definition 3.1 (Human Behavior Descriptor). For any receiver utility R prior X, sender information policy ग (and signal space )) human behavior descriptor is denoted by H. A. (a, a), representing the probability for human receiver choose action E A when seeing realized signal Σ.
When the context is clear, we omit the subscripts and write Ha A,, (,, a) as H a) for notational simplicity. With the above definition, we can rewrite the sender's information design problem as:
Σ X(0) Σ IE
T 7 H((ra))u(aa
(P")
OE:
aEAA
Below we give a few examples human behavior descriptors.
Bayesian rational (BR). In standard Bayesian persuasion, the receiver updates his posterior in Bayesian manner and takes action that maximizes the expected utility. Following the definition in Section the human descriptor can be written as H a) = 1( = (a)).
Probability weighting and discrete choice (TH-Model). We present another human behavior descriptor based on the work by Tang and Ho 70 (denoted as the TH-model in the description of this work). In partieular, they combine probability weighting, assuming the receiver's posterior distorted based on function w(.): (()) - A()), and discrete choice model, assuming the receiver's 's action is stochastic, with higher probability in taking an action with higher expected utility (based on the distorted posterior belief).
Formally, let (() be the receiver's distorted posterior belief after seeing signal o and 8y be a parameter in the discrete choice model that tunes how stochastic the receiver's action is (when 3H + oc, the discrete choice model reduces to standard expected utility maximization), the human behavior descriptor for this model can be written as:
exp (3x Eoco w (Olo)u 0)) H(a, a) Σ.. exp (3n Eoco (a)lo)u
Data-driven human behavior descriptor. Note that in our formulation, we use the function H (a, a) represent human behavior. Suppose we have access to sufficient human behavioral data, instead of expressing Ho, a) using a closed-form analytical expression, we can train a machine learning model approximate this function and utilize the learned model the human behavior descriptor.

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3.2 HAIDNet Framework and Optimization
We now introduce the framework of HAIDNet and explain how we utilize to optimize the sender's information policy for given human descriptor.
HAIDNet framework. As presented in Figure 1 HAIDNet consists of two modules: the sender's optimization module and the human descriptor. The sender's 's optimization module is a neural network responsible for optimizing the sender's optimal information policy. It takes the information design problem instances as input, including the prior distribution À over the states and the payoff functions u n for all players. The module outputs an information policy which consists of of conditional probabilities [T(-)))eee over the signal space for each state 0 E θ.
The human descriptor can either be model- based (e.g., Bayesian rational model or TH model in Equation ①. ) or data-driven (e.g.. a neural network modeling the receiver's 's behavior). The human descriptor is treated as black box from the perspective of the sender's 's optimization module, and is fixed before HAIDNet begins training. The input of the descriptor consists of the receiver utility u the prior distribution X, and the information policy (i.e. the output of the sender's optimization module), while the output is the receiver's 's response strategy H(a, a) = H ". x. a).
Optimization procedure. For the sender's optimization, we follow the recent line of research on using deep learning for auction design [20]: we randomly draw problem instances from a pre-specified distribution and perform stochastic gradient descent to minimize the loss function in the training process. The loss function is defined to be the negative of the sender's expected utility, since the goal of the sender is to find the optimal information policy that maximizes her expected utility.
( H) - Σ EOO T(O Σ H(o (a,0)) (a,0)
OEG
aEA
Our work differs from previous works in that we incorporate the human behavior descriptor in the definition of the loss function. The requirement is that the human descriptor H(o, a) needs to be differentiable. This requirement is naturally satisfied in many cases, c.g., when the human descriptor follows the model defined in (I) or is a neurallnetworr-bassee However, in the Bayesian rational model, since the receiver chooses the action that maximizes his expected utility, this argmax operation makes the human model not differentiable. To overcome this issue, we approximate the Bayesian rational model using softmax instead of argmax with a sufficiently large softmax scale parameter B. More concretely. let u(a) be the expected utility for action a. The softmax operator approximates the receiver's 's behavior by using exp exp(Su(a))/ E.. exp(Bu(cl to denote the probability of choosing action a. As sanity check, when ,3 -> oo, this expression reduces to argmax, choosing the action maximizing the expected utility.
model, the gradient always exists.
Optimization implementation. To optimize HAIDNet, we train a neural network with 3 fully connected layers employing ReLU activation functions and the Adam optimizer. The model is trained on 100 batches of size 1024, for total of 102, 400 uniformly drawn problem instances (i.e., data points for training). Evaluation of the model is conducted on test set consisting of 1000 problem instances. The specification of hyperparameters and implementation details are included in the appendix. We have also included the source code in the supplementary materials.
Experiments
4.1
Simulations
We have conducted extensive simulations to evaluate HAIDNet. Our results demonstrate that HAIDNet can find the near-optimal information policy in various settings. Specifically, we show its effectiveness in settings where efficient methods exist to obtain the optimal information policy (Section 4.1.T and in computationally challenging settings where finding the optimal information policy is difficult (Section 4.1.2). Moreover, even in settings where no known solutions exist in general, HAIDNet can generate information policy with good performance (Section 4.1.3). We have conducted additional simulations, including examining the convergence of the training, investigating
'The notation 3 here is different from In used to model human behavior in
5

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the scalability of the approach, accounting for varying number of receivers, comparing with random policy, and examining empirical run-time. Due to space constraints, these additional simulation results are included in Appendix B.
4.1.1 Settings with efficient solutions
We start our evaluations with a simple setting where there exist efficient solutions to find the optimal policy. In this setting, we leverage the efficient solutions as ground truth to examine whether our approach can also identify the optimal information policy.
In particular, we consider the setting with a single Bayesian rational receiver. In this setting, when there are only two actions available for the receiver and there are only two states, there exists a closed-form characterization of the optimal information policy. When the numbers of actions and states are finite constants, the optimal information policy can still be computed efficiently [39. Therefore, we can evaluate the performance of our approach by comparing the information policy generated by HAIDNet with the optimal policy.
Binary actions and binary states. We first examine the simplest setting with binary actions and binary states (a classical setting in Bayesian persuasion (39)) namely, the action space A = {0. 1 and the state space O {0, 1}. and observe whether HAIDNet produces near-optimal information policies. For the sender utility, we adopt a stylized setting where the sender obtains utility when the receiver takes action and utility 0 when the receiver takes action 0. The receiver aims to take the action that aligns with the true state, i.e., u =²(0, 1) u (1, 0) O, and we randomly draw each value for u R (0, 0) and uR( (1, 1) from [0. 1]. plain words, the receiver prefers action when the state is 1 and action O when the state is O, and the goal of the sender is to persuade the receiver to take action 1. The prior distribution À is drawn from a Dirichlet distribution. We then simulate data using the setting above and optimize HAIDNet.
We first examine whether the policy generated by HAIDNet matches the known optimal policy. Note that in this simple setting, via revelation principle [39, an information policy can be characterized by two signals, i.c., o E {0, 1), where each signal corresponds to a recommended action. Moreover, in the optimal policy, we have x (a 10 = 1) = 1, and therefore the optimal policy can be characterized by a single parameter "t (a = 1 0 = 0). To examine whether HAIDNet generates the same policy as the optimal policy, we compare the value of this parameter on different scenarios.
To showcase our results, we present two settings where we have fixed prior distributions: low prior with A(O = 0) = 0.3 and medium prior with A(O = 0) 0.5.
37 For each prior distribution, we vary the receiver utilities and report the parameter x² (a = 1 0) both from the optimal policy and from the output of HAIDNet. As visualized in Figure 2 the policy learned by HAIDNet essentially
recovers the optimal information policy in almost all scenarios.
1.0
Optimal
1.0
HAIDNet
Optimal
1.0
HAIDNet
0.8
0.8
08
7ac
0.4
5 20A
0.4
0A
02
9
0.0
0.2
10,00
10,00
(0.0)
L000,
(a) Optimal vs. HAIDNet the low prior case.
(b) Optimal vs. HAIDNet the medium prior case.
Figure 2: Comparing the optimal information policy and the policy generated by HAIDNet in the setting with binary actions and binary states.
Multiple actions and multiple states. To examine whether our approach scales with the size of the problem instances, we increase the number of states and the number of action&5 The performance is measured using the average sender utility. We report both the training performance (e.g., average sender utility for 1,000 instances drawn from instances used for training HAIDNet) and testing performance (c.g., average sender utility for newly drawn 1 1,000 instances) p The results, as shown
'The results are the same for a wide range of prior distributions.
'The results for scaling up both simultaneously are qualitatively the same and are included in the appendix. 'We have included additional comparisons to the performance of a simple baseline, random policy, in the appendix. The performance for the random policy is around 0.5 in all scenarios in this setting.

