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\begin{document}
\title{The impact of AI and digital platforms on the information ecosystem\thanks{We are thankful to Anton Korinek, Wei Li, Andrew Souther and numerous participants at the Economics of TAI Workshop Fall 2025, Sciences Po and Columbia University for their useful comments. Ventura-Bolet gratefully acknowledges financial support from the Bank of Spain through a doctoral fellowship.}}

\author{
    Joseph E. Stiglitz\thanks{Columbia University and NBER. Email: \texttt{jes322@columbia.edu}}
    \and
    Màxim Ventura-Bolet\thanks{Columbia University. Email: \texttt{mv3010@columbia.edu}} 
  }
\date{September 24, 2025}
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\begin{abstract}
    \noindent We develop a tractable model to study how AI and digital platforms impact the information ecosystem. News producers — who create truthful or untruthful content that becomes a public good or bad — earn revenue from consumer visits. Consumers search for information and differ in their ability to distinguish truthful from untruthful information. AI and digital platforms influence the ecosystem by: improving the efficiency of processing and transmission of information, endangering the producer business model, changing the relative cost of producing misinformation and altering the ability of consumers to screen quality. We find that in the absence of adequate regulation (accountability, content moderation, and intellectual property protection) the quality of the information ecosystem may decline, both because the equilibrium quantity of truthful information declines and the share of misinformation increases; and polarization may intensify. While some of these problems are already evident with digital platforms, AI may have different, and overall more adverse, impacts.   
    \\
    \\
    \noindent \textit{JEL Classification}: D80, D83, O33.\\
\noindent \textit{Keywords}: AI, information economics, digital platforms, knowledge, misinformation.
\end{abstract}



\newpage
\section{Introduction}
\setcounter{page}{1}

AI has been heralded as opening up new frontiers in the accumulation of knowledge, as problems that would have taken years to solve—or were essentially insoluble—have been answered in short order.  AI, especially transformative AI, has the potential to significantly increase the pace of innovation, not only processing information more efficiently and rapidly, but even asking and answering new questions. 
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But for a wide range of areas, AI could potentially have large negative effects on the information ecosystem.  In areas of the social sciences and even in some areas of the sciences, such as evolutionary biology, the world is ever changing in ways that are hard to predict, at least at any level of AI currently or likely to be available.   In these areas, it is imperative to constantly add information about the current state of the world.  
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Information, though, is a public good, in the sense defined by Samuelson (Arrow (1962), Stiglitz (1975, 1986, 1999, 2021)). Even with strong intellectual property laws, there are knowledge spillovers, and this is as true of the kind of information that we think of as “news” as for other forms of information and knowledge.  Those who read an informative newspaper article, based on expensive investigative research, relay that information to those they talk to, and these individuals thus obtain such information (if only part of it) for free.  In a sense, all producers of information contribute, in some measure or another, to the pool of knowledge which is available.  Others take out from this pool, and neither those who use the pool of knowledge (Stiglitz (2014)) nor those who contribute to it pay or receive compensation commensurate with marginal values, and accordingly there is no presumption of optimality in the production, transmission and usage of information. Indeed, the argument we have just given suggests that there will \textit{typically} be an undersupply of information, unless there is public intervention.\footnote{In our complex decentralized interdependent and interrelated society/economy, information production and acquisition is widely dispersed, with virtually everyone in our economy engaged in the process in one way or the other.  Inevitably there is some duplicative research.  Because there are real costs of transmitting information and acquiring information, both from others and about what information others are gathering, in some instances, duplicative production of information may be efficient, and this is especially so when the output of research is not fully accurate (as is often the case). Further, much information acquisition is associated with rent seeking, for instance returns one obtains simply by having information earlier than others, or expenditures on innovation the profits from which originates mainly by acquiring some part of the (possibly monopoly) rents of others.  In these cases, there may in fact be excessive expenditures on information.  There is also a large literature looking at expenditures on information acquisition from this perspective. See Alchian and Demsetz (1972), Stiglitz (1975, 1989), Aghion and Howitt (1992).}
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There is a further problem:  one has to ascertain the quality of information, and not only may there not be incentives for honesty (an essential idea within the screening/signaling  literature), there can be incentives for dishonesty, unless curbed, for instance, by fraud laws.\footnote{Until the advent of social media, information economics spent little time discussing the problem of mis and disinformation or the role of fraud laws. It was simply assumed that the information that individuals told was true—they told the truth, nothing but the truth, but not necessarily the whole truth.  An exception is Greenwald and Stiglitz (1992).} Just as there is an undersupply of truthful information and an oversupply of distortionary information, there is an undersupply of efforts to “correct” mis and disinformation, for that too is a public good.

\subsection{AI, digital platforms, and a deterioration of the information ecosystem}

Even before  the advent of AI, there was a concern that the digital platforms (search engines and social media) might lead to a deterioration in the quality of the overall information eco-system, even if all information were truthful (i.e. in the absence of mis and disinformation).
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The underlying idea of our paper is simple: our economy provided a peculiar, but workable, solution to the public-good problem associated with the production of news. Newspapers and magazines produced information that attracted readers, and readers attracted advertising.\footnote{The business models typically combined subscriptions with advertising.  In the model presented here, we do not deal with subscriptions.  There is a broad consensus on the limitations of the subscription business model, but a discussion of this would take us beyond this paper.} The hope was that by providing high quality and timely news, an outlet would attract more readers, which would attract more advertisers.\footnote{That hope was partially dashed by Rupert Murdoch, who first showed that more readers could be attracted by entertainment that by informative and timely news coverage; and even more so by Fox, which demonstrated that becoming part of an identity—the right wing cause—was more profitable than adherence to truth and traditional journalistic standards.} The uncomfortable marriage between advertising and the production and dissemination of news information continued until the arrival of the platforms, which in short order were able to garner for themselves “eyeballs,” partially by appropriating information gathered by the legacy media. 
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The acquisition of “eye-balls” by AI or digital platforms would have, on its own, reduced incentives for legacy media to produce information.  And since the social media can and do acquire eye-balls without producing news, the reduced production of new information means that, even if the digital platforms improve the efficiency of the transmission of information, it may, under quite general conditions, result in a worse information ecosystem, one where individuals and firms actually have worse information. Just as in Grossman-Stiglitz (1976, 1980), where a more informationally efficient financial market reduces incentives to gather information, so too here, with the net result that there may be a deterioration in the information ecosystem. This may be referred to as the competitive effect of the new technology.
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Digital platforms like Google not only divert attention from the legacy media, but steal information from them, information which enhances traffic to them and is accordingly an important contributor to their market value.  This may be referred to as the theft effect.\footnote{We distinguish between these effects partly because there are different policy implications.} The resulting decrease in returns to the production of news leads to a diminution in the supply of information .

