Performative Power
Authors: Moritz Hardt
Meena Jagadeesan
Celestine Mendler-Dünner
Venue: NeurIPS 2022 Accept (2022)
Keywords: markets
competition
power
performativity
prediction

Abstract: 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.
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.

Performative Power
Moritz Hardt
Meena Jagadeesan"
Max-Planck Institute for Intelligent Systems, Tübingen hardt@is www. de
UC Berkeley
m mjagadeesan@brrkkleyyy. edu
Celestine Mendler-Dünner"
Max-Planck Institute for Intelligent Systems, Tübingen cmendler@ttuebingen. mmp..dppp.......
Abstract
We introduce the notion of performative power, which measures the ability of firm operating an algorithmic system, such as a digital content recommendation platform, to cause change in population of participants. We relate performative power the economic study of competition in digital economies. Traditional economic concepts struggle with identifying anti-competttive 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 performa- tive 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 benefit from steering the population towards more profitable behavior. We confirm in simple theoretical model that monopolies maximize performative power. firm's '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 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.
Introduction
Digital platforms pose well-recognized challenge for antitrust enforcement. Traditional market definitions, along with associated notions 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 analysis of market definition, and more emphusis on theories of harm and identification of anti-competiiive strategies." Crémer et al. 2019
Our work responds to this call by developing normative and technical proposal for reasoning about power in digital economies, while relaxing the reliance on market definition. Our running
Authors listed in alphahetical order.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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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 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 'rrms power. Performative power is causal statistical notion that directly quantifies how much participants change in response to actions by the platform, such updating a predictive model. In doing it avoids market specifics, such as the number of firms involved, products, and monetary prices. Neither does it require a competitive equilibrium notion as 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 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 price-taker
economic condition that arises under perfect competition in classical market models. We contrast this optimization strategy with a firm that performs ex- 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- rest-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 's predictor. Morcover, 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 perfect substitutes for each other, then even two firms can lead to 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 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
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process, a dependency ruled out by the traditional theory of supervised leamnin. A growing line of work on performative prediction, e.g.. Mendler-Dünner al. 2020) Drusvyatskiy and Xiao
2022
1770 et al. 2021 Dong and Ratlifi 2021; Miller et al. 2021 Brown et al. 2022 Li and Wai
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.
2021
The strategic classification setup we use for our case study was proposed in [Brückner 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 Goodharl 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 predictive model. The focus has been on describing participant behavior in response
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 simultancously applying retraining algorithms in performative environments and analyze convergence to equilibrium solutions. Ginartt et al. [2021]
study another model of feedback loops arising from competition between machine learning models.
There 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 [02T]] a discussion by Parker et al. [2020), the reports already
mentioned (Stigler Committee
et al. 2019 as well as a macroeconomic perspective
on the topic [Syverson 2019
2 Performative power
Fix set 2 of participants interacting with designated firm, where each u U is associated with data point zu)). Fix a metric dist(2, over the space of data points. Let F denote the set of actions a firm can take. We think of an action f E F as a predictor that the firm can deploy at a fixed point in time. For each participant u E ll and action f E F, we denote by 2 (u) the potential outcome random variable representing the counterfactual data of participant if the firm were to take action f. Definition (Performative Power). Given a population I, an action set F, potential outcome pairs (z(u),z((u) for each unit E U and action E F, and a metric dist over the space of data points, we define the performative power of the firm as
P : sup
Σ
E (dist (z(u),
JEF
weu
where the expectation 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 supremum over possible actions firm can take at specific point in time. We can therefore lower bound performative power by estimating the causal effect of any given action f E F.
Having specified the sets F and U4 estimating performative power amounts to causal inference involving the potential outcome variables zy(u) for unit e U and action f E 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 3 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 '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 zy(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.
