extfieldsofstudy:
Computer Science
Economics
Medicine

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Tilted platforms: rental housing technology and the rise of urban big data oligopolies

Abstract This article interprets emerging scholarship on rental housing platforms—particularly the most well-known and used short- and long-term rental housing platforms—and considers how the technological processes connecting both short-term and long-term rentals to the platform economy are transforming cities. It discusses potential policy approaches to more equitably distribute benefits and mitigate harms. We argue that information technology is not value-neutral. While rental housing platforms may empower data analysts and certain market participants, the same cannot be said for all users or society at large. First, user-generated online data frequently reproduce the systematic biases found in traditional sources of housing information. Evidence is growing that the information broadcasting potential of rental housing platforms may increase rather than mitigate sociospatial inequality. Second, technology platforms curate and shape information according to their creators’ own financial and political interests. The question of which data—and people—are hidden or marginalized on these platforms is just as important as the question of which data are available. Finally, important differences in benefits and drawbacks exist between short-term and long-term rental housing platforms, but are underexplored in the literature: this article unpacks these differences and proposes policy recommendations. Policy and practice recommendations As rental housing technologies upend traditional market processes in favor of platform oligopolies, policymakers must reorient these processes toward the public good. Long-term and short-term rental platforms offer different market benefits and drawbacks, but the latter in particular requires proactive regulation to mitigate harm. At a minimum, policymakers must require that short-term rental platforms provide the information necessary for cities to enforce current, let alone new, housing regulations. Practitioners should be cautious inferring market conditions from rental housing platform data, due to difficult-to-measure sampling biases.

Introduction
Information communication technologies (ICTs) shape how we live our lives and interact with one another, typically as black boxes and without explicit consent (Fields et al. 2020). The smart cities literature in critical geography has interrogated how these ICT platforms restructure capital and urban space, emphasizing the impacts of government actors contracting with corporations (Barns 2020;Caprotti and Liu 2020;Leszczynski 2016; van der Graaf and Ballon 2019). At the same time, a peer-to-peer platform urbanism has emerged in which individuals use corporations' software and algorithms to facilitate decentralized interactions. Such interactions are often unmediated by local, subnational, or national governments-but should they be?
When new technologies are introduced into society, the benefits are often obvious but the attendant drawbacks can be difficult to recognize. For example, recent consumer technologies such as smartphones and internet-connected doorbells captured the public imagination years before their privacy drawbacks became better known (Zuboff 2019). Platform urbanism is no different. Its benefits have been heralded but its drawbacks are only just starting to be identified (Koster et al. 2021;Shabrina et al. 2021). If such technologies entrench or exacerbate pre-existing inequalities, technoutopianism is ultimately harmful and the oligopolization of user-generated "big data" by for-profit corporations becomes a matter of broad public concern.
This concern is more acute in housing than other economic sectors since, for most people, finding a permanent residence is a life-sustaining activity in a sense not true for many other consumer transactions. When a housing search fails the consequences range from bad (e.g., being unable to accept a better job in a different city) to calamitous (e.g., homelessness). In urban housing markets, technology platforms-including Airbnb and Craigslist-have created particularly strong tensions between solving and causing housing problems (Rae 2019). They enable widespread access to high volumes of short-term rental (STR) and long-term rental (LTR) housing information and should, in theory, be a boon for both homeseekers and researchers (Boeing et al. 2020b). They offer frictionless information exchange, expanded housing search radii, flexible reuse of one's home space, and a potential cornucopia of data for planners monitoring affordability. In practice, however, it is more complicated. First, user-generated data often reproduce the systematic biases found in traditional sources of housing information, intensifying the factors that fuel residential segregation (Besbris et al. 2021;Boeing 2020). Second, these platforms curate and shape information according to their owners' financial and political interests, which may not align with local public interests. Third, STR and LTR housing platforms offer different benefits and drawbacks with which policymakers have only just begun to grapple (Jiao and Bai 2019;Wachsmuth and Weisler 2018;Wegmann and Jiao 2017).
It would be one thing if housing platforms did nothing but reproduce the preexisting biases, distortions, and unequal power dynamics inherent in the underlying housing market. However they have also introduced a new concern that is relatively unfamiliar in the housing market but all too familiar in the platform economy: oligopolization. In 2021 the largest owner of multifamily rental housing in the United States owned just over 100,000 units-0.2% of the 48 million rental units overall (National Multifamily Housing Council 2021). But with housing platforms interposing themselves between brick-and-mortar housing units and many of their end users-and with the platforms subject to the same winner-take-all dynamics seen more generally with the rise of Big Tech (Lanier 2014)-the same worrying tendencies toward oligopolistic gatekeeping seen in other sectors of the economy are coming to housing.
This article reviews emerging scholarship on rental housing platforms-focusing on Craigslist and Airbnb as these are among the most researched and used LTR and STR websites-and considers how the technological processes connecting both STRs and LTRs to the platform economy are transforming cities. We discuss potential policy approaches to more equitably distribute benefits and mitigate harms of peer-to-peer platform urbanism in the rental market. ICTs are not value-neutral. User-generated information on some platforms is systematically biased against people and places of color, and for-profit corporations precipitate changes to the housing market that reduce the availability and affordability of housing in certain communities. Rental housing platforms' data may benefit certain market participants or analytics, but the evidence suggests that the benefit for some comes at the expense of society overall. ICTs alone are unlikely to mitigate existing urban inequality and policymakers must develop proactive solutions to the problems of platform urbanism.