\subsection{AI}

AI puts these perverse incentives on steroids. While AI has the potential to lower information processing and dissemination costs, it undermines the incentives for private producers to acquire and process accurate, timely, and reliable content, as AI firms take information from those who have traditionally been engaged in these activities, such as the legacy media, and combine it with that produced by others.  At least, in the case of search engines and social media, there was typically attribution of the source of information, which the platforms claimed (with little evidence) drove more traffic to the legacy media, and thus they got some benefit, even if it did not entail a “fair sharing” of the benefits of the information that the legacy media had produced (Mateen, Tabakovic, Holder, Schiffrin (2023)).  But with AI, there may be no clarity about the source of the information, and the synthesis of multiple sources risks reducing the demand for seeing any particular source.\footnote{Matters may be even worse:  AI sometimes attributes to a legacy media things that were not in the media, and which undermine the credibility of the legacy media.  The absence of accountability means that the incentives of AI to curb such activities which undermine trust in the legacy media is limited.}  Thus, the likelihood that traffic will be driven to the legacy media will be greatly reduced.  If AI firms are allowed to appropriate without compensation the information produced, say, by the legacy media, they will obviously not have adequate incentives to produce such information.  In that case, in spite of the improvement in technology, again there could be a deterioration in the quality of the information ecosystem. To echo an old expression, ``garbage in, garbage out".
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AI may also worsen the problem of the quality of information—again leading to a deterioration in the information ecosystem.  There is a risk of flooding the information ecosystem with lower-quality, synthetic, or misleading content, effectively polluting it. The potential adverse consequences are already evident from social media; AI may exacerbate these—again like adding steroids to a problem that we have not yet learned how to manage. There may be a “war” between the ability to detect misleading content and the ability to create such content in ways that deflect AI’s ability to detect.  If the latter effect dominates, there will be a deterioration of the overall quality of the information ecosystem. And because removing pollution (e.g. through fact checking) is itself a public good, there will be (without public intervention) insufficient expenditures on “clean up,” reinforcing the conclusion that, without public intervention, there is a risk of a worsened information ecosystem.  

\subsection{Outline}

This paper is organized as follows.  Section 2 develops a model of the information ecosystem. Section 3 examines the effects of AI and digital platforms in an environment where all information is truthful. Section 4 extends the analysis to an environment in which there is also mis/disinformation. Section 5 concludes. We provide an  Appendix with all proofs and derivations.

\section{A model of the information ecosystem}

This section develops a tractable model of producers and consumers of information. We use it to study how digital platforms and the rise of AI is reshaping the information ecosystem.
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We broadly define producers as any individual or organization that meets two conditions. The first  is that they create truthful or untruthful content that contributes to the stock of information or misinformation, thereby
becoming either a public good or a public bad. There are economic incentives to generate untruthful content: producing truthful information is expensive and, if some consumers cannot distinguish between truth and lies, serving this segment with low-cost untruthful information can be profit-maximizing. There are also ideological and strategic incentives: untruthful content can promote certain political or social agendas or discredit competing ones. These non-monetary benefits can be incorporated into a profit function via their monetary equivalent. The second condition is that producers' business models rely on consumer visits to their physical or digital locations. These visits generate revenue through advertisements, subscriptions, or data collection. Traditional producers include newspapers, broadcasting media, and specialized journals, as well as internet websites, educational platforms, and others.
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Let $Q^{T}\left( t\right) $ and $Q^{L}\left( t\right) $ be the total amount (stocks) of truthful and untruthful information in the ecosystem at period $t$. These can be interpreted as the truthful and untruthful information that is in society’s “ether”. There is a continuum of $n\in [0,1]$ ex-ante identical producers that at time $t$ produce new truthful and untruthful information, $I^{T}(n,t)$ and $I^{L}( n,t)$. With the ever evolving economy/society, the value of information accumulated in the past becomes less relevant. For simplicity, we assume that both types of information have the same depreciation rate $\delta$. We can therefore write the dynamics of the stocks of information as
follows:%
\begin{equation}
\frac{dQ^{T}\left( t\right) }{dt}= I^T(t)-\delta
Q^{T}\left( t\right)   \label{LMT}
\end{equation}%
\begin{equation}
\frac{dQ^{L}\left( t\right) }{dt}= I^L(t)-\delta
Q^{L}\left( t\right)   \label{LML}
\end{equation}%
where $I^{T}\left( t\right) \equiv \int_{0}^{1}I^{T}(n,t)dn$ and $%
I^{L}\left( t\right) \equiv \int_{0}^{1}I^{L}(n,t)dn$. Implicitly we are assuming that producers create distinct information. Hence there is no redundant information.\footnote{More generally, we can model $I^j(t), j\in \{T,L\}$ as functions of a $I^j(n,t)$ and a measure of differential news, with limiting cases given by:  $I^{T}\left( t\right)= \int_{0}^{1}I^{T}(n,t)dn$ (perfect differentiation) and $I^{T}\left( t\right) = I(n,t)$ (no differentiation). } 
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We broadly define consumers as individuals or organizations that allocate attention to searching for and using information. Consider a unit mass of consumers. A fraction $\omega$ of them use a perfect screening technology at zero cost and can distinguish between the two types of information. We refer to these consumers as informed. The rest of the consumers, which we label uninformed, have no screening technology and cannot distinguish between both types of information. At each time $t$, each informed and uninformed consumer decides how much time (``minutes") to spend searching for new information. Let $M^I(t)$ and $M^U(t)$ be the minute functions of the representative informed and uninformed consumer, respectively. Then, we assume that:
\begin{equation}
M^I(t) =v^I
I^{T}(t)^{\sigma }  \label{MI}
\end{equation}
\begin{equation}
M^U(t) =v^U
(I^{T}(t)+I^{L}(t))^{\sigma }  \label{MI}
\end{equation}

\noindent where $0< \sigma <1$ and $v^I=v^I(Q^T(t),Q^L(t),\gamma)$ and $v^U=v^U(Q^T(t),Q^L(t),\gamma)$ are smooth functions that measure the value of new information.\footnote{One can think that each individual has some subset of the available information (or even a garbling of that information) that depends on $Q^T(t),Q^L(t),\gamma$. This is the information upon which an individual can easily draw from and that will determine their minutes function.} At this point,
we only assume that these values might be a function of the stocks of
truthful and untruthful information and the efficiency of the information
ecosystem measured by the parameter $\gamma $ (which can be thought of as a
vector, each element of which describes one aspect of informational
efficiency, e.g. processing, transmitting...). The functions $v
^{I}( \cdot) $ and $v ^{U}( \cdot) $ will
play a crucial role in the analysis, and we shall say much more about them
later. The minutes function of the informed consumers is increasing and
concave in $I^{T}(t) $, but it does not depend on untruthful information $I^{L}( t)$. Having a perfect screening technology
at their disposal allows informed consumers to disregard lies. The minutes
function of the uninformed consumers, however, is increasing and concave in $%
I^{T}(t) +I^{L}(t) $. This is a natural assumption since these consumers cannot distinguish between truths and lies.
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How are these minutes distributed across producers? Let $a^{I}\left(n,t\right) $ and $a^{U}\left( n,t\right) $ be the share of minutes visiting producers that the representative informed and uninformed consumers spend on producer $n$. Then, we assume that:

\begin{equation}
    a^I(n,t) = \frac{I^T(n,t)}{I^T(t)}
\end{equation} 

\begin{equation}
a^U(n,t) =\frac{I^{T}(n,t) +I^{L}( n,t) 
}{I^{T}( t) +I^{L}(t) }  \label{aU}
\end{equation}

\noindent In the case of the informed consumers, the share of minutes depends on the size of producer $n$'s flow of truthful information relative to that of
other producers. In the case of uninformed consumers, this share depends on the flow of total information relative to that of other producers.\footnote{More generally, we could allow non-proportional targeting of consumers (opinion leaders, high-centrality nodes), so producers choose whom to reach, not just how much to produce, mapping directly to strategic competition among influencers (Li and Tan (2025)).}
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Consumers do not pay for their access to new information. But advertisers pay information producers to post advertisements, knowing that consumers
will be exposed to them when searching/processing new information, and this will affect their purchases. Let $1-\lambda $ be the share of the time used
to search/process new information in which consumers are directly visiting producers. Assume that the income generated by visits is directly proportional to the time spent in visits. Therefore, the revenue of producer $n$ is given by the visits by informed consumers, $\left( 1-\lambda
\right) \omega a^{I}\left( n,t\right) M^{I}\left( t\right) $; plus the
visits by uninformed consumers, $\left( 1-\lambda \right) \left( 1-\omega
\right) a^{U}\left( n,t\right) M^{U}\left( t\right) $. This means that we
can write the revenue function of producer $n$ as follows:%
\begin{equation}   
R(n,t) =(1-\lambda) \bigg[
      \omega v^I I^{T}(t)^{\sigma -1} I^{T}(n,t)  +  (1-\omega) v^U
      \big(I^{T}(t) + I^{L}(t)\big)^{\sigma -1} 
      \big(I^{T}(n,t) + I^{L}(n,t)\big) \bigg] 
\end{equation}