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Example: Digital content recommendation. Consider a digital content recommendation platform, such as the video sharing services YouTube or Twitch. The platform aims recommend channels that generate high revenue, personalized to each viewer. Towards this goal, the platform collects data to build predictor f for the value of a channel c to a viewer with preferences p. Let x = (Iee Fp) be the features used for the prediction task that capture attributes Te of the channel and the attributes dp 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 channel to specific viewer. For concreteness, take the supervised learning loss {(f())) y) incurred by a predictor f to be the squared loss (f(z)- - y))
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 include in the data point 2, 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 UI be content creators and suppose that each content creator u. E U maintains a channel of videos. Let the data point =(u) correspond features ze characterizing the channel created by content creator u. Let dist be metric over features of content. The resulting instantiation of performative power measures the changes in content induced by potential implementations of the prediction function. In Section 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) D. Thus, by changing which content it recommends to user, the platform can induce changes in the target variable: how much time the users spends watching content on 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 zuu) 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 (Ney Tip)) where e is a channel within the genre of interest. Let dist(2, z') 2 - 2'| capture the difference in watch time. The resulting instantiation of performative power measures the changes in consumption
of 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 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 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 choose predictive model f that suffers small loss {(f((;), y) measured over instances (z. y). To make the role of steering explicit we distinguish between the ex-ante loss (=f()))) yu)) and the ex-post loss E( (ff(()))) y (y))). 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 population 24 is given by
Σ BEWI
(()(
(I)
U
Expression 11 is an instance of what Perdomo et al. 2020 call performative risk 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(0) maps a predictive model, characterized by model parameters 0, to distribution over data instances.

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To express the firm's optimization problem within the framework of performative prediction, we define a data instance as z(u) (z(w), y(u)) for U so we can capture performativity in the features as well as in the labels. Let the firm's s action correspond to choosing parameter vector 0 for its predictor so from an action set F = O. Then, the aggregate distribution over data ((0) corresponds to the distribution over the potential outcome variable =g =œ(u) after the firm takes action 0 E θ, where the randomness comes from u being uniformly drawn from as well as randomness in the potential outcomes. The firm's ex-post risk ① for deploying fo is the performative risk:
(R(() = E [e(0; 2)].
z~D(1))
where the loss typically corresponds to the mismatch between the predicted label and the true label: E(0; 2) = f(fo ()) for
(z, y).
Observe that arises in two places in the objective: in the distribution D and in the loss E(O; 2). Thus, for any choice of model o, we can decompose the performative risk (f( as:
PR(0) R(Ø, 0) + (R(0, 0) R(ő,0)
(2)
where R( 0) := E2-D(a) ((0: 2) denotes the loss of a model 0 on the distribution D()). 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 decom- position 2 For any Φ. the resulting minimizer can be computed statistically and corresponds to 0SL. == arg minøge R(Ø, 0) Let fo Let
be any previously chosen model, then employing supervised learning on historical data sampled from ((6) 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. trades-off the two terms in 2 and directly optimizes the performative risk 8po arg minøee PR(Ø). Solving this problem exactly, and finding the performative optimum 8po requires optimization over the distribution map (()).
In the context of digital content recommendation, ex-ante optimization corresponds to training 0 on historical data collected by the platform, whereas ex ex-post optimization selects 0 based on randomized experiments, A/B testing or explicit modeling of (()). It holds that PR(Oro) S PR((s..). because ex- 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 '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 firm's performative power
We show that the gain of ex-post optimization over ex-ante optimization can be bounded by the firm's 's performative power with respect to the set of actions and the data vector 2 (x, y). Intuitively. if the firm's 's performative power is low, then the distributions D(0) and D(O) for any 0, @ E O 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 near-ope opttmoltimal ex sosp 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 O. Let La be the Lipschitzness of the loss in z with respect to the metric dist. Let OPO the ex-post solution and Ost. be the ex-ante solution computed from D (b) for any past deployment à E O. Then, we have that
PR(OsL) S PR((po) + AL.P.