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Table 1: Comparing the average sender utility generated by the optimal policy and the policy from
HAIDNet in the setting with single Bayesian rational receiver.
(a) Increase the number of states with binary actions. (b) Increase the number actions with binary states.
Training
Testing
Training
Testing
M
N
HAIDNet
Optimal
HAIDNet
Optimal
HAIDNet
Optimal
HAIDNet
Optimal
2
0.7409
0.7498
0.7408
0.7451
2
0.7409
0.7498
0.7408
0.7451
3
0.7737
0.7782
0.7598
0.7669
3
0.7017
0.7214
0.7089
0.7227
5
0.8171
0.8209
0.8066
0.8225
5
0.6906
0.7113
0.6690
0.7064
10
0.8495
0.8699
0.8196
0.8686
10
0.6861
0.7084
0.6623
0.6963
Table 2: Comparing the average sender utility generated by the optimal policy and the policy from HAIDNet in the setting with Bayesian
Table 3: Comparing the average sender utility by the optimal policy and the policy from HAIDNet in the setting with a non Baaesiann-aatione re- ceiver parameterized by 3H.
rational receivers.
Training
Testing
Training
Testing
K
Bh
HAIDNet Optimal
HAIDNet
Optimal
HAIDNet Optimal
HAIDNet
Optimal
2
0.7887
0.7934
0.7756
0.7873
0.5043
0.5051
0.5041
0.5060
0.7508
0.7665
0.7379
0.7573
5
0.5512
0.5557
0.5506
0.5559
5
0.7217
0.7458
0.7209
0.7570
10
0.6045
0.6170
0.5986
0.6168
10
0.6971
0.7152
0.6790
0.6966
50
0.7002
0.7134
0.6800
0.7081
15
0.6553
0.6882
0.6621
0.6843
100
0.7187
0.7291
0,6964
0.7179
in Table T demonstrate that our approach works well for large-scale problem instances and also generalizes well to instances not used in training.
4.1.2 Settings without efficient solutions
Next, we examine the performance of HAIDNet under the setting where there are no known computa- tionally efficient solutions to characterize the optimal information policy. The goal is to illustrate that HAIDNet performs well even in complicated scenarios and could provide more efficient approach for settings without analytically tractable solutions.
We consider the setting with multiple receivers and binary actions. The goal is design uniform information policy for all receivers public persuasion (78).. This setting has been shown to be #P-hard find a policy that approximates the optimal sender utility within any constant multiplicative factor [19). This means that, unlike the single receiver case, finding the optimal solution for a given problem is practically impossible solve with a large set of receivers, and we intend HAIDNet be new, efficient solver for opar-optimal solutions. To examine whether HAIDNet finds the optimal policy, we utilize brute-force linear- perear-progrmmnng approach [19] (the time complexity is exponential in the number of receivers since the number of constraints in the program grows exponentially) to identify the optimal policy when the number of receivers is small. We then compare the information policy generated by HAIDNet and the optimal policy output from the linear programming approach. The receiver utility and prior distributions are generated in the same way as in the single receiver 1SI if setting. The sender utility is the fraction of receivers choosing action 1, i.e. her utility is given K there are |S] receivers choosing action out of a total K receivers.
The simulation results are shown in Table 2. We randomly draw 1, 000 problem instances from the training/testing set and report the average performance of the optimal policy and the HAIDNet policy. As we can see in the results, the performance of the information policy output from HAIDNet is near- optimal. Moreover, HAIDNet provides much more efficient approach when the number of receivers is large. As a comparison, solving the exact optimal information policy for each problem instance is time-consuming (e.g., it takes more than 23 hours to solve instance with 18 receivers). On the other hand, HAIDNet only needs to optimize the model once to generate the optimal information policies for all possible problem instances with the same number of receivers (e.g., training HAIDNet with 18 receivers takes slightly more than I hour, and generating information policy for a problem instance takes less than 1 second). The empirical run-time comparison is included in the appendix.
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4.1.3 Settings without known solutions
We now examine the performance of HAIDNet in set- tings where there are generally no known analytical solutions yet. The goal is to showcase that HAIDNet can be leveraged to address information design prob- lems when we do not have access to solutions.
97
76
0.55
Br for HAIDNet Policy
79
74
10
0.90
20
92
86
82
81
All our simulations so far have focused on settings which assume that receivers are Bayesian rational. To examine whether HAIDNet works for non- -on-Bayesian- rational receivers, we adopt a relaxation of human behavioral formulation as in Equation (T). While there are no known solutions for identifying the opti- mal policy in this setting in general, Tang and Ho [70] derived solution for the simple setting with binary actions and binary states. Therefore, we compare the performance of the optimal policy and the HAIDNet policy in this simple setting under different choices of 3y in the human descriptor in Equation ( Using the same setup as in previous simulations, we report the results in Table showing that HAIDNet works receiver.
50
100
36
S44
92
AN
94
2.D
19.3
200
500
97
94
91
87
83
85
80
94
1.0
96
34
32
0.70
4
500
Human Model Prr
Figure 3: The performance of HAIDNet in settings when the receiver is not Bayesian ra- tional. We train HAIDNet with non-Bayesian- rational receiver model parameterized by 9н then evaluate the learned information policy for all receiver models. The performance is normalized for each human model, the op- timal performance is 1.0 among all policies.
even for non-Baye Non-Bayesian-ratione
Next, would like to examine how HAIDNet per- forms in scenarios when there are no known solutions (c.g., in settings with more than binary actions/states). To demonstrate the results, we choose the setting with three states and three actions. The lack of an optimal solution means we cannot evaluate the performance of HAIDNet by comparing its performance with the optimal policy in the simulations above. Instead, take different method and provide evidence to support our approach: We evaluate the set of all learned policies THi against each of the human models By.
For each human model 3n = k, if TE is the best- rest-performing policy, this indicates that our approach generates a reasonably good information policy. Specifically, for each human model, we compute the performance of each policy available, and we then normalize the set these performances so that the best best-performing performance for each human model has value 1. If our HAIDNet indeed learns a good information policy, we would expect the best performing HAIDNet to be the one trained on the right human descriptor. The results, as shown in Figure 3 demonstrate this behavior and provides evidence that our HAIDNet generates good information policy even when the receiver is not Bayesian rational.
4.2 Real-World Human-Subject Experiments
In the simulations, we have assumed access to closed-fome behavior model of the receiver. However, in practice, human behavior complex and there may not exist single model that can perfectly rep- resent human behavior. Motivated by this practical concern, we conduct human-subject experiments to examine whether HAIDNet adapts to real-world human behavior. The goal to examine whether we can utilize data-driven approaches to learn human-behavio descriptors and examine whether HAIDNet performs well when it is paired with data-driven behavior descriptors.
Task description. In our human-subject experiments, we present the product purchasing example in Section to human participants. Each human participant is asked make multiple rounds of purchase decisions. In each round, the participant is presented a product with unknown binary quality (good or bad product). The participant is told that (noisy) inspection has been performed on the product, and is given the conditional distribution associated with the inspection (i.e., the probability to receive a good/bad signal given the product is good/bad). Finally, the participant given realization of the inspection signal and is asked to make binary decision of purchasing or not. The participant's payment depends on both their purchasing decisions and the true product quality. The task interface is included in Appendix C The experiment is approved by the IRB in our institution.
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Table 4: Test accuracy of different human behavior descriptors human human-subject experiments.
Model
Bayesian rational
TH-Model
Neural network
Test Accuracy
0.562
0.735
0.770
Experiment procedure. We have recruited 300 workers from Amazon Mechanical Turk. We set the base payment to be $0.50. Workers could earn additional bonuses depending their performance. The average hourly rate was around $11 USD. The experiment contains two phases as described next.
Learning human behavior descriptors. The goal of the first phase is to examine whether we could learn accurate human behavior descriptors from worker's response data. In this phase, recruited 100 workers, and each worker completed 20 rounds of product purchasing decisions. The parameters of each decision (prior, sender utility, receiver utility, and policy) was drawn uniformly at random. We split the collected data into training/test sets, with 80% of the data for training, and 20% for testing. We trained and examined the performance of three different human behavior descriptors.
Bayesian rational: This descriptor makes the standard assumption that humans are Bayesian rational. There no training needed for this descriptor.
TH-Model: We the parameters of the TH model, as described Section from data to minimize the least squares error.
Neural network: We use fully connected- layer neural network fit the data in the training set. We further split the training dataset and use 25% of the data as the validation set to implement carly-stopping during training.
We then examine how accurately each descriptor predicts human behavior in the test data. The test accuracy reported in Table As we can see from the results, the data-driven neural network model leads to the best prediction accuracy, and both TH-Model and the data-driven descriptor significantly outperform the Bayesian rational assumption, reaffirming the need to relax this common assumption.
Evaluating HAIDNet. In the second phase, we recruited 200 workers to examine the per- formance of different information policies. In
0.8
0.6
Senders usiity
particular, we examine the following four infor- mation policies:
0.4
Random:
This information policy is drawn
from Dirichlet distribution.
BR-policy: The optimal policy when the
ceiver is a Bayesian rational receiver.
0.0
Randem
BR-Policy
TH-policy
HAIDNet
Treatment
TH-policy: The optimal policy when the re- ceiver behavior follows the TH-Model, as in Section 3.1
Figure 4: Average sender utility of different poli- cies in numan-subjers experiments. The differ- ences between BR policy and TH-policy and be- tween BR-policy and HAIDNet are statistically significant (p < 0.01).
HAIDNet: The policy by HAIDNet when use the neural network learned from the first phase as the human model.
When each worker arrives, they are randomly assigned to one of these four policy treatments. They then presented with 20 rounds of purchase decisions (the parameters of each round randomly drawn from distributions fixed across all treatments) coupled with the associated information policy the treatment. We then measure the average sender utility in each treatment. The results, as shown in Figure 4 demonstrate that HAIDNet achieves the best performance. The results showcase the effectiveness of HAIDNet coupled with data-driven human behavior descriptors.
In addition to examining the sender's utility, we also measure the receiver's utility in each treatment (see the full information in Appendix C). We observe that, the policy of HAIDNet leads to average receiver utility of 0.532, which is the lowest of all four treatments. This creates the concern that when we incorporate the knowledge receiver behavior optimize the sender's 's utility in information design, we are potentially exploiting the knowledge of receiver behavior and hurting the receiver. We offer more discussion on this concern in the discussion section of the paper.

