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Disinformation, Stochastic Harm, and Costly Effort: A Principal-Agent Analysis of Regulating Social Media Platforms

The spread of disinformation on social media platforms is harmful to society. This harm may manifest as a gradual degradation of public discourse; but it can also take the form of sudden dramatic events such as the recent insurrection on Capitol Hill. The platforms themselves are in the best position to prevent the spread of disinformation, as they have the best access to relevant data and the expertise to use it. However, mitigating disinformation is costly, not only for implementing detection algorithms or employing manual effort, but also because limiting such highly viral content impacts user engagement and thus potential advertising revenue. Since the costs of harmful content are borne by other entities, the platform will therefore have no incentive to exercise the socially-optimal level of effort. This problem is similar to that of environmental regulation, in which the costs of adverse events are not directly borne by a firm, the mitigation effort of a firm is not observable, and the causal link between a harmful consequence and a specific failure is difficult to prove. For environmental regulation, one solution is to perform costly monitoring to ensure that the firm takes adequate precautions according to a specified rule. However, a fixed rule for classifying disinformation becomes less effective over time, as bad actors can learn to sequentially and strategically bypass it. Encoding our domain as a Markov decision process, we demonstrate that no penalty based on a static rule, no matter how large, can incentivize adequate effort. Penalties based on an adaptive rule can incentivize optimal effort, but counterintuitively, only if the regulator sufficiently overreacts to harmful events by requiring a greater-than-optimal level of effort. We prescribe the design of mechanisms that elicit platforms' costs of precautionary effort relating to the control of disinformation.


Introduction
Contemporary web and social media platforms provide a ripe ground for the spread of false news, hoaxes, and disinformation [25]. Compounding the problem, social platforms' business models often conflict with efforts that can mitigate these problems. Facebook, for instance, uses machine learning models to maximize user engagement; however, in doing so, these models also favor content that is toxic and filled with conspiracy, lies, and misleading, divisive information [18,2,32].
We use disinformation to refer to all such toxic content, including all kinds of false and fabricated news posing as truth, created with the intention to mislead [25]. The unmitigated spread of disinformation is harmful to society. The harm can be direct physical or emotional distress to an individual; it can also manifest as a negative externality affecting public discourse, or social welfare. Examples include the undermining of public health response due to Covid-19 false rumors [41,3], disease outbreaks due to anti-vaccination propaganda [39], violent conspiracy movements surrounding the 2020 US presidential elections [4,37], and horrific incidents such as the Pizzagate shooting [26] and ethnic violence in Myanmar [34].
The costs of these rare and dramatic events are borne exclusively by society, rather than the social platforms themselves. These events are inherently stochastic as it is impossible to predict with certainty that a given collection of content will cause a specific harm. Furthermore, deploying techniques to prevent the spread of disinformation is costly: filtering, demoting or assigning warning labels to associated content comprises both the direct costs of implementing classification algorithms or employing manual detection effort, and also the indirect opportunity costs of advertising revenue due to subsequent losses in user engagement [22,33]. Therefore, platforms such as Facebook and Twitter face no compelling incentives to prevent the spread of disinformation; thus, relying on platforms to police themselves will not work [9]. The only reason for a profit-motivated platform to control the spread of disinformation is to avoid penalties imposed either by users or a public regulator.
Techniques for mitigating disinformation must leverage tools in artificial intelligence (AI), which further complicates the issue of misaligned incentives. The sheer scale at which users generate and share content on social platforms mean that any form of content moderation must rely, to some degree, on the automation afforded by AI in order to handle the vast volume of data. This aspect is different from traditional publishing, television, and print media where humans are involved in the editorial feedback loop before any content is allowed to be published. The problem of assigning liability for content is therefore much simpler in traditional media. With user-generated content on social platforms, however, the quality and accessibility of data determine whether AI will be effective at moderating content. But because only platforms have full, real-time access to their data, the problem is thus to motivate their use of AI to proactively mitigate disinformation -in spite of their selfinterest in not doing so [9] -without having the same expertise or access to data.
The principal-agent framework of microeconomics models the interactions between an agent, who can influence the probability of an outcome by incurring costly effort, and a principal, who has preferences over the outcome. We model the domain of disinformation prevention through this lens, with the platform as an agent who has the ability but not the incentive to undertake costly precautions against the spread of disinformation, and a regulator as the principal who seeks to balance the cost of the precautions against the harm caused by disinformation.
We begin by reviewing related work on techniques for mitigating disinformation and background on the principal-agent approach to modeling the regulation of stochastic externalities in Section 2. We then lay out our modeling assumptions in Section 3. We make three key assumptions. First, any attempt to regulate the actions of a platform before harm occurs (using a so-called negligence standard) must specify what level of effort for mitigating disinformation is adequate. Even if this specification is left implicit, we can model its effect as being a public model, operated by the regulator, requiring some level of effort. Second, any given public model will, in practice, require less effort over time as disinformation authors can learn to circumvent it; the platform is thus able to get away with expending less effort policing disinformation in order to save on costs, since the data and expertise needed to continuously re-train a model of content harmfulness is possessed by the platforms but not the regulators. Third, the public standard for content that ought to be prohibited on the platform will increase after a harmful event.
We formalize these assumptions as a Markov decision process (MDP) in Section 4, and use the model to derive our main results. We show that that no level of fines based on a static public model can induce the socially-optimal effort for mitigating disinformation.
However, in the presence of a public model that reacts to a harmful event by increasing the required level of effort, the platform's individually-optimal level of effort may exceed the level currently required by the public model. In particular, the platform may be incentivized to continue exerting effort at a specific threshold when the public model becomes less stringent over time. However, perhaps counterintuitively, this effort threshold will fall short of the socially-optimal level unless the public model sufficiently overreacts as a response to any harmful event, by requiring a level of effort that is greater than socially optimal.
Finally, to further demonstrate the complexity of this incentive problem, we show that even under a simpler, more stylized setting -where the regulator has the same technical ability as the platform and the costs of harm from disinformation are known -absent knowledge of the platform's costs of effort, there is no specification of the public model's required effort that will always incentivize the socially-optimal level of effort. We thus conclude that the design of mechanisms that may elicit the costs of foregone engagement incurred by the platform in policing disinformation is one of the promising directions for future work. 4 2 Background and Related Work