Reproduction of systematic biases
Information asymmetries define housing markets, and information environments shape housing search outcomes and in turn residential sorting dynamics (Akerlof and Shiller 2015;Besbris 2020). Information access is determined by social networks, life experiences, available technology, real estate agents, and demographic characteristics such as race and wealth (Besbris and Faber 2017). These combine to circumscribe homeseekers' knowledge of places and available units and exacerbate residential segregation (Krysan and Crowder 2017).
In recent years, a variety of corporations, such as RentJungle, Craigslist, and Facebook, have created centralized platforms to facilitate online information exchange in rental housing markets. Centralized exchanges could expose everyone to the same information, hypothetically erasing traditional information inequalities that correlate with race/ethnicity, language, age, and wealth. But, in reality, user-generated online data often reproduce the systematic biases of traditional information sources (Brannon 2017;Stephens 2013). Rental housing markets offer a clear example of this. Craigslist significantly over-represents whiter, wealthier, and better-educated census tracts (Boeing 2020). In addition, a tract's relative over-or under-representation in online rental listings is significantly associated with demographic differences in age, language, college enrollment, median rent, poverty rate, and household size (ibid.).
Beyond the volume of information, variation also exists in the type and quality of information. Craigslist rental listings in whiter or less-poor census tracts tend to include more words (i.e., information) than listings in less-white or poorer tracts (Boeing et al. 2020a). They also provide more information about available amenities, such as the presence of a washer/dryer or whether pets are allowed, that homeseekers routinely use to filter their searches. Conversely, Craigslist listings in less-white or poorer tracts have more text that focuses on prospective tenants' disqualifications-such as eviction history, criminal history, and proof of income-than listings in whiter and wealthier tracts, which tend to provide more text describing the housing unit and local neighborhood amenities (Besbris et al. 2021).
These patterns take on particular importance when considering Craigslist against the broader context of platform urbanism. Craigslist is a free, low-bandwidth web site that is easily accessible for homeseekers as well as landlords or building managers advertising available units. Yet even this platform cannot eliminate the structural factors that influence information supplies and user self-selection into certain information channels. Other platforms demonstrate similar information segregation and biases. For example, the US Department of Housing and Urban Development recently sued Facebook, claiming its rental housing platform violates fair housing law by targeting specific listings to specific users based on race, ethnicity, and religion (Porter et al. 2019). Corporate entities such as Facebook have total control over what types of information they share and where, and their platforms are the sites of clear biases that may harm consumers and benefit already advantaged groups and places. The problem is acute across the globe. For example, rental housing platforms in China frequently include inaccurate information to either disguise informal market activity or lure prospective tenants (Harten et al. 2020).
The rapid consolidation of rental housing information in the hands of a few corporations has created black boxes around information exchange, making it difficult for researchers or regulators to understand or address platform-mediated biases, steering, and sorting. These big-data oligopolies' platforms now dominate rental housing markets and the evidence suggests they amplify long-standing inequities. The ways these platforms shape market interactions and information access are inherently driven by their own corporate interests (Zuboff 2019). Planners and policymakers seeking to use them to understand housing affordability and other neighborhood processes such as gentrification will struggle to quantify and account for selection biases in community over-and under-representation. Meanwhile, homeseekers will not receive equal or holistic information about available housing, exacerbating existing sociospatial inequalities.
The non-neutrality of housing platform data The word "platform" conjures the image of a level surface or playing field which a disinterested party provides for others to use. In the platform economy, however, this image is misleading. Instead of providing a level playing field, technology firms curate and shape their platforms' information flows to advance their own financial and political interests. In the platform economy, data are a source of monopoly rents (Birch 2020)-a scarce factor that is economically valuable to the extent that it remains scarce.
Most consequentially, these firms exert their control over information flows to advance their financial and political interests by withholding the platform data that policymakers need to enforce current regulations, let alone craft new ones. Accordingly, a growing body of research argues that effective local regulation of STR platforms is at a minimum fraught and potentially unachievable (Ferreri and Sanyal 2018;Kerrigan and Wachsmuth 2021;Leshinsky and Schatz 2018;Wegmann and Jiao 2017;Yeon et al. 2020). For example, these scholars argue that the information necessary to regulate Airbnb hosts-minimally, their identities, the addresses of their STR listings, and the volume of activity of those listings-cannot be feasibly obtained by governments without the cooperation of Airbnb itself or private third-party firms such as Host Compliance. Airbnb remains reluctant to cooperate and has met regulation attempts with court challenges and intransigence (Holpuch 2014). At the same time, the company releases carefully curated slices of data which paint its impacts on cities and communities as favorably as possible.
Platform firms also use their control over information flows in ways that benefit the firm at the expense of users. For example, Airbnb's pricing tool recommends a nightly price to hosts. Airbnb engineers built this tool by applying "Aerosolve," the company's in-house machine learning library, to its enormous private dataset of bookings on its platform (Yee and Ifrach 2015). Airbnb does not share details of how different factors are weighted in its pricing recommendation algorithm, but industry observers widely believe that the firm systematically recommends prices lower than those that would maximize hosts' revenue (e.g. Airbnb Smart 2021; Get Paid for Your Pad 2019).
Why would Airbnb do this? As a firm, it competes with hotels and other types of tourist accommodations, and thus has an interest in keeping listing prices low, regardless of the impacts on its hosts. While individual hosts may make more money if they charge a higher nightly rate (i.e., the increased revenue per night will outweigh the decreased number of bookings which higher prices will lead to), the platform as a whole will generate more revenue if prices are lower and bookings more frequent. Airbnb wields its control over the information flowing through its platform to maximize its own utility at the expense of hosts. 1 And it does so in an opaque fashion, extracting digital rents from its control of information. This behavior is not limited to Airbnb and housing platforms. Notoriously, in April 2020, the Wall Street Journal reported that Amazon was covertly using the data it collects about third-party sales on its Amazon Marketplace to competitively guide its own product development decisions for its AmazonBasics line of consumer goods (Mattioli 2020).
Airbnb and many other housing platform firms are profit-maximizing corporations and it comes as no surprise that they flex their oligopoly power over data toward this end. The implication, though, is that housing researchers should view the data that these platforms expose to the public as a highly curated, biased fragment of a larger whole. The question of which data are made public by housing platforms must be accompanied by the trickier but equally important question of which data-and social actors-are hidden or obscured.

Why we should be more worried about STR than LTR platforms (for now)
Platform urbanism has reshaped urban living, but nowhere are its impacts as dire as in rental housing. Housing as shelter represents a basic human need and it is therefore hard to overstate the importance of housing platforms' impacts on markets, access, and the sorting and distribution of people.
Although both LTR and STR platforms raise concerns, current research makes clear that policymakers should be more concerned, for the time being at least, by those purveying STRs rather than LTRs. Before STR platforms existed, accommodation options 1 A more benign interpretation of Airbnb's price-setting behavior is that the firm is acting in the collective interests of its hosts by resolving a collective action problem around price setting (i.e. Airbnb suggests prices which may lose hosts income in the short run but will increase the competitiveness of the STR sector relative to hotels-and thus STR host income-in the long run). But, even if this were true, Airbnb's monopolistic control of the relevant information flows, and its lack of transparency around the precise aims of its pricesetting tool and other related host services, means that the firm is not acting as a neutral "platform" connecting buyers and sellers.
in cities throughout the developed world were abundant, varied widely in price and quality even in the most popular destinations, and were generally only in short supply briefly, such as during major sporting events or large conferences. If Airbnb, Vrbo, or their competitors disappeared tomorrow, out-of-town travel would continue. LTR platforms, by contrast, bring new visibility and scope-akin to what has long existed in residential sales-to the heretofore opaque rental housing market (Boeing et al. 2020b). For the time being, they do so without imposing significant costs on rental homeseekers. For instance, while Craigslist charges small fees in a small number of municipalities to landlords listing available housing, it does not profit from brokering lease agreements, and therefore does not extract excess profits from people who fulfill this basic need.
While LTR platforms like Craigslist reproduce systematic biases that have long existed in rental markets, STRs amplify existing inequalities and create altogether new ones. This is a consequence of how the STR and LTR rental markets interact with each other. The former exists primarily through parasitically absorbing a portion of the latter. STR industry rhetoric, as implied by the phrase "homesharing," holds that guests absorb otherwise slack residential capacity by staying in spare bedrooms of permanent housing units or occupying the entire unit while the resident is out of town. In the early days of Airbnb, this was a fair characterization of the STR market. However, robust evidence now exists that a substantial-and continually growing-portion of the STR market comprises entire units that otherwise would be used for LTR housing (Wachsmuth and Weisler 2018;Combs et al. 2020;Garcia-López et al. 2020). Housing space originally created for a city's residents is now reapportioned for use by its visitors through these platforms. This comes at a time when low-income rental housing, in the United States at least, is scarce not only within the usual list of high-priced coastal cities, but in every type of urban, suburban, and rural setting (Joint Center for Housing Studies 2020).
Although STR platforms' spokespeople claim, when arguing in the policy arena, that they act as mere facilitators of transactions between guests and hosts, they tell a different story to their users. They burnish a brand with certain connotations, such as the culturally astute Airbnb frequent traveler ("don't call her a tourist") who learns about her destination by staying and interacting with a local host. LTR platforms are different. Finding permanent housing is not something someone is eager to do frequently. A typical person seeking rental housing is looking for the right type of place in the right location at the right price, and likely cares little whether the transaction is brokered via Trulia, Craigslist, or analog means. LTR platforms therefore behave more like genuine marketplaces and less like brands.
This distinction has real consequences. Brands entail exclusivity in direct tension with transparency. In this light, it is unsurprising that Airbnb opposes fully sharing its data with researchers and governments. By contrast, Zillow readily shares its rental listing data with researchers, enabling entirely new lines of research such as questions about whether and how new market-rate multifamily buildings affect the rents of nearby existing rental housing (Asquith et al. 2019;Damiano and Frenier 2020), and whether and how STRs impact rents and housing prices (Barron et al. 2020).
Yet sharing data does not make Zillow or any other platform, like Facebook or Redfin, inherently virtuous. Zillow currently profits by selling advertising to and providing some marketing services for agents and property owners/managers, so it faces less urgency to commodify its data. But in theory, given its market power, it could create tiered access to properties such that consumers would have to pay more to see more listings. Moreover, Zillow Group, which in addition to Zillow also owns Trulia, StreetEasy, RealEstate.com, and other websites, could begin to provide other housing services (e.g., mortgage lending, direct brokerage), which would only increase the corporation's power and monopoly potential.
Like Zillow and unlike most STR platforms, LTR platforms tend not to profit from individual transactions, so the question remains how to induce STR platforms to behave more transparently. This would benefit researchers, but more urgently, it would allow cities to act on escalating calls by their residents for more robust monitoring and enforcement of new regulations to manage disruptions to quality of life and safeguard scarce rental housing.