---

5 Conclusion and Discussion
We initiate the study of behavioral information design that encodes human behavior into the design process. We propose HAIDNet, neural-network-basee optimization framework for information design that can adjust to multiple forms of human behavior. Through extensive simulations and human-subject experiments, we demonstrate the effectiveness of HAIDNet in response to different human behavior descriptors, Below we discuss the generalization / limitations and highlight the potential social impacts of this work.
While this work has focused on integrating human behavior in automated information design, we believe the methodologies are generalizable to design mechanisms for general cemman-in-the-lope systems, explicitly encoding realistic human behavior and/or human responses to the system when designing the system. Moreover, our current investigations have adopted the most standard deep learning setup (e.g.. full-connected neural networks coupled with stochastic gradient descent). It would be interesting to examine whether the performance could be further improved with carefully crafted network architecture and optimization procedure.
Generalization and limitations.
We would like to note the potential limitations of our approach. Our optimization procedure, based on applying stochastic gradient descent on neural networks, does not guarantee to lead to globally optimal solutions in general. Moreover, compared with analytical solutions that are guaranteed to be optimal for all problem instances if the receiver behavior follows the assumption, HAIDNet a data- driven approach that optimizes the expected utility, which requires training data to be representative to ensure generalizability. While our results suggest that HAIDNet recovers the near-optimal policy (e.g., the results in Figure 2) examining the impacts of different training data distributions and whether the results are robust to distributional shifts are potential important future research directions.
Another limitation pertains to the scalability of our proposed approach. While our method exhibits better scalability than exact solvers that utilize linear programming (more detailed discussion is included in Appendix [B.). our current results primarily focus on discrete action/state spaces. As the number of states and actions expands, so does the input size for HAIDNet. It could require much more training iterations to reach convergence. Furthermore, in scenarios with continuous action/state spaces, our approach is not immediately applicable. While discretization might be employed to address the setting with continuous spaces, such an approach requires additional smoothness assumptions to ensure small discretization errors. Overall, understanding and improving the scalability of HAIDNet is an important next step for increasing its practical applicability.
Potential negative social impacts. Finally, we highlight the potential negative social implications of the usage of information design frameworks. In information design, the sender often represents the party in power (e.g.. the government, social networking platforms), while the receiver is in a less advantageous position (e.g., the general public, users) due to the asymmetry of information access. While is possible to use information design for social good, guiding the receiver towards actions that are beneficial for himself or the public, the vast majority of information design literature - including our work focuses on optimizing the sender's utility. When the interests of the sender and receiver are not aligned, optimizing the sender's 's utility could result in a negative impact on the receivers, who are often the general public. In other words, with an ill-specified objective in information design, the sender could exploit the information advantage and create significant negative social impacts. This concern is further amplified when we obtain more accurate knowledge about the receiver. is therefore important to consider the impacts and potential regulations on information design.
In light of the concerns raised, to initiate the discussion, we discuss two potential risk mitigation methods. Firstly, on the technical front, we could employ differential privacy techniques 22 21 to control the amount of private human behavior being incorporated into receiver models. Differential privacy provides a means to balance privacy with utility, typically by introducing controlled noise into the data. This mechanism might be helpful in mitigating the exploitation of marginalized groups, an issue that might be exhibited in our approach. Secondly, from policy perspective, once we develop a comprehensive understanding of the capabilities of information design with data-driven human models, we, as a society, could and should weigh the utility gains from this method against potential harm. This discussion could then pave the way for the development of regulations and policies for deploying information design. For instance, we might impose constraints ensuring that the deployed information policy does not significantly reduce receiver utility, especially when compared to policies designed assuming standard models such Bayesian rationality.
10

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Acknowledgements
We thank the anonymous reviewers for the helpful comments. This work is supported part by the Office of Naval Research Grant N00014-20-1-2240.
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A Related work
Our work joins a growing number of studies that leverage computational tools for automated mecha- nism design [1] (5], the problem of utilizing computational approaches learning-hase -hasi techniques for finding maxiii-max mechanisms in auction settings. One strand of works [0] 66] in this line of research has focused using learning approaches for mechanism design where only samples of bidder valuations are used to design arrenue-mxxmmizen deep neural networks has been utilized for the automated design of optimal auctions (20), in which the
mechanisms. More recently,
authors propose multiple neural-network architectures for learning approximately optimal auctions.
Several works has extended this study in various applications [27 [3 [ 42
61
[88 45 12] Our
work differs from this line of works in two ways. First. we extend the approach beyond auction design to address the automated information design problem. Second and more importantly, we have incorporated human behavior descriptors in our design, while prior works mostly require standard rationality assumptions.
Our information design formulation builds on top of the seminal work of Bayesian persuasion [39,, which initiated rich theoretical literature on communication games in which sender can design information to persuade a receiver to take certain actions. Their work has provided theoretical foundations and inspired an active line of research in information design [e.g.. see the recent surveys by 38 Our work builds on top of this line of work through integrating human behavior in the design of information policy, while existing works mostly assume the receiver Bayesian rational. In particular, our proposed HAIDNet can dynamically adjust to various forms of model-based or data-driven human behavior descriptors. For the model-based receiver behavior, as an example, we have included the probability weighting function [77, 60, 64] for belief updating and the discrete
for decision making under uncertainty. Non-Baye non-Bayesian belief updating in information design also appears in earlier works [14), and the receiver's 's behavior following the
choice model [22
68 7
discrete choice model also appears in previous works 770 26). Our work generalizes the above in that our framework can adapt to both the above form and the data-driven form of human behavior.
The problem of information design and persuasion has received increasing attention both in research and in practice. For example, researchers have argued that one-quarter of the GDP in the United States persuasion (51). Due to its practical relevance, this problem getting attention more broadly in the general research community, as demonstrated by the recent papers in machine learning and artificial intelligence venues, studying various problem settings such as in security [7)], human language interactions [2). data marketplace design [8), algorithmic recourse [3), online recommendation [25]. and market competitions [5]. Our work joins this line of study and aims to develop more efficient approaches for information design under more realistic settings of human behavior.
On a conceptual level, our work is related the growing attention in understanding, modeling, and accounting for human behavior in computational systems, especially in the context of human-robot or human- AI interactions
9 67 7 63 54 [55] 74. Moreover, our work joins the recent research theme that incorporates human models in computational and machine learning frameworks (28] (7] 69 70 49 50 72 80
There have been other lines of research that includes humans in the loop of learning frameworks, such as inverse reinforcement learning [56] 24, 67 36, 8II that infers the reward functions in Markov decision process through (potentially human) demonstrations. Our work differs in that we focused on the information design problem with realistic human receiver models.
Lastly, in this study, we incorporate insights from human behavior into information design. Ex- tensive literature from psychology and behavioral economies has been devoted to deepen our un- derstanding of human behavior. Examples include studies examining deviations from the standard Bayesian assumption in processing information [57 40 3 and the rationality assumption in decision-
making 37 52 68
73
35 While these classical models, often grounded in human data from
behavioral experiments [48 16 [7], offer interpretable behavioral insights, they tend to lack in terms of predictive accuracy. Recently, given the advancements of machine learning techniques and the avaialability of a larger amount of human data, there has been a growing effort to integrate behavioral insights from these classical models with machine learning techniques to enhance predictive accu- racy [5] 59). These models developed in this line of effort are directly applicable in our framework. Moreover, as outlined in Section S integrating human behavioral insights into information design can raise concerns about exploiting human irrationality. One potential solution is to incorporate the concept of differential privacy [22 21 71. This would control the amount of personalized information that can be used, preventing undue exploitation.
16

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B Experimental results and details
In this section, discuss additional sets of simulation results highlight the properties and performance of HAIDNet. We also provide details of the optimization process of HAIDNet.
B.1 Additional experiment results
B.1.1 Convergence of training
In this set of simulations, we have examined the convergence of training with respect to the number of training iterations and also with respect to the softmax parameter 3 when dealing with Bayesian rational receivers. Overall, HAIDNet converges to finding the optimal policy within reasonable setup.
To illustrate the results, here we present the simplest setting with binary actions and binary states, namely, the action space A = {0, } and the state space e = {0. 1}. and observe whether HAIDNet can produce near-optimal information policies. For the sender utility, we adopt a stylized setting where the sender obtains utility 1 when the receiver takes action 1 and utility 0 when the receiver takes action O. We randomly draw each value the receiver utility R from (0, 1]. The prior distribution is drawn from Dirichlet distribution. We then simulate data using the setting above and optimize HAIDNet.
We compare the performance of the policy learned by HAIDNet with the closed-form optimal policy, Recall that when the receiver is rational (expected utility maximizer), he chooses the action that maximizes his expected utility given his belief about the state. As introduced in Section (.2, to enable the gradient- based method in optimizing HAIDNet, we replace this argmax operation as softmax using a softmax scale parameter B. Therefore, we first examine the impact of this choice of 3 and the amount of training (# iterations in gradient descent) in optimizing the information policy. As shown in Figure 5] when ,9 is large enough and when we optimize over large enough number of data batches, the learned information policy from HAIDNet converges to the information policy that achieves near-optimal performance.
0.80
Sender's utility
Sender's utility
0.75
0.75
0.70
0.70
0.05
0.65
0.60
Optimal HAIDNet
0.60
Optimal HAIDNet
0.50
5000
10000
15000
20000
0.50
5
20
50
100 200 500 1000
Iterations
B
(a) Training iterations.
(b) B.
Figure 5: The convergence results, with respect to # training iterations and 3, of the sender's utility derived from the information policy generated by HAIDNet.
B.1.2 Scalability: Empirical run-time comparison
One of the benefits of HAIDNet is provide efficient solutions for settings when it is computational challenging to derive the optimal policy exactly (e.g., in settings with multiple receivers).
To demonstrate this benefit empirically, we first record the time for computing the exact optimal policy for problem instance with receivers via Linear Programming approach [9]. As we can from Table the amount of time to compute the optimal information policy grows significantly (the computational complexity grows exponentially as the number of constraints is exponential the number of receivers in the linear programming approach) as the number of receiver increases. This reaffirms the computational barriers to computing the exact optimal policy. Note that Xu (78) has shown that it is #P #a-hard to approximate the optimal sender utility within any constant multiplicative factor. So this computational barrier is backed by theoretical analysis.
For HAIDNet, for each class of problems (i.e., given number of receivers), we only need to train HAIDNet once. For each new problem instance (different priors, sender/receiver utilities, etc), we only need to make a test-time prediction (one pass of forward propagation) to generate information
17