\noindent The marginal effects on revenue of both types of information are positive and depend on the choices of other producers: $I^{T}(n' ,t)$ and $I^{L}(n' ,t)$ for $n' \neq n$. Since $\sigma <1$, the marginal revenue of both types of information is decreasing on the information produced by other producers. Thus, on the marginal revenue side, producer choices are strategic substitutes: as the flow of information of other producers grows, the incentives to produce information declines.\footnote{It would be interesting to explore network effects and other forces that can make producer actions strategic complements. We leave this for future research.}
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Lies are cheaper to produce than the truth. In particular, we assume that:%
\begin{equation}
C\left( n,t\right) =\dfrac{c}{%
\varepsilon }\left[ I^{T}\left( n,t\right) +\theta I^{L}\left( n,t\right) %
\right] ^{\varepsilon }  \label{C bis}
\end{equation}%

\noindent where $\varepsilon > 1$, $0<\theta <1$ and $c=c (Q^T(t),Q^L(t),\gamma)$ is a smooth function that measures the cost of producing new information. At this point, we only assume that it might depend on the
stocks of truthful and untruthful information and the efficiency of the
information ecosystem. We shall say much more about this function later. The
parameter $\theta $, measures the cost of producing untruthful information
relative to truthful information. Since $0<\theta <1$, lies are always
cheaper to produce than the truth. As $\theta \rightarrow 1$,lies become
arbitrarily as expensive as the truth. As $\theta \rightarrow 0$, lies
become arbitrarily cheap. The cost of producing new information is
increasing and convex in both flows of information, and it is not affected
by the choices of other producers. The latter means that, on the cost side,
producer actions are neither substitutes nor complements from a strategic
viewpoint.\footnote{In a richer model, an increase in the flow of information by other producers raises the prices of inputs (including labor) and, as a result, the costs of producer $n$. Thus, we would have that producer actions are also strategic substitutes on the cost side.}
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Summing up, producers maximize the net present value of profits by choosing
how much truthful and untruthful information to produce, $I^{T}(n,t)$ and $I^{L}(n,t)$. We have made two assumptions that simplify the analysis
dramatically. First, future profits of producers do not depend on the flow
of information that they create today, $I^{T}\left( n,t\right) $ and $
I^{L}\left( n,t\right) $.\footnote{
This would not be the case, for instance, if previously produced information
by the producer is absorbed perfectly, while that produced by others is not.
Alternatively, it would also not be the case if producers had reputation
costs. Then, each producer in making its decisions about the flow of new
information would take into account effects at later dates.} Second,
producers are atomistic and they take as given the aggregate stocks of
information, $Q^{T}(t)$ and $Q^{L}\left( t\right) $. Jointly, these
assumptions imply that individual producer choices at time $t$ have no
effects on their future profits and, as a result, their dynamic maximization
problem breaks down into a sequence of independent static problems. Thus,
the flow of information created in period $t$ is the outcome of a simple
static producer game.
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What types of Nash equilibria arise in this producer game? Let $P\left( t\right) $, be the share of untruthful information created in period $t$, that is, $
P\left( t\right) \equiv \dfrac{I^{L}\left( t\right) }{I^{T}\left( t\right)+I^{L}\left( t\right) }$. This share can be interpreted as a measure of how polarizing new information is.\footnote{If $P(t)=0$, the flow is not polarizing because all consumers receive the same information. If $P(t)=1$, the flow's polarizing effect is maximized as
informed and uninformed consumers receive totally different information at
time $t$.} It turns out that the information ecosystem induces two regimes. For some parameter values, only truthful information is produced ($P(t)=0$). In this case, we say we have a truthful equilibrium. For the remaining parameter values, both truthful and untruthful information are produced  ($P(t)>0$). We call this the truth-lies equilibrium. Within this second regime, there are some limiting cases in which the share of lies approaches one. This can occur either because no truthful information is produced or because the production of lies grows without bound. We now examine both regimes separately.

\section{Truthful equilibrium}

We start by analyzing the truthful equilibrium, which serves as a benchmark
on how AI and digital platforms would affect the information ecosystem in a
world without misinformation.\footnote{%
The Appendix contains the proofs of all propositions.}
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\textbf{Proposition 1 (Truthful equilibrium)} Suppose $\dfrac{\omega v ^{I}}{\left( 1-\omega \right) v ^{U}}%
\dfrac{\theta }{1-\theta }>1$. Then the \textit{unique} Nash equilibrium is such that  $P\left( t\right) =0$, and:
\[
I^{T}\left( t\right) =\left( \left( 1-\lambda \right) \frac{\omega v
^{I}+\left( 1-\omega \right) v ^{U}}{c}\right) ^{\frac{1}{\varepsilon
-\sigma }} 
\]
\[
I^{L}\left( t\right) =0 
\]

\noindent We attain a truthful equilibrium when: informed consumer demand for information is high (high $\omega v ^{I}$), uninformed consumer demand for information is low (low $\left( 1-\omega \right) v ^{U}$) and/or lies are not that cheap relative to truths (high $\theta $). In such an environment, all producers will want to solely produce truthful information. The flow of new information $I^{T}\left( t\right) $ is high when: many consumers obtain their information directly from producers (low $\lambda $), the value of information is high (high $\omega v^{I}+\left( 1-\omega \right) v ^{U}$) and the cost of information is low (low $c$).
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If the information ecosystem starts with a positive stock of lies, these
lies depreciate over time and we reach a steady state with $Q^{L}(t)=0$.
Thus, it makes sense to simplify the analysis by assuming that information
ecosystem already starts with $Q^{L}(t)=0$. Hence, in the remainder of this
section, whenever we say ``information"
without making any distinction, we are referring to ``truthful information". It also makes sense that, in an
environment in which there is only truthful information, the value of
information is the same for informed and uninformed consumers. Thus, in the
remainder of this section, we assume that $v ^{I}=v
^{U}=v $.