Toward better housing platform policy
Across both STR and LTR housing platforms, evidence of harm is growing. While research to date has overwhelmingly focused on Airbnb and Craigslist, many housing platforms operate similarly and, as a result, reproduce systematic biases and residential sorting patterns, filter and conceal information flows for their own benefit at the expense of users and communities, reapportion housing stock, and exacerbate affordability challenges and information asymmetries.
Cities can and should push back against the negative impacts of these technology firms while working with them to actually deliver the positive benefits they promise. In particular, cities should require that rental housing platforms share their listings data while protecting privacy. This is critical for understanding LTR housing affordability as the vast majority of market information now flows through online platforms. But it is also critical for STRs, where policy is more urgently needed due to their more pernicious impacts. Currently, it is nearly impossible for regulators to identify or track violations of STR regulations. Cities should require these platforms to register units, provide data on each listed unit, and report when each is rented. Some municipalities already require landlords to file leases and many counties require sellers to file deed transfers. Platforms that broker STRs should be subject to similar requirements.
Several cities offer useful models to point others toward better policy. In Los Angeles, local officials estimated that thousands of the city's Airbnb listings were illegal, but could not enforce existing regulations without better data: in response, the city recently co-developed with Airbnb a system to identify and remove illegal STR listings from the platform (Reyes 2020). The development process was contentious. City officials accused Airbnb of dragging its feet, committing insufficient resources, and unnecessarily delaying the development timeline. Airbnb in turn complained that the city failed to approve and provide the guidelines necessary to design the system in a timely manner (ibid.). Nevertheless, the system finally launched in mid-2020, allowing the city to more efficiently flag and remove noncompliant STR listings.
Likewise, Boston now requires Airbnb and similar platforms to share information monthly on listings' locations, nightly occupancy, and whether they consist of a single room or an entire unit (Enwemeka 2019). It also banned "investor units" by requiring that owners register with the city and reside in any listed STR units: Airbnb sued the city over these regulations before finally agreeing to terms (ibid.). More dramatically, Lisbon and Barcelona both took or threatened to take possession of empty short-term units and transfer them back into the long-term market at subsidized rates during the COVID-19 pandemic (Minder and Abdul 2020).
Notwithstanding these examples, cities with fewer resources may face greater challenges regulating large platforms. State or federal government intervention is likely necessary for effective policy outside of large, wealthy cities. For example, US governmental agencies could take charge of collecting, cleaning, and publicly disseminating LTR spot market data collected from Craigslist, given its value for understanding affordability conditions and the limitations of existing sources of rental market data (Boeing et al. 2020b). On the STR side, legal scholars have argued the US should modernize 1960s-era laws enacted to end racial discrimination by hotels, but that today need updating to address the well-documented discrimination against African American guests by racist hosts on Airbnb (Leong and Belzer 2017). In response to growing pressures, Airbnb has recently introduced a "City Portal" to share local indicators with cities, but specifics are scarce as the tool is brand new. Given the history of concealment and lawsuits, this likely will not solve any significant problems.
Regardless of the exact approaches ultimately taken, only with these data publicly available can policymakers begin to design platform regulations that put human lives first in the rental housing market. Cities must enforce the laws they pass and they must be willing to shut down housing platforms that refuse to comply.


Extended Beings: Screendance as a Reflective and Interrogative Tool During Covid-19

In response to the Covid-19 pandemic and controlling its spread, the first school closures began in mid-March 2020 and classes shifted to home-based distance-learning models in Washington state where my children go to elementary school. Originally from South India, I have lived in the United States for 19 years and have worked closely with different communities and cultures in the context of traditional and folk dance. As a family, we are naturally curious about the world around us and our children enjoy being part of Indian cultural traditions as well as the larger milieu of cultures they are exposed to, growing up and going to school in the US. Our family spent a year (2018-2019) living and traveling within South India where the children walked through the several of the national parks, wildlife sanctuaries, and protected reserves in the Western Ghats, learning about the wildlife, trees, and the people who live in these protected forest areas. The Western Ghats is among the world’s key biodiversity hotspots. During the year in India, my children also learned about traditional practices which centered on cross-species empathy and learned from environmental biologists about the issues that needed to be resolved to conserve biological diversity in the Western Ghats. As school closures dampened a sense of normalcy, my children went back and forth remembering the year in India. As a parent suddenly taking on a bigger teaching responsibility with school closures, I decided to use dance and embodied approaches to teach some basic scientific principles and develop resilience at a time of uncertainty and global climate change. In the following essay, I have outlined our engagement, storytelling for reflexive learning and using screendance as a tool for bringing in a deeper focus and understanding to special topics in childhood education/pedagogy.

Reflecting on resilience
The few months post-March were intense as my two children grappled with what Covid-19 means, listening to special CNN news segments for children and trying to figure out how a virus can impact our immune system. This was complicated with concerns about family and friends living in other countries. We played with balls to understand how Covid-19 spreads. We drew T-cells and talked about innate and adaptive immunity. We pretended to be T-cells fighting 'coronavirus balls' with light sabers. As a mom, I wanted my children to learn resiliency. As an educator, I believe in reflective pedagogy particularly in early childhood education.
For example, here is a short dialogue: My kindergartener asked many times each day, "Will I get the coronavirus, will I have to be in hospital?" I asked her in turn, "Do you know your body is strong and it has an immune system?" Which led to more questions: "What is an immune system, where is the immune system in my body, does my sister have an immune system too?" I explained that, "just like the digestive system helps you eat yummy food, get vitamins and make energy for playing all day, the immune system keeps you safe and fights bad germs." I encouraged her by asking if she would like to draw or maybe dance out what she learnt. I asked, "Can you tell me what characters we need to have for this drawing?"; she would pick herself, a germ, and the immune system. Then I would break down the immune system into smaller components such as a T-cell which would give me an opportunity to explain the idea of binding sites. At the end I asked, "how do you feel about Covid-19 now?" Her answers were calmer and focused on the spiky pattern of the virus, touching multiple surfaces which can lead to contact with the virus, MHC Complex, a T-cell etc. An engagement like this could last for a couple of hours and then be left behind for a whole week and picked back up another day. The advantage of this approach to teach some concepts in biology is that it gives young children the time to fulfill their curiosity and ask lots of questions. Children can also understand that there are many players in immunity and that it is not exactly a linear process or outcome. Children comprehend that there is the time to infection, the immune response and aligned processes with variable outcomes that depend on multiple underlying factors.
Performing coronavirus and immunity: coronavirus light-up balls and T-cell light sabers, April 2020.
When we perform this with a spiky light-up ball, it is easy to denote 2 states: an initial virus entry into host and the lit-up ball showcasing active viral replication mechanisms inside host.
While the complexity of this entire biological process cannot be taught to a kindergartener, it is still possible to develop a systems-thinking approach from a very young age.
In my work with teaching my own children as well as working with elementary school children using dance to teach science, I have found that enacting or embodying the concept goes a really long way in how they understand the concept and begin to apply it. When I work with elementary school children, this learning and application of concepts is particularly enhanced when I take videos of children in groups and play it back to them asking them to explain in words what action they were performing, how did this action make them feel, what was the outcome of this action and could they explain why the action ended up with a certain result. In a follow-up session, the video would allow me to teach the scientific method.