The gain achievable through ex-post optimization is bounded by performative power Hence, 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 I studies the optimization strategy of single firm in isolation. In this section, we investigate the interaction between the strategies of multiple firms that optimize si- multaneously over the same population. We consider an idealized marketplace where C' firms
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all engage in ex post optimization and we assume all exogenous factors remain constant. Let DC ..... gi-11 0², fff 8C) be the distribution over 2(u) induced by each firm i [C] de- ploying model fe. Let & denote the loss function chosen by firm i. We say a set of predictors (for...., [oo] Nash equilibrium if and only if no firm has an incentive to unilaterally deviate from their predictor using ex ex-post optimization:
O' E arg min le
E 1,00 ,0.,,00)
(c;(0; =)]-
First, we show that at the Nash equilibrium, the suboptimality of each predictor for on the induced distribution depends on the performative power of the respective firm.
Proposition 2. Suppose that the economy is in a Nash equilibrium (0)..... (0'....,00) and firm has performative power P, with respect to the action set O. Let L, be the Lipschitzness of the loss & in with respect to the metric dist. Then, it holds that Ez>D|e,(0% 2)] S ming BenD(e 2) L.Pi. where D DD (0'..... (o.....,0)) 00) is the distribution induced at the equilibrium.
Proposition 2 iimplies that if the performative power of all firms is small (P; + 0 Vi), then the equi- librium becomes indistinguishable from that of a static, non non-performative economy with distribution D over content. Interestingly. there is an important distinction with the static setting: if the firms were collude-foe for example, because common ownership [Azar et al. 2018) -then they would be able to significantly shift the distribution. In particular, even if the performative power P, of any given firm i small, the aggregate performative power of 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 l and performative power is uniformly distributed across firms (we formalize the construction of mixture economy in Appendix B.T.. We can apply Proposition 2 to analyze the equilibria as C + co.
Corollary 1. Suppose that all firms i E [C] share the same loss function & = f. Let 6 be symmetric Nash equilibrium in the mixture economy with C platforms. As C oc, it holds that:
E DD 9")
[e(0"; 2)] + min
DC-1 (8*) [8(0; 2)].
E
Corollary [ demonstrates that symmetric equilibrium approaches a performatively stable point of the monopoly economy where C C= 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 monopoly platform and instead choose a performatively optimal point of DD=11 A competitive economy can thus exhibit very different equilibrium from that of the monopoly or collusive economy.
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 '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 æ(u) be the features and y(u) the binary label describing a participant E U. A firm chooses a binary predictor f T {0, 1} and incurs loss (f(x),, 3/) [f() w/. Let Dorig denote the base distribution over features and labels (Fong(u), Yonig (u)) absent any strategic adaptation, which we assume continuous and supported everywhere. Let D(f) be the distribution over potential outcomes (1)(1), y (u)) that arises from the response of participant to the deployment of model f. We assume that participant u incurs a cost c(Fargg( (u), x²) for changing their features to x'. line with the standard setup, the cost for feature changes is measured relative to the original features. We assume a metric and any feature change that deviates from the original features results in nonnegative cost for participants. Further, assume the label does not change, i.c.. (y(u) Yoorg Vorig(u).

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Since performative effects surface changes in participant features, we measure performative power over 2 = r(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 7 > to its participants for positive prediction. Participants want to the service regardless of what classifier the firm chooses. In addition, assume there is an outside options at utility level / > that does not demand any extra effort from participants. This decreases the budget they willing invest to their surplus utility A = max (0, B). We adopt the following standard rationality assumption on participant behavior.
Assumption 1 (Participant Behavior Specification). Let Br 2 0 be the surplus utility that participant can expect from a positive classification outcome from classifier f over any outside option. Then, participant U with original features Torig (u) will change their features according to y (2) = arg max, (A'f(x') c(Torig (u), x²)).