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policy. Again, in 5 we report the time for training HAIDNet and the time for generating the information policy for each problem instance. To provide the number comparisons, when the number of receivers is 18, traditional linear program method of solving the information policy for problem instance takes more than 23 hours. On the other hand, for HAIDNet, we only need little more than 1 hour to train HAIDNet for all problem instances with 18 receivers, and it takes less than 1 second to generate the information policy for each receiver. The reported numbers are performed on the machines with Intel(R) (R) Gold 6148 CPU (2.40GHz) and Tesla V100-SXM2-32GB GPU,
Table 5: Comparing run-time between HAIDNet and linear programming methods. K is the number of receivers. The reported run run-times are in seconds.
Training Time Testing time per instance of HAIDNet
Optimal policy per instance via Linear Programming
K
of HAiDNet
2
1082
0.184
0.323
3
1291
0.189
0.367
5
1571
0,221
0,371
10
2174
0.270
4.820
15
3284
0.299
235.0
17
3713
0.333
14290
18
4030
0.352
84280
B.1.3 Additional results for a single Bayesian rational receiver
In Section 4.1.1 we compare the performance between the policy from HAIDNet and the optimal policy in the single Bayesian rational receiver setting with an increasing number of states with binary actions, and an increasing number of actions with binary states. To further complete the results, we have also run simulations when we simultaneously increase the number of actions and the number of states at the same time. To put the performance of HAIDNet into context, we also include the performance of random policy, which provides random signals all the time. This random policy serves as the naive bascline setting. The results shown in Table [6c| The average sender utility obtained by HAIDNet policy is close to optimal policy in both training and testing problem instances (averaged over 1,000 instances) even in cases with large action and state numbers. We also evaluated
the model training error for binary action case and binary state case in Table 6 which shows that HAIDNet works well for large-scale problem instances.
B.1.4 Additional results for multiple receivers and non- non-Bayesian-rationa receivers
Due to space constraints, do not include all results in these two settings in the main paper. Here we provide the full results, including the performance of random baseline as well. The results for settings with multiple receivers are included in in Table and the results for settings with a single non-Bayesian-ratione receiver are included in Table [ HAIDNet performs well in multiple receiver settings and non- non- Bratiian- receiver settings.
Recall that for all the results presented in this work, both the training and testing performance are the average performance for 1, 000 data points sampled from the training and testing datasets.
B.1.5 Varying number of receivers, actions and states
In our main paper, each HAIDNet can accommodate any problem instance (i.e., different specifica- tions of priors, receiver utility, and sender utility) with a fixed number actions, states, and receivers. It is then natural to wonder whether varying numbers of receivers, actions, and states. As a proof of concept, in this set of simulations, we attempt to address this question and present approach that can work with varying numbers of receivers, actions, and states when the numbers are upper bounded.
extend the HAIDNet structure that it can work with
We first examine the relaxation of a fixed number of receivers. In particular, we can generalize our approach to address varying numbers of receivers when the number of receivers is upper bounded. One straightforward approach is to maintain multiple HAIDNet, one for each fixed number of receivers, for generating the optimal information policy. Another approach is to train a HAIDNet that can generate information policy for the maximum number of receivers. In settings when the
18

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Table Comparing the average sender utility generated by the optimal policy and the policy from HAIDNet in the setting with single Bayesian rational receiver.
(a) Increase the number of states M with binary actions.
Training HAIDNet
Testing HAIDNet
M
Random
Optimal
Random
Optimal
2
0.4901
0.7409
0.7498
0.4909
0.7408
0.7451
0,5009
0.7737
0.7782
0.4819
0.7598
0.7669
5
0.4898
0.8171
0.8209
0.5227
0.8066
0.8225
10
0.4841
0.8495
0.8699
0.4838
0.8196
0.8686
(b) Increase the number of actions N with binary states.
Training HAIDNet
Testing HAIDNet
N
Random
Optimal
Random
Optimal
2
0.4901
0.7409
0.7498
0.4909
0.7408
0.7451
3
0.4911
0.7017
0.7214
0.5064
0.7089
0.7227
5
0.4919
0.6906
0.7113
0.5119
0.6690
0.7064
10
0.4907
0.6861
0.7084
0.4861
0.6623
0.6963
(c) Increase both the number of states and actions. M N represents state number equals action number.
Training HAIDNet
Testing HAIDNet
M N
Random
Optimal
Random
Optimal
0.4901
0.7409
0.7498
0.4909
0.7408
0.745I
3
0.4791
0.7352
0.7679
0.4854
0.7199
0.7587
0,5029
0.7771
0.8113
0.5101
0.7755
0.8121
10
0.4799
0.8613
0.8971
0.4872
0.8323
0.8994
50
0.4903
0.9247
0.9550
0.7058
0.9166
0.9545
Table 7: Comparing the average sender utility generated by the optimal policy and the policy from HAIDNet in the setting with Bayesian rational receivers.
Training
Testing HAIDNet
K
Random
HAIDNet
Optimal
Random
Optimal
0.5158
0.7887
0.7934
0.5195
0.7756
0.7873
0.5050
0.7508
0.7665
0.4898
0.7379
0.7573
0,4920
0.7217
0.7458
0.4980
0.7209
0.7570
10
0.5063
0.6971
0.7152
0.5192
0,6790
0.6966
0.5007
0.6553
0.6882
0.4841
0.6621
0.6843
17
0.5037
0.6166
0.6503
0.5004
0.6160
0.6497
Table 8: Comparing average sender utility by the optimal policy and the policy from HAIDNet in the setting with con-Bayesiannrational receiver parameterized by 8T..
Training
Testing HAIDNet
8h
Random
HAIDNett
Optimal
Random
Optimal
0,4986
0.5043
0.5051
0.5006
0.5041
0.5060
0.4929
0.5512
0.5557
0.5036
0.5506
0.5559
10
0.4904
0.6045
0.6170
0.5064
0.5986
0.6168
0.4919
0.7002
0.7134
0.5093
0.6800
0.7081
100
0.4931
0.7187
0.7291
0.5097
0.6964
0.7179
500
0.4946
0.7418
0.7468
0.5096
0.7354
0.7396
number of receivers is less than the maximum number, we can include "null receivers" who always choose action (by setting the receiver utility such that the utility for taking action 0 always larger than taking other actions in both states). By including this in the training process, we can have single HAIDNet that can generate policies for bounded variable number of receivers. proof of concept, we have implemented the above approach and trained HAIDNet that can work with up
19

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to 10 receivers. We then examine its performance when the number of receivers is smaller than 10. As we can see from Table 9 this approach achieves reasonable performance and shows promising results.
Table 9: Comparing the average sender utility by the optimal policy and the policy from HAIDNet in the setting with at most 10 Bayesian rational receivers.
Training
Testing HAIDNet
K
Random
HAIDNet
Optimal
Random
Optimal
0.5042
0.7830
0,8018
0.4986
0.7538
0.7921
0,5066
0.7337
0.7586
0.4866
0.7139
0.7450
0,5032
0.7245
0.7451
0.5071
0.7121
0.7387
10
0.4944
0.6911
0.7118
0.5009
0.6650
0.6901
We now examine whether this approach also works for extending the number of states M and the number of actions N. As proof of concept, we adopt the approach above and train HAIDNet for maximum of 5 actions and states. We then examine the performance of HAIDNet for problem instances with less or equal actions or states. As shown in Table I0] this approach also works in addressing varying numbers of actions and states.
Table 10: Comparing the average sender utility by the optimal policy and the policy from HAIDNet in the setting with most states and 5 actions, for single Bayesian rational receivers.
Training HAIDNet
Testing
(M, N)
Random
Optimal
Random
HAIDNet
Optimal
0.4994
0.6564
0.7308
0.5276
0.6517
0.74I1
(2, 4)
0,4852
0.6450
0.7134
0.5236
0.6535
0.7329
(2,
0.4898
0.6498
0.7042
0.5111
0.6641
0.7258
(3
0.5094
0.6856
0.7735
0.4731
0.6462
0.7574
(3,3)
0.5128
0.7072
0.7791
0.4832
0.6689
0.7615
(3,4)
0.5343
0.7165
0.7729
0,5322
0.6940
0.7672
(3, 5) 0.4798
0.6922
0.7453
0.5308
0.6990
0.7492
(4, 2)
0.4898
0.6849
0.7701
0.5216
0.6922
0.7883
(4,3)
0.4721
0.7051
0.7761
0.4796
0.6940
0.7844
(4,4)
0.5032
0.7239
0.7812
0.5143
0.7186
0.7962
(4.5)
0,4700
0.7347
0.7807
0.5144
0.7421
0.7925
(5
0,4883
0.7147
0.7915
0.5186
0.7137
0.8038
(5.3)
0.5394
0.7736
0.8398
0.4928
0.7318
0.8184
(5, 4)
0.4998
0.7810
0.8289
0.4951
0.7494
0.8242
(5, 5)
0.4819
0.7722
0.8159
0.4863
0.7605
0.8079
B.2 Data generation
Here provide the details in generating data instances for training HAIDNet in settings.
Single receiver, binary actions and binary states. In the simplest setting with binary actions and binary states, the action space is A {0,, 1} and the state space is θ {0, 1). We adopt stylized setting for binary actions where the sender obtains utility when the receiver takes action 1 and utility 0 when the receiver takes action 0 33). The receiver utility & is uniformly drawn from [0. and prior distribution draw from Dirichlet distribution. We filter out trivial problem instances where the receiver will always choose action whatever the information policy, e.g., the receiver always chooses action when receiver utility (11 0) > u RB 0), vo E θ. Total 102,400 instances are generated for training. 1,000 for validation and 1,000 for testing.
Single receiver, multiple actions, and multiple states. the setting with N actions and M states, and the state space is θ {0. I,..,, M 1}. The sender utility is set to u (a, 0) = N vo i 2 3, and the same as above binary actions
the action space is A = 1, 1,...
The receiver utility n. is uniformly drawn from [0, 1] and prior distribution is drawn from Dirichlet distribution. We also filter out trivial cases where the receiver will always choose one action whatever the information policy is.
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Multiple receivers, binary actions, and binary states. The receiver utility and prior distributions are generated in the same way as in the cases of single receiver, binary actions, and binary states. The sender utility is the fraction of receivers choosing action 1, i.e., her utility is given [S] |S] number of receivers choosing action and K is the total number of receivers. We also filter out trivial cases where the receiver will always choose one action whatever the information policy is.
K if there are
Problem instances in human-subject experiments. In our human-subject experiment, the problem setup is the same as the setting with a single receiver, binary actions, and binary states. To make the setting easier to understand for experiment participants, the receiver utility is drawn from {1, 2, 4, when the participant chooses to purchase a good product or chooses to not purchase a bad product, and the participant utility is 0 for other cases. The sender utility is set to when the receiver chooses to buy, and 0 otherwise. The prior distribution is drawn from the Dirichlet distribution, however, we round all probability in the prior distribution and the information policy to the nearest tenth digit, {0%, 10%.... 100%}. to make it easier to interpret for human participants.
B.3 HAIDNet optimization procedures
Here we provide more detailed parameter setups for our simulations and human subject experiments.
Optimizing HAIDNet in simulations. We build a 3 fully connected layer neural network with ReLU activation functions as the sender's 's optimization module in HAIDNet. Network parameters are initialized by Glorot uniform initializer. When optimizing HAIDNet, we use the Adam optimizer and batch gradient descent. Batch size is 1,024, batch number is 100, and maximum training epoch (each epoch contains 100 batches) is 1,000.
The hyperparameters are tuned by using the validation dataset. We then report the per- formance on the test dataset. The number of nodes for each hidden layers is tuned in the range of (64, 128, 256, 512, 1024], and the initial learning rate is tuned in the range of {0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1). When the human descriptor is Bayesian rational, we use softmax to smoothen the argmax operator. Empirically we find directly assigning large value to 33 leads to bad performance. So instead, we increase /3 gradually from 10 to 1000 exponentially in first 100 epochs of training, that is 3, = 10 io in i'th epoch, and maintain , = 1000 for remaining training process. In the setting with multiple actions and multiple states, softmax approximation of Bayesian rational behavior leads to higher errors. We further adapt the approach of iteratively training networks, reweighting data distributions, and aggregate learned neural networks to reduce the error. Empirically, aggregating three models are enough to reach promising performance.
Optimizing HAIDNet in human-subject experiments. After collecting human responses in the first phase of human experiments, we fit TH-Model and train neural network model for human behavior descriptors. 31 in TH-Model is fitted by minimizing a square error hetween model prediction and human data, and BH = 20 fits the best. For the neural network model, we use 3 fully connected- layer neural network with ReLU activator. We split the data into training/testing sets, with 80% of the data for training. and 20% of the data for testing. We further split the training dataset and use 25% of the training dataset as a validation set to implement carly-stopping during training. The neural network for fitting human behavior is trained with batch number 12, batch size 100, and maximum epoch 100. The number of nodes for each hidden layers is tuned in { {16, 32, 64, 128, 256, 512, 1024] and the initial learning rate is tuned in (0.001, 0.002, 0.005. 0.01). We select the typer-paramettr based on average performance of validation sets Because of 5-fold splitting, we have trained neural networks for human descriptors, and we train HAIDNet corresponding each of the human model. The learned policy are close in terms of expected utility, in range of [0.697, 0.726), and we select the model of the highest performance in simulation to design the information policy in the second phase experiment.
C Details of Human-Subject Experiments
We provide more detailed information about our human-subject experiments here, We compare average sender utility of different policies in human-subject experiment Figure and we also compute the receiver utility in each treatment, included in Table As we can see from the table, while HAIDNet helps find a policy that leads to the highest sender utility, it comes at the cost of reducing the receiver utility, a demonstration of the ethical concerns as discussed in Section 5
21