Using Screendance tools and Embodied inquiries into sustainability
Dance is an immanent aspect of my children's lives. From birth they have been witness to dance, to solkattu (the rhythmic syllables of Bharatanatyam footwork), and been privy to my conversations and collaborations with dancers from different cultures. They have watched videos of dance and the practice of dance is not a new concept or meant only for a dance lesson.
With remote learning, I structured our day around necessary school work but emphasized outdoor time hiking, biking, or simply watching birds at the bird bath. I encouraged them to express their observations through journaling, drawing, as well as play-acting or dancingsimple activities that my children enjoy-and used these expressions to create photo stories and musicals. One topic discussed often at our home is climate change, environment, and sustainability. During the remote learning months, we researched several children's books that talked about sustainability and also decided to use dance movements to interrogate what we learned more deeply. We learned about extinct and endangered species and the environmental modifications and stressors that lead to the challenges these species face. This provided a segue into conversations about many questions relevant to sustaining our planet and its biodiversity for the future children of the world. It also allowed them to enact three different human-animal conflicts, which I discuss in the sections that follow. The sections about vultures and tigers in India were directly inspired by their experience of living in India for a year and waiting to see a tiger for many hours at Bandipur National Reserve in Karnataka, India. The section about ocean pollution comes from their love for dolphins and swimming.
Working with movement exercises and using technology to capture these movements provided a lot of opportunities and time to discuss and ask several questions during this process. While my kindergartener enthusiastically embodied the animals in different scenarios, my fifth grader used the Apple iPhone 11 Pro Max 12MP camera; ƒ/2.2 aperture panorama setting to take photographs. Her favorite is the panorama feature. In her words, "I can create new and magical scenes like my imagination. I can change the perspective and make doubles or triples of the same person while they are doing different actions. When I see the pictures again, I think about many different choices."

Plastic pollution in the ocean
A dolphin swims through the ocean playfully in the opening scene surrounded by dappled light on the water. On its way it encounters plastic waste and ingests them.
Embodying a dolphin swimming in the ocean and encountering plastic waste, May 2020. Prompts: Looking at documentaries of ocean life and plastic pollution. Discussing memories of watching ocean life from our different trips to Maldives, Hawaii and San Francisco and aquariums in India and the US. We inquire into two modes of being: 1) Cross-species empathy and 2) 'Becoming' another species. In the empathy mode, they are human beings trying to help an animal in distress while in becoming another species they dance as if they are themselves the animal experiencing the distress. Questions from the children: Why did the dolphin eat the plastic? Why does it not know that plastic is not yummy food? What will happen to the dolphin if it eats a lot of plastic? Will it get sick? Will it die? Why is there so much plastic in the ocean? I asked them to try different movements in response to the two modes of being: 1) Crossspecies empathy and 2) 'Becoming' another species. The children explored different movements such as writhing, struggling, curling up into a ball, distress calls, carrying a hurt animal, etc. They took photographs of each other in different postures and facial expressions and actively discussed which movements looked like a dolphin, what if they could try this in the swimming pool and photograph underwater with a debate about how iPhone pictures can really show what they were feeling when they were dancing. When they shifted to dancing outdoors, they started adapting to the site, adding picnic blankets over the black stones to allow for a gliding movement and altering the length of the movements. In an interesting way, using panorama technology functioned in integrating their little minds and bodies to make sense of themselves in the context of the environment. In reflecting on the dolphin series of images, the two most important aspects of using technology and the screen that stood out in how they were learning and how their thinking was changing as a dynamic response to the screen are: Negotiating time: The juxtaposition of the abstract inner self with variations in 'dolphin' movement captured within a single image provided the children with a human-technology interface that could be explored in many ways for deeper learning. Instead of a linear progression of time, there are multiple temporal frames that the children discovered and navigated. Using technology, the children broke boundaries of the self and 'became' another species, bridging the human world and the animal world with technological realities.
Multicentered gaze: Navigating the rich imagination, memories of travel and reality of their remote schooling in the US brought a multicentered gaze to how the children reflected on the panorama images. Instead of just seeing a dolphin trapped in plastic waste, the picture lent itself to the imagination of three dolphins and a variety of plastic waste making its way from around the world into different oceans. This led to further questions about rising temperatures, bleached corals, and endangered species and the need to address climate change. In the multicentered gaze, many places gathered in the location of their dance. The notion of place and identity gathered multiple layers of meaning through the eyes of my children as the real and virtual transposed and emerged new possibilities for the role of screendance in educational pedagogy.

Impact of veterinary drugs on vultures
In India, drugs such as diclofenac and ketoprofen were used widely for treating inflammation in cows. However, both these drugs cause liver damage in vultures and India saw a steady decline in all of its nine vulture species through the 1990s until the drugs were banned in 2006. Research following the ban showed that population decline had slowed or ceased in some parts of the Indian subcontinent between 2007 and 2011. 1 However, complete elimination of diclofenac from the vultures' food supply has not yet been achieved. In the photograph, one child portrays a vulture spreading its wings to fly towards carrion. After feeding, the vulture feels sick and unable to fly. This was a tricky situation for the children to resolve because they felt torn between helping a sick cow with the medicine it needs and saving vultures from becoming endangered.
Enacting a vulture flying to feed on carrion and feeling sick later. May 2020. Prompts: What should we do if there are two animals that need to be saved, do we pick the most useful one? How should we think when we have interconnected issues? Questions from the children: Do animals that become endangered always become extinct? Is it possible to help both the cows and the vultures? Why do you say that the earth needs biodiversity?

Human-wildlife conflict
There are increasing instances of human-wildlife conflict throughout South Asia, particularly in buffer zone areas adjacent to key wildlife reserves, which are heavily utilized by farming communities or forest food-gatherers. One of the biggest concerns is declining populations of tigers as human-tiger conflicts remain unresolved. These conflicts manifest as attacks by tigers on livestock or people, or as people harming tigers. Once again, this scenario was both provocative and sad for the children as they understood that losing livestock could be crippling to poor farmers and wild plants are important sources of food, materials, and medicines for the gatherers. As they looked at their images again, the fact that each of the images captured different actions and timepoints stood out. My 5-year-old described the image in her own words for example: "The tiger is hiding in the green buffer zone. It is thinking, wait what is the tiger thinking? It must be thinking about lunch. Zoom… it is moving so fast! Oh no! it caught a poor little lamb; it is going to eat it. Mmmmm … so yummy." Embodying a tiger crossing the buffer zone to prey on a lamb. May 2020. Prompts: Why does the tiger attack the farmer's lamb? How much space does an adult tiger need in the forest? Questions from the children: Can we save the tiger? What can children do to help our planet? Am I a tiger or am I myself?
An interesting question that my 5-year-old posed was: "Am I a tiger or am myself?" As a child, she was easily able to move back and forth between a 'tiger state' and 'herself' and examine her movement in digital and real time and space. Being part of a generation that is so tuned to technology, my kindergartener would look at the images and tell her sister, see I look like this (making an expression or a posture) and then she would playfully come up with more jumps or rolls or more facial expressions asking, "do I look like the vulture is dying?" or "hmm how long should the tiger wait, I am hungry now-now (roar)." Her sentences conflated her identity with the identity of the animal. Also, interestingly she tried to respond to the excitement of her movement as a digital presence. This is quite a contrast with my teaching Bharatanatyam to her when I emphasize perception rooted in the physicality of the body and the tuning to inner and outer rhythm. Her conflated identities paved ways for me to think more about her movements embodying a tiger and her experiences of responding to technological capture of the scene. As an artist, understanding how, what, and why the intersection of embodiment and technology could enable the creative process is fascinating.