Fighting Falsity Online
Detecting disinformation. A popular approach towards limiting online disinformation is to develop tools or frameworks that are effective in detecting associated content. This process aims to identify disinformation in its initial stages so that mitigating efforts thereafter may restrict or eliminate exposure to users of social media. Zhou and Zafarani [42] survey some techniques that make false news detection efficient and explainable. These techniques are categorized into four areas: knowledge-based methods that involve fact-checking, stylebased methods that focus on studying linguistic features of false content, propagation-based methods that analyze how such content spreads in the social network, and source-base methods that investigate the credibility of sources that generate false news. The goal of studying these and other characteristic features of the false news ecosystem, such as those surveyed by Kumar and Shah [25], is to develop algorithms and tools for early detection.
Knowledge-based methods mainly involve fact-checking, which in turn can be either manual or automatic. Fact-checking is the process of extracting claims made in a given piece of content that is to be verified and checking these against known facts [42]. Manual fact-checking can either be crowd-sourced from users on social platforms, similar to Facebook or Twitter provisioning its users with the ability to report hoax content [27,11]; or it can also be conducted via third-party websites such as Snopes 1 , PolitiFact 2 , or FactCheck 3 that employ domain experts dedicated to serving the public by debunking disinformation.
While knowledge-based and style-based detection techniques focus on analyzing the textual content of disinformation -so that predictive classifiers might be trained to readily and effectively flag false news -propagation-based techniques study how such content disseminates amongst users in a given social network. Vosoughi, Roy, and Aral [40] have conducted an empirical study of tweets on Twitter to analyze the differences between the spread of true and false news stories. It is shown that false news stories travel faster, farther, and more widely than true news stories [40]. Such an analysis not only aids in investigat-ing the causes and consequences of disinformation proliferation, but it also helps formulate propagation-based false news detection as a classification problem.
Our work starts from the assumption that the platform has the ability to detect and limit the spread of objectionable content [30]. Our focus instead is on modeling the incentives faced by the platform to not exercise this ability.
Mitigating the effects of disinformation. Once effective technology for detecting disinformation content and diffusion networks is implemented, the next step is to mitigate or limit the impact such content may have on users. A straightforward approach is to simply remove associated content from the platform entirely; another is to demote or down-rank content so that it is less likely to be served on users' feeds. These approaches rely on platforms to undertake action to reduce the spread of disinformation since the recommendation algorithms serving content to users are proprietary. However, there are also studies conducted by third-party researchers offering other solutions for mitigating the effects of disinformation once it has entered the social network.
One such technique draws from concepts in human cognitive psychology to study deception cues that influence users' decision-making process related to sharing content in the social network [24]. The goal here is to provide users with informative cues so that they are less likely to share disinformation. Another, more proactive intervention is the "Facts Before Rumors" campaign [19], where the focus is to preempt the kinds of rumours that are likely to spread on a social network -based on user locations and localized news content, for example -and counteract these in advance by employing certain users to spread truthful news. Other interventions focus on curing the effects of disinformation instead of preventing it initially. For example, the "Correct the Record" initiative proposes a visual correction that may be sent to users exposed to false content on Facebook [5].
Again, our work focuses less on advocating a particular mitigation technique; rather, our goal is to determine the conditions under which platforms can be induced to actively implement any such technique to prevent harm from the spread of disinformation.

Hidden-Action Principal-Agent Model
Many economic interactions involve two parties, a principal and an agent, where the agent's choice of action imposes some form of (negative or positive) externality on the principal.
In most realistic scenarios, the principal cannot directly monitor or observe the agent's action, but instead only observes a stochastic outcome resulting from it. For example, in the interaction between a property insurer (principal) and a property owner (agent), if the insurer bears the costs of any damages to the property, the owner might not be incentivized to maintain it and might engage in risky behaviors (e.g., leave the kitchen unattended while cooking). This situation exemplifies the problem of moral hazard, which is an important feature of the principal-agent interaction because it precludes straightforward incentive schemes. Many employment settings also share this characteristic. For example, the CEO (principal) of a small startup company -whose income is directly related to the company's growth and product sales -would want their employees (agent) to undertake effort that profits the company (e.g., a UI/UX developer improving the company's website leading to increased traffic and sales). But if the employees are simply compensated at a fixed hourly rate, they might not be incentivized to put in their best effort to benefit the company.
Naturally, it will be in the principal's interests to influence the agent's choice of action.
The principal may therefore be invested in drafting a contract for such influence in order to guard against the problem of moral hazard [15]. The need for a contract arises due to information asymmetry between the two parties -i.e., the agent has more information or expertise about their actions than the principal. For property insurance, the hidden information is the agent's act of not maintaining the property and engaging in some risky behavior; for the startup company example, the hidden information is the UI/UX developer's expertise in developing clean, functional websites; whereas for our setting, AI is the source of asymmetric information: only platforms possess the expertise, models and data to promptly flag and mitigate disinformation.