As a result of Assumption M a participant will change their features if and only if the cost of feature change is no larger than Ay. Furthermore, if participants change their features, then they will expend the minimal cost required to achieve positíve outcome. This specification of participant behavior allows us to bound the firm's 's performative power terms of the cost function e and the distance function dist. Namely, the set of potential values zy z can take on is restricted to (u)((u) : {z e(Forig ()), X) Ar). Thus, the effect of change in the decision rule on participant can be upper bounded by the distance between æ(u) and the most distant point in X'y Aggregating these unilateral effects yields bound on performative power:
Lemma 1. The performative power P of the firm with respect to any set of predictors F and population U can be upper bounded as
P
sup dist(r(u),z').
wEll ('EXar(u)
Tightness of Lemma I If the firm action space F is restricted a parametric family of predictors, the upper bound in Lemma need not be tight. A typical decision rule, such as linear threshold classifier. does impact all participants that the firm can induce with a decision rule f on an individual u depends the relative position of their features Forig (u) to the decision boundary. (In Appendix B.3 we provide a precise bound for threshold classifiers).
24 equally. More specifically, the amount of change
Interestingly, the ability to personalize decisions to each user maximizes performative power.
Proposition 3. Consider a population Ut of users. Let F be the set of all functions from Ih" to {0, 1}. Suppose that the first coordinate of 2z æ²u) is immutable, and uniquely identifies user U: that is, [1]] = xy (u)[11] = for all F. Then, the performative power of the firm matches the upper bound in
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 Ay. Let us now investigate the role of A the upper bound 3 Recall that user behavior is determined by the cost of changing features relative to Torig Zorig(u), performative power measured as the distance from the current state x(u) with respect to dist. The Lipschitz dist(z.z") allows us relate the two metrics and derive simpler upper bound: constant L sup, o(2,z')
Corollary The performative power of a firm in the monopoly setup can be bounded P s 2L Ar, where A measures the surplus utility offered by the service of the firm over outside options.
Corollary 2 makes explicit that 2y > 0 is prerequisite for firm to have any performative power. This qualitative behavior is line with common intuition economics that monopoly power relies on the firm offering service that is superior to existing options (i.c. 7 > 3).
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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 natural constraint on the firm's action set, each firm's performative power can drop to zero equilibrium, regardless of how much utility participants derive from the firm's 's service.
To model competition in strategic classification, specify participant behavior as follows: Given that the first firm deploys fi and the second firm deploys f2, then participant u will choose fi max,' (x²) cr (u), x²)) > maxg (f2(z') - o(Fong (u), z²)), and choose f2 otherwise. We assume that a participant tie- breaks in favor of the lower threshold, randomizing the two thresholds equal. After choosing firm E (1,2), they change their features according Assumption
| aas
y (u) arg m (='f((z))= - c(Torig (u) x²)), where 7 is the utility of a positive outcome.
We assume that the firm chooses their classifier based the following utility function. For a rejected participant, the firm receives utility 0 and for an accepted participant, the firm receives utility a > 0 if they have positive label and utility -a if they have negative label. We assume that the firm's s action set is constrained to models for which it derives non-negative utility. More specifically, if fo denotes the model deployed by the competing firm, let the action set F+ (0) of this firm denote the set of models that yield non- negative utility for the firm.
For simplicity, focus on 1-dimensional setup where F is the set of threshold functions. We assume that the cost function c(x,,²) continuous in both of its arguments, strictly increasing x' for x' > x, strictly decreasing for x' <, and satisfies lim c(z, x') co. Furthermore, we assume
that the posterior p(r) = P [Y- =1 1 X = x] is strictly increasing in x with 1im p(x) = 0, and lim p(x)
We show that the presence of competition can drive the performative power of each firm to zero
Proposition 4. Consider the I-dimensional setup with two competing firms specified above. Suppose that the economy is a symmetric Nash equilibriam (0" )²). If L < co, then the performative power of either firm with respect to the action set F+ (0*) is P 0.
The intuition behind Proposition is that performative power of firm purely arises from how much larger the current threshold 0 is than the minimum threshold firm can deploy within their action set (0). 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 g bears an intriguing resemblance to well-known results on market power under Bertrand competition 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 propose an observational design to measure the recommender system's 's ability to shape consumption patterns.