Reflective learning using panorama images
My fifth grader with her inquisitiveness in exploring technology has been experimenting with iPad and iPhone features to create composite, layered, and complex images of movement and locations and both my children have improvised their techniques over time to capture the stories they wish to tell. In this activity I observed her actively trying to come up with a question or an answer or an opinion before she took a photo. This attempt to have an intent was the hardest exercise. Her final image choices were the ones that she felt contained the narrative she wanted for her audience, however during the entire process, she used the panorama feature and took many images of the site and of the characters that were obscure, tilted, collapsed, distorted, elongated, or crooked and panned the frame to clone the characters (dolphin, vulture, tiger) into multiples. Tilting the phone to change the perspective of the image was another area of experimentation. She decided to use the panorama feature (without the add-on 360° app) based on the fact that it allows for a variety of modulations. In her words, "When I tilt, I think of the iPhone like a magical machine and it lets me take pictures that are always changing. The photos can be moved, cut, copied, colored, given effects, made into an iMovie or one long attached picture for telling the stories I want. I can even tell a story in many different ways." Improvising 'tiger' habitats, May 2020 The children reflected with the panorama photographs of themselves in new ways they had never done before. Their perception and experience were challenged as they integrated the dimension of technology into their thinking. Much like looking through a kaleidoscope, they use of technology allowed them to reimagine scenarios and locations in multiple ways to tell their stories. Based on their research with digital storytelling tools in early childhood education, O'Bryne et al. write, "This medium and the associated tools incorporate higher order thinking skills while also strengthening social connections in and out of the classroom." 2 Panorama mode works by stitching together many individual images into a single image. In screen dance, videos or images may be edited and or modified for a differentiated perspective. One advantage with the panorama mode is that the children could have each other appear in both the left and right sides of a panorama by panning past the person and then asking them to shift their position or change their action. In the figure above, as the child enacting the tiger runs to grab the lamb, three images get stitched together to form one panorama image. The 'tiger' is prowling in the buffer zone followed by a very blurry 'moving tiger' and finally the tiger having captured the lamb. These images compressed and expanded time all at once while their little bodies surfaced a complexity of meanings. Douglas Rosenberg writes, "The raw data of the dancing body is stitched together in the editing process of either film or video, resulting in an impossible body." 3 As the children reflected on this image, they were able to re-imagine a new 'tiger-self' for themselves on the iPhone screen and a sense of urgency in asking, what can I do to help? Blending narratives with dance movements, technological capture and active interpretation of the documentation led to a complex and nuanced understanding of humananimal conflict issues, concepts of sustainability, and inquiries into personal lifestyle and ethical choices. Combining information from categories that do not display a direct relationship, and understanding that stimuli are not independent between timepoints in the different scenarios, are essential skills for incorporating and using technology to solve problems in creativity as well as larger global issues of climate change and sustainable development. These kinds of skills can be successfully developed in young children using a combination of embodied and image processing technologies.

Conclusion
Technology allows my children to intensify their experiences and find new ways to express their cognitive processing. The screen extends their bodies into a hyper virtual space-time, capturing an inner reality that is processing a complex understanding of the world. At a time when we are all forced to engage with the virtual world as a primary model for both learning and communication, screendance offers a dual space of engaging with our bodies and extending the experience into the screen, making physical experience primary to learning and communication. It also helps to re-corporealize our bodies in relation to technology.
I also see screen dance offering a challenging pedagogical tool for teachers across the world to engage with screen time without screens become invasive or overwhelmingly cognitive, providing substantial space for embodied enquiries that extend the screen as a collaborative aspect of the mind-body complex.

Biography
Sandhiya Kalyanasundaram is a dance educator, choreographer and poet. She partners with local communities to develop sustainable foodscaping and urban restoration. Before her work with the environment, she was a researcher in the field of Neuroscience. Trained in Bharatanatyam, Butoh and Flamenco, Sandhiya has led and performed in several collaborative performances between dance styles, served on the Jury Panel for the San Francisco Ethnic Dance Festival and used dance therapy to work with survivors of domestic violence. She enjoys working at the intersection of science, technology and art and is currently expanding her research into the role of dance in science education. Sandhiya's focus areas include delving into the tensions between word and meaning through dance, reflection of artistic processes, particularly imagination, improvisation and abstraction of form.


A Simple Model of Monetary Policy under Phillips-Curve Causal Disagreements

I study a static textbook model of monetary policy and relax the conventional assumption that the private sector has rational expectations. Instead, the private sector forms inflation forecasts according to a misspecified subjective model that disagrees with the central bank's (true) model over the causal underpinnings of the Phillips Curve. Following the AI/Statistics literature on Bayesian Networks, I represent the private sector's model by a direct acyclic graph (DAG). I show that when the private sector's model reverses the direction of causality between inflation and output, the central bank's optimal policy can exhibit an attenuation effect that is sensitive to the noisiness of the true inflation-output equations.