Contract Theory Meets Computer Science
The principal-agent model is central to contract theory, which is an important field in microeconomics. This area has recently gained traction in the algorithmic game theory community, primarily through works such as [15,13,14], where the aim is to concisely represent principalagent settings and computationally characterize the design of optimal contracts 4 permitted by such settings.
Our work is similar to these studies in that we consider the optimal design problem of maximizing the utility of the principal, who in our setting is a social welfare-maximizing regulator. Yet, instead of a computational complexity analysis, we represent disinformation prevention as a principal-agent problem through our descriptive MDP model, which to our knowledge is a unique approach towards modeling the incentives faced by social platforms pertaining to the mitigation of false news and other toxic content. Unlike those cited works, the outcome space for our setting is simply the realization of a single harmful event due to the unmitigated spread of disinformation; our focus as such is specifically on the design of penalty contracts or schemes enforced by some regulatory agency in order to contain this stochastic externality, or harm from disinformation.

Regulating Stochastic Externalities
The hidden action principal-agent model can also be applied to the regulation of firms that generate stochastic externalities as a result of their operations. Examples include harmful accidents such as medical product failures, oil spills, nuclear waste leakages and other forms of pollution [21]. Moral hazard exists in these settings because firms (agent) might not be incentivized to take a costly precaution (unobservable action) to reduce accident risk, which is where a regulatory authority (principal) steps in to specify a penalty contract to guarantee some form of enforcement.
Cohen [10] explores optimal enforcement strategies for the regulation of firms that stochastically pollute the environment in the form of oil spills. It is shown that under a strict liability standard, where a polluting firm is always penalized if an oil spill occurs regardless 4 An optimal contract is one that maximizes the principal's expected reward assuming that the agent best responds to the contract [14]. of its level of precautionary effort, the firm can be induced to exercise the socially-optimal or first-best level of effort. However, this requires that a specific firm can be identified as being responsible for a spill.
When a strict liability standard is impractical -for example because the perpetrator of harm cannot be reliably identified -a regulator might prefer to expend resources to monitor a firm's effort directly. In these situations, a negligence standard can be preferable, in which a firm is not held responsible for an accident if it can demonstrate that it took adequate precautions. Naturally, the quality of information available for regulatory monitoring is a consideration for enforcing such a standard [31].
Our domain shares many of the features of the oil spill prevention domain: there are stochastic externalities associated with the spread of certain kinds of content on social platforms (harm from disinformation), as there are with firms transporting oil (oil spills); the likelihood or severity of such harm may be reduced to some degree if platforms exercise responsible and proactive content moderation, but not completely eliminated as the harm is ultimately a direct outcome of individual actions -akin to a spill that occurs because of inclement weather and not due to the oil tanker being faulty.
However, our domain is also importantly different from that of oil spill regulation, or environmental regulation more generally. The following subsection expands on these differences. In Section 3 that introduces our formal, descriptive model, we will elaborate on the similarities and highlight how these key differences prevent the application of standard enforcement strategies.

Why Online Disinformation is Different
Disinformation prevention via regulatory mechanisms has its own unique challenges. First, there are ongoing debates around assigning liability for content hosted by social platforms [9], particularly due to editorial control being different for the social media setting. As discussed previously, it is infeasible to implement human-in-the-loop feedback for every piece of realtime, user-generated content shared on online platforms, as this medium is unlike traditional forms of media; there exist as such not only the issue of scalabilty for any disinformation mitigation technology, but also the question about whether similar liability rules for harmful content should apply to social media as they would for traditional media.
Second, in order to handle the vast volume of content, AI must be utilized for the proactive and automated flagging of disinformation. This aspect complicates regulation because the data powering such AI is only accessible to the social platforms themselves. Moreover, the recommendation algorithms that filter and serve content to users are also proprietary. Therefore, unlike for the environmental regulation domain, mandating exact precautions against the spread of disinformation for social platforms is likely to be an involved process for any regulatory authority -especially in comparison to, for example, specifying precise conditions that render an oil tanker safe for the transport of oil, or promoting adequate technology that will reduce emissions causing air pollution.
Third, and also different from pollution regulation, there exists the issue of malicious actors responding strategically to any explicitly fixed rules or precautions against the spread of disinformation. Authors and purveyors of disinformation are constantly coming up with new, sophisticated methods to ensure that their fabricated stories disseminate online: techniques include obfuscation strategies to hide disinformation propagating networks and the origins of propagandist content; and also changing the content itself via constructing new falsehoods, or targeting different groups [1,18]. Any successful attempts to moderate such users or content at scale must therefore utilize all the technical expertise and data required to counteract efforts of these bad actors. Regulation becomes challenging because only social platforms have access to such resources and data, and they are not necessarily incentivized to undertake action at the expense of losses in user engagement [22,9].