\subsection{Dynamics and equilibrium}

With these simplifications, we can now write the dynamics of the stock of
information as follows:%
\begin{equation}
\frac{dQ^{T}\left( t\right) }{dt}=\left( \left( 1-\lambda \right) \frac{%
v \left( Q^{T}\left( t\right) ,\gamma \right) }{c\left( Q^{T}\left(
t\right) ,\gamma \right) }\right) ^{\frac{1}{\varepsilon -\sigma }}-\delta
Q^{T}\left( t\right)
\end{equation}%
where now we write the functions describing the value and cost of new
information explicitly as functions of the stock of information, $Q^{T}\left(
t\right) $, and the efficiency of the system, $\gamma $. Up to now, we have not made assumptions about these functions because these were not needed. But now we are forced to make assumptions about them,
because the dynamics of the system crucially depend on the ratio of value to cost of information.
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The first assumption we make is that $\lim\limits_{Q^{T}\left( t\right)
\rightarrow 0}\dfrac{v \left( Q^{T}\left( t\right) ,\gamma \right) }{%
c\left( Q^{T}\left( t\right) ,\gamma \right) }=0$. The cost of information grows without bound as the stock of information goes to zero. One cannot simply produce information without any knowledge. We assume that even if the value of information increases, it does so less rapidly. This assumption implies that the information ecosystem always has a steady state with zero information (information collapse). 
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The second assumption is that $\lim\limits_{Q^{T}\left( t\right)
\rightarrow \infty }\dfrac{v \left( Q^{T}\left( t\right) ,\gamma
\right) }{c\left( Q^{T}\left( t\right) ,\gamma \right) }=0$. The value of new information shrinks to zero as the stock of information goes to infinity. One does not need more additional information when one's knowledge is unbounded. The same does not happen with the cost of information. This assumption ensures that the stock of information does not grow without a bound.
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The number of steady states and their location depends on the behavior of
the ratio of value to cost of information. Our assumptions ensure that this ratio starts at zero when $Q^{T}\left( t\right) =0$ and eventually returns to zero when $Q^{T}\left( t\right) =\infty $. What happens in the middle depends on the relative growth in the value and cost of information as $%
Q^{T}\left( t\right) $ grows. We assume that initially $\dfrac{v _{Q}%
}{v }>\dfrac{c_{Q}}{c}$ and the ratio increases. The value of
information rises fast relative to its cost because, if the stock of
information is low, additions to this stock raise the marginal value of new
information (Radner and Stiglitz (1984)) and lowers marginal costs by improving the information acquisition technology. At some point the growth in this ratio starts to decline and eventually reverses $\dfrac{v _{Q}}{v }<\dfrac{c_{Q}}{c}$. The cost of information now rises fast relative to its value because there are diminishing returns to information and there are less ``low-hanging fruits" available for producers to find.
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To sum up, our assumptions ensure that the ratio of value to costs exhibits
an inverse-U shape starting at zero when the stock of information goes to
zero, and returning to zero as the stock of information goes to infinity. Our final assumption is that there exists a value $Q$ such that:%
\begin{equation}
\left( \left( 1-\lambda \right) \frac{v \left( Q,\gamma \right) }{%
c\left( Q,\gamma \right) }\right) ^{\frac{1}{\varepsilon -\sigma }}>\delta Q
\label{NIC}
\end{equation}%

\noindent This additional assumption ensures that there is always a steady state besides the information collapse. Moreover, this steady state is stable.
We refer to Condition (\ref{NIC}) as the `` no information
collapse" condition. If the ratio of value to cost of
information grows slowly when $Q^{T}\left( t\right) =0$ (at a rate lower
than $\varepsilon -\sigma $), there are an odd number of steady states, and
the information collapse is a stable equilibrium. This is the case we focus from now on (to illustrate we focus on the case of three steady states).
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Figure 1 shows the dynamics of the information ecosystem.\footnote{In drawing the figure, we calibrate equation (9) as follows: $\varepsilon = 1.5$, $\sigma = 0.5$, $\delta = 0.75$, $\lambda=0.2$, $\gamma = 1$, $\frac{v(Q^T(t),\gamma)}{c(Q^T(t),\gamma)}=\frac{v(\gamma Q^T(t))}{c(\gamma Q^T(t))}= \max \{0,\gamma Q^T(t)^3  e^{-0.6 \gamma Q^T(t)}\} $.} If the initial stock of information is high,
the information ecosystem converges to the highest steady state (from now on referred to as the equilibrium and denoted as $Q^{T*}$). If the initial stock of information is low, the ecosystem experiences an information collapse. The model shows that there is path dependence, that is, even temporary aberrations in the information ecosystem (a sufficiently large drop/increase
in $Q^{T}\left( t\right) $) can have permanent consequences.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xmin=-1, xmax=12,
    ymin=-1, ymax=7,
    ticks=none
]
% S-shaped curve
\addplot[domain=0:8, samples=500, thick] {0.8 * x^3 * exp(-0.6 * x)};
% Straight line
\addplot[domain=0:8, thick] {0.75 * x};

% Points and labels
\addplot[only marks, mark=*] coordinates {(0,0)(1.5,1.15)(6.15,4.65)};
\addplot[dashed] coordinates {(6.15,4.65)(6.15,0)};

\node at (axis cs:9.1,3.4) {$I^T(t)$};
\node at (axis cs:9.4,6.2) {$\delta Q^T(t)$};
\node at (axis cs:11,-.5) {$Q^T(t)$};
\node at (axis cs:6.15,-.5) {$Q^{T*}$};
\end{axis}
\end{tikzpicture}
\caption{Equilibrium level of $Q^{T*}$.}
\label{fig:equilibrium-illustration}
\end{figure}

\newpage
\subsection{Welfare and comparative statics}

There is no presumption that the steady state stock of information is socially optimal. But it would be inaccurate to measure welfare simply as the total stock of information in society, $Q^{T}(t)$, given that we would not be taking into account the efficiency of the information ecosystem. More precise is to measure welfare as some function $q\left( t\right) =q\left(
Q^{T}\left( t\right) ,\gamma \right) $ which could be interpreted as the personal information set of the representative consumer. That is, the information upon which an individual can easily draw from.
\\
\\
Our enquiry here, is: how will AI affect the information ecosystem. AI improves the ways in which we acquire, process, and transmit information produced by others. This
can be modeled as how an increase in $\gamma $ affects $q\left(
Q^{T}\left( t\right) ,\gamma \right)$. At the same time, if consumers spend more of their time
searching/processing new information through AI intermediaries, producers of
information will obtain less direct visits, endangering their business
model. This can be modeled as an increase in $\lambda $.\footnote{For purposes of most of this section, digital platforms (traditional information aggregators) have similar qualitative consequences as that of AI, although quantitatively these will differ. AI is likely to increase $\gamma $ by a larger magnitude than digital platforms as it operates across a far broader set of disciplines and draws on a more diverse range of informational inputs in the process of generating and disseminating information. AI is also likely to increase $\lambda $ by a greater magnitude than digital platforms since they offer specific and detailed information and obscure attribution more thoroughly. Although the overall effect is evident - many information producers have shut down or downsized in recent years - this effect varies across types of information and producers (Lyu et al (2025)) and can depend on the relative strength of two opposing forces (as found in Jeon and Nasr (2016)). There is a substitution effect, whereby information aggregators divert traffic away from original news sources, and an expansion effect, whereby these aggregators enhance overall demand for information, potentially increasing exposure and visits to original producers (as found in Calzada and Gil (2020) and Athey et al (2021)).} We proceed to study each of these effects separately.\footnote{We are simply considering the impact of AI as a technological shock. Although we will not go down this path, further insights can be gained by adding AI firms as players who choose $\gamma $ and $\lambda $ to maximize discounted profits. In such an extension, the nature of competition among AI firms would be an important determinant of the effects of AI on the
information ecosystem. For instance, a monopolist would strategically decide
how much consumer attention/information to steal from producers, taking into
account that too much stealing eliminates the incentives to produce information in the future. But competitive AI firms might not take this into account and lead to an information collapse. We leave for further research
the study of the optimal market structure of AI firms.}
\\
\\
To determine the effects of an improvement in efficiency (increase in $%
\gamma $), we need to make assumptions about the effects of $\gamma $ on the
ratio of the value to cost of information. Here, we have two opposing
forces. On one hand, an increase in $\gamma $ reduces the value of new
information. If information is transmitted faster (more efficiently)
throughout society, the value of obtaining new information before others is lower (Alchian and Demsetz (1972)). This is analogous to how (it is alleged
that) weaker intellectual property rights weaken incentives to innovate or how information efficiency of financial markets lowers incentives to acquire information. On the other hand, an increase in $\gamma $ reduces the cost of new information. It is reasonable to assume that higher efficiency in managing information lowers production costs. Thus, the effect of $\gamma $ on the ratio of value to cost could go either way. Naturally, an increase in $\gamma $ could lead to greater information production and an overall improvement of the information ecosystem. But it could also lead to a worsening of the information ecosystem if information production falls. It is hard
to tell which force will predominate in this setting.
\\
\\
Figure 2 illustrates this point with an example in which the force that
predominates depends on $Q^{T}\left( t\right) $. In particular, we assume
that $v =v \left( \gamma Q^{T}\left( t\right) \right) $ and $%
c=c\left( \gamma Q^{T}\left( t\right) \right) $.\footnote{We set $\gamma_L=1$ and $\gamma_H = 1.4$. All other parameters are calibrated as in Figure 1.} Hence, an increase in $%
\gamma $ shifts the curve $I^{T}(t)$ upwards for small $Q^{T}(t)$ but
downwards for large $Q^{T}(t)$. Interestingly, this case leads to a
reduction in the steady state stock of information. This dynamic mirrors the logic of the Grossman-Stiglitz paradox (1980).\footnote{If markets (or AI-intermediaries) were perfectly informative (fully informationally efficient markets, in the traditional sense (see, e.g. Fama (1970)), then no agent would have an incentive to acquire or produce costly information. Markets would be fully efficient in transmitting the information that exists within the information ecosystem, but the only
information within the system would be that which could be obtained costlessly -with appropriate normalization, we can think of as a zero
information system - and therefore a zero knowledge ecosystem.}