5.1 The causal effect of position
We assume that there are C pieces of content C = 1,, C- 1} that the platform can present in m display slots. We make the convention that item O 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 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 E (0. 1) be the action of flipping the content in the first and second display slots for viewer u, and let the potential outcome variable (u)
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indicate whether, under the treatment T = t. viewer consumes the content that is initially in the first display slot. We call the corresponding average treatment effect
- E [Y Yo]
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 number of ordered slots whenever a keyword 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 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 's action corresponds to discrete decision 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 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 mostly technical exercise, the second step relies on 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 extrapolate the effect from single viewer to the population DDD relies on a inteerfeeeeeece 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 I: Instantiating performative power. Let the units 2/ be the set of viewers. For each viewer e U let 2(u) IR be the distribution over content items C consumed by viewer u, represented as a histogram. More formally, let 2u) be vector in the C-dimensional probability simplex where the ith coordinate is the probability that viewer u consumes content item The metric dist(z, the
& diditttecee dist(2, (2, E |
2/60]--
The decision space F of the firm corresponds to its decisions of how to arrange content in the m 2 display slots. is natural to decompose this decision into continuous score function 8 followed by discrete conversion function that maps scores into an allocation. The score function 8: U + RC maps the viewer to vector of scores, where each coordinate an estimate of the quality of the match between the viewer and the corresponding piece of content. The conversion function k: 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 's action space with the set of feasible score functions S C u RC.
To define the reference state 2(u), we think of Scur as being the score function currently deployed by the platform. Let 5 be the maximum difference in the highest score and second highest score for any user under Scur- Consider the set S of scoring functions defined
S : f= RC Vu E W: [|s(u) (uur( S 55.
Notably, there exists Sswap S that is capable of swapping the order of the first and second highest scoring item under Seurr for any user u E l simultaneously. We denote the counterfactual variable corresponding to score function E 8 as (u). Given this specification, performative power with respect to the action set S can be bounded by the causal effect of Sawap as follow
P sup SES
1
Σ NEW
I (u) 2, (u)ll2 Z
| (u)
2ss (w)l12-
U
wEW
Step Lower bounding performative power. To relate the lower bound on performative power from to the causal effect of position, let (u) = K Scur (u)[1] denote the coordinate of the
item displayed to user the first display slot under Seur- Then, we can lower bound each term
in the sum 4 as ||Z (u) Zzawop (u) /eee (2) (eop Zesmap (w) (u)]l. Now to enable

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us to study the effect of changing Scurr to Syaap 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 E l and any pair of scoring functions $1:82 82 S, if (f(((((())))))))())))))())))(( (u)) = (()2(u)), it also holds that Esi (u) Feg (i).
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 other viewers. The effect of unilateral change to the consumption of item (u) under Sywap 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 can be found in Appendix C.12
Theorem Let P be performative power as instantiated above. If Assumption holds, then performative power is at least as large as the causal effect of position
P P2B.
Let return to the search advertisement marketplace study of Narayanan and Kalyanam 2015 to demonstrate how we can leverage Theorem It 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 keyword is searched. They found that the effect of showing an ad in display slot 1 versus display slot corresponds to 0.0048 clicks per impression (see Table in their paper). By treating each incoming keyword query as a distinct "viewer" u. this number exactly corresponds what we defined as the causal effect of position. Thus, we can apply Theorem I to get / 2 0.0048. Putting this into context; the mean click-throug! rate in display slot 2 0.023260. Hence, the lower bound 0.0048 is 21% percent increase relative the baseline. The firm thus has a substantial ability to shape what advertisements users click on.
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 evaluat- ing and regulating these economies. One challenge is that traditional measures market power- such as the Lerner Index [Lerner] 1934], the Heerindahh-HHirshhmal Index (HHl) based on classical markets for homogeneous goods, but these markets map poorly to digital economies. In particular, these measures struggle 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 addiction, and the impact of framing and nudging 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, 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 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 regulator wishes to draw fine- ine-grained conclusions about consumer harm, it is crucial to appropriately instantiate the choice of action set F, the definition of a population 24, and the specification of the features 2. 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.