Mechanism Design
Another closely related body of work is the economic theory of mechanism design, where the goal is to design protocols or procedures that mediate interactions between strategic agents in order to achieve some desired objective. Naturally, the outcome is subject to the constraint that agents behave selfishly, in that they act according to their rational self-interests; and also that agents hold some private information, i.e., their hidden types. A mechanism seeks to attain the desired outcome by incentivizing agents to report their private types. Mechanism design theory contrasts with the standard principal-agent model with respect to where the information asymmetry exists: it is the agents' type information that is hidden from the mechanism designer; whereas, for the principal-agent model, the principal cannot directly observe an agent's action(s), which form(s) the source of asymmetric information.
Because our work is concerned with setting up a regulatory policy in order to achieve a desired social outcome -that is, the socially-optimal level of control of disinformation -mechanism design is a pertinent framework for our domain. Yet, it does not directly apply to our setting since the regulator cannot reliably observe a social platform's efforts, or action, to curb the spread of harmful content. We therefore utilize the principal-agent framework to model the regulation of disinformation. The goal for the regulator (principal) is to incentivize a platform (agent) to use its proprietary expertise and AI technologywhich are not available to the regulator -to responsibly limit toxic and harmful content in order to control the harm from disinformation. The following section introduces our formal descriptive model.

Modeling the Regulation of Disinformation
We have the following scenario: A regulator (principal) would like the platform (agent) to limit the amount of disinformation spread to control the likelihood of harm, which is a stochastic and observable event. The underlying assumption is that the unmitigated spread of disinformation on social platforms makes the occurrence of harm more likely.
We assume the platform possesses a proprietary classification model that accurately assigns for every a piece of content the probability of it causing harm [30,22]. Thus, extremely violent, graphic, or objectionable content, which contains nudity, racism, child pornography, or any form of human/animal abuse, is tagged by the model with a very high harm probability value. Other, benign forms of content, such as cute photos of pets or birthday greetings, are assigned with a very low harm probability value.
Using this model, the platform can flag content exceeding some chosen harm probability threshold as being unacceptable and in violation of their community standards of acceptable postings. Thus, the platform's effort constitutes first detecting such content, and thereafter employing methods to either filter it entirely, downgrade it so that it appears on fewer user feeds, or label it with a warning invoking users' discretion. As discussed previously, the exact choice of technique is not important for this analysis; any and all such methods effectively count as the platform exercising effort to prevent the spread of harmful content and, by extension, disinformation.
Let H be a binary random variable indicating whether harm occurs with density function h(e) = Pr[H | e] ∈ (0, 1] representing the probability that harm occurs if the platform exerts effort e. Similar to [10], we assume that although the platform is unable to control this externality directly, the platform can make it less likely for harm to occur by exercising more effort. In line with the standard economic model of unilateral accidents [20,21,38], we assume that there are diminishing returns to effort; that is, effort reduces risk of harm at a decreasing rate: h ′ (e) < 0 and h ′′ (e) ≥ 0. 5 The business model of most social platforms is primarily based on advertising. For instance, advertising accounted for 98% of the Facebook's $86 billion revenue in 2020 [17].
Essentially, platforms monetize users' attention by optimizing their engagement for content, a subset of which includes disinformation. Thus, in addition to direct costs, limiting disinformation is also costly for platforms in terms of these indirect costs of losing potential ad revenue. Let c(e) denote the cost of exerting effort e. We assume effort is increasingly costly; i.e., c ′ (e) > 0 and c ′′ (e) > 0, which is also standard under the unilateral accident model.
Given their behavioral advertising business model, platforms face no incentives to moderate attention-grabbing content, toxic or otherwise, especially because they do not directly incur the costs of any societal harm [22,9]. Under this scenario of misaligned incentives, a social welfare-maximizing regulator aims to incentivize the platform to exercise adequate precautions against the spread of disinformation. Concretely, the regulator wishes to maximize the expected social welfare, where D is the cost of damages as a result of any harm due to disinformation, assumed to be constant here for simplicity.
The socially-optimal or first-best effort maximizing (1) is given by e * = arg max e EW (e).
At e * , the sum of the total expected costs of harm, or h(e * )D, and the platform's costs of exerting this effort, or c(e * ), is minimized; thus, e * by definition is the platform's precautionary effort at which the cost of any additional effort is balanced by the expected cost of damages due to harm.
We now discuss possible methods by which the platform may exert effort e * , and further expand on domain specific features for our descriptive model.

Strict Liability
Under the strict liability standard, the platform is held completely liable for any harmful event, irrespective of its precautionary effort. To incentivize the first-best level of effort e * , the strict liability fines T must equal D, the societal cost of harm [10]; thus, the platform's expected utility is given by, which equals the expected social welfare equation (1).
A regulator might pick this enforcement standard because it does not require expending resources to monitor the platform's effort, which is only imperfectly observable because of the difficulty in identifying the exact mechanics of the platform's proprietary algorithms.
However, strict liability is impractical for a few reasons. Most importantly, the direct causal links between any harmful event and the platform are sufficiently loose for this standard not to work, since the perpetrators are ultimately individuals; the platform can claim plausible deniability, or point to efforts at prohibiting dangerous content after the harm has already occurred, akin to when Facebook and Twitter banned groups like "QAnon" or "Proud Boys" after the insurrection on Capitol Hill [29]. Furthermore, it is also difficult to estimate D a priori as the harm could manifest in different forms.

