# performative_power__b79060c4.pdf

Performative Power

Moritz Hardt Max-Planck Institute for Intelligent Systems, Tübingen

hardt@is.mpg.de

Meena Jagadeesan

UC Berkeley mjagadeesan@berkeley.edu

Celestine Mendler-Dünner Max-Planck Institute for Intelligent Systems, Tübingen

cmendler@tuebingen.mpg.de

We introduce the notion of performative power, which measures the ability of a ﬁrm operating an algorithmic system, such as a digital content recommendation platform, to cause change in a population of participants. We relate performative power to the economic study of competition in digital economies. Traditional economic concepts struggle with identifying anti-competitive patterns in digital platforms not least due to the complexity of market deﬁnition. In contrast, performative power is a causal notion that is identiﬁable with minimal knowledge of the market, its internals, participants, products, or prices. We study the role of performative power in prediction and show that low performative power implies that a ﬁrm can do no better than to optimize their objective on current data. In contrast, ﬁrms of high performative power stand to beneﬁt from steering the population towards more proﬁtable behavior. We conﬁrm in a simple theoretical model that monopolies maximize performative power. A ﬁrm s ability to personalize increases performative power, while competition and outside options decrease performative power. On the empirical side, we propose an observational causal design to identify performative power from discontinuities in how digital platforms display content. This allows to repurpose causal effects from various studies about digital platforms as lower bounds on performative power. Finally, we speculate about the role that performative power might play in competition policy and antitrust enforcement in digital marketplaces.

1 Introduction

Digital platforms pose a well-recognized challenge for antitrust enforcement. Traditional market deﬁnitions, along with associated notions of competition and market power, map poorly onto digital platforms. A core challenge is the difﬁculty of precisely modeling the interactions between the market participants, products, and prices. An authoritative report, published by the Stigler Committee [2019], details the many challenges associated with digital platforms, among them: Pinpointing the locus of competition can also be challenging because the markets are multisided and often ones with which economists and lawyers have little experience. This complexity can make market deﬁnition another hurdle to effective enforcement. Published the same year, a comprehensive report from the European Commission calls for less emphasis on analysis of market deﬁnition, and more emphasis on theories of harm and identiﬁcation of anti-competitive strategies. [Crémer et al., 2019]

Our work responds to this call by developing a normative and technical proposal for reasoning about power in digital economies, while relaxing the reliance on market deﬁnition. Our running

Authors listed in alphabetical order.

36th Conference on Neural Information Processing Systems (Neur IPS 2022).

example is a digital content recommendation platform. The platform connects content creators with viewers, while monetizing views through digital advertisement. Key to the business strategy of a ﬁrm operating a digital content recommendation platform is its ability to predict revenue for content that it recommends. Often framed as a supervised learning task, the ﬁrm trains a statistical model on observed data to predict some proxy of revenue, such as clicks, views, or engagement. Better predictions enable the ﬁrm to more accurately identify content of interest and thus increase proﬁt.

A second way of increasing proﬁt is more subtle. The platform can use its predictions to steer participants towards modes of consumption and production that are easier to predict and monetize. For example, the platform could reward consistency in the videos created by content creators, so that the popularity of their videos becomes more predictable. Similarly, the platform could recommend addictive content to viewers, appealing to behavioral weaknesses to drive up viewer engagement. How potent such a strategy is depends on the extent to which the ﬁrm is able to steer participants, which we argue reveals a salient power relationship between the platform and its participants.

1.1 Our contribution

We introduce the notion of performative power that quantiﬁes a ﬁrm s ability to steer a population of participants. We argue that the sensitivity of participant behavior to algorithmic changes in the platform provides an important indicator of the ﬁrm s power. Performative power is a causal statistical notion that directly quantiﬁes how much participants change in response to actions by the platform, such as updating a predictive model. In doing so it avoids market speciﬁcs, such as the number of ﬁrms involved, products, and monetary prices. Neither does it require a competitive equilibrium notion as a reference point. Instead, it focuses on where rubber meets the road: the algorithmic actions of the platform and their causal powers.

We build on recent developments in performative prediction [Perdomo et al., 2020] to articulate the difference between learning and steering in prediction. We show that under low performative power, a ﬁrm cannot do better than standard supervised learning on observed data. Intuitively, this means the ﬁrm optimizes its loss function ex-ante on data it observes. We interpret this optimization strategy as analogous to the ﬁrm being a price-taker, an economic condition that arises under perfect competition in classical market models. We contrast this optimization strategy with a ﬁrm that performs ex-post optimization and takes advantage of performative power to achieve lower expected risk.