10

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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13


Reviews:
[{"review_id": "zxRX6UQnAcn", "reviewer": "Reviewer_hRaw", "summary": "The paper proposes and formalizes a novel framing to understand a decision maker’s ability to affect the behavior of the population it is making decisions on. The paper relates this notion of “performative power” to classical notions in the economics literature: they draw parallels with the notions of “market power”, of being a price-taker vs a price-maker or a monopoly, and with Bertrand competitions in competitive settings.", "questions": "- Can the authors provide more examples of standard economic/learning settings with strategic agents and relate the performative power there to the actual ability to steer the population in a beneficial way? The performative power is defined to be best-case and it would be useful to understand practically when this best case is realistic.  ", "limitations": "The limitations of the paper seem discussed. In particular, the second paragraph of Section 5 takes potential harms into account, and echoes some of the weaknesses discussed in this review (performative power measures *potential* harm or benefits from affecting population behavior, not necessarily *realized* harms/benefits.)", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- The main strength of the paper comes in the framing. Using a statistical measure like performative power abstracts away from market specifics. It is useful in the sense that modelling the specifics of a given problem often gives rise to complex problems with intractable equilibria, versus performative power can be measured from simply looking at the outcomes of said markets. \n- The notion of performative power is a measure of/proxy for a firm’s ability to affect its users. By extension, it provides some useful understanding of when it is vs is not possible to benefit from affecting the target population’s behavior.\n- The parallels with economics are nice and useful to understand the value of the framework. I think the monopoly and the Bertrand competition are especially nice settings where the notion of performative power is “tight”: competition implies 0 performative power, meaning the firms have no influence on the populations and might as well solve the ex-ante problem, now equivalent to the ex-post one; for monopolies, the performative power is closely bounded by the surplus utility.\n\n\nWeaknesses:\n- Overall, the main theorem remains fairly weak. Theorem 1 seems almost immediate given the Lipschitzness assumption, and it only gives an upper bound on the difference between the performances of \\theta_PO and \\theta_SL. In turn, it says that with low performative power, the performance of \\theta_PO is unlikely to me much better than \\theta_SL, given that we have little power to steer the population hence the ex-ante and ex-post optimization do not differ by much anyway. It does not however say that high performative power guarantees significant differences in performance between \\theta_PO and \\theta_SL. \n- Related to the point above: bcause the notion of performative power looks at the supremum over possible actions/decision rules f deployed by the firm, higher performative power seems to only guarantee that the population can be steered in some, not all cases. In turn, it is not clear that performative power means that you can steer the population *in a way that is beneficial to the firm*. It would be nice to have more insights on situations in which perfomative power is “tight” in that sense. \n\nOverall, I think the approach and setting of the paper are valuable and practically useful. As such, I think this paper would be a good addition to the program and spark discussion. However, I think the results could be pushed further and it may be worth illustrating the power (pun intended) of the framework a bit more. \n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "mop2Ey3xu74", "reviewer": "Reviewer_rFMx", "summary": "The paper proposes a new measure of the power of a firm in a market, namely the performative power.  The idea is the firm's actions can affect the behavior of the participants of the market, causing them to deviate from the \"natural\" state without the presence of the firm.  And the performative power of the firm is the maximum average deviation it can cause by performing some action.  The authors show how this notion connects to a number of issues, including (1) whether a firm should focus on learning customers' preferences stastistically or changing their preferences to align better with the firm's goal, and (2) how much benefit the firm can achieve by optimizing the objective post the response, vs optimizing statistically without considering the effect of the response.  They also show how to bound the performative power in some concrete settings.", "questions": "(Also putting some detailed comments here.)\n\nLine 40, \"a second way\": any evidence that this actually happens?  And to what extent?  I know this might be hard to measure -- just curious.\n\nParagraphs between line 114 and line 130: I find part of the argument a bit shaky.  