Negligence
Under this standard, a regulator must specify a duty of care that the platform must follow in order to avoid liability for any harm. Monitoring the platform's effort is thus necessary to determine liability.
Although monitoring is imperfect, the platform's content moderation efforts are not completely unobservable: there exists a crude public notion about the kinds of content that ought to be limited on social platforms. From an incentive standpoint, a negligence standard already exists in the sense that there is not a lot of nudity or child pornography, or content with explicit death threats, vile or racist remarks on most social platforms -platforms like Facebook and Twitter expend ample resources to enforce their community standards via active content moderation [30,16]. Presumably, platforms do not want public outrage, or to be charged with trafficking or any other forms of liability for such content, which if not controlled would be reported extensively in popular press.
We model this descriptive situation with the presence of an explicit public model, operated by a regulator, that fixes a required level of precautionary effort for mitigating disinformation. In reality, there is no concept of an explicit public model specifying effort, but rather an implicit public notion about the types of content that ought to be moderated by the platform. Nonetheless, regardless of what the public standards are at any given moment, these standards imply a certain level of precautionary effort, which we encode with an explicit public model to simplify our formal analysis.

Performative Prediction of Disinformation
When predictions about the actions of an agent change the outcomes for that agent, there is a risk that the predictive model will cease to be accurate [28]. We say that predictions that exhibit this problem are performative. 6 For example, a certain keyword that is extremely predictive of a message being spam may cease to be predictive once we filter based on it, as spammers will now have an incentive to stop using that keyword. Classifying disinformation is performative in this sense because bad actors can learn to bypass any detection model with new forms of disinformation [1,18].
We assume the platform has sufficient technical resources and the data to retrain its proprietary model in order to counterbalance performativity; i.e., the platform is able to successfully classify future modifications of disinformation via predicting true harm proba-bilities of associated content. The same is not true for the public model as the regulator does not possess the same expertise or access to data. The regulator in theory could utilize open-source, state-of-the-art disinformation detection learning models to effectively flag false content as not satisfying the public standard [43,23]. Yet, to the extent that platform data is not completely accessible to the public [6], these open-source models will be susceptible to performative prediction of new, evolved forms of disinformation unless retrained with the same, easily accessible data that is available to the platform.
Consequently, because it is publicly accessible, the public model weakens over time due to performativity as disinformation authors strategically learn to circumvent it. We encode this feature effectively as a gradual downward drift or decrease in the public model's required effort if harm does not occur. However, if harm occurs, we see a public backlash in that the public's tolerance of content linked to the harmful event gets lower ex post. This is akin to when Facebook and Twitter began suspending accounts, content, and hashtags linked to the Capitol Hill riots [7]. We encode this backlash as an effective increase in the public model's required effort as a response to a harmful event.

Formal Model
We formalize our model as a MDP incorporating descriptive features of our domain as Consistent with MDP literature [35], we use π : S → A to denote an arbitrary, deterministic policy specifying the platform's choice of effort e ∈ A for all e c ∈ S. The state value function v π (e c ) = R e + γE[v π (s t+1 )] is the expected discounted value of following policy π from state e c ; the state-action value function q π (e c , e) = R e + E[v π (s t+1 )|A t = e] is the expected discounted value of choosing effort e in state e c , and then following policy π thereafter.

Optimal Effort Under a Static Public Model
In the first analysis, we assume a static standard model with no downward drift of the Note that effort e c is considered adequate because it is specified by the regulator. Thus, by definition, P f (e | e c ) = 0 for all e ≥ e c : the platform never fails its audit by at least following the public model's prescribed effort (full compliance). Further note that for any given e c and fine structure F , there exists an individually-optimal level of effort that maximizes (3). By inspection, this individually-optimal level will never exceed e c , irrespective of how large the size of fines F is.
Proposition 1. Given a fixed adequate effort level e ′ , there exists no fine scheme F that can incentivize the platform to exert more effort than e ′ .
All proofs are deferred to the appendix.
This result trivially follows from the specified conditions: for any two adequate effort levels, the platform will pick lower effort because that will maximize (3). Thus, with the static public model, no amount of fines solely based on ex ante regulation, no matter how large, can induce the platform to exercise more effort than the public model's specified e c .
Only if e c = e * , and if the regulator can guarantee full compliance with the public model, can this scheme incentivize socially-optimal effort.

Optimal Effort Under an Adaptive Public Model
We now consider an adaptive MDP setup. The current state represents the required level of effort e c ; if no harm occurs, the required effort reduces over time due to performativity; and if harm does occur, then the required effort increases to e h , representing public backlash.
Assumption 1. Given a fixed fine structure F and the effort required by the public model e c , the platform's individually-optimal effort level is at least e c .
This assumption is without loss of generality as we will label the states of the MDP according to the individually-optimal static effort required. 7 where d(π, e c ) = g(e c )v π (χ(e c )) + (1 − g(e c ))v π (e c ).
Given e c , Lemma 1 specifies the condition under which the platform's picks one effort level over another from the continuous action set A, expressed via the state-action value function of the MDP. Proposition 2. For all threshold strategies π τ , we have that v π τ (e h ) ≤ v π τ (e c ) holds for all e c ∈ S.
Proposition 2 establishes e h as the worst state for the platform following a threshold strategy. Intuitively, because it encodes the public backlash, e h by definition is the highest effort the public model will require and thus it must also yield the lowest expected discounted reward for the platform. Crucially, this guarantee of the lowest reward in state e h acts as the incentivizing mechanism for the platform to exert more effort than explicitly required by the public model.
Given these MDP dynamics of performativity and public backlash as a response to harm, we characterize the platform's individually-optimal effort policy at any state e c . The following existing results support our main result in Theorem 1.

Lemma 3 ([8]). Suppose f is a differentiable function of one variable in dom(f ). Then f is convex if and only if
holds for all x, y ∈ dom(f ). And analogously for strict convexity, for all x = y.

Lemma 4 ([35]
). Given a pair of deterministic policies π and π ′ such that for all states s ∈ S q π (s, π ′ (s)) ≥ v π (s), We now specify the platform's individually-optimal policy under the adaptive MDP setup: the public model's prescribed effort e c increases to the high effort state e h if harm occurs; and e c decreases over time conditional on harm not occurring. The following theorem demonstrates that the optimal policy for the platform under these dynamics is to follow a threshold strategy.
The result follows from the first-order condition of convexity [8] and the policy improvement theorem [35]. Intuitively, the theorem statement holds because past a certain level of effort, the gain to the platform of not exerting more effort is traded off against the increased probability of transitioning to the e h state, which yields the lowest expected reward as shown in Proposition 2.
The primary takeaway from Theorem 1 is that the platform is incentivized to exert more aggressive effort at thresholdê, despite an over-time reduction of the public model's prescribed effort e c due to the performative prediction of disinformation. Thus, the platform's optimal effort level is stable atê for all states e c ≤ê. The regulatory scheme that induces more aggressive effort is the ex post public backlash, i.e., when the required effort increases to e h , which effectively poses as stricter future ex ante regulation as a response to a harmful event.
This result is also important because with the correct choice of public backlash e h , the platform can in theory be induced to exert the socially optimal level of effort e * . We formalize this claim in the following proposition.
Proposition 3. For any given socially optimal level of effort e * , there exists a MDP consistent with our given conditions such that the optimal policy for the platform is a threshold strategy with threshold τ = e * .
The existence proof for e h directly follows from the defining constraint of the platform's optimal policy in (6) and the continuity assumptions of the cost and harm functions. An interesting consequence of this result, however, is captured in the following proposition, where we effectively specify a strict lower bound on the public backlash as a necessary condition to induce the socially-optimal effort e * .
Proposition 4. The platform's optimal stable effort is guaranteed to be socially suboptimal unless the public model overreacts by requiring effort e h > e * if harm occurs.
Our ancillary result in Proposition 4 captures the counterintuitive nature of the penalty scheme according to our model: it is not sufficient to set the ex post required effort to the optimal effort e * , assuming e * were known; instead, to incentivize optimal effort, the public model must overreact and mandate suboptimal effort e h > e * as a response to any harmful event.

Incentivizing Socially-Optimal Effort Under a Robust Public Model
Our descriptive model places an emphasis on overreacting to harmful events in order to incentivize socially-optimal effort. But since mandating suboptimal effort via such an overreaction is undesirable, we consider a simpler problem setting: suppose that the regulator has access to the platform's proprietary model and its underlying data, which can now be used as the public model robust to performativity. The regulator thus has knowledge of the harm function h. Suppose further that the societal costs of harm D are also given; the only missing information is the cost function c, or the platform's costs of effort to mitigate disinformation.

Proposition 5.
There is no way of adjusting the effort e c required by the public model, purely as a function of the harm function h and the cost of damages D, without regard to the cost function c, such that the platform is always incentivized to exert the socially-optimal level of effort.
This result shows that even if a regulator has precise control over the public model, without knowledge of the platform's costs, there is no way to set up the public model's effort threshold such that the platform's individually-optimal effort level is always socially optimal. Thus, since social platforms' costs of precautionary effort underpin the incentive problem, it is crucial to model these costs in more detail to better understand their incentives relating to the control of disinformation. Determining how engagement translates to money, therefore, serves as an important avenue for future exploration, as platforms risk losing out on engagement revenue with content moderation.

Conclusion
Events like the Covid-19 "infodemic" or the Capitol Hill riots are recent examples of the harm associated with disinformation. There is increasing evidence that the failure of social media platforms to control the spread of disinformation is due to incentive issues rather than a lack of technical ability [22,12,9]. This work provides a formal analysis of these incentive issues that adapts the standard principal-agent framework to incorporate the unique features of the domain. Our formal model, although stylized, includes what we take to be key aspects of the setting, including the performativity of disinformation classification and public backlash as a response to harmful events. Our formal results provide insights for the effective regulation of social media platforms.
We argue that although a strict liability standard would theoretically align the platform's incentives with those of society, it is unlikely to be practical given the difficulty of assigning responsibility for harmful events to specific instances of disinformation after the fact. Using our formal model, we derive a number of results relating to the use of a negligence standard.
Most importantly, we show that in the absence of a public backlash to harmful events, there is no monitoring scheme that can induce the platforms to perform a socially-optimal level of control of disinformation. However, a platform can be incentivized to exert more diligent Disinformation is one of the most urgent problems facing society. But it is a problem driven by incentives as much as by technology. This work takes a first step toward explicitly modeling the incentive issues that must be accounted for by any effective solution to the problem.