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Table 11: Comparing sender and receiver utility of different policies in human-subject experiments.
Information Policy
Random
BR-Policy
TH-Policy
HAIDNet
Sender Utility 0.489
0.524
0.621
0.667
Receiver Utility 0.663
0.634
0.565
0.532
In experiment setup, given the sender's goal is to have the receiver purchase the products regardless of the product quality, when the sender more successful, it leads to a lower receiver utility in general and implies the potential negative social impacts.
C.1 Demographic of Workers
We have recruited 300 workers from Amazon Mechanical Turk in total for our experiments. Table contains the demographic information of the 300 workers.
Table 12: Demographic information of the participants in experiment. Group
Category
Number
20 to 29
30 to 39
111
Age
40 to 49
50 to 59
25
60 or older
Female
131
Gender
Male
168
Other
Caucasian
240
Black or Aetericnmmei
18
American Indian/ Alaskan Native
5
Race / Ethnicity
Asian or American
22
Spanish/Hispanic
6
Other
9
High school degree
12
Some college credit, no degree
9
Associate's degree
24
Education
Bachelor's degree
223
Graduate's 's degree
29
Other
3
C.2 Task Interface and Description
In our human-subject experiments, we simulate the setting with binary actions and binary states. In particular, we present the product purchasing example as we discussed in Section 2 The task interface about our human- experiments is shown in Figure
Each human participant is asked to make multiple rounds of purchase decisions. In each round, the participant is presented a product with unknown binary quality (either good product or bad product). The participant is told that a (noisy) inspection has been performed the product, and is given the conditional distribution associated with the inspection (i.e., the probability to receive good/bad
signal given the product is good/bad). Finally, the participant given a realization of the inspection signal and is asked to make a binary decision of purchasing or not. The participant's 's reward depends on both their purchasing decisions and the product quality. When collecting human response in the first phase, random policy are presented to all participants. In the second phase, different policies are presented: [Random, BR-policy, TH-policy. HAIDNet policy). ). The policies are designed with the assumption that the sender is persuading human receivers to purchase the product, and we caleulate the probability of participants choosing to purchase and report it as the sender utility to evaluate performance of different policies.
22

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following scenario:
1. You are presented with a product (9. product could be Good or Bad @, and you need to decide whether to purchase
bonus depend your purchese
product quality shown
you "Buy" Good product, you will get a bonus of cents.
you "Don't Buy" Bad product, you will get a bonus cents.
Otherwise, you won't get bonus this round.
Buy
't Buy
3
0
2
Bonus table
about product you decision.
The product randomly selected from a pool of products. For all products, 70% are Good and 30% are Bad.
After given the product, you will perform noisy inspection.
product good, will receive
rár with 20% chance bad with 80% chance.
product will good signal rdr chance signal
20%
Product Quality
Noisy Inspection
20%
70%
80%
Good
80%
30%
20%
3. Which action will you take when the inspection result is
Your Decision Is:
O Buy
Don't Buy
Figure 6: Human experiment interface.
23