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xmin=-1, xmax=10,
    ymin=-1, ymax=7,
    ticks=none
]
% S-shaped curve
\addplot[domain=0:7, samples=500, thick] {0.8 * x^3 * exp(-0.6 * x)};
\addplot[domain=0:7, samples=500, thick] {0.8 * (1.4 * x)^3 * exp(-0.6 * (1.4* x))};
% Straight line
\addplot[domain=0:7, thick] {0.75 * x};

% Points and labels
\addplot[only marks, mark=*] coordinates {(5.2,3.9)(6.15,4.65)};
\addplot[dashed] coordinates {(6.15,4.65)(6.15,0)};
\addplot[dashed] coordinates {(5.2,3.9)(5.2,0)};

\node at (axis cs:8.2,4.2) {$I_ {\gamma_L} ^T(t)$};
\node at (axis cs:8.2,2) {$I_ {\gamma_H} ^T(t)$};
\node at (axis cs:8.4,5.5) {$\delta Q^T(t)$};
\node at (axis cs:9,-.5) {$Q^T(t)$};
\node at (axis cs:6.35,-.5) {$Q_{0}^{T*}$};
\node at (axis cs:4.9,-.5) {$Q_{1}^{T*}$};
\end{axis}
\end{tikzpicture}
\caption{An increase in $\gamma$ can lower $Q^{T*}$.}
\label{fig:equilibrium-illustration}
\end{figure}

\newpage
\noindent Does this mean that welfare declines as a result of AI? Let us assume that $q\left( t\right) =\gamma Q^{T}\left( t\right) $. In this
case, the effect of AI on welfare is ambiguous. On one hand, $\gamma $
increases. On the other hand, $Q^{T}\left( t\right) $ declines. A simple way to think about this result is that the total stock of information in
society, $Q^{T}\left( t\right) $, is a ``pie" and $\gamma $ is the share of the pie that any individual in society has access to. An increase in $\gamma $ has the immediate effect of increasing the share of the pie. But it then slowly reduces the size of the pie as the economy transitions to the new steady state. Whether ex-post
knowledge is greater or lower than ex-ante is therefore unclear. Figure 3 illustrates the case where the fall in the size of the pie predominates and welfare drops (the opposite is, of course, possible too). Point A is the ex-ante equilibrium, associated with a low level of $\gamma $. When $\gamma $ increases, we immediately jump to point B (the share of the `` stock of information``pie" that gets translated into `` meaningful knowledge"
increases). Everyone praises the transformative role of the new technology. However, as $Q^{T}\left( t\right) $ falls and we move towards point C, societal welfare declines.


\begin{figure}[ht]
\centering

\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xmin=-2, xmax=10,
    ymin=-1.5, ymax=8,
    ticks=none
]

% Straight line
\addplot[domain=0:7, thick] {0.9 * x};
% Straight line
\addplot[domain=0:7, thick] {0.4 * x};

% Points and labels
\addplot[only marks, mark=*] coordinates {(4.65,1.9)(4.65,4.15)(1.15,1)};
\addplot[dashed] coordinates {(4.65,4.15)(4.65,0)};
\addplot[dashed] coordinates {(4.65,1.9)(0,1.9)};
\addplot[dashed] coordinates {(1.15,1)(0,1)};
\addplot[dashed] coordinates {(1.15,1)(1.15,0)};

\node at (axis cs:8.4,6.5) {$\gamma_H Q^T(t)$};
\node at (axis cs:8.4,3) {$\gamma_L Q^T(t)$};
\node at (axis cs:4.65,4.65) {$\textbf{B}$};
\node at (axis cs:1.15,1.5) {$\textbf{C}$};
\node at (axis cs:4.65,2.4) {$\textbf{A}$};


\node at (axis cs:9.1,-.5) {$Q^T(t)$};
\node at (axis cs:-1,9.5) {$\gamma Q^T(t)$};
\node at (axis cs:4.65,-.5) {$Q_0 ^*$};
\node at (axis cs:1.15,-.6) {$Q_1 ^*$};
\node at (axis cs:-1,2.25) {$\gamma_L Q^{T*}$};
\node at (axis cs:-1,0.85) {$\gamma_H Q^{T*}$};

\end{axis}
\end{tikzpicture}
\caption{An increase in $\gamma$ can decrease $\gamma Q^{T*}$.}
\label{fig:equilibrium-illustration}
\end{figure}

\newpage
\noindent It is much easier to determine the effects of a deterioration of the
business model (increase in $\lambda $). An increase in $\lambda $ unambiguously reduces the incentives to produce new information and the stock of information declines. Figure 4 illustrates this.\footnote{We set $\lambda_L=0.2$ and $\lambda_H = 0.4$. All other parameters are calibrated as in Figure 1.} It
also follows that societal welfare  unambiguously declines.

\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xmin=-1, xmax=10,
    ymin=-1, ymax=7,
    ticks=none
]
% S-shaped curve
\addplot[domain=0:7, samples=500, thick] {0.8 * x^3 * exp(-0.6 * x)};
\addplot[domain=0:7, samples=500, thick] {0.6 * x^3 * exp(-0.6 * x)};
% Straight line
\addplot[domain=0:7, thick] {0.75 * x};

% Points and labels
\addplot[only marks, mark=*] coordinates {(5,3.7)(6.15,4.65)};
\addplot[dashed] coordinates {(6.15,4.65)(6.15,0)};
\addplot[dashed] coordinates {(5,3.7)(5,0)};

\node at (axis cs:8.2,4.2) {$I_ {\lambda_L} ^T(t)$};
\node at (axis cs:8.2,2.7) {$I_ {\lambda_H} ^T(t)$};
\node at (axis cs:8,5.5) {$\delta Q^T(t)$};
\node at (axis cs:9,-.5) {$Q^T(t)$};
\node at (axis cs:6.35,-.5) {$Q_{0}^{T*}$};
\node at (axis cs:4.9,-.5) {$Q_{1}^{T*}$};