Future Work
Our model makes a number of simplifying assumptions. Treating public standards as an explicit model implies that a platform can guarantee a given probability of escaping punishment if it conforms to an explicit standard, which is an oversimplification of reality. The assumption that the platform can perfectly tune its proprietary model to flag toxic content is also unrealistic; technical challenges, although they may not pose the main obstacle to the practical control of disinformation, are nevertheless a real issue [18]. Extending the model to more richly model these aspects are important directions for future work. users.

Homogeneous Harm
Yet, in reality, just as content is not homogeneous in terms of the varying degrees of harmfulness of each item, content will also differ in the levels of user engagement attained.
Therefore, modeling this heterogeneous relationship of the harm and benefit of content in future work will likely drive different conclusions. For example, with such explicit modeling, one question we might hope to answer is whether highly toxic content is more likely to produce high levels of user engagement (in the form of likes, shares, retweets, comments etc.) than less toxic content, thereby being more valuable to the platform. This will shed light on the degree to which the incentives of social platforms relating to the control of disinformation are misaligned with those of society, which, in turn, will inform the nature of any regulatory interventions required to realign these incentives.
Taxing toxicity A Pigouvian tax is a tax on a market transaction that generates a negative externality borne by individuals not directly involved in the transaction [36]. Social platforms exhibit the precise criterion of generating negative externalities that calls for the levying of this tax: the more users a platform has, the more lucrative it is for advertisers to pay for the platform's services to target them with ads; and furthermore, the more time these users spend engaging with other users and content on the platform, the greater the opportunity for the platform to cater to the precise needs of advertisers. Thus, a platform benefits from more engagement than less, irrespective of whether such engagement is induced from harmful or benign content. But because it is only society that incurs the costs of harmful content, the idea behind taxation is to internalize the costs of toxicity to the original transaction between the platform and an advertiser.
Devising a taxation scheme requires a good harm model to measure content toxicity, using an access to data and expertise that is only available to platforms. Therefore, a mechanism designer (regulator) might instead impose taxation in a more crude manner. For example, the regulator can ask the platform report its cost function for moderating content and then tax the platform based on its report. Since the indirect costs of effort essentially capture the value of engagement, this mechanism levies a tax on user engagement on the platform more generally, rather than on the harmfulness of hosted content, which is the entity we wish to control on social platforms.
As shown by Proposition 5, the platform's costs of effort underpin the incentive problem for disinformation mitigation; thus, any effective mechanism must in some way be responsive to these costs. Naturally, such a mechanism must also factor in incentives that might prevent the platform from misreporting its true cost function for moderating content, in the hopes of attaining a lower tax rate, for instance.

Ethical considerations
Regulating social media is an especially sensitive issue. Although allowing disinformation to spread unchecked is clearly unsustainable, disinformation control always runs the risk of becoming censorship. In this work, we take the existence of a "public model" of acceptable postings for granted. However, the content of this public standard is a question of societal standards that can be settled only by public debate. Similarly, we analyze the use of "monitoring" without specifying its exact form. A naively implemented monitoring scheme would run the risk of serious privacy violations.

A Appendix
Here we recall our main results and include proofs omitted from the main body of the paper.
Proposition 1. Given a fixed adequate effort level e ′ , there exists no fine scheme F that can incentivize the platform to exert more effort than e ′ .
Proof. By contradiction. Suppose the platform prefers to exert effort e > e ′ . Thus, the following must hold: By definition, P f (e ′ |e c ) = 0 and therefore P f (e|e c ) = 0. Thus, which does not hold for e > e ′ because by assumption c ′ (e) > 0 for all e (contradiction).
Lemma 1. Fix a state e c representing the current effort required by the public model, and an arbitrary policy π, and let e h > e c be the effort that the public model will require if harm occurs. For all e 2 > e 1 ≥ e c , where d(π, e c ) = g(e c )v π (χ(e c )) + (1 − g(e c ))v π (e c ).
Proof. At e c , the state-action value function for some effort e is given by: By substituting in d(π, e c ) = g(e c )v π (χ(e c )) + (1 − g(e c ))v π (e c ) we have: Thus, for q π (e c , e 2 ) > q π (e c , e 1 ), we have: .
Lemma 2. Given a threshold strategy π τ , the state value function v π τ (e c ) is fixed for all e c ≤ τ .
Proof. The state value function for some arbitrary e c ≤ τ is given by, )v π τ (e c ))]. (8) Note that the platform's policy specifying effort for all e c ≤ τ is fixed by definition; that is, π τ (e c ) = τ for all e c ≤ τ . Thus, the transition to state e h is also fixed because the transition probability h(τ ) is fixed. And similarly, the probability that harm does not occur is also fixed at (1 − h(τ )).