Reviews:
[{"review_id": "4c1O9sCO7d", "reviewer": "Reviewer_AWce", "summary": "This paper proposes a neural network based framework for automated information design that can optimise both the senders and the receivers behaviour. In situations where the receivers behaviour cannot be approximated analytically, a neural network that learns the preferences from existing data can be utilised. In all situations the sender's behaviour is optimised via minimising a differentiable loss. In experiments they show that this framework can closely approximate the optimal solution in smaller scale problems, and can also be utilised with real human participants where the receivers behaviour must be learnt, and outperforms the less data-driven baselines. ", "questions": "(I have put my important questions in the section above - apologies)\n", "limitations": "Yes, limitations and potential negative societal implications have been discussed.", "rating": 5, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "As somebody not familiar with the field of automated information design, I still enjoyed reading the paper. \nI liked the presentation of the motivations and the preliminaries on Bayesian persuasion.\nRelaxing the assumption of Bayesian rationality and demonstrating that a more data-driven approach (backed by ML) can be utilised is hugely useful, especially since it is demonstrated on a study with real human participants. \n\nThat a learning based approach (utilising a small MLP) can be trained significantly quicker and still recover good approximations of optimal solutions is a big plus for the method (provided the practitioner has the appropriate hardware to train a neural network fast).", "weaknesses": "(Again not very familiar with the field) I am struggling a little bit to understand the exact contributions of this paper. \nProposing to use a neural network and that machinery to solve Eqn 2 doesn't seem to be novel (as mentioned in the paper lines 168-169).\nThere does seem to be in attempting to learn human behaviour in these scenarios from data with a neural network.\nI am a little unsure as to whether there have been ML (specifically neural network based) approaches to this problem in this field.\nNonetheless, since the ML setup+framing is such a crucial part of the proposed method, the paper would benefit from a more in depth description and discussion of the different components. \n1. What exactly is the input to the neural network, what size is it (it looks to be a table that is presumably flattened?), how does that scale wrt problem instances.\n2. What architectures were considered/studied for this particular problem.\n3. How can this approach scale when a tabular representation doesn't suffice. \n\nRelated to the previous point, from my understanding of the paper a big contribution is that human behaviour in these tasks is much better approximated with a learning based approach. Thus, how exactly that module is setup is very important and should be discussed in more detail.\nIn particular, how exactly the sender's information is policy is represented (and how this might or might not scale). In addition, the human's behaviour is approximated using existing data. However, as soon as the sender's policy changes to something that has not already been observed then we are reliant on the generalisation of the neural network to accurately capture the human's behaviour. How well this part of the framework generalises to the optimised sender policies should also be looked at. \n\nI like Figure 3, testing the generalisation. However, I would like a more absolute measure of the performance as well. Does the best\nperforming HAIDNet policy actually perform well on the task?\nAdditionall,y I would like more discussion on these results. Particularly for cases like $\\beta_H = 100$, where it seems to 'generalise' much better than say $\\beta_H=1$. What is the difference in the learned policies? Why for $\\beta_H=\\infty$ does the performance dip around 20?\n\nI like Table 4 a lot, some more discussion on it would be good to really drive home the differences between the Bayesian rational assumption and real observed human behaviour. Also a comparision between the NN and TH-Model to try and identify what about the TH-Model is preventing it from learning a better solution. Also, where does the neural network make errors? It's performance is indeed higher than the others, but there is still quite a bit of headroom.\nAlso for Figure 4, a more qualitative discussion on the resulting policies learned by the methods.\n\nMinor:\n\n> \"HAIDNet essentially recovers the optimal information policy in almost all scenarios\"\nWould be nice to quantify this a bit more, since there do look to be small differences in Figure 2.\n\nYou mention a boosting and aggregation approach is taken, please described this in more detail. \n\nIs the information in C.1 required/does it need to be reported in the paper (Race/Ethnicity in particular)?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful review. \n\n**[Contributions]** We first highlight our contributions. Firstly, we introduced a data-driven optimization framework for information design. Note that even in standard settings where humans are Bayesian rational, designing optimal information policy is known to be #P hard. Our proposed approach can more efficiently identify near-optimal information policies compared with standard techniques. Secondly, our approach can accommodate different representations of human behavior beyond standard assumption of Bayesian rationality. This flexibility also makes it straightforward to incorporate recent research trends that combine domain knowledge with data-driven models for human behavior. Lastly, we have validated our methods through human-subject experiments, showcasing the practical viability of our approaches.\n\nBelow we respond to your other comments/questions. We will incorporate your suggestions and the following clarifications in the revision.\n\n**[Scalability]** Please see our responses to Reviewer 5W54.  \n\n**[Network input and continuous input space]** The input to the neural network (NN) specifies the problem's specifications: the prior distribution, sender utility, and receiver utility. For instance, in a single-receiver setting with $M$ states and $N$ actions, the input size is given by $M * (2N+1)$. In scenarios with continuous action/state spaces (i.e., the input can't be tabulated), the information design problem is generally challenging to solve, both in the traditional information design literature and in our approach. To address this challenge, one common approach is to employ discretization. However, this approach requires some smoothness assumption (e.g., small deviation of actions wouldn’t lead to significant different outcomes) to ensure the discretization error can be bounded.\n\n**[Network architecture]** As described in Section 3.2 and Appendix B.3, we utilized a 3-layer fully connected neural network with ReLU activation functions. We want to emphasize that the objective of our study is to demonstrate the viability of a data-driven approach for automated information design. We are not positing that our chosen architecture is the best possible. Nevertheless, even with this relatively standard choice, our results have already been promising.\n\n**[Receiver modeling without representative data]** We want to highlight that our framework is designed to accommodate various representations of human behavior. As shown in our evaluations, it can handle both standard closed-form representations of human models (e.g., Bayesian rationality or TH-model) and data-driven forms (e.g., neural networks). The reviewer's concern regarding the challenge of developing human models when data is not representative (e.g., encountering situations not in the historical data) pertains specifically to data-driven human models. Indeed, if our human model is entirely data-driven, we would require the training data to be representative to ensure generalizability, as is commonly required in supervised learning.  \n\nThat said, the reviewer’s concern can be mitigated by incorporating the recent studies in utilizing existing close-form human models as priors and leverage behavioral data to improve the models [Bourgin et al. ICML 2019, Peterson et al. Science 2021]. The main benefit of this approach is that in cases when the data is insufficient, the model behavior would fall back to existing human models. Our framework can seamlessly integrate with human models developed using such methodologies.\n\n- Bourgin et al. \"Cognitive model priors for predicting human decisions.\" ICML.  2019.\n- Peterson et al. \"Using large-scale experiments and machine learning to discover theories of human decision-making.\" Science. 2021.\n\n**[Explain Figure 3]**  The parameter $\\beta_H$​ in this human model represents the stochasticity level of human behavior. A smaller value of $\\beta_H$​ indicates more random human behavior. Consequently, human models with $\\beta_H=1$ depict behavior that is close to random. As demonstrated in the left-most column of Figure 3, in such scenarios, all information policies lead to similar outcomes. Conversely, when $\\beta_H$ approaches infinity, human behavior becomes deterministic. In these instances, an accurate understanding of human behavior is essential for effective performance, as illustrated in the rightmost column of Figure 3. To explain the relative performance dips observed around $\\beta_H=20$ for policies trained on the human model with $\\beta_H=\\infty$: For smaller values of $\\beta_H$, the performance across different policies tends to be similar. However, for considerably large values of $\\beta_H$, a $\\beta_H$ value of infinity serves as a reasonable approximation. Thus, the most significant relative performance gap for a policy trained on $\\beta_H=\\infty$ occurs with moderate values of $\\beta_H$.\n\n**[Test accuracy of human models]** In our human-subject experiments, we are learning a model to predict the behavior for a population of humans. Together with the intrinsic stochasticity of human behavior, it is not possible to reach perfect accuracy in predicting human behavior. In fact, Tang and Ho [56] have estimated that close to 20% of worker behavior in a similar setting is random. \n\n**[Optimization procedure]** The boosting procedure is only used in settings with Bayesian rational receivers. Since this human model is not differentiable, we adopt the common practice of using differentiable softmax function to approximate its behavior, which leads to higher errors. We address this by taking a boosting approach of iteratively training networks, reweighting data distributions, and aggregate learned neural networks to reduce the error.\n\n**[Worker demographics]** It is a common practice to report the demographics of the user studies, for transparency and generalizability reasons."}, {"review_id": "riJ3VAvjxn", "reviewer": "Reviewer_7rN5", "summary": "The paper focuses on the problem of automated information design, where the sender strategically reveals information to persuade the receiver to take specific actions. The main contribution of this paper is addressing the challenge of modeling human behavior when individuals do not act as Bayesian rational agents. The problem is formalized as an optimization problem, and the authors propose a neural network architecture (HAIDNet) to solve this optimization problem. The effectiveness of their approach is evaluated through experiments conducted in both simulation and real world settings. In their simulations, they show that HAIDNet can solve settings _without_ efficient solutions, including multiple receivers and non-Bayesian receivers. They then fit a neural network model to model how human subjects update their preferences in practice, and show that HAIDNet significantly improves on baselines in convincing human subjects to change their behavior. ", "questions": "1. I'd be interested in hearing more about the relationship between this work and Reddy, Levine, and Dragan 2021*. In that work, they also fit a neural network policy (albeit a larger, recurrent one) to synthesize observations to influence the actions of a human users. Similarly, I think there's some amount of related work in the human-assistance literature in general that feature similar algorithms (where a blackbox neural network is used to learn a policy that performs well according to a utility function with human subjects).\n2. I'm somewhat puzzled about why you call it an architecture, when HAIDNet seems to me to be more of an algorithm for training arbitrary neural network architectures. In my case, I was a bit confused by this, and expected a special neural network architecture more directly designed for modeling information policies, but instead the paper uses 3-layer MLPs. I'd appreciate if this was made more clear in the introduction and in section 3.2. \n3. Similarly, I would appreciate more discussion on the core assumptions of the HAIDNet framework. What do you think are the crucial parts of the algorithm related to the key insights, and which parts are contingent on implementation details? For example, is it important that you optimize the policy with SGD instead of an RL algorithm like PPO or DDPG? Is it important that the neural network is a 3-layer MLP?\n\n\\* Reddy, S., Levine, S. and Dragan, A., 2021, October. Assisted perception: optimizing observations to communicate state. In Conference on Robot Learning (pp. 748-764). PMLR.", "limitations": "The authors describe the key limitations of HAIDNet as 1) possibly finding only locally optimal solutions as it uses Adam (a local first-order optimizer) to update the information policy and 2) only getting good performance _in expectation_, as opposed to uniformly across all cases. However, I think in comparison to these two limitations, the scalability, differentiability, and interpretability issues described in the weaknesses section above are more important for using their method in practice. \n\n**(Negative) Social impacts**\nThe authors correctly note that their method (as with most other work in this space) can be used for disinformation or user manipulation. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The method proposed in the paper is intuitive and make sense.\n* The writing is clear and effectively communicates the ideas, with Figure 1 providing a helpful visual illustration of the concepts.\n* The experimental analysis is extensive and noteworthy, particularly with the inclusion of results with real human subjects. This adds credibility to the findings.", "weaknesses": "* It is unclear how scalable this approach is. To fit the neural network, a large amount of data needs to be collected, and such data is often scarce. For example, in their experiments, they used 2000 worker-rounds of product purchasing decisions to fit a small 3-layer neural network on a binary tasks , more realistic human models \n* The proposed method requires differentiable human models, which can be hard to come by in practice, especially in cases where the human models are e.g. designed by hand or feature random components. \n* The learned human models and information policies are hard to interpret, which is compounded by the issue of scarce data. It's not clear what exactly the small neural networks have learned in their experiments, and the authors do not discuss this in their paper. In practice, it seems important to ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the valuable feedback and comments!  \n\n**[Scalability]** Please see our responses to Reviewer 5W54. \n\n**[Differentiable human models]** While the standard rationality model is not differentiable, many of the other models are, e.g., discrete choice models and data-driven models. The differentiable assumption essentially requires human behavior to be smooth, which is often approximately satisfied if we consider the stochasticity of human behavior or focus on dealing with a population of humans.  In cases when the human model is not differentiable, we can utilize standard approaches, e.g., softmax relaxations, to approximate human behavior. We will add discussion in the revision.\n\n**[Interpretability]** For the interpretability of information policy, we would like to first note that this is an open question even in the standard information design literature. While in some settings, e.g., when the receiver’s optimal action depends only on the expected state, an interpretable optimal information policy can be characterized with certain mild assumptions (see, e.g., Arieli et al EC’20; Kolotilin et al, Theoretical Economics'22), to the best of our knowledge, there are no existing characterizations on interpretable information policy for general settings.  In our work, to increase interpretability, one potential approach is to add constraints in the sender’s optimization problem to force the information policy to satisfy certain interpretability conditions. We will add more discussions on this point in the revision.  \n\nFor the interpretability of human models, we want to first highlight that our framework is designed to accommodate various representations of human behavior, including the standard closed-form representations of human models (e.g., Bayesian rationality or TH-model) and data-driven forms. While the former is interpretable, the latter is not in general. Developing an accurate and interpretable human model is actually an open and ongoing research question at the intersection of cognitive science and ML (e.g., Bourgin et al. ICML 2019 and Peterson et al. Science 2021). Our framework can seamlessly integrate with human models developed using such methodologies, mitigating the interpretability concerns.\n\n- Bourgin et al. \"Cognitive model priors for predicting human decisions.\" ICML.  2019.\n- Peterson et al. \"Using large-scale experiments and machine learning to discover theories of human decision-making.\" Science. 2021.\n\n**[Comparison to prior work]** As discussed in the expanded related work in the appendix, on a conceptual level, our work is related to the growing attention in understanding, modeling, and accounting for human behavior in computation systems, especially in the context of human-robot or human-AI interactions. We have cited several works from the same lab of the work mentioned by the reviewer on human-robot interactions, and we will include this one as well. More specifically, our works are similar in the sense of aiming to develop realistic human models and design AI, ML, or the environment to work with humans. However, depending on the problem setting, the format of interactions and objectives are different for different works in this literature. These differences result different problem formulations and challenges, including the need to abstract proper forms of human bebavior, identifying the objectives to be optimized with data-driven approaches, etc.\n\n**[Architecture]** We called HAIDNet an architecture to highlight that we have included human models in the design. However, we agree with the reviewer that it might confuse readers with the architecture of neural networks. We will call it a framework in our revision.\n\n**[Assumptions]** Thanks for the suggestions. We will more explicitly list the assumptions. In particular, we require two assumptions: (1) the behavior of humans used for training HAIDNet is draw from the same distribution of the human behavior during deployment (i.e., the standard training/test assumption in ML), and (2) human models are differentiable (this assumption is soft, as it can be approximated using soft-max relaxations if it’s not).\n\n**[Importance of implementation details]** Our goal is to show that a data-driven approach can be used to find a near-optimal information policy for general human receiver models. Our investigation has adopted the most standard setup: Fully-connected MLP + SGD, and this setup already shows promise in improving the outcomes. We did not examine the other architecture and approaches but agree it would be interesting future work.\n\n**[Expanding discussion]** Thanks for the suggestions. We agree with the reviewer’s comment and will discuss the scalability, differentiability, and interpretability in our revision."}, {"review_id": "d7BwhWjjQw", "reviewer": "Reviewer_5W54", "summary": "Information design is the problem in which a sender would like to optimize the information sent to a receiver, such that the receiver takes actions that the sender likes. As a simple example, a company may want to optimize what information they communicate about their product, to get consumers to buy the product. The typical Bayesian-rational model assumes that the receiver knows the policy by which the sender chooses what information to communicate, performs an appropriate Bayesian update, and then chooses an action that maximizes their expected utility (which can be different from the sender’s utility). The authors consider a more general setting in which the receiver is given the same information, but no constraints are placed on how they act on this information.\n\nIn general, information design problems are very difficult to solve (one result shows that a particular setting is #P-hard). The authors suggest that we instead solve information design problems using deep learning: in particular, we train a neural network (HAIDNet) to take in a specification of an information design problem, and output a policy for information to communicate. Crucially, HAIDNet takes in a (differentiable) specification of the receiver’s policy, and so can also work with models of the receiver that are not Bayesian-rational – including a neural network trained to mimic human behavior.\n\nExperiments show that in small problems where the optimal policy is known, HAIDNet produces results that are nearly optimal (achieve a sender reward that is single-digit percentages away from optimal, well above that achieved by a random policy). HAIDNet continues to produce good-looking results in larger settings where the optimal policy is not known, though it is hard to tell how good the results are. Finally, the authors run an experiment with real humans, where they first elicit the human policy, train a neural network to imitate the humans, and then train HAIDNet to solve the information design problem with the neural network as the receiver policy. They show that HAIDNet produces a solution that, when tested with real humans, significantly outperforms the optimal solutions under the assumption of Boltzmann rationality, and non-significantly outperforms the TH-model (a model from prior literature), illustrating the benefit of using data-driven models of human behavior.", "questions": "Do you have any plans or ideas for scaling up HAIDNet to much larger problems, where the full state space is implicit and the receiver policies cannot be enumerated in full?\n\nCan you elaborate more on the ethical implications / broader impacts of this research? (See discussion above; in particular I’m interested in a response to the point about exploiting systematic human biases.)\n\nMinor suggestions:\n\nLine 305: Mention that in the human experiment you tell the humans the prior in addition to the conditional probabilities (currently it only mentions the conditional probabilities).", "limitations": "The authors have some discussion of potential negative societal impacts, but I think it is insufficient – see Weaknesses.\n\nI would also recommend that the authors briefly discuss the limitation of scalability.\n\nTo get additional space, I think it would be fine to remove the discussion about how gradient descent does not find optimal solutions, and how the resulting solutions may not be robust to distributional shift: the NeurIPS audience will tend to assume these two limitations by default for any paper applying deep learning, unless the paper explicitly claims otherwise.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "1. By far the biggest strength of this paper is its experiment with humans. Showing strong performance with real humans is a significantly higher bar than showing good results in simulations with human stand-ins; this paper meets this higher bar.\n2. The setting of information design is scientifically interesting, and to my knowledge deep learning has not been applied to it before (though I am not familiar with the literature).\n3. The authors show two major benefits from using deep learning: the ability to approximate answers with arbitrary (differentiable) human behavior models, as well as the ability to compute approximate solutions much faster than with e.g. linear programming. (That being said, I do not know the classical state of the art for approximately solving information design problems.)\n4. The authors’ approach is simple and natural, suggesting it would be easy to replicate.", "weaknesses": "1. Scalability: Information design problems require networks to take full policies as an input to the network. This will be a challenge for large problem instances, and isn’t tackled in this paper.\n2. Ethics: Encoding realistic models of human behavior into information design seems particularly likely to have negative social impact.\n\n**Scalability**\n\nInformation design problems require networks to take full policies as an input to the network: in particular, the network modeling human behavior must take the full stochastic policy that the sender uses to make decisions as an input. (The HAIDNet does not need to take the full human policy as input, it just samples from the policy on appropriate inputs as needed.)\n\nThis can be done when there are a small number of possible states and possible signals to send, as in the simple experiments done in this paper. However, in realistic settings such as recommender engines deciding which ratings to display, or politicians deciding how to design policy experiments, there will be a huge number of possible states and signals, which cannot even be enumerated. It’s not clear how HAIDNet would scale to these situations.\n\n(That being said, it does not look impossible, or even necessarily very hard. Ultimately there is a bi-level optimization problem, and while those are hard to solve, there are many techniques that get traction on the problem.)\n\nI do think the contribution of the paper is significantly more limited due to the unscalable nature of the current architecture.\n\n**Ethics**\n\nI am worried that this research will primarily contribute to negative impacts on society, and have flagged it for ethics review below. If I were making the decision based only on the information I currently have, I would not publish the paper and would encourage the authors to set aside this research area and move on to other work.\n\nHowever, this is not one of the areas mentioned in the NeurIPS code of ethics, so I am not incorporating this into my evaluation for the paper, which I will base just on the scientific merits of the paper. I will leave it to the area chair / ethics reviewers to decide how to incorporate ethical considerations into the decision.\n\nThe authors note that typically in information design, the sender is the party in power, while the receiver is not, and so optimizing for the sender’s utility can result in using a sender’s informational advantage to exploit the receivers. I agree with this, but in addition I think moving from a Bayesian-rational model to a data-driven human behavioral model makes the problem much worse\n\nFor a Bayesian-rational receiver, we have the following property: the receiver’s beliefs over the underlying state are well-calibrated (they follow from an appropriate Bayesian update that incorporates how the information was generated). Relative to getting no signal at all from the sender, this can never harm the receiver (in expectation), since for a rational Bayesian agent the value of information is always positive. This bounds the amount of harm that can be done from information design.\n\n(It is of course still possible that the receiver does much worse than if the sender simply provided the information most useful to the receiver, which we would count as a harm in many cases, e.g. the sender may probabilistically choose not to reveal that a more expensive product is actually lower quality, so that the receiver buys the more expensive product some of the time; we would plausibly consider this unethical since the sender could have provided better quality information allowing the receiver to make a more informed decision.)\n\nIn reality, human receivers will not be Bayesian-rational. However, a sender policy that was chosen based on a Bayesian-rational receiver model is still going to produce actions whose intended effects are to give the receiver true information. In particular, it will _not_ choose actions based on their propensity to mislead or trick the receiver, because it is impossible to trick the Bayesian-rational receiver.\n\nHowever, once we move to data-driven models of human behavior, we lose these guarantees. A data-driven model of human behavior will likely also include many human biases, and so a sender policy optimized against such a data-driven model will learn to exploit these biases. For example, it may:\n\n1. Appeal to irrational fears (e.g. selling products by making some risks emotionally salient, even if the risks are tiny in practice – maybe an expensive shark repellant to ward off shark attacks)\n2. Understand which claims humans can check (i.e. which aspects of the sender’s policy the receiver can observe and is sensitive to), and lie about any claims that can’t be easily checked\n3. Identify markers of credibility, and design signals that exploit credibility marker (e.g. endorsements from celebrities or scientific experts with strong public reputations, but who give their endorsement freely to even poor quality products).\n\nI’m sure there are many more such applications, these are just for illustration. We already see some of these effects with advertising and politics, and that is with relatively unsophisticated models of human behavior.\n\nAll of these effects could be significantly exacerbated if there are significantly more accurate data-driven models of human behaviors, or if these models could be personalized to specific individuals. In the case of personalized models, I would also expect to see significant inequality effects, where more elite, well-connected, and knowledgeable people will better understand how automated information design results in manipulation, and will be better able to counteract its effects, and meanwhile worse-off people will be more vulnerable to manipulation.\n\nThere are of course potential positive applications as well: for example, this could be used to design more effective campaigns to improve public health (e.g. leveraging knowledge of human biases to get people to wash their hands more often). However, it seems like the negative applications are both more numerous and more likely to happen.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the insightful comments! We will incorporate the clarification suggestions. Below we respond to the two major comments.