SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To showcase our results, we present two settings where we have fixed prior distributions: low prior with  $\lambda (\theta = 0) = 0.3$  and medium prior with  $\lambda (\theta = 0) = 0.5$ . For each prior distribution, we vary the receiver utilities and report the parameter  $\pi^{*}(\sigma = 1|\theta = 0)$  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.
(a) Optimal vs. HAIDNet in the low prior case.
Figure 2: Comparing the optimal information policy and the policy generated by HAIDNet in the setting with binary actions and binary states.
(b) Optimal vs. HAIDNet in the medium prior case.
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 actions<sup>5</sup>. 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 (e.g., average sender utility for newly drawn 1,000 instances).<sup>6</sup> The results, as shown
4The 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.
Table 1: Comparing the average sender utility generated by the optimal policy and the policy from HAIDNet in the setting with a single Bayesian rational receiver.
M	Training	Testing
HAIDNet	Optimal	HAIDNet	Optimal
2	0.7409	0.7498	0.7408	0.7451
3	0.7737	0.7782	0.7598	0.7669
5	0.8171	0.8209	0.8066	0.8225
10	0.8495	0.8699	0.8196	0.8686
(a) Increase the number of states  $M$  with binary actions. (b) Increase the number of actions  $N$  with binary states.
N	Training	Testing
HAIDNet	Optimal	HAIDNet	Optimal
2	0.7409	0.7498	0.7408	0.7451
3	0.7017	0.7214	0.7089	0.7227
5	0.6906	0.7113	0.6690	0.7064
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  $K$  Bayesian rational receivers.
K	Training	Testing
HAIDNet	Optimal	HAIDNet	Optimal
2	0.7887	0.7934	0.7756	0.7873
3	0.7508	0.7665	0.7379	0.7573
5	0.7217	0.7458	0.7209	0.7570
10	0.6971	0.7152	0.6790	0.6966
15	0.6553	0.6882	0.6621	0.6843
Table 3: Comparing the average sender utility by the optimal policy and the policy from HAIDNet in the setting with a non-Bayesian-rational receiver parameterized by  ${\beta }_{H}$  .
βH	Training	Testing
HAIDNet	Optimal	HAIDNet	Optimal
1	0.5043	0.5051	0.5041	0.5060
5	0.5512	0.5557	0.5506	0.5559
10	0.6045	0.6170	0.5986	0.6168
50	0.7002	0.7134	0.6800	0.7081
100	0.7187	0.7291	0.6964	0.7179
in Table 1, 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 computationally efficient solutions to characterize the optimal information policy. The goal is to illustrate that HAIDNet performs well even in complicated scenarios and could provide a more efficient approach for settings without analytically tractable solutions.
We consider the setting with multiple receivers and binary actions. The goal is to design a uniform information policy for all receivers (i.e., public persuasion [78]). This setting has been shown to be #P-hard to 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 to solve with a large set of receivers, and we intend HAIDNet to be a new, efficient solver for near-optimal solutions. To examine whether HAIDNet finds the optimal policy, we utilize a brute-force linear-programming 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 setting. The sender utility is the fraction of receivers choosing action 1, i.e., her utility is given  $\frac{|S|}{K}$  if there are  $|S|$  receivers choosing action 1 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 nearoptimal. Moreover, HAIDNet provides a 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 an 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 1 hour, and generating information policy for a problem instance takes less than 1 second). The empirical run-time comparison is included in the appendix.
4.1.3 Settings without known solutions
We now examine the performance of HAIDNet in settings where there are generally no known analytical solutions yet. The goal is to showcase that HAIDNet can be leveraged to address information design problems when we do not have access to solutions.
All our simulations so far have focused on settings which assume that receivers are Bayesian rational. To examine whether HAIDNet works for non-Bayesian-rational receivers, we adopt a relaxation of human behavioral formulation as in Equation (1). While there are no known solutions for identifying the optimal policy in this setting in general, Tang and Ho [70] derived a 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  $\beta_{H}$  in the human descriptor in Equation (1). Using the same setup as in previous simulations, we report the results in Table 3, showing that HAIDNet works even for a non-Bayesian-rational receiver.
Figure 3: The performance of HAIDNet in settings when the receiver is not Bayesian rational. We train HAIDNet with non-Bayesianrational receiver model parameterized by  $\beta_{H}$  then evaluate the learned information policy for all receiver models. The performance is normalized so for each human model, the optimal performance is 1.0 among all policies.
Next, we would like to examine how HAIDNet per
forms in scenarios when there are no known solutions (e.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 as in the simulations above. Instead, we take a different method and provide evidence to support our approach: We evaluate the set of all learned policies  $\pi_{\beta_H}$  against each of the human models  $\beta_H$ .
For each human model  $\beta_{H} = k$ , if  $\pi_{k}$  is the best-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 of these performances so that the 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 a closed-form behavior model of the receiver. However, in practice, human behavior is complex and there may not exist a single model that can perfectly represent human behavior. Motivated by this practical concern, we conduct human-subject experiments to examine whether HAIDNet adapts to real-world human behavior. The goal is to examine whether we can utilize data-driven approaches to learn human-behavior 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 2 to human participants. 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 (good or bad product). The participant is told that a (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 is given a realization of the inspection signal and is asked to make a 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.
Table 4: Test accuracy of different human behavior descriptors in 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  $0.50. Workers could earn additional bonuses depending on their performance. The average hourly rate was around$ 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, we 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 is no training needed for this descriptor.
- TH-Model: We fit the parameters of the TH model, as described in Section 3.1, from data to minimize the least squares error.
- Neural network: We use a 3 fully connected-layer neural network to 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 early-stopping during training.
We then examine how accurately each descriptor predicts human behavior in the test data. The test accuracy is reported in Table 4. 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 performance of different information policies. In particular, we examine the following four information policies:
- Random: This information policy is drawn from a Dirichlet distribution.
- BR-policy: The optimal policy when the receiver is a Bayesian rational receiver.
- TH-policy: The optimal policy when the receiver behavior follows the TH-Model, as in Section 3.1.
HAIDNet: The policy by HAIDNet when we use the neural network learned from the first phase as the human model.
Figure 4: Average sender utility of different policies in human-subject experiments. The differences between BR-policy and TH-policy and between BR-policy and HAIDNet are statistically significant  $(p < 0.01)$ .
When each worker arrives, they are randomly assigned to one of these four policy treatments. They are then presented with 20 rounds of purchase decisions (the parameters of each round are randomly drawn from distributions fixed across all treatments) coupled with the associated information policy in 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 an average receiver utility of 0.532, which is the lowest of all four treatments. This creates the concern that when we incorporate the knowledge of receiver behavior to optimize the sender'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.
We initiate the study of behavioral information design that encodes human behavior into the design process. We propose HAIDNet, a neural-network-based 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.
Generalization and limitations. While this work has focused on integrating human behavior in automated information design, we believe the methodologies are generalizable to design mechanisms for general human-in-the-loop 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.
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 is 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.1), 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 it 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 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. It 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 a 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 as Bayesian rationality.
We thank the anonymous reviewers for the helpful comments. This work is supported in part by the Office of Naval Research Grant N00014-20-1-2240.
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