But then
for all g(e k ) ∈ [0, 1], and we are done.
Proposition 2. For all threshold strategies π τ , we have that v π τ (e h ) ≤ v π τ (e c ) holds for all e c ∈ S.
Proof. For ease of notation, let S = {e 0 , e 1 , . . . , e h } denote the set of all states with e 0 < · · · < e h , and let π = π τ with τ = 0. Note that this specification is w.l.o.g.; for τ > 0, we will consider a subset of S such that the first state of this subset e 0 = sup{e ∈ S | e ≤ τ }, since from Lemma 2 we know that the state value function for all e ≤ τ is fixed.
Now we move on to the proof. Suppose the claim is false. Then {e | v π (e) < v π (e h )} = ∅.
Let e z = min{e | v π (e) < v π (e h )} and d(e j ) = (1 − g(e j ))v π (e j ) + g(e j )v π (e j−1 ) for all First, observe that d(e z ) = (1 − g(e z ))v π (e z ) + g(e z )v π (e z−1 ) where the inequality follows from combining the assumptions v π (e h ) > v π (e z ) with v π (e z−1 ) ≥ v π (e h ), both from the definition of e z . Note that if e z = e 0 , then the same result holds, since g(e 0 ) = 0.

It then follows that
.
We now show inductively that v π (e k ) ≥ v π (e h ) for all z ≤ k < h.
The base case is e h−1 . Suppose the contrary that v π (e h−1 ) < v π (e h ). Then we have contradicting the definition of e z .
Therefore, v π (e k ) ≥ v π (e h ) is true for all z ≤ k < h, which in particular implies that the initial claim {e | v π (e) < v π (e h )} = ∅ must be false, thus completing the proof.
Proof. By contradiction. Suppose πê is suboptimal. Then by the process of policy improvement, there must exist a state e c where some effort e = πê(e c ) guarantees a higher expected reward than πê(e c ). Thus, we apply the policy improvement theorem (Lemma 4) to find any such e c where q πê (e c , e) > v πê (e c ) holds, which would imply that a greedy deviation from πê exists as the better policy.
Case 1 (∀e c ): Less aggressive effort than e c The first deviation from πê at any e c might be to exert less aggressive effort e < e c . Suppose that less aggressive effort e guarantees a higher expected reward than the required effort e c . However, we know that lower effort than e c does not guarantee a higher expected reward for all e c because e c by definition is the platform's individually-optimal level of effort. Thus, we have a contradiction and this deviation does not work.
Case 2 (e c ≤ê): Less aggressive effort thanê Suppose that the platform prefers to exert less aggressive effort e 1 such that e c ≤ e 1 <ê. Then q πê (e c , e 1 ) > q πê (e c ,ê) must be true.
From the definition in (6), note that becauseê is the supremum taken over a closed interval, it satisfies the following equation (intermediate value theorem): Now consider the L.H.S of (9) and of (10). Recall that d(πê, e c ) = g(e c )v πê (χ(e c )) + (1 − g(e c ))v πê (e c ). Since πê(e c ) =ê is fixed for all e c <ê, the value functions v πê (χ(e c )) and v πê (e c ) must be equal (Lemma 2). Thus, d(πê, e c ) = v πê (e c ) as 0 ≤ g(e c ) ≤ 1. Furthermore, because s −1 (ê) <ê by definition, q πê (s −1 (ê),ê) = v πê (e c ) must be true. Moreover, v πê (e h ) = q πê (e h , e h ) as e h ≥ê. Thus, the L.H.S of (9) and of (10) are equal, or Suppose that the following is true of the R.H.S of (9) and (10): Thus, Notice that the final inequality is always true: we know by assumption that c is strictly convex (c ′′ (e) > 0) and so from equation (5)  Since h ′ (e) < 0 and e 1 <ê, it follows that the L.H.S fraction overall is strictly greater (less negative) than the R.H.S fraction (more negative).
If (9) holds, the contrapositive statement is false, and so the original statement must also be false; thus, the platform instead prefers to exactly exert effortê, and no less, for all e c <ê, a contradiction.
The process of policy improvement must give us a strictly better policy except when the original policy is already optimal [35]. Since there exists no greedy deviation e = πê(e c ) such that q(e c , e) > q(e c , πê(e c )) is true for any e c , the proposed policy πê must be optimal, thus completing the proof.
Proposition 3. For any given socially optimal level of effort e * , there exists a MDP consistent with our given conditions such that the optimal policy for the platform is a threshold strategy with threshold τ = e * .
Thus, we have  Proof. This result directly follows from the defining condition of the platform's stable effort e in (6). Forê to equal e * , the required effort e h must be strictly greater than e * . By definition,ê is the supremum over the closed interval [0, e h ] and so if e h < e * , thenê < e * is also true.
If e h = e * , thenê < e * is also true; the L.H.S of (6) equals zero for e = e h , or q π e h (s −1 (e h ), e h ) − q π e h (e h , π e h (e h )) = 0, Thus, e h > e * must be true in order for the platform's stable effortê to equal e * .

Proposition 5.
There is no way of adjusting the effort e c required by the public model, purely as a function of the harm function h and the cost of damages D, without regard to the cost function c, such that the platform is always incentivized to exert the socially-optimal level of effort.