\n\n**[Scalability]** \n\nIn our framework, there are two separate scalability considerations:\n\n*Scalability of optimizing information policy*: We'd like to begin by noting that scalability is a recognized challenge in the traditional study of information design. A key motivation for our work is the result by Dughmi and Xu [15] that computing the optimal information policy is #P-hard. \n\nWe have empirically examined the scalability of our approach in Appendix B.1.2. Specifically, as depicted in Table 5, for the more computational heavy setting with multiple receivers, the standard linear programming (LP) method of solving the information policy scales poorly, taking 0.3 seconds to solve an instance with one receiver and over 23 hours to address an instance with 17 receivers. In contrast, our approach demonstrates better empirical scalability. It takes 0.3 hours to train a neural network (NN) for one receiver and 1.12 hours for 17 receivers. It's noteworthy that the trained NN can generate information policies to all problem instances, and the testing time to generate a policy for an instance takes less than 0.352 seconds, even for cases with 17 receivers. These empirical results suggest that our method offers a much more scalable solution for designing information policies compared to traditional methods. \n\nIn scenarios with continuous action/state spaces, the information design problem is generally challenging to solve, both in the traditional information design literature and in our approach. To address this challenge, one potential approach is to employ discretization. However, this approach requires some smoothness assumption (e.g., small deviation of actions wouldn’t lead to significant different outcomes) to ensure the discretization error can be bounded.\n \nWe will add discussion in the main text in the revision.\n\n*Scalability of approximating human behavior*:  Our framework is designed to accommodate various representations of human behavior. As shown in our evaluations, it effectively handles both the standard closed-form representations of human models (e.g., Bayesian rationality or TH-model) and data-driven forms. While we concur with the reviewer's concern regarding the scalability of training a purely data-driven human model, recent studies at the intersection of cognitive science and ML have shown that we can utilize established human models (e.g., Bayesian rationality or the TH-model) as priors and harness data to refine these models [Bourgin et al. ICML 2019, Peterson et al. Science 2021]. The main benefit of this approach is that it provides a reasonable model to begin with even when we do not have enough data and could lead to a more accurate model with more data. Our framework can seamlessly integrate with human models developed using such methodologies, mitigating the scalability concerns.\n\n- Bourgin et al. \"Cognitive model priors for predicting human decisions.\" ICML. 2019.\n- Peterson et al. \"Using large-scale experiments and machine learning to discover theories of human decision-making.\" Science. 2021.\n\n**[Ethics]** \n\nWe wholeheartedly agree with the reviewer’s concern, which is the reason we brought up this point during our discussion. The information design literature, and more broadly, the principal-agent problem or Stackelberg game in economics, presents a scenario where one advantaged party might exploit its informational advantage over the disadvantaged party. As the reviewer noted, this concern could be amplified as we acquire more accurate knowledge about the disadvantaged party. \n\nThat said, we still believe it is important to conduct our research. First, as also pointed out by the reviewer, this concern of exploiting irrational human behavior is ubiquitous and is already being deployed by private sectors in the domain of marketing or political campaigns. In order to develop risk mitigation methods, it is our belief that we first need to understand the capacities of this approach. Secondly, there are also benign ways of utilizing this approach, e.g., encouraging the general public to take actions promoting social good or preventing individuals from taking self-sabotaging decisions. Without a public grasp of both the potential risks and benefits, we risk leaving the utilization of these techniques to the private sectors exclusively, which, arguably, could present an even more concerning situation.\n\nWe also want to discuss two potential risk mitigation methods in light of the concern raised. Firstly, on the technical front, we might employ differential privacy techniques to control the amount of personal data used in training human behavior models. Differential privacy offers a means to balance privacy with utility, typically by introducing controlled noise into the data. This mechanism is potentially helpful in mitigating the exploitation of marginalized groups, an issue raised by the ethics reviewer. Secondly, from the policy perspective, after we develop a deep understanding of the capability of information design with data-driven human models, we as a society could and should discuss the tradeoff of the utility we can gain from this method and the harm we might suffer. This tradeoff discussion can then lead to the developments of regulations and policies. For example, we could add constraints requiring that the deployed information policy shouldn’t lead to much lower receiver utility, compared with deploying information policy designed assuming Bayesian rational models. This is similar to one common approach of imposing fairness constraints in the recent fairness literature. But again, these discussions requires us to develop a good understanding of the capability of this approach in the first place\n\nWe will provide more discussion on the above in the revision.\n"}, {"review_id": "UKImtoWML8", "reviewer": "Reviewer_34E5", "summary": "The study demonstrates commendable efforts in employing supervised learning techniques and utilizing Amazon Mechanical Turk to acquire data to develop a human behavior descriptor. The authors further utilize neural networks to optimize the sender's signaling scheme based on the fitted human decision-making model, aiming to maximize their expected returns. The experiments conducted are extensive and encompass a broad range of participants that avoids bias.", "questions": "N/A", "limitations": "N/A", "rating": 4, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "The work is one of the early works that attempt neural network-based approaches for information design. The work also heavily involves human behavior which is meaningful and potentially useful.", "weaknesses": "Major issues:\n\n1. In line 27, the assumption of Bayesian rationality does not appear to be a limitation of previous works. Prior research primarily focused on studying information design problems, such as equilibrium between agents and when senders can benefit from persuasion. The authors' research, however, addresses a different direction, representing a distinct problem. Therefore, the assumption of previous works is not supposed to be claimed as a \"limitation\".\n2. This paper implies that $H$ must have encountered all possible $\\pi$, for the two-stage process employed to be reasonable. Otherwise, the sender and receiver should adapt to each other iteratively in the search for equilibrium, rather than being divided into two distinct stages. This is a strong assumption because $\\pi$ is continuous. But the paper does not emphasize this assumption adequately. Furthermore, treating continuous variables $\\pi$ and priors as features is a simplistic neural approach that may perform worse when the state and action spaces become larger.\n\nMinor comments:\n\n1. This paper explores information design problems in a sequential manner with multiple receivers. From the view of the settings, the receivers are set to be independent and they may have different priors, and the sender will persuade them through a public communication channel. It would be better if the authors emphasize these assumptions. Also, I notice that there are several previous works that study this kind of task, e.g., [Interactive Information Design](https://shs.hal.science/halshs-01791918/file/Publi_2.pdf) and [State-Dependent Central Bank Communication with Heterogeneous Beliefs](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3923047).\n2. In line 54, the author claims to be the first to incorporate neural networks. However, we have come across some arXiv preprints that have already explored this area, such as: [Learning to Persuade] (https://openreview.net/forum?id=0oSM3TC9Z5a).\n3. The example in line 110 may require more description. For example, the reward functions of both agents is not formally introduced.\n4. In lines 218-223, does the author provide a more detailed description of this example? Why $\\pi^*(\\sigma=1\\mid\\theta=1)$? As far as I know, the optimal signaling scheme would be symmetric if the seeds of the experiment codes are random.\n5. In line 285, why are there no known solutions for three states and three actions? Shouldn't this be solvable in a similar manner? \n6. The human experiments are intriguing, and I am curious about some issues regarding these experiments. For example, during the experiments, did the human participants undergo a learning process? Did the players have any prior experience with trial runs? Were the players allowed to communicate with each other? Also, I noticed that the questionnaires provided in the appendix appear to be quite professional. How did the authors ensure that the participants understood the game?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful review.  We first respond to your major concerns.\n\n**[The assumption of Bayesian rationality as a limitation]** While Bayesian persuasion has provided an elegant framework for studying information design, it has made several assumptions that limit the practical impacts. One of them is the assumption that the receiver is Bayesian rational. Meanwhile, extensive empirical works (e.g., Camerer 1998; Benjamin 2019) have shown that people often systematically depart from Bayes’s rule when confronted with new information. By “limitation” we mean that directly applying the techniques assuming Bayesian rational receivers when the receiver is not would lead to suboptimal outcomes. As also noted in recent work by de Clippel & Xu, JPE’22, new treatments need to be developed to account for receivers who make mistakes in probabilistic inference and decision making. We will make our descriptions more clear to reflect this.  \n\n**[Assumptions on human models]** We want to first highlight that our framework is designed to accommodate various representations of human behavior. As shown in our evaluations, it can handle both standard closed-form representations of human models (e.g., Bayesian rationality or TH-model) and data-driven forms.  The reviewer's concern is mostly surrounding the data-driven human models. Indeed, if our human model is entirely trained on historical data, we would require the training data to be representative. Note that even in training data-driven human models, we do not require the data to encounter “all” scenarios, as the goal of ML is to generalize beyond the training data. \n\nWe also want to note that this concern can be mitigated by leveraging the recent developments at the intersection of cognitive science and ML that have shown we can utilize established human models (like Bayesian rationality or the TH-model) as priors and harness data to refine these models [Bourgin et al. ICML 2019, Peterson et al. Science 2021]. The main benefit of this approach is that it provides a reasonable model to begin with even when we do not have enough data and could lead to a more accurate model with more data. Our framework can seamlessly integrate with human models developed using such methodologies.\n\n- Bourgin et al. \"Cognitive model priors for predicting human decisions.\" ICML. 2019.\n- Peterson et al. \"Using large-scale experiments and machine learning to discover theories of human decision-making.\" Science. 2021.\n\nWe now respond to your minor concerns. \n\n**[Multiple receivers]** There seems to be misunderstandings of our settings. In the multi-receiver setting that we are addressing, our goal is to design a single information policy for multiple (potentially heterogeneous) receivers simultaneously. We do not interact with each receiver in a sequential manner. Our multi-receiver setting is known to be public persuasion [60]. We will provide clarification of our settings and also include discussion to the additional related work as the reviewer suggested.\n\n**[Related work]** Thanks for the pointer! We will include the work in our revisions. We didn’t notice this paper since it doesn't seem to be published yet and doesn’t have an public arXiv version (albeit it has an openreview record). Our work is generally a more comprehensive investigation of automated information design and has addressed general forms of human behavior.\n\n**[Explain the optimal information policy of binary actions/states]** In this example, we adopt the most classical setting with binary states/actions introduced in the original Bayesian persuasion paper [32].  It’s our oversight that we didn’t specify all the details. In particular, the receiver utility $u^R(0,1)=u^R(1,0) = 0$, and we randomly draw values from $[0,1]$ for $u^R(0,0)$ and $u^R(1,1)$. In plain words, in this setting, the receiver prefers action 1 when the state is 1 and action 0 when the state is 0, and the goal of the sender is to persuade the receiver to take action 1.  We will fix the description and provide a full description.\n\n**[No known solutions for non-Bayeisan-rational receivers]** To the best of our knowledge, the studies on deriving optimal information policies when the receiver is not Bayesian rational is limited. The solution for the setting with binary action/states is from Tang and Ho [56]. However, their approach analytically derives a closed-form solution for the optimal information policy for binary actions/states, and their derivation doesn’t seem trivial to extend to multi-state and multi-action settings. \n\n**[Clarifications of human experiments]** The participants cannot discuss with each other, and each individual is allowed to participate in the study only once. During the human experiments, participants were provided with instructions and an illustrative example of the problem instance to help them understand the task setting. We also offered incentives in the form of bonus payments for those who correctly inferred the latent state and made the right decision. However, despite these measures, we cannot guarantee that every participant fully understands the task or puts in a reasonable effort. Nonetheless, this is consistent with many real-world human decision-making scenarios. Furthermore, as mentioned in our response to Reviewer 7rN5, our framework is effective as long as the human behavior used to train the neural network (NN) for the information policy comes from the same distribution as the human behavior observed during deployment. Our experiments aim to meet this condition by recruiting first-time participants from the MTurk platform for both phases."}]

Author rebuttal:

Meta-review: The reviews and discussion give mixed support, acknowledging the technical contributions but also raising potential ethical concerns. We encourage the authors to use the feedback to revise and strengthen the technical and ethical discussions in this work.
Decision: Accept (poster)