\end{axis}
\end{tikzpicture}
\caption{An increase in $\lambda$ lowers $Q^{T*}$.}
\label{fig:equilibrium-illustration}
\end{figure}

\subsection{Information collapse}

Finally, we note that, by increasing $\gamma $ and $\lambda $, AI might lead to the violation of Condition (\ref{NIC}), posing an existential threat to the information ecosystem. At the current state of AI, hallucinations and inaccuracies
serve as a natural brake on user substitution away from primary sources.
Because users cannot fully rely on AI outputs, they continue to engage with
information producers preserving the incentives for information production.
However, as AI intermediaries become increasingly accurate and contextually
fluent, with the ability to automatically check and verify the original sources to which human actors refer, the need to consult original sources
diminishes. In the limit, if AI becomes perfectly reliable-or reliable
enough and all information consumption is mediated by such systems,
incentives to produce new information collapse. No one invests in producing
accurate information when their work is instantly absorbed and intermediated
by an AI that captures all downstream attention and value.\footnote{%
Much the same applies to the deeper problem of information aggregation and
processing. While the platforms champion their role in information
transmission, AI does that as well as acquiring and transmitting information
produced by others. Grossman and Stiglitz (1976) showed that (futures)
markets, on their own, do not do a good job in information aggregation,
contrary to the assertions of Hayek (1944) and others. Forecasting accuracy
can be increased when such information is combined with other information,
but if that kind of analysis is costly, and if the fruits of that kind of
analysis can be obtained at a low enough cost, then there will be diminished
incentives to acquire the additional information and/or use such additional
information in combination with the future price information to make better
forecasts. Again, the amount of usable information within society is
diminished.}




\section{Truth-lies equilibrium}

In this section we study the presence of mis/disinformation. We start with the following proposition:
\\
\\
\textbf{Proposition 2 (Truth-lies equilibrium)} Suppose $\frac{\omega v^I}{(1-\omega)v^U} \frac{\theta}{(1-\theta)} \leq 1$. Then the \textit{unique} producer Nash Equilibrium is such that $P(t) = 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }} \geq 0$, and:
      
    \[
        I^{T}\left( t\right) =\left( 1-P(t) \right) \bigg[(1-\lambda) \frac{\omega v^I(1-P(t))^{\sigma -1}+(1-\omega)v^U}{c
(1-P(t) \left( 1-\theta \right)) ^{\varepsilon -1}}\bigg] ^{\frac{1}{\varepsilon -\sigma 
}}
    \]
  \[
        I^{L}\left( t\right) =P(t) \bigg[(1-\lambda) \frac{\omega v^I(1-P(t))^{\sigma -1}+(1-\omega)v^U}{c
(1-P(t) \left( 1-\theta \right)) ^{\varepsilon -1}}\bigg] ^{\frac{1}{\varepsilon -\sigma 
}}
    \]

\noindent We can uniquely pin down $P(t)$. If the degree of polarization in the information ecosystem, $P(t)$, is high enough, all producers prefer to only produce truthful
information. The reason is that truthful information is scarce and its value
(in terms of minutes) is high. Hence $P(t)$ falls as a result. The opposite
occurs if $P(t)$ is low enough. In this case, untruthful information is
relatively scarce and all producers prefer to only produce untruthful
information thereby increasing $P(t)$. These stabilizing dynamics leads the information ecosystem towards a unique $P(t)$. This also pins down the flows of truthful and untruthful information.\footnote{%
We cannot pin down individual producer flows $I^{T}(n,t)$ and $I^{L}(n,t)$,
because in equilibrium producers are indifferent between producing truths or
lies. Thus, the model is not inconsistent with producers specializing in different types of information ex-post even though they are identical
ex-ante. For more details, see the Appendix.}
\\
\\
The flows of new information $I^{T}\left( t\right) $ and $\ I^{L}\left(t\right) $ are high when: many consumers obtain their information directly
from producers (low $\lambda $), the value of information is high (high $%
\omega v ^{I}\left( 1-P\left( t\right) \right) ^{\sigma -1}+\left(
1-\omega \right) v ^{U}$) and the cost of information is low (low $c%
\left[ 1-P\left( t\right) \left( 1-\theta \right) \right] ^{\varepsilon -1}$%
). Relative to the previous regime, we note three differences. The first is that a fraction of the new informational flows consist of lies. The second is that the value of information as perceived by consumers is lower, as informed consumers disregard lies. The third is that the cost of information is lower because producing lies is cheaper. 
\\
\\
Now, the dynamics of the information ecosystem can be obtained by
substituting the expressions in Proposition 2 into the laws of motion in
Equations (1) and (2). This is a complicated dynamical system, and a full
analysis of it is not possible in such a short paper. But we can derive most
of the relevant insights using a simple case. Assume that $v
^{I}=v ^{U}=v $, and let $v $ and $c$ be only a
function of $Q^{T}\left( t\right) $ and $\gamma $, but not a function of $Q^{L}\left( t\right) $. Admittedly, our main justification for these
assumptions is that they provide a tractable benchmark. But one could also
justify them by arguing that lies are short-lived in the sense that, even
though uninformed consumers cannot recognize lies on impact, they are able
to do so without too much delay. In this case, uninformed consumers cannot
distinguish between flows of truthful and untruthful information, but they
can distinguish between stocks of truthful and untruthful information.
\\
\\
Under these assumptions, we have that:
\begin{equation}
P\left( t\right) =P\equiv 1-\left( \dfrac{\omega }{1-\omega }\dfrac{\theta }{%
1-\theta }\right) ^{\frac{1}{1-\sigma }}
\end{equation}
\begin{equation}
\frac{dQ^{T}\left( t\right) }{dt}=\left( 1-P\right) \left( \left( 1-\lambda
\right) \frac{v \left( Q^{T}\left( t\right) ,\gamma \right) \left[
\omega \left( 1-P\right) ^{\sigma -1}+1-\omega \right] }{c\left( Q^{T}\left(
t\right) ,\gamma \right) \left[ 1-P\left( 1-\theta \right) \right]
^{\varepsilon -1}}\right) ^{\frac{1}{\varepsilon -\sigma }}-\delta
Q^{T}\left( t\right)   \label{LM}
\end{equation}
\begin{equation}
\frac{dQ^{L}\left( t\right) }{dt}=P\left( \left( 1-\lambda \right) \frac{%
v \left( Q^{T}\left( t\right) ,\gamma \right) \left[ \omega \left(
1-P\right) ^{\sigma -1}+1-\omega \right] }{c\left( Q^{T}\left( t\right)
,\gamma \right) \left[ 1-P\left( 1-\theta \right) \right] ^{\varepsilon -1}}
\right) ^{\frac{1}{\varepsilon -\sigma }}-\delta Q^{L}\left( t\right) 
\end{equation}