Introduction
Monetary policy is a classic example of an economic interaction in which policy makers' ability to achieve their objectives depends on the accuracy of agents' forecasts of certain economic variables. Consider a textbook static model of the type originated by Kydland and Prescott (1977) and Barro and Gordon (1983). In such a model, the central bank controls a policy variable that affects inflation. The private sector forms an inflation forecast, possibly after observing some signal regarding the central bank's information or its decision. Private-sector expectations are relevant because they affect the joint realization of inflation and output, via some kind of an "expectationsaugmented" Phillips curve. It follows that monetary policy involves "expectations management". To quote Woodford (2003, p. 15): ". . . successful monetary policy is not so much a matter of effective control of overnight interest rates as it is of shaping market expectations of the way in which interest rates, inflation and income are likely to evolve. . . " Conventional models constrain the central bank's ability to manage expectations by assuming that the private sector has "rational expectations" -i.e., it fully understands the statistical regularities in its environment, and thus forms an unbiased inflation forecast conditional on its information. To put it differently, the private sector shares the central bank's (true) model of the macroeconomy and uses it to form its beliefs. In Evans and Honkapohja (2005), Thomas Sargent refers to this feature as "communism of models". In this paper, I revisit the textbook model and relax "model communism", by assuming that the private sector forms its forecast according to a different model than the central bank's (true) model. The key difference between the two models is in their causal treatment of the Phillips Curve.
The modern history of macroeconomic thought has included a number of forms of Phillips Curves, which differ in their dependent and R.H.S variables. Popular descriptions of the Phillips Curve often impose a causal interpretation on these forms. For instance, when the dependent variable is inflation, the causal story is about how changes in unemployment affect inflation. When inflation expectations enter the R.H.S, they are presented as another cause of inflation. When the dependent variable is unemployment or output, the story is about how inflation (possibly in addition to inflationary expectations) affects real variables. Cogley and Sargent (2005), following King and Watson (1994), describe this as a distinction between two econometric identification procedures, which they term "Keynesian direction of fit" and "classical direction of fit". Taking a broader perspective, Hoover (2001) interprets debates in the history of monetary theory (and other areas of macroeconomics) in terms of disagreements over causality.
Therefore, if we wish to model how the private sector forms inflation forecasts using some kind of a Phillips Curve, it seems reasonable to assume that it will involve a causal interpretation. Even if the true model is not fundamentally causal (in the sense that the variables are determined in equilibrium by some system of simultaneous equations with influences in both directions), popular conceptions of the Phillips Curve and how it is integrated with a broader macroeconomic model are likely to be causal in nature.
Following Spiegler (2016Spiegler ( ,2020a, I adopt the formalism of Bayesian networks (Cowell et al. (1999), Pearl (2009), Koller and Friedman (2009)) and represent the private sector's causal model with a directed acyclic graph (DAG). Nodes in the graph represent economic variables and directed links represent postulated causal influences. The DAG can be interpreted in at least two ways. First, it may capture intuitive narratives about causal relations between the relevant variables. Second, it may represent an explicit formal model of the kind that professional forecasters sometimes employ, consisting of a recursive system of equations. The private sector fits its subjective model to the steady-state distribution, and then relies on this "estimated model" to produce an inflation forecast. When the model is misspecified, the inflation forecast systematically distorts the true expected inflation conditional on the private sector's information. I incorporate this model of private-sector expectations into a variant on the familiar static textbook model of central-bank policy. The central bank aims to minimize the expectation of a quadratic loss function of the devia-tions of output and inflation from their exogenous targets. The output target is fixed, whereas the inflation target is random. The central bank chooses a policy instrument in response to its inflation target, trading off the deviation of inflation from its moving target against the deviation of output from its fixed target. A Phillips Curve links these two deviations. Output deviations depend on how the private sector's inflation forecast responds to its information. The key question is therefore how the private sector's narrative regarding the Phillips Curve affects the responsiveness of the private sector's inflation forecast, and in turn the responsiveness of the central bank to the inflation target.
Two qualitative insights emerge from the analysis. First, the private sector's wrong causal model can result in rigid inflation forecasts, which in turn leads to monetary-policy attenuation -that is, the central bank's equilibrium response to the inflation target is weaker than in the rationalexpectations benchmark. Second, the variance of the noise terms in the equations for inflation and output (one of which is a Phillips Curve) can play an important role in the rigid-forecast effect, when the private sector's model reverses the direction of causality between these two variables.
When macroeconomists talk about "flatness" of the Phillips relation, there is ambiguity about whether this means a weak yet precise relation or a noisy relation. In a finite sample, a noisy Phillips relation will appear as a "cloud" of data points, such that the Phillips effect may end up being statistically insignificant -hence, effectively regarded in much the same way as a weak, precise relation. However, when the private sector has rational expectations, the two have very different effects: the variance of the noise terms is irrelevant for the private sector's forecast, hence also for the central bank's optimal policy. This is no longer the case when the private sector's forecasts are based on a misspecified model. Using examples, I show that the variance of the noise terms in the inflation-output equations has a subtle effect on the private sector's inflation forecast and consequently on central-bank policy.

Related literature
The Bayesian-network approach to modeling equilibrium non-rational expectations was presented in Spiegler (2016). In Spiegler (2020a), I presented a simple monetary-policy example to illustrate the question of whether causal misperceptions can lead to systematically biased expectations. In this paper, I adhere to the standard linear-normal parameterization, such that the private sector's expectations are correct on average. Therefore, systematically biased inflation forecasts are ruled out; the departure from rational expectations lies in the under-sensitivity of the private sector's inflation forecast to its information.
Within the macroeconomic theory literature, there are a few precedents for modeling monetary policy when the rational-expectations assumption is The most closely related equilibrium concept that is employed in the macroeconomic literature is known as "restricted perceptions equilibrium" (Evans and Honkapohja (2001)), which is based on a notion of coarse beliefs in the same spirit as Piccione and Rubinstein (2003) and Jehiel (2005 (2020) for critical discussions). This paper contributes to this literature by proposing a new possible motive for monetary-policy distortion: the private sector's misperception of the causal relation between inflation and output.

The Model
The following is a variant on a familiar textbook static model of monetary policy in the tradition of Kydland and Prescott (1977) and Barro and Gordon (1983). It most closely resembles Sargent (1999) and Athey et al. (2005).
All variables in the model get real values. A central bank observes an exogenous, normally distributed variable θ and chooses an action a. Subsequently, the private sector observes a Gaussian signal t of (θ, a) and forms an inflation forecast e. Following the realization of θ, a, t, e, actual inflation π and actual output y are realized according to the following pair of equations: where λ ∈ (0, 1) is a constant that represents the extent to which anticipated inflation offsets the real effect of actual inflation; and ε and η are statistically independent noise terms, ε ∼ N(0, σ 2 ε ) and η ∼ N(0, σ 2 η ). The central bank's objective is to minimize a standard quadratic loss function: Thus, the exogenous state variable θ is interpreted as the central bank's ideal inflation target. The central bank's strategy is a function (a(θ)) θ that assigns an action to every realization of θ. The private sector's strategy is the forecast function (e(t)) t that assigns an inflation forecast to every signal t.
Given the central bank's strategy and private sector's forecast function, we can define a joint distribution p over all six variables θ, a, t, e, y, π. The exogenous components of p are the distribution over θ, the conditional distribution over t (given θ, a) and the conditional distribution of π and y (given θ, a, t, e). The endogenous components are the pair consisting of the central bank's strategy (a(θ)) θ and the private sector's forecast (e(t)) t .
We now arrive at the key definition of how the private sector forms its inflation forecast. The private sector is endowed with a subjective causal model, which is represented by a directed acyclic graph (DAG) G = (N, R), where N is a set of nodes and R is a set of directed links. Each node in N represents a variable. Some variables may be omitted from the causal model, but we assume that t, y and π are always represented. A link represents a perceived direct causal influence.
It will be convenient to enumerate the six variables x 1 , ..., In other words, R(i) represents the set of variables that are considered to be direct causes of the variable x i . I sometimes use boldface notation of a variable to represent its label, such that for any variable z,its label is z. For instance, when R postulates that y is a direct cause of π, we can rewrite this as y ∈ R(π).
To arrive at an inflation forecast, the private sector "estimates" its model by performing measurements that quantify the postulated causal links, and combining the results of these measurements in accordance with the causal model. In other words, the private sector fits the causal model G to the joint distribution p. Formally, the private sector's belief is defined as follows: This is a Bayesian-network factorization formula: it factorizes the objective joint distribution p (defined over x N ) into a product of conditional-probability terms. Each term is extracted from p, but the terms are put together in a way that may result in p G = p. As long as p has full support, p G is a welldefined distribution, which induces well-defined marginal and conditional distributions. We say that a distribution p is consistent with G if p G = p. Indeed, the objective joint distribution p is consistent with the following One interpretation of p G is that it is the limit belief of a discrete-time process of Bayesian updating, where the prior has full support over all distributions that are consistent with G, and the private sector observes an independent draw from p at every period (see Spiegler (2020b)). Another interpretation, which is appropriate when p is multivariate normal, is that G represents a recursive system of linear regression equations, where every term p(x i | x R(i) ) in (1) corresponds to a linear regression equation for x i , in which the set of regressors is R(i). In this case, (1) defines a so-called Gaussian Bayesian network (Koller and Friedman (2009, Ch. 7)). Each equation is estimated via OLS against an arbitrarily large sample, and the estimated equations are put together in accordance with the assumption that the noise terms in the regression equations are independent. This assumption is wrong when G is a misspecified causal model.
Because the private sector perceives statistical regularities through the prism of an incorrect model, the subjective belief p G may systematically distort the correlation structure of the actual distribution p.
The private sector uses its belief p G to obtain the inflation forecast: As we shall see, because G is a misspecified model, e may be sensitive to all features of the joint distribution of p, including those that would not matter under rational expectations.
Definition 2 (Linear equilibrium) An equilibrium (a(θ)) θ and (e(t)) t is linear if a(θ) and e(t) are linear functions.
Throughout the paper, I restrict attention to linear equilibria. This ensures that the joint distribution p is multivariate normal.

Rational-expectations benchmark
As a benchmark, suppose that the private sector has rational expectations.
That is, its subjective DAG is G * itself. Furthermore, suppose it is fully informed of θ, a. Then, its inflation forecast conditional on θ, a is e = E(π | θ, a) = E(π | a) = a Therefore, The central bank's expected cost from an action a given the realization of θ is: Therefore, the central bank's optimal action given θ is The solution a * (θ) is not fully responsive to changes in θ. However, as λ tends to one -such that only unanticipated inflation matters for output -the central bank's policy approaches fully flexible targeting. Another key observation is that a * (θ) is invariant to σ 2 ε and σ 2 η .

Discussion: Model-based forecasts
The private sector's forecasting method consists of two steps: if first estimates a causal model and then uses the estimated model to generate an inflation forecast. The exact interpretation of this process depends on whether we think of private-sector agents as lay persons who reason about macroeconomic processes informally and intuitively, or as professional forecasters. Under the former interpretation, the idea is that agents have some qualitative causal perceptions of how macroeconomic variables relate to one another, and they interpret observational data in light of these perceptions.
As to the latter interpretation, it is clear that not all professional forecasting is model-based. However, models do play a role in forecasting. One virtue of models is that they ground the forecast in a "story" that can be Thus, we should not think of the private sector as being exclusively interested in predicting inflation -otherwise, there would be no need to rely on a model; the private sector could simply measure the steady-state conditional expectation E(π | t). Instead, the private sector employs a model as a multi-purpose vehicle for making sense of statistical regularities in the macroeconomic environment, and for making conditional predictions as the occasion arises. The model is a knowing simplification of a complex environment, and therefore the private sector expects it to get some things wrong, and would be unfazed if some of the model's assumptions and predictions turned out to be incorrect (defending this stance with familiar statements like "every model is wrong" or "it takes a model to beat a model").

The Basic Result
The model assumes that the private sector has a correct causal model of the joint behavior of the variables θ, a, t, e. The private sector's error is that it gets the direct causes of π or y wrong. In the true model G * = (N * , R * ), y / ∈ R * (π) -that is, output is not a direct cause of inflation, because causality runs in the opposite direction. Our first result establishes that if the private sector's causal model G shares this property, its inflation forecast in a linear equilibrium is unaffected by σ 2 ε and σ 2 η . Moreover, the equilibrium can be "rationalized": it can be obtained by assuming that G = G * while modifying the private sector's signal function, given by (p(t | θ, a)).
. Then: (i) Any linear equilibrium is invariant to (σ 2 ε , σ 2 η ). (ii) Any linear equilibrium can be obtained from a specification of the primitives in which the private sector's causal model is G * , and the private sector is either fully informed or entirely uninformed of (θ, a).
It follows that we only need to examine three cases. First, suppose that a ∈R(π) (or, equivalently, θ ∈R(π)). Then, where the second equality follows from the fact that under the objective distribution, π ⊥ (θ, t, e) | a. It follows that if a ∈R(π), the private sector's inflation forecast is consistent with rational expectations, even if we do not change its signal function.
Second, suppose that {a, θ} ∩ R(π) is empty, but t ∈R(π) (or, equivalently, e ∈R(π)). Then, by definition, Then, as in the first case, the private sector's inflation forecast is consistent with rational expectations, even if we do not change its signal function.
The only remaining case is that R(π) is empty. In this case, This inflation forecast can be generated in a rational-expectations model, in which the private sector is entirely uninformed of θ, a -i.e., we modify

Examples
In this section I analyze in detail two examples in which the private sector's causal model violates the condition in Proposition 1. As a result, the noise terms in the equations for inflation and output play a non-trivial role in the private sector's inflation forecast, and therefore also in the central bank's equilibrium policy.
In both examples, I assume that t = a -i.e., the private sector observes a before forming its inflation forecast. (The private sector does not observe θ, but this is immaterial.) Since t coincides with a, we can remove it from the model, such that G * can be written as: That is, the central bank's action a is an immediate consequence of the exogenous variable θ; actual inflation π and the private sector's inflation forecast e are conditionally independent consequences of a; and these two variables are the direct causes of real output y.
We will now analyze two specifications of the private sector's DAG G.

Case 1:
This DAG tells a causal story, according to which inflation is a consequence of two independent causes: real output and the central bank's policy. This causal model postulates absolute monetary neutrality: it admits no path of causal links from a to y. Thus, the private sector's causal model distorts the true model G * in two ways: it omits the expectations variable e and inverts the causal link between inflation and output. The private sector's inflation forecast after observing the central bank's action a is Compare this with the following way to write the rational-expectations inflation forecast: The discrepancy arises because p G (π | a) involves an implicit expectation over y without conditioning on a. (Note that since π, y ⊥ θ | a according to both G * and G, θ does not feature in either of the two expressions.) Because the term p(y) in (4) is actually not independent of a, a change in the central bank's strategy can lead to a change in E G (π | a).
Proposition 2 There is a unique linear equilibrium, given by Proof. Let us begin by deriving a formula for the private sector's inflation forecast. Denote

y)
Since π = a + ε, E(π | a, y) = a + E(ε | a, y) By the Phillips Curve, For given a and y, the R.H.S is a constant, whereas the L.H.S is a sum of two independent variables that are normally distributed with mean zero. Therefore, to calculate E(ε | a, y), we can apply the standard formula for E(X | X + Z) when X and Z are independent normal variables, and obtain We can now write Plug this into the expression for e(a), and get: Plugging the expression for β, we obtain Plugging the expression we obtained for e(a), the expected output conditional on a is We can now write down the central bank's expected payoff from a given θ as follows: Our definition of linear equilibrium adopts the interim approach -i.e., when finding the optimal a given θ, we need to take E(a) as given. (This does not matter in fact; we could also take the ex-ante approach and the result would be the same, although this is an artifact of the linearity of the model.) The first-order condition with respect to a is Taking the expectation with respect to a and recalling that E(θ) = 0, it is immediate that E(a) = 0. Therefore, We have now obtained the precise formulas for the unique linear equilibrium, given by (6).
Thus, when λ = 1, e(a) = a = E(π | a) -i.e., the private sector's inflation forecast coincides with the rational-expectations benchmark. Despite the private sector's misspecified Phillips Curve, it ends up having correct expectations and therefore the central bank's strategy coincides with the rational-expectations benchmark. However, when λ < 1 -i.e., when anticipated inflation has real effects -the inflation forecast is "rigid" in the sense of being a convex combination of the correct conditional expected inflation E(π | a) and the ex-ante, unconditional expected inflation E(π). As a result, the central bank's equilibrium strategy is less responsive to θ than in the rational-expectations benchmark.
To understand the reason for the private sector's partially responsive expectations, recall that according to the DAG G : θ → a → π ← y, the private sector erroneously regards y as an exogenous variable that affects π, and therefore assigns some weight to the ex-ante expected value of y when forming its inflation forecast. Because y is in fact a consequence of a, the private sector ends up assigning weight to the ex-ante expectation of a, thus failing to fully condition on the actual realization of a.
The extent of this failure depends on the relative magnitudes of σ 2 ε and σ 2 η . As the Phillips relation becomes less noisy (relative to the relation between the central bank's action and inflation), the erroneous weight on E(a) increases and the deviation from rational expectations is exacerbated. This "expectational rigidity" leads to a policy-attenuation effect that increases with σ 2 ε /σ 2 η . The intuition behind the comparative statics is as follows. The private sector tries to account for fluctuations in π (conditional on a) by the variation in y, mistakenly treating the latter as exogenous. The variables π and y are normally distributed. When σ 2 η /σ 2 ε is low, the definition of conditional expectation of normal variables implies that y gets a large weight in the private sector's inflation forecast, and therefore the model misspecification error generates a larger forecast error -which in turn impels the central bank to adopt a more rigid policy.

Case 2:
This DAG tells a causal story, according to which the central bank's action has a causal effect on real output, which in turn is the sole direct cause of inflation. As in Case 1, this specification of G inverts the true causal link between output and inflation. However, unlike Case 1, this causal model regards y as an endogenous variable that mediates the effect of monetary policy on inflation.
The private sector's inflation forecast after observing the central bank's action a is Compare this expression with (5). The error in (8) lies in the term p(π | y), which fails to condition π on a. This failure reflects the assumption, embedded in G, that π is independent of a conditional on y. This assumption is inconsistent with the true model G * . A linear equilibrium is strongly linear if e(a) is proportional to a and a(θ) is proportional to θ. The next result restricts attention to strongly linear equilibria, in order to avoid the minor technical distraction of proving that every linear equilibrium must be strongly linear (this could be demonstrated transparently in Case 1, but unfortunately not in the present case).
Proposition 3 There is a unique strongly linear equilibrium, given by e(a) = γa (9) a(θ) = δθ where γ and δ are uniquely given by the pair of equations: Proof. Let us begin by deriving a formula for the private sector's inflation forecast. Denote Guessing a strongly linear equilibrium, suppose e(a) = γa and a(θ) = δθ. Therefore, Given θ, the central bank effectively chooses a to minimize The first-order condition with respect to a pins down the expression for δ given by (10). Our remaining task is to derive the expression for γ. To do so, let us first obtain an expression for E(π | y). By the true equations for inflation and output, which is equal to The expression (1 − γλ)a + ε + η is a sum of independent, normally distributed variables with mean zero. Denote Then, using the conditional-expectation rule for such variables, Therefore, Now, plug this expression in (11) and obtain By the inflation and output equations, Therefore, where γ is given by (10). It remains to establish uniqueness of (γ, δ). Clearly, δ is a monotone function of γ in the relevant range. Therefore, it suffices to show that γ is unique. Plugging the equation for δ in the equation for γ, we obtain It can be checked that the R.H.S of this equation is continuously and monotonically decreasing in γ in the range [0, 1]. Moreover, it is strictly positive when γ = 0 and strictly below 1 when γ = 1. Therefore, there is a unique intersection between the curve given by the R.H.S and the line given by the L.H.S.
As in Case 1, the variance of the noise terms in the inflation and output equations plays a role in the strongly linear equilibrium in the present case. However, this role is different in a number of ways.
First, the effect of the noisiness of the Phillips Curve (i.e. σ 2 η ) on expectational rigidity goes in the opposite direction. Expectations become more rigid as the Phillips relation becomes less precise. The intuition is that in Case 2, G posits that the effect of a on π is mediated by y. As the statistical relation between π and y becomes noisier, the perceived mediation effect is attenuated, and this means that the perceived indirect effect of a on π becomes weaker. Second, unlike Case 1, the private sector's expectations depart from the rational-expectations benchmark even when λ = 1. That is, even when anticipated inflation fully offsets the effect of inflation on output, the private sector's causal model leads to incorrect, rigid inflation forecasts. In particular, when λ = 1 and σ 2 ε (which measures the noisiness of the mapping from a to π) gets large, γ approaches 1 2 -i.e., the inflation forecast is only "half-responsive" to changes in a.
Finally, unlike Case 1, the variance of the exogenous inflation target θ plays a role in the characterization of the linear equilibrium. As σ 2 θ becomes larger, the private sector's expectations become less rigid, and as a result the central bank's policy approaches the rational-expectations benchmark. The intuition is that larger fluctuations in θ translate to larger fluctuations in a.
This in turn implies that the private sector's interpretation of fluctuations in inflation and output becomes dominated by the fluctuations in a, which is the correct interpretation; the misperception of the role of σ 2 η and σ 2 ε becomes less important for the private sector's expectations.

Discussion
The comparison between Cases 1 and 2 shows that the details of the private sector's causal misinterpretation of the Phillips Curve matter for the qualitative properties of the equilibrium in the interaction between the central bank and the private sector. However, the broad lesson that σ 2 η and σ 2 ε matter for the equilibrium holds in both cases because they share the reverse-causality error that misperceives the direction of causality between output and inflation.
Note, however, that not every G that exhibits this reverse causality will give rise to this effect. For example, suppose R(π) = {y} and R(y) = ∅. Then, G admits no causal path from a or e to π. As in earlier examples that exhibited this feature, the private sector's equilibrium expectations will be entirely rigid: e = 0 with probability one, leading to the most rigid centralbank policy that is possible in this model, without any sensitivity to σ 2 η and σ 2 ε . Another example is R(π) = {a, y} and R(y) = {a}. In this case, G induces rational expectations. This can be shown algebraically, or using basic equivalence results from the Bayesian-network literature (see Spiegler (2016Spiegler ( ,2020b for details). Thus, reverse causality is a necessary condition for the noise-sensitivity of central-bank policy, but not a sufficient one.
The noise-sensitivity of linear equilibria is of interest in light of the "caution principle" attributed to Brainard (1967), which has been suggested as an explanation of monetary-policy attenuation. A high σ 2 ε captures a situation in which the central bank is highly uncertain about the effect of monetary policy on inflation. However, given our conventional linear-quadratic parameterization of the model, this uncertainty has no effect on the central bank's policy under rational expectations. However, when the private sector's model reverses the causal link between inflation and output, high σ 2 ε can lead to an attenuated policy -although, as we saw, the extent of this effect depends on details of the private sector's model. The reason is that σ 2 ε affects the private sector's interpretation of inflation fluctuations, and therefore its inflation forecast conditional on its information. The lesson is that Phillips-Curve disagreements between the central bank and the private sector can rationalize policy distortions due to uncertainty about the consequences of monetary policy -even in settings that otherwise could not give rise to policy distortions.
The policy distortion that emerged in this paper takes the form of attenuation -i.e., the policy is less responsive to exogenous shocks than in the rational-expectations benchmark. However, I do not want to stress this point too strongly. The important effect that our analysis has highlighted is the attenuated response of the private sector's inflation forecast, as a result of its causally misperceived Phillips Curve. The fact that this effect translated to a policy-attenuation effect is an artifact of how the model treated the central bank's private information θ -namely, as an exogenous inflation target. Under alternative specifications of the central bank's preferences, rigid private-sector forecasts would lead to over-responsive central-bank policy.
One could also imagine a model in which θ has no direct payoff relevance, and instead plays a role as a parameter in the inflation/output equations. 2 Depending on how the private sector incorporates this dependency into its causal model, such a specification could lead to a more complicated, less transparent relation between the private sector's forecast errors and the central bank's policy distortion. Since paper is a first attempt to incorporate causal disagreements into familiar models of monetary policy, I prioritized pedagogical considerations and refrained from adding these complications.
However, I believe they merit further study.
2 I thank Stephane Dupraz for making these observations.