First the authors say that the actions that we should consider are those which \"increase profit for the firm\".  It seems this should really be interpreted as actions that \"align with the objective of the firm\".  But later, the authors argue that it's \"not necessary to know the firm's objective function ...\"  I wonder how the two claims should be reconciled.\n\nLine 216: \"let \\phi by any ...\"\n\nSec 4.2, \\gamma vs \\Delta \\gamma: while this is conceptually interesting, I wonder how much technical difference there is when \\gamma is replaced by \\Delta \\gamma.  If I understand it correctly, one can simply replace every appearance of \\Delta \\gamma in sec 4.2 with \\gamma, and then we are back in the world where conceptually there isn't an outside option?  In that case I wouldn't overemphasize the effect of outside options.  Conceptually, the outside option formulation also implicitly assumes that a participant can always get utility \\beta if they choose so, no matter what their features are.  This in a sense creates asymmetry between the \"inside\" option and the outside option (one has to \"qualify\" to get the inside utility, but can get the outside utility no matter what).  So perhaps the outside option is more like unemployment benefits...\n\nPresentation in general: the paper is generally quite well written and polished, but a small thing that I think can be improved is the (somewhat unnecessary) use of jargon.  Some examples: I'd replace \"performativity\" with \"best response\" or something like that; in line 363, it's not clear what \"standard microfoundations\" are and why we need them if the reader is not familiar with \"alternative microfoundations\", and it seems totally fine to just say \"Assumption 1\".  Overall I think the presentation of the paper is heavily influenced by a particular line of work (which is of course fine), but maybe the paper would be more enjoyable for the general audience if presented in a more self-contained way.", "limitations": "Overall I'm satisfied (see detailed comments above for some minor suggestions).", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\nThe contributions of the paper are predominantly conceptual, so my evaluation is inevitably subjective.  I like the idea of measuring the power of a firm in a clean and well-defined way.  In particular, I think it's very nice that the definition doesn't rely too much on the specific form of the market (we still need some prior knowledge in order to properly choose the set of actions, the dist function, etc.).  The authors are able to confirm some natural phenomenon within the new framework and draw connections to other quantities of interest, which makes the new measure more credible.  They also show how to bound the measure (albeit in somewhat simple settings), which makes it more useful.\n\n\nWeaknesses\n\nAlthough the new measure definitely makes sense in some ways, it's not immediately clear to me it's the best way to measure the potential harm a firm can do to the market.  One potential criticism is that the measure is too pessimistic, and a rational and profit-driven firm wouldn't actually cause nearly as much harm.  Also, the authors present a number of applications of the new measure, but I'd be more excited if there's a \"killer\" application where the new measure helps solve a real problem that was previously beyond reach (it doesn't even have to be a technical one).  Also see detailed comments for some minor points.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Og3bbNwRkpE", "reviewer": "Reviewer_BiVM", "summary": "The authors study the degree to which the deployment of predictors can influence, or cause changes in, a population interacting with said predictor.  This paradigm is framed through the lens of recommender systems and strategic classification in which a firm deploys a model $f$ which makes decisions on a population $U$, where each $u\\in U$ has original data $z(u)$ and modified data $z_f(u)$ in the presence of model $f$. A firm's ability to shift a population is formalized in the proposed definition of Performative Power which measures the largest average deviation to agents' data that the firms choice of $f$ can have.  Several theoretical results are shown such as demonstrating how performative power changes with and without competition, as well as linking  performative power to the well established notion of performative prediction. ", "questions": "- **Framing of performative power**: Throughout the paper performative power is framed as a potentially negative phenonium, meaning we may want to be wary of firms with large performative power. While this certainly seems true, there is also be a positive side to performative power which appears to be absent from the paper. For example, in the context of lending, agents can manipulate their features to appear more creditworthy to the firm, and through the lens of Strategic Classification, it is assumed that these are non-productive manipulations such as lying on applications (i.e. an agents true creditworthiness is not effected). However, some \"manipulations\" may be productive in the sense that not only do they cause the agent to appear more credit worthy, but they also cause the agent to actually be more creditworthy. For example, decreasing one's debt and closing excessively opened credit cards will have both effects. In a setting like this, where the alterations to an agents features have a causal relationship on their true label in a \"positive\" way, the performative power of a firm is desirable in the sense that the firm could help induce stronger financial literacy and true creditworthiness throughout the population.