\noindent This benchmark case is tractable for two reasons. The first is that the share of lies in the new flows of information is constant over time $P\left( t\right) =P$. Thus, in any non-zero steady state, the share of lies in the information ecosystem converges to $P$, that is, $\dfrac{Q^{L}\left(
t\right) }{Q^{T}\left( t\right) +Q^{L}\left( t\right) }\rightarrow P$. This means that, if we know the evolution of the stock of truthful information, we also know the evolution of the stock of lies. The second reason is because the evolution of the stock of truthful information is independent of
the stock of lies in the system, that is, the stock of lies does not appear
in Equation (\ref{LM}). This means that we can study the evolution of the
stock of truthful information using a simple one-dimensional dynamical
system.
\\
\\
The properties of this system depend again on the assumptions we make about
the ratio of the value to the cost of information. We adopt the assumptions
of the previous section, except that we now re-write Condition (\ref{NIC})
by saying that there exists a value $Q$ such that:%
\begin{equation}
\left( 1-P\right) \left( \left( 1-\lambda \right) \frac{v \left(
Q,\gamma \right) \left[ \omega \left( 1-P\right) ^{\sigma -1}+1-\omega %
\right] }{c\left( Q,\gamma \right) \left[ 1-P\left( 1-\theta \right) \right]
^{\varepsilon -1}}\right) ^{\frac{1}{\varepsilon -\sigma }}>\delta Q
\end{equation}%
If this condition holds, there are non-zero steady states for the stock of
truthful information, and the dynamics of $Q^{T}\left( t\right) $ are
qualitatively similar to those shown in Figure 1. If this condition fails,
there is only one steady state which has zero truthful information. We saw
in the previous section that an increase in $\gamma $ and/or $\lambda $ could lead the ecosystem to zero information steady state. An interesting novelty here is that a reduction in $\omega $ and/or $\theta $ can also generate a steady state without truthful information as $P$ increases.
\\
\\
So what is new here about the effects of AI and digital platforms? As
discussed in the previous section, these technologies improve efficiency
(raise $\gamma $) and deteriorate the business model of producers (raise $%
\lambda $). All the effects discussed in the previous section apply also
here. The presence of mis/disinformation in the ecosystem does not affect
this part of the analysis. But the presence of mis/disinformation creates
two additional channels through which AI and digital platforms might affect
the information ecosystem: the ability of consumers to detect lies ($\omega $%
) and the relative cost of producing lies ($\theta $). These are key
determinants of polarization and the amount of mis/disinformation in the
system. We focus on these new channels next.
\\
\\
AI has an ambiguous effect on the proportion of informed/uninformed
consumers ($\omega $). On the one hand, AI can be used by producers to make
untruthful information harder to detect. On the other hand, AI can also be
used by consumers to improve detection of untruthful information. We refer
to these conflicting effects of AI on $\omega $ as the ``drone war effect". Which force predominates? Although this
is an open question, there are reasons to believe that the the first force
dominates. It is undeniable that AI is helping producers create untruthful
information (creation of malicious bots, propagation of ``fake news", targeted and personalized news feed
algorithms...). However, the extent to which AI is currently helping
consumers detect untruthful information is unclear. AI-intermediaries can
hallucinate, and more importantly, when users search for information they
are not constantly verifying everything they consume with AI. So, we presume
that the effect of AI on $\omega $ is likely negative. This leads to an
increase in polarization and a reduction in the steady state stock of truthful information.
\\
\\
The effect of AI on relative cost of producing lies ($\theta $), however, is
more convincingly unambiguous: AI reduces the cost of producing untruthful
information relative to the cost of producing truthful information,
effectively making it very cheap to produce lies. This also leads to an
increase in polarization and a reduction in the steady state stock of
truthful information. As we have just discussed, this could even lead to a steady state with no truthful information. There is mounting evidence (AI-generated
``news" sites, social media clickbait
images, ``AI slop"...) that unless
producers are held accountable for producing untruthful information, we will
be heading in this direction.
\\
\\
Digital platforms also incentivize the production of mis/disinformation and
increase polarization. But the channels are different: digital platforms do
not affect $\omega $ and $\theta $ directly. Despite this, they might
generate effects that could be interpreted along similar lines. Although not modeled here, both AI and digital platforms reduce producer accountability by obscuring attribution. Hence, digital platforms do indirectly reduce the relative cost of lies (but certainly not as much as AI). Suppose also that consumers are especially engaged by extreme events or information that enrages them (``engagement through
enragement"). This can incentivize producers to create this type of information, even if untruthful, to satisfy the demand from
digital platforms. Also, digital platforms such as social media target
consumers individually and give different information to different consumer types, increasing polarization. This is not the case of AI-intermediaries
who, for the moment, generate synthesis and a more neutral narrative giving
each consumer similar information, reducing polarization.


\section{Conclusions}

This paper is a first attempt to formalize the analysis of the consequences of digital platforms and AI on the information ecosystem.  We have questioned the nirvana promised by the champions of these new technologies, if left to their own devices:  competition in this arena does not necessarily lead to the well-being of society.  We have provided a variety of reasons that the quality of the information ecosystem may deteriorate—even as these innovations increase the ability to process information and decrease the costs of disseminating information. In addition, our framework opens up a new research agenda on measurement, aimed at quantifying the key channels of our model, such as the relative cost of producing lies ($\theta$), the fraction of informed consumers ($\omega$), and the share of advertising revenue captured by intermediaries ($\lambda$). Finally, we emphasize that an ``information collapse” does not require the advent of transformative AI (TAI); it is enough for AI to be merely “good enough” at synthesizing existing information. This means that the period between today’s systems and the potential arrival of TAI is itself fraught with risk, as AI platforms can undermine incentives for information production without yet replacing it. 
\\
\\
\textit{Policy responses }
\\
\\
The design of appropriate intellectual property, accountability, competition and regulatory frameworks will be critical in mitigating the adverse effects on the information ecosystem we’ve described. Our legal frameworks clearly need to adapt to the challenges posed by these new technologies. For instance, in the absence of accountability, there is little incentive for digital platforms not to circulate mis and disinformation, and if such mis and disinformation garners more eyeballs, there are even incentives for expanding the reach of such information.  The willingness of AI firms to share their profits with those who produce the information they rely on may be affected by the competitive structure in AI itself, and by the terms of intellectual property legislation.  Current frameworks were obviously created before these challenges presented by AI and the digital platforms were on the scene.  Critical constructs like fair use were not designed to address whether such use of others’ intellectual property would increase or decrease the quality of the information ecosystem in the presence of AI or the digital platforms, and courts that have naively extended the reach of such constructs may be imposing real harms on our society.   A full analysis would, however, take us beyond the scope of this paper.

\newpage 
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\newpage
\section*{Appendix}

This Appendix proves Proposition 1 and Proposition 2. 
\\
\\
Profit maximization by producer $n$ implies that:%
\begin{equation}
\left( 1-\lambda \right) \left[ \omega v^I I^{T}(t)^{\sigma -1}+\left( 1-\omega
\right)v^U \left[ I^{T}(t)+I^{L}\left( t\right) \right] ^{\sigma -1}\right]
\leq c\left[ I^{T}\left( n,t\right) +\theta I^{L}\left( n,t\right) \right]
^{\varepsilon -1}  \label{FOC1}
\end{equation}%
\begin{equation}
\left( 1-\lambda \right) \left( 1-\omega \right) v^U \left[ I^{T}(t)+I^{L}%
\left( t\right) \right] ^{\sigma -1}\leq \theta c\left[ I^{T}\left(
n,t\right) +\theta I^{L}\left( n,t\right) \right] ^{\varepsilon -1}
\label{FOC2}
\end{equation}%

\noindent Conditions (\ref{FOC1}) and (\ref{FOC2}) state that profit-maximizing choices must be such that the marginal revenue of additional units of truthful or untruthful information cannot exceed their marginal cost. If Condition (\ref{FOC1}) holds as a strict inequality, then $I^{T}\left(
n,t\right) =0$. If Condition (\ref{FOC2}) holds as a strict inequality, then 
$I^{L}\left( n,t\right) =0$.


\subsection*{Proposition 1} Suppose $\dfrac{\omega v ^{I}}{\left( 1-\omega \right) v ^{U}}%
\dfrac{\theta }{1-\theta }>1$. Then the \textit{unique} Nash equilibrium is such that  $P\left( t\right) =0$, and:
\[
I^{T}\left( t\right) =\left( \left( 1-\lambda \right) \frac{\omega v
^{I}+\left( 1-\omega \right) v ^{U}}{c}\right) ^{\frac{1}{\varepsilon
-\sigma }} 
\]
\[
I^{L}\left( t\right) =0 
\]
 \\
 \\
\noindent \textbf{Proof:} 
\\
\\
Suppose $\frac{\omega v^I}{(1-\omega)v^U} \frac{\theta}{(1-\theta)} > 1$. 
\\
\\
First, we will establish that $I^{L}\left( n,t\right) = 0, \forall n, t $. Suppose, by contradiction, there is a producer $n$ such that $I^{L}\left( n,t\right) >0$. Then, since producer $n$ is maximizing, it must be that condition (\ref{FOC2}) holds with equality. Then, substituting condition (\ref{FOC2}) into condition (\ref{FOC1}), and rearranging the expression in terms of $P(t)$, we find that:%
\begin{equation*}
\frac{\omega v^I \theta }{(1-\omega)v^U(1-\theta)} \leq (1-P(t))^{1-\sigma}
\end{equation*}%
which is a contradiction since $P\left( t\right) \in %
\left[ 0,1\right]$ and $\sigma <1$.
\\
\\
This implies that $I^T(n,t)>0, \forall n,t$. This is because the costs are convex and start at 0 while producers have a positive and constant marginal revenue. Therefore, it will always generate some positive profit to produce some $I^T(n,t)$. Hence, condition (\ref{FOC1}) holds with equality. Thus, given our assumptions, the unique equilibrium is for all producers to follow the same strategy: $I^{T}\left( n,t\right) =I^{T}\left( t\right) $ and $%
I^{L}\left( n,t\right) =I^{L}\left( t\right) $ for all $n\in \left[ 0,1%
\right] $, and the producer game has a unique Nash equilibrium given by:%
\begin{equation}
I^{T}( t) =\Bigg[ \frac{( 1-\lambda) 
}{c}(\omega v^I + (1-\omega)v^U)\bigg] ^{\frac{1}{\varepsilon -\sigma }}  
\end{equation}% \notag
\begin{equation}
I^{L}\left( t\right) =0 
\end{equation}%  \notag

\noindent $\square$


\subsection*{Proposition 2} Suppose $\frac{\omega v^I}{(1-\omega)v^U} \frac{\theta}{(1-\theta)} \leq 1$. Then the \textit{unique} producer Nash Equilibrium is such that $P(t) = 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}$, and:
      
    \[
        I^{T}\left( t\right) =\left( 1-P(t) \right) \bigg[(1-\lambda) \frac{\omega v^I(1-P(t))^{\sigma -1}+(1-\omega)v^U}{c
(1-P(t) \left( 1-\theta \right)) ^{\varepsilon -1}}\bigg] ^{\frac{1}{\varepsilon -\sigma 
}}
    \]
  \[
        I^{L}\left( t\right) =P(t) \bigg[(1-\lambda) \frac{\omega v^I(1-P(t))^{\sigma -1}+(1-\omega)v^U}{c
(1-P(t) \left( 1-\theta \right)) ^{\varepsilon -1}}\bigg] ^{\frac{1}{\varepsilon -\sigma 
}}
    \]
\\
\\
\noindent \textbf{Proof:}  
\\
\\
Suppose $\frac{v^I}{v^U} \frac{\omega \theta}{(1-\omega)(1-\theta)} \leq 1$. 
\\
\\
First, we will establish that $P(t) = 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}$ by discarding the alternatives: 
\\
\\
i) Suppose, by contradiction, that $P(t) > 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}$. This implies that $I^L(n,t)= 0,  \forall n,t$. Suppose not, and let $I^{L}\left( n,t\right) >0$. Then, condition (\ref{FOC2})
holds with equality. Then, substituting condition (\ref{FOC2}) into condition (\ref{FOC1}) we find that:%
\begin{equation*}
P(t) \leq 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}
\end{equation*}% 
which is a contradiction. But if $I^L(n,t)= 0, \forall n,t$ we then have that $P(t)=0$ which is a contradiction.
\\
\\
ii) Suppose, by contradiction, that $P(t) < 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}$. This implies that $I^T(n,t)= 0,  \forall n,t$. Suppose not, and let $I^{T}\left( n,t\right) >0$. Then, condition (\ref{FOC1})
holds with equality. Then, substituting condition (\ref{FOC1}) into condition (\ref{FOC2}) we find that:%
\begin{equation*}
P(t) \geq 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}
\end{equation*}% 
which is a contradiction. But if $I^T(n,t)= 0, \forall n,t$ we then have that $P(t)=1$ which is a contradiction.
\\
\\
Let us now consider the case in which $P(t) = 1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}$. Since $0< P(t) < 1$, we now have that both (\ref{FOC1}) and (\ref{FOC2}) hold with equality. Then, (\ref{FOC1}) implies that  each producer $n$ is producing a combination of $I^{T}\left( n,t\right) \geq 0$ or $I^{L}\left(
n,t\right) \geq 0$ such that:%

\begin{equation}
I^{T}\left( n,t\right) +\theta I^{L}\left( n,t\right) = \bigg(\frac{\left(
1-\lambda \right)}{c} \bigg[\omega v^I I^{T}(t)^{\sigma -1}+\left( 1-\omega
\right)v^U ( I^{T}(t)+I^{L}\left( t\right))^{\sigma -1} \bigg]\bigg) ^{\frac{1}{\varepsilon -1}}
\label{IT+thetaIL}
\end{equation}%

\noindent That is, if the aggregate
degree of polarization is exactly as assumed, producer $n$ is indifferent
about producing truthful information or lies, provided that the total flows satisfy equation (\ref{IT+thetaIL}).
\\
\\
Since individual productions are not determined, there are many Nash
equilibria for the producer game. Each of them correspond to a different distribution of flows among producers. But in all these equilibria, aggregate flows of information must be such that they satisfy equation (\ref{IT+thetaIL}) and:
\begin{equation}
\frac{I^{L}\left( t\right) }{I^{T}\left( t\right) +I^{L}\left( t\right) }
=1-\bigg[\frac{\omega v^I}{(1-\omega )v^U}\frac{ \theta} {( 1-\theta)}\bigg]^{\frac{1}{1-\sigma }}  \label{E22}
\end{equation}
\\
\\
We now use \eqref{IT+thetaIL} and \eqref{E22} to find
 to find $I^T(t)+I^L(t)$ and express $I^T(t)$ and $I^L(t)$ as fractions of the total sum. Thus, it follows that, if $\frac{v^I}{v^U} \frac{\omega \theta}{(1-\omega)(1-\theta)} \leq 1$, the equilibrium flows of truthful and untruthful information are given by:
\\
\\
 \[
        I^{T}\left( t\right) =\left( 1-P(t) \right) \bigg[(1-\lambda) \frac{\omega v^I(1-P(t))^{\sigma -1}+(1-\omega)v^U}{c
(1-P(t) \left( 1-\theta \right)) ^{\varepsilon -1}}\bigg] ^{\frac{1}{\varepsilon -\sigma 
}}
    \]
    \\
    \\
  \[
        I^{L}\left( t\right) =P(t) \bigg[(1-\lambda) \frac{\omega v^I(1-P(t))^{\sigma -1}+(1-\omega)v^U}{c
(1-P(t) \left( 1-\theta \right)) ^{\varepsilon -1}}\bigg] ^{\frac{1}{\varepsilon -\sigma 
}}
    \]
\\
\\
\noindent $\square$





\end{document}