To better understand our deﬁnition, we study performative power in the concrete algorithmic market model of strategic classiﬁcation. Strategic classiﬁcation models participants as best-responding agents that change their features rationally in response to a predictor to achieve a better prediction outcome. We study the role of different economic factors by extending the standard model to incorporate competing ﬁrms and outside options. Our ﬁrst observation is that a monopoly ﬁrm can derive signiﬁcant performative power because participants are willing to incur a cost up to the utility of using the service in order to adjust to the ﬁrm s predictor. Moreover, performative power is maximized if the ﬁrm can personalize decisions to individual users. Our second observation is that performative power decreases in the presence of competition and outside options. When ﬁrms compete for participants, offering services that are perfect substitutes for each other, then even two ﬁrms can lead to zero performative power. This result stands in analogy with classical Bertrand competition.

On the empirical side, we propose a causal design to identify performative power in the context of a recommender system arranging content into display slots. This design, we call discrete display design (DDD), establishes a connection between performative power and the causal effect of display position

on consumption. To derive a lower bound on performative power, DDD constructs a hypothetical algorithmic action that aggregates the causal effects of display position across the population. This allows us to repurpose reported causal effects of display position as lower bounds on performative power. It also charts out a concrete empirical strategy for measuring power in digital economies.

Finally, we examine the role of performative power in competition policy. We contrast performative power with traditional measures of market power, describe how performative power can capture complex behavioral patterns, and discuss the potential role of performative power in antitrust debates.

1.2 Related work

Our notion of performative power builds on the development of performativity in prediction by Perdomo et al. [2020]. Performativity captures that the predictor can inﬂuence the data-generating

process, a dependency ruled out by the traditional theory of supervised learning. A growing line of work on performative prediction, e.g., [Mendler-Dünner et al., 2020; Drusvyatskiy and Xiao, 2022; Izzo et al., 2021; Dong and Ratliff, 2021; Miller et al., 2021; Brown et al., 2022; Li and Wai, 2021; Ray et al., 2022; Jagadeesan et al., 2022; Wood et al., 2022], has studied different optimization challenges and solution concepts in performative prediction. Rather than viewing performative effects as a challenge for the learning algorithm, we argue that performativity reveals a salient power relationship between the decision maker and the population. From an optimization perspective, our work demonstrates that sufﬁciently high performative power is necessary for performative optimization approaches to be beneﬁcial compared with standard supervised learning.

The strategic classiﬁcation setup we use for our case study was proposed in [Brückner et al., 2012; Hardt et al., 2016] and is closely related to a line of work in the economics community [Frankel and Kartik, 2022; Ball, 2020; Hennessy and Goodhart, 2020; Frankel and Kartik, 2019]. A line of work on strategic classiﬁcation makes the assumption that performative effects are the result of individuals manipulating their features so as to best respond to the deployment of a predictive model. The focus has been on describing participant behavior in response to a single ﬁrm acting in isolation. Our extensions incorporate additional market factors into the model, such as outside options or the choice between competing ﬁrms, which we believe are helpful for gaining a better understanding of interactions in digital economies. Beyond the case of a single classiﬁer, recently, Narang et al. [2022] and Piliouras and Yu [2022] consider multiple ﬁrms simultaneously applying retraining algorithms in performative environments and analyze convergence to equilibrium solutions. Ginart et al. [2021] study another model of feedback loops arising from competition between machine learning models.

There is extensive literature on the topic of competition on digital platforms that we do not attempt to survey here. For starting points, see, for example, recent work by Bergemann and Bonatti [2022], a survey by Calvano and Polo [2021], a discussion by Parker et al. [2020], the reports already mentioned [Stigler Committee, 2019; Crémer et al., 2019], as well as a macroeconomic perspective on the topic [Syverson, 2019].

2 Performative power

Fix a set U of participants interacting with a designated ﬁrm, where each u 2 U is associated with a data point z(u). Fix a metric dist(z, z0) over the space of data points. Let F denote the set of actions a ﬁrm can take. We think of an action f 2 F as a predictor that the ﬁrm can deploy at a ﬁxed point in time. For each participant u 2 U and action f 2 F, we denote by zf(u) the potential outcome random variable representing the counterfactual data of participant u if the ﬁrm were to take action f. Deﬁnition 1 (Performative Power). Given a population U, an action set F, potential outcome pairs (z(u), zf(u)) for each unit u 2 U and action f 2 F, and a metric dist over the space of data points,

we deﬁne the performative power of the ﬁrm as

E [dist (z(u), zf(u))] ,

where the expectation is over the randomness in the potential outcomes.

The expression inside the supremum generalizes an average treatment effect, corresponding to scalar valued potential outcomes and the absolute value as metric. We could generalize other causal quantities such as heterogeneous treatment effects, but this avenue is not subject of our paper. The deﬁnition takes a supremum over possible actions a ﬁrm can take at a speciﬁc point in time. We can therefore lower bound performative power by estimating the causal effect of any given action f 2 F.

Having speciﬁed the sets F and U, estimating performative power amounts to causal inference involving the potential outcome variables zf(u) for unit u 2 U and action f 2 F. In an observational design, an investigator is able to identify performative power without an experimental intervention on the platform. We propose and apply one such observational design in Section 5. In an experimental design, the investigator deploys a suitably chosen action to estimate the effect. Neither route requires understanding the speciﬁcs of the market in which the ﬁrm operates. It is not even necessary to know the ﬁrm s objective function, how it optimizes its objective, and whether it successfully achieves its objective. In practice, the dynamic process that generates the potential outcome zf(u) may be highly complex, but this complexity does not enter the deﬁnition. Consequently, the deﬁnition applies to complex multisided digital economies that defy mathematical speciﬁcation. To make this abstract concept of performative power more concrete, let us instantiate it in a concrete example.

Example: Digital content recommendation. Consider a digital content recommendation platform, such as the video sharing services You Tube or Twitch. The platform aims to recommend channels that generate high revenue, personalized to each viewer. Towards this goal, the platform collects data to build a predictor f for the value of a channel c to a viewer with preferences p. Let x = (xc, xp) be the features used for the prediction task that capture attributes xc of the channel and the attributes xp of the viewer preferences. Let y be the target variable, such as watch time, that acts as a proxy for the monetary value of showing a channel to a speciﬁc viewer. For concreteness, take the supervised learning loss (f(x), y) incurred by a predictor f to be the squared loss (f(x) y)2.

When deﬁning performative power, participants could either be viewers or content creators. The deﬁnition is ﬂexible and applies to both. By selecting the units U, which features to include in the data point z, and how to specify the distance metric dist, we can pinpoint speciﬁc power relationships.

Content creators. The predictor f can affect the type of videos that content creators stream on their channels. For example, content creators might strategically adjust various features of their content relevant for the predicted outcome, such as the length, type or description of their videos, to improve their ranking. Thus, by changing how it predicts the monetary value of a channel, the platform can induce changes in the content on the channel. To measure this source of power, we let the participants U be content creators and suppose that each content creator u 2 U maintains a channel of videos. Let the data point z(u) correspond to features xc characterizing the channel c created by content creator u. Let dist be a metric over features of content. The resulting instantiation of performative power measures the changes in content induced by potential implementations F of the prediction function. In Section 4, we investigate this form of performative power from a theoretical perspective by building on the setup of strategic classiﬁcation.

Viewers. The predictor f can shape the consumption patterns of viewers, since viewers tend to follow

recommendations when deciding what content to consume (e.g. [Ursu, 2018]). Thus, by changing which content it recommends to a user, the platform can induce changes in the target variable: how much time the users spends watching content on a given channel. Let s suppose that we wish to investigate the effect of the predictor on viewer consumption of a certain genre of content (e.g. radical content). To formalize this source of power, we let the units U be viewers. Let the data point z(u) correspond to how long the viewer u spends watching content in the genre of interest. More formally, let z(u) for a user u with preferences p be equal to the cumulative watch time over pairs (xc, xp) where c is a channel within the genre of interest. Let dist(z, z0) = |z z0| capture the difference in watch time. The resulting instantiation of performative power measures the changes in consumption of a given genre of content induced by a set of prediction functions F. In Section 5, we propose an observational design to identify this quantity.

3 Learning versus steering

Performative power enters the ﬁrm s optimization problem and has direct consequences for their optimization strategies. Instead of identifying the best action f while treating data as ﬁxed, high performative power enables the ﬁrm to steer the population towards data that it prefers. In the following, we elucidate the role of performative power in optimization and study the equilibria attained in an economy of performative predictors.

3.1 Optimization strategies

Let us focus on predictive accuracy as the optimization objective of the ﬁrm. Hence, the goal of the ﬁrm is to choose a predictive model f that suffers small loss (f(x), y) measured over instances (x, y). To make the role of steering explicit we distinguish between the ex-ante loss (f(x(u)), y(u))

and the ex-post loss (f(xf(u)), yf(u)). The former describes the loss that the ﬁrm can optimize when building the predictor. The latter describes the loss that the ﬁrm observes after deploying f. More formally, the ex-post risk that the ﬁrm suffers after deploying f on a population U is given by

(f(xf(u)), yf(u)) . (1)

Expression (1) is an instance of what Perdomo et al. [2020] call performative risk of a predictor. That is the loss a predictor incurs on the distribution over instances it induces. To simplify notation we adopt their conceptual device of a distribution map: D( ) maps a predictive model, characterized by model parameters , to a distribution over data instances.

To express the ﬁrm s optimization problem within the framework of performative prediction, we deﬁne a data instance as z(u) = (x(u), y(u)) for u 2 U so we can capture performativity in the features as well as in the labels. Let the ﬁrm s action correspond to choosing a parameter vector for its predictor f from an action set F = . Then, the aggregate distribution over data D( ) corresponds to the distribution over the potential outcome variable z (u) after the ﬁrm takes action 2 , where the randomness comes from u being uniformly drawn from U as well as randomness in the potential outcomes. The ﬁrm s ex-post risk (1) for deploying f is the performative risk:

PR( ) := E z D( )[ ( ; z)],

where the loss typically corresponds to the mismatch between the predicted label and the true label: ( ; z) = (f (x), y) for z = (x, y).

Observe that arises in two places in the objective: in the distribution D( ) and in the loss ( ; z). Thus, for any choice of model φ, we can decompose the performative risk PR( ) as:

PR( ) = R(φ, ) + (R( , ) R(φ, )) (2) where R(φ, ) := Ez D(φ) ( ; z) denotes the loss of a model on the distribution D(φ). This tautology highlights the difference between learning and steering and we differentiate between the following two optimization approaches:

Ex-ante optimization. Ex-ante optimization focuses on optimizing the ﬁrst term in the decomposition (2). For any φ, the resulting minimizer can be computed statistically and corresponds to SL := arg min 2 R(φ, ) . Let fφ be any previously chosen model, then employing supervised learning on historical data sampled from D(φ) corresponds to what we call ex-ante optimization.

Ex-post optimization. In contrast to ex-ante optimization, ex-post optimization accounts for the impact of the model on the distribution. It trades-off the two terms in (2), and directly optimizes the performative risk PO := arg min 2 PR( ). Solving this problem exactly, and ﬁnding the performative optimum PO requires optimization over the distribution map D( ).

In the context of digital content recommendation, ex-ante optimization corresponds to training on historical data collected by the platform, whereas ex-post optimization selects based on randomized experiments, A/B testing or explicit modeling of D( ). It holds that PR( PO) PR( SL), because in ex-post optimization the ﬁrm can choose to steer the population towards more predictable behavior. Remark (Generalizing to other objectives). We thus far have focused on the ﬁrm s prediction problem when describing performative effects. Nonetheless, the conceptual distinction between learning and steering applies to general optimization objectives. Ex-ante optimization corresponds to optimizing on historical data, whereas ex-post optimization corresponds to optimizing over the counterfactuals.

3.2 Gain of ex-post optimization is bounded by a ﬁrm s performative power

We show that the gain of ex-post optimization over ex-ante optimization can be bounded by the ﬁrm s performative power with respect to the set of actions and the data vector z = (x, y). Intuitively, if the ﬁrm s performative power is low, then the distributions D( ) and D(φ) for any , φ 2 are close to one another. Coupled with a regularity assumption on the loss, this means that the second term in (2) should be small. Thus, using the ex-ante approach of minimizing the ﬁrst term produces a near-optimal ex-post solution, as we demonstrate in the following result: Proposition 1. Let P be the performative power of a ﬁrm with respect to the action set . Let Lz be the Lipschitzness of the loss in z with respect to the metric dist. Let PO be the ex-post solution and SL be the ex-ante solution computed from D(φ) for any past deployment φ 2 . Then, we have that

PR( SL) PR( PO) + 4Lz P.

The gain achievable through ex-post optimization is bounded by performative power. Hence, a ﬁrm with small performative power cannot do much better than ex-ante optimization and might be better off sticking to classical supervised learning practices instead of engaging with ex-post optimization.

3.3 Ex-post optimization in an economy of predictors

The result in Proposition 1 studies the optimization strategy of a single ﬁrm in isolation. In this section, we investigate the interaction between the strategies of multiple ﬁrms that optimize simultaneously over the same population. We consider an idealized marketplace where C ﬁrms

all engage in ex-post optimization and we assume all exogenous factors remain constant. Let DC( 1, . . . , i 1, i, i+1, . . . , C) be the distribution over z(u) induced by each ﬁrm i 2 [C] deploying model f i. Let i denote the loss function chosen by ﬁrm i. We say a set of predictors [f 1, . . . , f C] is a Nash equilibrium if and only if no ﬁrm has an incentive to unilaterally deviate from their predictor using ex-post optimization:

i 2 arg min

E z DC( 1,..., i 1, , i+1,..., C)[ i( ; z)].

First, we show that at the Nash equilibrium, the suboptimality of each predictor f i on the induced distribution depends on the performative power of the respective ﬁrm.

Proposition 2. Suppose that the economy is in a Nash equilibrium ( 1, . . . , C), and ﬁrm i has performative power Pi with respect to the action set . Let Lz be the Lipschitzness of the loss i in z with respect to the metric dist. Then, it holds that Ez D[ i( i; z)] min Ez D[ i( ; z)] + Lz Pi , where D = DC( 1, . . . , C) is the distribution induced at the equilibrium.

Proposition 2 implies that if the performative power of all ﬁrms is small (Pi ! 0 8i), then the equilibrium becomes indistinguishable from that of a static, non-performative economy with distribution D over content. Interestingly, there is an important distinction with the static setting: if the ﬁrms were to collude for example, because of common ownership [Azar et al., 2018] then they would be able to signiﬁcantly shift the distribution. In particular, even if the performative power Pi of any given ﬁrm i is small, the aggregate performative power of a set of ﬁrms S can be much larger.

To illustrate this, we consider a mixture economy, where all of the ﬁrms share a common loss function and performative power is uniformly distributed across ﬁrms (we formalize the construction of a mixture economy in Appendix B.1). We can apply Proposition 2 to analyze the equilibria as C ! 1.

Corollary 1. Suppose that all ﬁrms i 2 [C] share the same loss function i = . Let be a symmetric Nash equilibrium in the mixture economy with C platforms. As C ! 1, it holds that:

E z DC( ,..., )[ ( ; z)] ! min

E z DC=1( )[ ( ; z)].

Corollary 1 demonstrates that a symmetric equilibrium approaches a performatively stable point of the monopoly economy where C = 1 as the number of ﬁrms in the economy grows and the performative power of each individual ﬁrm diminishes. In contrast, if all C ﬁrms collude, they would obtain the performative power of a monopoly platform and instead choose a performatively optimal point of DC=1. A competitive economy can thus exhibit a very different equilibrium from that of the monopoly or collusive economy.

4 Performative power in strategic classiﬁcation

We now turn to a stylized market model and investigate how performative power depends on the economy in which the ﬁrm operates. Speciﬁcally, we use strategic classiﬁcation [Hardt et al., 2016] as a test case for our deﬁnition. In strategic classiﬁcation, participants strategically adapt their features with the goal of achieving a favorable classiﬁcation outcome. Hence, performative power is determined by the degree to which a ﬁrm s classiﬁer can impact participant features. We use this concrete market setting to examine the qualitative behavior of performative power in the presence of competition and outside options.

4.1 Strategic classiﬁcation setup

Let x(u) be the features and y(u) the binary label describing a participant u 2 U. A ﬁrm chooses a binary predictor f : Rm ! {0, 1} and incurs loss (f(x), y) = |f(x) y|. Let Dorig denote the base distribution over features and labels (xorig(u), yorig(u)) absent any strategic adaptation, which we assume is continuous and supported everywhere. Let D(f) be the distribution over potential outcomes (xf(u), yf(u)) that arises from the response of participant u to the deployment of a model f. We

assume that participant u incurs a cost c(xorig(u), x0) for changing their features to x0. In line with the standard setup, the cost for feature changes is measured relative to the original features. We assume c is a metric and any feature change that deviates from the original features results in nonnegative cost for participants. Further, we assume the label does not change, i.e., yf(u) = yorig(u).

Since performative effects surface as changes in participant features, we measure performative power over z(u) = x(u). In Appendix B.2, we describe how this instantiation of performative power can be interpreted in the context of content recommendation platforms and hiring platforms.

4.2 Performative power in the monopoly setting

Consider an economy with a single ﬁrm that offers utility γ > 0 to its participants for a positive prediction. Participants want to use the service regardless of what classiﬁer the ﬁrm chooses. In addition, assume there is an outside options at utility level β > 0 that does not demand any extra effort from participants. This decreases the budget they are willing to invest to their surplus utility γ = max(0, γ β). We adopt the following standard rationality assumption on participant behavior.

Assumption 1 (Participant Behavior Speciﬁcation). Let γ 0 be the surplus utility that a participant can expect from a positive classiﬁcation outcome from classiﬁer f over any outside option. Then, a participant u 2 U with original features xorig(u) will change their features according to

xf(u) = arg maxx0 ( γf(x0) c(xorig(u), x0)).

As a result of Assumption 1, a participant will change their features if and only if the cost of a feature change is no larger than γ. Furthermore, if participants change their features, then they will expend the minimal cost required to achieve a positive outcome. This speciﬁcation of participant behavior allows us to bound the ﬁrm s performative power in terms of the cost function c and the distance function dist. Namely, the set of potential values xf(u) can take on is restricted to X γ(u) := {x : c(xorig(u), x) γ}. Thus, the effect of a change in the decision rule on participant u can be upper bounded by the distance between x(u) and the most distant point in X γ. Aggregating these unilateral effects yields a bound on performative power:

Lemma 1. The performative power P of the ﬁrm with respect to any set of predictors F and a population U can be upper bounded as

sup x02X γ(u)

dist(x(u), x0). (3)

Tightness of Lemma 1. If the ﬁrm action space F is restricted to a parametric family of predictors, the upper bound in Lemma 1 need not be tight. A typical decision rule, such as a linear threshold classiﬁer, does not impact all participants u 2 U equally. More speciﬁcally, the amount of change that the ﬁrm can induce with a decision rule f on an individual u depends the relative position of their features xorig(u) to the decision boundary. (In Appendix B.3 we provide a precise bound for threshold classiﬁers).

Interestingly, the ability to personalize decisions to each user maximizes performative power.

Proposition 3. Consider a population U of users. Let F be the set of all functions from Rm to {0, 1}. Suppose that the ﬁrst coordinate of x(u) is immutable, and uniquely identiﬁes user u 2 U: that is, xorig(u)[1] = xf(u)[1] = i for all f 2 F. Then, the performative power of the ﬁrm matches the upper bound in (3).

The intuition behind Proposition 3 is that personalization enables the ﬁrm to simultaneously extract the maximum utility from each individual participant.

Role of surplus utility γ. Let us now investigate the role of γ in the upper bound (3). Recall that user behavior is determined by the cost c of changing features relative to xorig(u), performative power is measured as the distance from the current state x(u) with respect to dist. The Lipschitz constant L := supx,x0 dist(x,x0)

c(x,x0) allows us to relate the two metrics and derive a simpler upper bound:

Corollary 2. The performative power of a ﬁrm in the monopoly setup can be bounded as P 2L γ, where γ measures the surplus utility offered by the service of the ﬁrm over outside options.

Corollary 2 makes explicit that γ > 0 is a prerequisite for a ﬁrm to have any performative power. This qualitative behavior is in line with common intuition in economics that monopoly power relies on the ﬁrm offering a service that is superior to existing options (i.e. γ > β).

4.3 Firms competing for participants

We next consider a model of competition between two ﬁrms where participants always choose the ﬁrm that offers higher utility. We demonstrate how the presence of competition reduces the performative power of a ﬁrm. In particular, we will show that for a natural constraint on the ﬁrm s action set, each ﬁrm s performative power can drop to zero at equilibrium, regardless of how much utility participants derive from the ﬁrm s service.

To model competition in strategic classiﬁcation, we specify participant behavior as follows: Given that the ﬁrst ﬁrm deploys f1 and the second ﬁrm deploys f2, then participant u will choose f1 if maxx0 (f1(x0) c(xorig(u), x0)) > maxx0 (f2(x0) c(xorig(u), x0)), and choose f2 otherwise. We assume that a participant tie-breaks in favor of the lower threshold, randomizing if the two thresholds are equal. After choosing ﬁrm i 2 {1, 2}, they change their features according to Assumption 1 as xf(u) = arg maxx0 (γfi(x0) c(xorig(u), x0)), where γ is the utility of a positive outcome.

We assume that the ﬁrm chooses their classiﬁer based on the following utility function. For a rejected participant, the ﬁrm receives utility 0 and for an accepted participant, the ﬁrm receives utility > 0 if they have a positive label and utility if they have a negative label. We assume that the ﬁrm s action set is constrained to models for which it derives non-negative utility. More speciﬁcally, if f denotes the model deployed by the competing ﬁrm, let the action set F+( ) of this ﬁrm denote the set of models that yield non-negative utility for the ﬁrm.

For simplicity, focus on a 1-dimensional setup where F is the set of threshold functions. We assume that the cost function c(x, x0) is continuous in both of its arguments, strictly increasing in x0 for x0 > x, strictly decreasing for x0 < x, and satisﬁes limx0!1 c(x, x0) = 1. Furthermore, we assume that the posterior p(x) = PDorig[Y = 1 | X = x] is strictly increasing in x with limx! 1 p(x) = 0, and limx!1 p(x) = 1.

We show that the presence of competition can drive the performative power of each ﬁrm to zero.

Proposition 4. Consider the 1-dimensional setup with two competing ﬁrms speciﬁed above. Suppose that the economy is at a symmetric Nash equilibrium ( , ). If L < 1, then the performative power of either ﬁrm with respect to the action set F+( ) is P = 0.

The intuition behind Proposition 4 is that performative power of a ﬁrm purely arises from how much larger the current threshold is than the minimum threshold a ﬁrm can deploy within their action set F+( ). At the Nash equilibrium (where both ﬁrms best-respond with respect to their utility functions taking their own performative effects into account), the ﬁrms deploy exactly the minimum threshold within their action set. The formal proof of the result can be found in Appendix C.9.

Corollary 4 bears an intriguing resemblance to well-known results on market power under Bertrand competition in economics (see e.g., [Baye and Kovenock, 2008]) that show how a state of zero power is reached with only two competing ﬁrms.

5 Discrete display design

Now that we have examined the theoretical properties of performative power, we turn to the question of measuring performative power from observational data. We focus on our running example of digital content recommendation and we propose an observational design to measure the recommender system s ability to shape consumption patterns.

5.1 The causal effect of position

We assume that there are C pieces of content C = {0, 1, 2, . . . , C 1} that the platform can present in m display slots. We make the convention that item 0 corresponds to leaving the display slot empty. We focus on the case of two display slots (m = 2) since it already encapsulates the main idea. The ﬁrst display slot is more desirable as it is more likely to catch the viewer s attention. For example, the ﬁrst display slot could be the ﬁrst ad slot on a page of search results. Researchers have studied the causal effect of position on consumption, often via quasi-experimental methods such as regression discontinuity designs, but also through experimentation in the form of A/B tests.

Deﬁnition 2 (Causal effect of position). Let the treatment T 2 {0, 1} be the action of ﬂipping the content in the ﬁrst and second display slots for a viewer u, and let the potential outcome variable Yt(u)

indicate whether, under the treatment T = t, viewer u consumes the content that is initially in the ﬁrst display slot. We call the corresponding average treatment effect

""E [Y1 Y0]

the causal effect of position, where the expectation is taken over the population of viewers and the randomness in the potential outcomes.

For example, Narayanan and Kalyanam [2015] estimate the causal effect of position in search advertising, where advertisements are displayed across a number of ordered slots whenever a keyword is searched. They measured the causal effect of position on click-through rate of participants.

5.2 From causal effect of position to performative power

The identiﬁcation strategy we propose, called discrete display design (DDD), derives a lowerbound on performative power by repurposing existing measures of the causal effect of position. Note that we focus on content recommendation in this section, the design however can be generalized to other settings where the ﬁrm s action corresponds to a discrete decision of how to display content. Setting up the DDD involves two steps: First, we need to instantiate the deﬁnition of performative power with a suitable action set so that the ﬁrm s actions result in swapping the position of content items, and second, we plug in the causal effect of position to lower bound performative power.

While the ﬁrst step is mostly a technical exercise, the second step relies on a crucial assumption. In particular, it involves relating the unilateral causal effect of position to performative power that quantiﬁes the effect of an action on the entire population of viewers. Thus, for being able to extrapolate the effect from a single viewer to the population DDD relies on a non-interference assumption. In the advertising example, this means that the ads shown to one viewer do not inﬂuence the consumption behavior of another viewer. Let us investigate the two steps in greater detail:

Step 1: Instantiating performative power. Let the units U be the set of viewers. For each viewer u 2 U let z(u) 2 RC be the distribution over content items C consumed by viewer u, represented as a histogram. More formally, let z(u) be a vector in the C-dimensional probability simplex where the ith coordinate is the probability that viewer u consumes content item i. The metric dist(z, z0) is the 1-distance dist(z, z0) = PC 1

i=0 |z[i] z0[i]|.

The decision space F of the ﬁrm corresponds to its decisions of how to arrange content in the m = 2 display slots. It is natural to decompose this decision into a continuous score function s followed by a discrete conversion function that maps scores into an allocation. The score function s: U ! RC maps the viewer to a vector of scores, where each coordinate is an estimate of the quality of the match between the viewer and the corresponding piece of content. The conversion function : RC ! C2 takes as input the vector of scores and outputs an ordered list of items with the top 2 scores. We assume the platform displays these 2 items in order and the conversion function is ﬁxed. Hence, we identify the ﬁrm s action space with the set of feasible score functions S U ! RC.

To deﬁne the reference state z(u), we think of scurr as being the score function currently deployed by the platform. Let δ be the maximum difference in the highest score and second highest score for any user under scurr. Consider the set S of scoring functions deﬁned as

s: U ! RC | 8u 2 U : ks(u) scurr(u)k1 δ

Notably, there exists an sswap 2 S that is capable of swapping the order of the ﬁrst and second highest scoring item under scurr for any user u 2 U simultaneously. We denote the counterfactual variable corresponding to a score function s 2 S as zs(u). Given this speciﬁcation, performative power with respect to the action set S can be bounded by the causal effect of sswap as follow

kzscurr(u) zs(u)k1 1

kzscurr(u) zsswap(u)k1. (4)

Step 2: Lower bounding performative power. To relate the lower bound on performative power from (4) to the causal effect of position, let itop(u) = scurr(u)[1] denote the coordinate of the item displayed to user u in the ﬁrst display slot under scurr. Then, we can lower bound each term in the sum (4) as kzscurr(u) zsswap(u)k1 |zscurr(u)[itop(u)] zsswap(u)[itop(u)]|. Now to enable

us to study the effect of changing scurr to sswap independently for each user we place the following non-interference assumption on the counterfactual variables which closely relates to the stable unit treatment value assumption (SUTVA) [Imbens and Rubin, 2015] prevalent in causal inference. Assumption 2 (No interference across units). For any u 2 U and any pair of scoring functions s1, s2 2 S, if (s1(u)) = (s2(u)), it also holds that zs1(u) = zs2(u).

The assumption requires that there are no spill-over or peer effects and the content a viewer consumes only depends on the content recommended to them and not the content recommended to other viewers. The effect of a unilateral change to the consumption of item itop(u) under sswap exactly corresponds to what we deﬁned as the causal effect of position. Aggregating these unilateral causal effects across all viewers in the population we obtain a lowerbound on performative power. The proof of Theorem 1 can be found in Appendix C.12. Theorem 1. Let P be performative power as instantiated above. If Assumption 2 holds, then performative power is at least as large as the causal effect of position

Let us return to the search advertisement marketplace study of Narayanan and Kalyanam [2015] to demonstrate how we can leverage Theorem 1 to relate the ﬁndings of their observational causal design to performative power. In particular, they examine position effects in search advertising, where ads are displayed across a number of ordered slots whenever a keyword is searched. They found that the effect of showing an ad in display slot 1 versus display slot 2 corresponds to 0.0048 clicks per impression (see Table 2 in their paper). By treating each incoming keyword query as a distinct viewer u, this number exactly corresponds to what we deﬁned as the causal effect of position. Thus,

we can apply Theorem 1 to get P 0.0048. Putting this into context; the mean click-through rate in display slot 2 is 0.023260. Hence, the lower bound 0.0048 is a 21% percent increase relative to the baseline. The ﬁrm thus has a substantial ability to shape what advertisements users click on.

6 Discussion

We discuss the potential role of performative power in competition policy and antitrust enforcement.

The complexity of digital marketplaces has made it necessary to develop new approaches for evaluating and regulating these economies. One challenge is that traditional measures of market power such as the Lerner Index [Lerner, 1934], or the Herﬁndahl Hirschman Index (HHI) are based on classical markets for homogeneous goods, but these markets map poorly to digital economies. In particular, these measures struggle to appropriately capture the multi-sided nature of digital economies and to account for the role of behavioral weaknesses of consumers such as tendencies for single-homing, vulnerability to addiction, and the impact of framing and nudging on participant behavior [e.g. Thaler and Sunstein, 2008; Fogg, 2002]. We further expand on this discussion in Appendix A.

By focusing on observable statistics, performative power could be particularly helpful in markets that resist a clean speciﬁcation. Performative power is sensitive to the market nuances without explicitly modeling them. For example, suppose that as a result of uncertainty about market boundaries, a regulator failed to account for a competitor in a marketplace. Performative power would still implicitly capture the impact of the competitor and indicate the market is more competitive than suspected.

We leave open the question of how to best instantiate performative power in a given marketplace. Conceptually, we view performative power as a tool to ﬂag market situations that merit further investigation, since it corresponds to potential for harm to users . However, if a regulator wishes to draw ﬁne-grained conclusions about consumer harm, it is crucial to appropriately instantiate the choice of action set F, the deﬁnition of a population U, and the speciﬁcation of the features z. As an example, we show in Appendix B.4 how to closely relate performative power into consumer harm for strategic classiﬁcation. In general, however, harm and power are two distinct normative concepts, and going from performative power to consumer harm thus requires additional substantive arguments.

Acknowledgments

We re grateful to Guy Aridor, Dirk Bergemann, Emilio Calvano, Gabriele Carovano, and Martin C. Schmalz for helpful feedback and pointers.

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