\n\n\n- **Definition 1**: Performative power is defined via supremum over the firm's action space $\\mathcal{F}$. While I understand the motivation for the supremum, I am curious how the results or interpretations change when performative power is defined in terms of an average over $\\mathcal{F}$. Are we missing anything of value by using a supremum instead of an average?\n\n- **Proposition 2**: How reliant is Proposition 2 on the fact that the classifiers from each firm are thresholds? It seems that when we move away from thresholds to more complex models, the idea of  threshold predictors can still somewhat apply in the sense that  $x$ can be thought of as the outcome of the classifier's score function $h$, and we are now thresholding on that score. However, in this setting the cost function should now depend on the underlying features, not on the model's scores, and each firm now has the option to choose both the score function and the associated threshold . It is unclear to me how to extend this result past the 1-D case. Can we say anything for more complex models?\n\n\n- **Minor comments/issues**:\n\n    - Line 102: It may help readability to specify the domain of $z(u)$. For example, does $z(u)$ contain only predictive features? Is there a label associated with $z(u)$. This is cleared up later in the paper, but referring to $z(u)$ simply as a data point may be too vague for some readers.\n\n    - Line 102: Using $d$ instead of $\\text{dist}$ may be more in line the notation of prior literature.\n\n\n", "limitations": "The authors clearly outline their limitations and are overtly transparent about the applicableness of their work.", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- This paper is well placed in the literature and lays a solid foundation for an interesting problem. The idea of performative power is an insightful extension of performative prediction and opens many directions for future work.\n\n- The paper is extremely well written.\n\n- The authors provide several strong theoretical results which serve to outline the types of questions/observations one may expect to see in the setting of performative power.\n\n- Intuition and interpretation for each result is provided.\n\n- Example 2.1 provides both a concrete way to understand the problem, as well as a highly topical and realistic setting which the authors work is directly applicable.\n\n- Section 4 demonstrates that performative power is relevant  in several well studied fields/settings.\n\n- Performative power is examined in both isolation (single firm) and competition (multiple firms).\n\n## Weaknesses\n\n- Proofs being deferred to the supplement can cause parts of the paper to feel incomplete. I would have liked to see at least a proof sketch or some intuitive reason as to why the results hold. Some results, such as Theorem 1 have an intuitive explanation preceding their formal statement, but I would still prefer to see proof sketches. \n\n- Similar to the comment of proofs being deferred to the supplement, I think the section on display discontinuity design could benefit from more details. My first thought when looking at Definition 1 (as pointed out on line 131) is that performative power is likely to be intractable, or impossible, to measure in any type of realistic setting. The idea of display discontinuity design lends credence to performative power not only as a worthwhile metric and analytical tool, but as something that may be feasible to examine in practice. With that said I understand that at a certain point things must be deferred to the supplement, and this paper covers a lot of material so I'm sure the choice of what to defer is not straightforward.\n\n- There is a slight disconnect between the definition of Performative Power and some of the motivation/examples given by the authors. Specifically, in the context of a firm steering a population in a desired direction. The notion of performative power measures the worst case average shift that a firm can induce in a population. This shift is measured in terms of magnitude only. I understand that results such as Theorem 1 are trying to tie performative power to a firms ability to steer the data in a desired direction, but there should perhaps be some clarification that performative power does not necessarily, or directly, measure a firm's ability to steer a population in a desired direction. In the language of Adversarial Machine Learning, the current definition of performative power takes an \"untargeted\" perspective. Perhaps examining performative power from a \"targeted\" perspective is also worth investigating.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept