On the Impact of Performative Risk Minimization for Binary Random Variables
Authors: Nikita Tsoy
Ivan Kirev
Negin Rahimiyazdi
Nikola Konstantinov
Venue: ICML 2025 poster (2025)
Keywords: performative prediction
impact
sequential decision making
Primary area: social_aspects
TL;DR: We provide a binary model for studying the impact of performative risk minimization under linear shifts on the data distribution and model predictions

Abstract: Performativity, the phenomenon where outcomes are influenced by predictions, is particularly prevalent in social contexts where individuals strategically respond to a deployed model. In order to preserve the high accuracy of machine learning models under distribution shifts caused by performativity, Perdomo et al. (2020) introduced the concept of performative risk minimization (PRM). While this framework ensures model accuracy, it overlooks the impact of the PRM on the underlying distributions and the predictions of the model. In this paper, we initiate the analysis of the impact of PRM, by studying performativity for a sequential performative risk minimization problem with binary random variables and linear performative shifts. We formulate two natural measures of impact. In the case of full information, where the distribution dynamics are known, we derive explicit formulas for the PRM solution and our impact measures. In the case of partial
information, we provide performative-aware statistical estimators, as well as simulations. Our analysis contrasts PRM to alternatives that do not model data shift and indicates that PRM can have amplified side effects compared to such methods.

On the Impact of Performative Risk Minimization for Binary Random Variables
Nikita Tsoy! Ivan Kirev
Negin Rahimi"
Nikola Konstantinov
Abstract
deployed model 0 induces data distribution D(O), which gives rise to new learning objective, the performative risk Be--(()) (L(0,2) 2)), with l being a loss function and 2 be- ing a sample from the model-induced distribution. Much progress has been achieved in performative risk minimiza- tion (PRM), i... finding performatively-ppiinn points (see Hardt & Mendler-Dünner 2023 for a recent survey).
Performativity, the phenomenon where outcomes are influenced by predictions, is particularly preva- lent in social contexts where individuals strategi- cally respond to a deployed model. In order to preserve the high accuracy of machine learning models under distribution shifts caused by per- formativity. PPedoomo et al. (2020) introduced the concept of performative risk minimization (PRM). While this framework ensures model accuracy, it overlooks the impact of the PRM on the un- derlying distributions and the predictions of the model. In this paper, initiate the analysis of the impact of PRM, by studying performativity for sequential performative risk minimization problem with binary random variables and linear performative shifts. We formulate two natural measures of impact. In the case of full informa- tion, where the distribution dynamies are known, we derive explicit formulas for the PRM solution and our impact measures. In the case of partial information, we provide performative-aware sta- tistical estimators, as well as simulations. Our analysis contrasts PRM to alternatives that do not model data shift and indicates that PRM can have amplified side effects compared to such methods.
While PRM is preferable to standard risk minimization (RM) for the sake of test-time accuracy, the broader impact of PRM remain elusive. In particular, using PRM instead of RM leads to different predictions deployed by the learner and also changes the evolution of the data distribution, and these effects are compounded when deploying multiple mod- els over time. This limited understanding of the impact of PRM on the predictions and distribution may be partially due to the mathematical challenges arizing from analyzing the long-term dynamics of the learned models, in the pre- sense of intricate dependencies of the data distribution on all previous models.
Contributions In this work, we initiate the analysis of the broader impact of PRM, by studying a sequential per- formative mean estimation problem for binary variables, in the presence of linear performative distribution shifts. The simplicity of the learning setup enables us to derive the long-term dynamics of PRM, despite the complicated downstream impact of each deployed prediction on the fu- ture data distributions. This in turn allows us to quantify the evolution of the predictions and the data distribution.
1. Introduction
Predictions can significantly influence everyday life (Ro- damar 2018), an effect known as performativity. For in- stance, traffic predictions alter people's i daily routes, crime predictions can affect police resource allocation, and stock price predictions can steer traders' decisions. These changes can lead to shifts in the underlying data distribution, making the original predictions less accurate.
Within this model, we formulate two measures of impact. The first measure concerns the model predictions and corre- sponds to the usual statistical notion of bias of an estimator. The second measure quantifies the shift in the mean of the bi- nary random variable, relative to the mean in the case of lack of performative effects, and thus allows us to understand the evolution of the data distribution under PRM.
To capture these effects, Perdomo al. (2020) introduced the concept of performative prediction. In this framework,
We analyze PRM and the two measures one-period (sin- gle model deployment) and an infinite horizon (sequential model deployment) setting. In each case, we first study a full information setting, where all problem parameters (c.g. strength of performativity and initial distribution) are known to the model provider, in order to isolate the effects of perfor- mativity from exploration. We then analyze performativity
'INSAIT, Sofia University "St. Kliment Ohridski". Correspon- dence to: Nikita Tsoy <nikita.tsoy @ @i>..
Proceedings of the 4dnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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On the Impact of Performative Risk Minimization for Binary Random Variables
and exploration jointly via theory and simulations.
Long- Terg-Termmaa Fairness The line of works on long-ferm fair- ness also studies the evolution of distribution in social con- texts. [nniign et al. (2018b); Bechavod et al. (2019) focus
Our results indicate that, compared to RM, PRM may select more biased estimators and/or ones that shift the mean to extreme values. This happens in particular because mini- mizing the PRM loss suggests trading-off the usual mean squared error (MSE) for reduced aleatoric uncertainty in the future data distribution. Such effects occur when the distri- bution responds positively model predictions or when the distribution responds negatively, but the model is updated rapidly and the performativity is high.
on social feedback loops. Williams & Kolter (2019); Liu et al. (2020) propose models for performative responses mo- tivated by their learning context. While these works model performativity, they focus on finding fair models. In con- trast, we focus on performatively optimal algorithms and analyze their impact on the data distribution and predictions.
Instances of Performativity Performativity arises in many social contexts. Economic agents respond to the actions of the government (Lucas) 1976). Performative policing affects the distribution of observed crime rates (En- sign et al. 2018a). Traffic predictions reroute drivers to new areas (Macfarlane] 2019 Cabannes] 2022). Recommenda- tion systems affect the consumption of new content (Brown) & Agarwal 2022 Dean & Morgenstern) 2022). Since per- formativity is so widespread, it is important to study op- timization formulations in such settings and the effects of performativell-optime solutions on their environment.
Finally, we use two example scenarios to interpret our mea- sures and technical results in a social context |
2. Related Work
Performative Prediction In machine learning, performa- tivity is often studied within the framework of performative prediction, where the goal is to find model with good per- formance on the distribution that it induces. The setting was introduced by |Perdomo et al. (2020) and was inspired by works on strategic classification (Hardt et al., [0166] Dalvi
at. 2004). Numerous works study methods for finding per-
3. Model
formatively optimal/stable models (Mendler-Dünner
al.
2020
Miller et al.
Jagadessan et al. 2022 Izzo et al.
We now discuss the sequential performative prediction framework we study, the specific instance considered in our analysis and the impact metrics we focus on.
2022
Ray et al.
2022b Lin & Zrnic 2024), see Hardt &
Mendler-Dünner
(2023) for a recent overview. Brown et al.
(2022): Ray et al. (202aa) Mandal et al. (2023) extend this
framework to stateful environments, where previous model
3.1. Performative Prediction Framework
deployments impact the data distribution at later stages.
Optimization Problem This work analyzes how optimiz- ing for performative accuracy influences the model provider actions and the underlying probability distribution. To an- swer this question, we consider the following sequential performative prediction problem inspired by [Perdomo et al. (2020); [Brown et al. (2022); Ray al. (2022a). Denote by θ the space of models, by 0g E O the model parameters at time f, by D the space of all data distributions on data space Z, by De E DD the data distribution after the response to f and by V: D x θ + D the model of performa- tive response, such that D, = VD 1, 0r-1). The model provider is interested in minimizing discounted loss
In contrast to the works above, we focus on the impact of PRM on the data distribution and on the predictions made by these models. To our awareness, the only work that studies properties beyond performative loss in the context of performative prediction is that of Jin et al. (2024), who, however, focus on the fairness and polarization properties of PRM instead.
Distribution Steering Our work analyzes the secondary effects of PRM on the distribution and outcomes, which were not intended by the model provider. Several related works could allow the model provider to encode penalties for these unintended changes into its optimization task. Kim & Perdomo (202)) investigate how to steer distributions towards a more desirable outcome by using omnipredictors. Similarly, Golowich et al. (2024) study the task of distri- bution steering in population dynamics context. However, these results do not inform about the unintended distribution changes due to PRM, which is the focus of our work.
Σ n2t I. Dz (2000
min E (OO (B.)
(I)
where l is the loss function, 7 E (0,1), T T and O, depends only on the information up to time L. We denote the solution to this problem by (0:) and refer to it as the PRM path. Conceptually, this path can be seen as the "performatively-oppinaa sequence of models (Perdomo) et al. 2020).
Remark 3.1. Notice that our problem falls within the frame- work of reinforcement learning under partial observability, where Og corresponds to action and Dr corresponds to state.
Pleasse see the replication files for our paper at https://qithub.. wttp://ww..comiiaatttiiiaaaaaaaa
2

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On the Impact of Performative Risk Minimization for Binary Random Variables
The objective ① depends on the model of test distributions Dist In the standard performative set- ting (Perdomo et al. 2020), D Dr+1, the model is tested in an environment adapted to it. This property holds when the speed of model deployment is slower than that of societal adaptation. Thus, we call this case the slow deploy- ment case. For example, drug efficacy estimates can only be updated after consuminn clinical trials.
Test Distribution
estimates might have negative feedback on the congestion: when the model predicts that one route is less busy than others, people might use it more. On the other hand, drug efficiency estimates might have positive feedback on the drug efficacy due to the well-known placebo effect.
We capture these effects using a linear response model
P++1 = al, + (1 - |o|)s++1;
We also consider the case of Distt Dr. when the environ- ment adapts to the predictions with delay. Such delays arise whenever models are updated frequently. Therefore, we call this case the rapid deployment case. For example, the predictions of road congestion can be updated "on the fly", so people may not be able to adapt to the latest predictions.
where St-1 Ap: + (1 A)x, a E (-1,1), X [0. 1), and π E (-1/2, Here, 8;-1 is the next period mean in the absence of performativity, a controls the strength and direction of performativity, À controls the friction in the dis- tribution update, and π is the equilibrium (long-term) mean in the absence of model influence. Positive a describes posi- tive feedback situations. Negative a describes negative feed- back situations. We also use the notation 3 = (1 Jal) under which Pc+1 = al, + 3p, (1 |a - Ir.
3.2. Instance of Performative Problem
Distribution We assume that D, describes binary random variables 2 ~ Dr with mean ps.
Limitations This work considers a specific instance of our general performative framework to get a comprehensive theoretical description of the considered impact metrics (see Section (.3). While the considered form of the problem limits generality, we believe that our analysis could be in- formative for real-world situations. Therefore, we discuss the potential applications and limitations of our analysis in detail in Section 6
11/2, 1/2,
w.p. 1/2 Pt.
w.p. 1/2 + Pt-
Note that is Bernoulli random variable shifted by 1/2 for mathematical convenience. For these variables, a positive outcome could mean that a drug is effective for treating a patient or that a certain route free from traffic.
3.3. Measuring the impact of PRM
Loss At time t. the model provider deploys 0, E 1-2/2, to minimize mean squared error (MSE) 2(0,.2) (0t- 2)?. We denote the expected loss (w.rt. all randomness) by
The distribution consider is determined by its mean. This property allows us to formulate two natural "impact" met- ries, bias and mean shift. Bias captures the impact of PRM on the learner's 's predictions, while mean shift describes the impact of PRM on the underlying distribution. We discuss the generalizations of these metrics to other learning tasks in Appendix A.I
loss; = EE ((0g - 2))
We denote the means produced by the PRM path (0;)T by (pi )T=o We also denote by p the mean of the distribution Diess Note that पt is equal to p++1 and pr in the slow and rapid deployment cases, respectively.
Bias Consider an arbitrary path (sequence of predictions) (0.) Inspired by the classic notion of bias, at each time t we study the expected error in the estimate of the mean
Lemma 3.2 (Error-Uncertaie squared error of Or on Diet is
Tradeoff). The mean
bias, == EE(0c
Pt
test
(4)
1E(0,- - 2)² p( = (01--pt-) + (1/1-(p/30)) (2). (2)
Intuitively, the bias captures how far (on average) are the predictions of the path from the true mean at a given time.
Here, the first term corresponds to the standard mean squared error (MSE). The second term corresponds to the aleatoric uncertainty of Diest (note that such decompositions are valid for big class of distributions, Gupta et al. (2022). Thus, under performativity, the model provider is also in- centivized to decrease the environment uncertainty, while in the perforrforeeeive case they only minimize the MSE.
Mean Shift Here we compare the mean pe of the distribu- tion under the path (8,)%_ and corresponding mean in the absence of performativity pf (i.e., when a 0). Formally,
(5) The mean shift measures the amount (and direction) of de- viation of the mean of the distribution under the considered path (p compared to mean pp at time t if the distribu- tion was not affected by the predictions.
shift, = E(p: -P))
Performative Response Performativity manifests differ- ently in different contexts. For example, route congestion
3

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On the Impact of Performative Risk Minimization for Binary Random Variables
Analyzing the impact of PRM We study the bias and shift of the PRM path (0r 10 which we denote by bias; and shift;, respectively, Additionally, we compare the PRM path to naive path, 6 which ignores the performa- tivity when making predictions. Formally, of is defined as the mean of the previously observable distribution
We visualize this solution in Figure subsection we assume that 0% * 1/2.
For the rest of the
0.45
-0.45
0.5
0.5
0.0
%
0.0
0-
Test Pt-1-
(6)
Pi
0..6
This corresponds to the usual approach to prediction in which one minimizes the loss with respect to the currently observable distribution (akin to the usual ERM principle). In the rapid case, where distribution is observed in the first period, we define Og = O. We will denote the bias and shift of the naive path by bias"" and shift", respectively.
-0.5
0.0
0.5
-0.5
0.0
0.5
Po
po
Figure I. The dependence of 6 (blue), pi (orange), and 81 (green) pu for X 0.8 and 0.2 in slow T = 1 case. Columns correspond to the different a.
Interpreting bias and shift Sections 4 and 5 derive ex- plicit formulas for our impact measures under the PR. and naive paths, allowing us to reason about the quantitative behaviour of these metries. In general, high bias can be interpreted as an undesirable property, even from solely statistical standpoint (Young al., (200)). However, as we will see, the PRM path is biased due to the trade-off with distribution uncertainty (Lemma 3.2). In contrast, mean shift interpretation is usually context-dependent. Section 6 provides two example scenarios to illustrate these points.
Loss
We get that
(1 2
|s|| < 1/2-a a
loss
7
1-2a
I- [c
1-a - (1
|o 1111
|s 2 1- [o]
1/2
2
Bias We have that biasi a(1-lal)ss Thus, the PRM 1-2a
path is generally biased. This bias does not arise from the usual bias-variance trade-off in statistics. Instead, the performativity incentivizes the model provider to reduce the uncertainty in the distribution. To see this, notice that the unbiased predictor. which minimizes the error term in 1-jal
4. One-Period Model
MSE 2). exists: f8 = T-a 81. For positive feedback, the prediction is biased towards extreme values (i.e., -1/2 or 1/2). For negative, the prediction is biased towards 0. The absolute value of bias increases in |a] if a > v3-1
This section analyzes the case of = 1. First, we discuss the full information case where Po. a, A. and I are known to the model provider. This allows us to separate the ef- fects of PRM from the hardness of designing of algorithms that achieve PRM (due to exploration/fnite-sampp effects). Next, we assume that the initial mean Po is unknown and study how this uncertainty affects our previous results. Fi- nally, in episodic RL setting, study via simulations the case where no information about the parameters is available.
Shift Once 6 is deployed, it induces shift shift; a lalltoaal 81. The direction of the shift depends on sign(a) (a)) in the same way as the bias. The effect increases with lal). While a no-shift prediction exists, 0o = (a))(a)s), ,, it differs from the unbiased prediction in the negative feedback case. This shows that unbiasedness and the absence of shift can not be achieved simultaneously under negative feedback.
Notice that the slow deployment case for T' = 1, which is the main focus of this section, corresponds to the standard setting of (Perdomo et al. 2020).
Discussion We can see that, in general, the PRM predic- tion is biased (even though the model provider has perfect information about the distribution), and its impact on the mean of the distribution is not zero. In the positive feedback case, the model provider benefits from shifting the mean to extreme values. Even though this strategy increases the error term, decreases the uncertainty. In the negative feedback case, the performative response to the unbiased prediction shifts the mean closer to O. So, the provider employs a biased prediction to reduce this drop in the uncertainty.
4.1. Perfect Information
4.1.1. SLOW DEPLOYMENT
Proposition 4.1 (Proof in Appendix B.2. The solution to
the problem 7 in the T = slow deployment case is
05 c
(1-10))s1
1. 2)
1a > 0,
1 1-2a
(/)
1 < 0.
Whenever [8%1 + 1/2, we get
Comparison with Naive Path Now, we consider the naive path, where, for the clarity of exposition, we as- sume that the system was initially equilibrium, i.e.
(1 - a1 lo:)
31 = 1-a a
pr=
(8po +(1-|al-18)))
- 2x
1 - 2a
4

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On the Impact of Performative Risk Minimization for Binary Random Variables
Po 81. We get
Figure 6 middle row, in Appendix visualizes 0g. If 10%] 1/2, we get
pi (1 + |al)81,
bias = va(a + 3) + ya((1 |o|| - -3)Tr
loss 1/4 (1 + 2a 2/al)s}
1-ya²
shift? biass" (a |al))))
shiftj (a-lalA+ra Alpo - (lal-7a" ( (1-2))111)))))) (1-
If a > O, the bias and shift of the naive path is zero, which might be more desirable compared the PRM path. How- ever, the naive loss is worse than the PRM loss by T-2a si- At the same time, if a I O, the bias and shift of the naive path are higher in absolute values than the bias and shift of the PRM path, i.e. RM increases our measures in the negative feedback case compared to PRM. Moreover, in the negative case, the loss penalty increases to 1-2a s7.
1-aa2
bias = 0.
Compared to the case of T = 1. the mean shifts to more extreme values due to the denominator, and the first-period bias becomes non-zero. However, the final prediction re- mains unbiased. The long-term loss incentivizes the model provider to actively manipulate the mean, even if the short- term loss suffers from such manipulation.
4.1.2. RAPID DEPLOYMENT
Summary Similarly to the slow case, the bias and shift of the PRM path are generally not zero and increase in |a|. In contrast to the slow case, only the long-term effects incentivize uncertainty optimization in the rapid model.
One-Period Case
Equation
1 reduces to
min A 1//
00 20.Po.
which results in 0g po. (We visualize this solution in
4.2. Imperfect Information
Figure top row, in Appendix.) Additionally, we get
This section analyzes how uncertainty affects our full infor- mation results in the slow deployment case.
pi (a + 3)po + (1 -1--0 |a| -),,
bias = 0, shifti = (ax |a|))pp - |a|(1 - A)TT
4.2.1. UNKNOWN MEAN
First, we analyze the case where and X are known to the model provider but po is unknown. Thus, the model provider needs to simultaneously learn Pu and adjust for per- formativity. For simplicity, we focus on the equilibrium case where Po = π. To learn po. the model provider observes m i.i.d. samples So = {
If a O, the PRM prediction shifts the mean closer to po relative to π. If < 0, the effect is hard to interpret. We only consider the case of π po- In this case, sign(po) of sign(p} $1). The mean is shifted away from Po in the direction of 0. Also note that the absolute value of the shift increases in |a| under both negative and positive feedback.
1 ~ D and uses an estimator Oo: RT + 1/2, 1/2] to get an estimate (s(S))
Comparison with Two-Period Case To see whether the prediction remains unbiased once the distribution changes, compare the one- and two-period models. For T = 2, get the following two-period problem:
Estimators To study the extent which the results of the previous section transfer, we introduce the analogues of the optimal and naive predictions. For the naive case, we use the empirical mean 0g := 1. > Po.: =: Po- For the optimal case, we use the result from Proposition 4.1 in which we replace 81 with Po. which results in estimator 0z-
(0p - 20(p))
min
0..01.PEE! 1/2,
1=0
Bias and Shift Figure 2 depicts the bias and shift of 0g with one standard deviation confidence intervals. For a > 0. the confidence intervals shrink very fast with m for big values of pn due to the shrinking introduced by clip function.
such that P1 = =00 + 3po + (1 |al)(( -A)T.
Proposition 4.2 (Proof in Appendix B.3). The solution to the problem
the T rapid deployment case is
0 = =clip
(1 + v0:3)
+ 7a |a| - 3)T
1 1
Now, we analyze the loss of 03- Theorem 4.3. The expected loss of 67 for $ 0 is
Loss
1 2a²
2'2
0- = pi-
4(m + 1)P0 Ez-D; ((05 =)²) (1 (1- (al1a)) 1
Whenever 1051 # 1/2, we get
21x
4ma.
p (a + 9)po + (1- - |a| 8)
For all values of a, the expected loss converges to the opti- mal expected loss: limm-+oo Ei(07 - =)2) E( 2) 2)2)
1 1-702
5

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On the Impact of Performative Risk Minimization for Binary Random Variables
Blas for m = 10, a 0.3
Blas for m 1000, d= 0.3
Shift for m 10, a' 0.3
Shift for m 1000. 0 0.3
62
Biss
Biss
ShR
0.075
Skift
0.10
0.1
0.00
69
0.50
0.0
0.000
-0.000
-07
-0.077
-0.4
-0.2
-02
0.0
-0.4
+5.2
0.
D2
0.4
-02
Bias
-0.4
Bias for 1000, -0.4
Shift -0.4
Shift 1000, -0.4
0.3
0.08
0.02
0.1-
0.0
-0.03
-0.2
-02
Po
Pr
Pn
Ph
Figure 2. The dependence of bias (0G) (left) and shift(0ā) (right) and corresponding variances po- The upper corresponds
a = 0.3, the lower row corresponds to a 0.4. Columns correspond to the different yn..
We discuss the loss for all values of or in Appendix To vi- sualize the results of Theorem 4.3 we plot the difference in expected losses between Oj and 6 in Figure 3 Due to ran- dom sampling, we observe region where 6 outperforms 0G. However, for larger values of m, this region diminishes.
1, Slow
00, Slow
0.50
0.25
0.25
0.00
9
0.00
8
P1
0.25
8g
-0.25
P1 -0.50
Pt
50
00.50
Episode
Figure 4. The predictions, 0,, (blue) means, Pt. (orange) and their theoretical equilibrium values (red and green, respectively) in RL setting over episodes (left) or time (right) for ग = 0.2, 100. where is the number of
-04
0.15, 7 0.9, and samples observed from test distribution at each step. The left and right plots correspond to the 1 slow episodic setting (with 0) and T = ac slow setting (with X = 0.3), respectively.
Figure 3. The dependence of the differences in expected losses, E(loss (ft) loss (0 )) on po and ax for different m.
5. Infinite Horizon Model
4.2.2. RL SIMULATIONS
Now, study the long- effects of performativity by analyzing our model for T We first theoretically study the perfect information case and then use simulations to analyze the case of unknown problem parameters.
In this section, check whether results in the per- feet information transfer the general performative prediction problem with information restrictions. In this setting, we consider episodic exploration and additionally assume that À = 0 known to the provider: In this case, the samples from the second period after deployment allow the model provider to estimate the performativity parameters.
5.1. Perfect Information
5.1.1. SLOW DEPLOYMENT
We implement Algorithm of Liu et al. (2022) with hy-
Theorem 5.1 (Proof in Appendix B.5
Assume that the
perparameter 3 2 to find the optimal predictions. We visualize the prediction path of the algorithm in Figure 4 (left). After some exploration episodes, the predictions of the model provider and the means of the distribution quickly converge to the theoretically predicted values, which vali- dates our results the perfect information case.
PRM path does not take extreme values Vt. 10 1/2 and
2a 2 VTB. Then, the solution to the problem 7 in the T oc slow deployment case satisfies
0: - 0-
2(1 - -|aa))
P P = wi, where
po P
1 2a
+
Po
p
6

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On the Impact of Performative Risk Minimization for Binary Random Variables
0.85
0.15
-0.15
-0.85
0.4
0.4
0.4
0.4
0.2
0.2
0.0
6
0.0
0.2
P1
-0.2
-0.2
0.2
0.4
Poo
-0.4
0.50
-..2
0.00
0.25
0.50
0.50
-0.25
0.00
0.25
0.50
-0.50
-0.25
0.00
0.25
0.50
-0.50
-0.25
0.00
0.25
0.50
0.4
0.4
0.2
0.0
0.0
0.0
0.0
0.2
0..50
-0.25
0.00
0.25
0.50
-0.50 -0.25
0.00
0.25
0.50
-0.50
-0.25
0.00
0.25
0.50
-0.50
-0.25
0.00
0.25
0.50
Po
Po
Po
Pg
Figure 5. The plots depiet the dependence of 0 (bluc), pi (orange), (green), and Ptc (red) pu for X = 0.8, = 0.2, and y 0.5 in oo case. Columns correspond to the different a, the top and bottom rows correspond to the slow and rapid cases, respectively.
- (1- 3)
0% 1 2a - + 73(1 3)
uncertainty because the biased prediction shifts the mean to more extreme values. On the other hand, if < 0 and |o| is small, the bias negative. In the negative feedback case, the negative bias again shifts the mean to more extreme values than the unbiased prediction, reducing uncertainty,
(1 - - |a| -8
P
1 2a 8 + aB 73(11 a B)
2aH3
4a(1 a))
w = 8+ - 2a +
3 V
1- 182
Long- -eng-Term Shift
The long-term shift of 0r is non-zero:
Notice that the restriction 1t. 10;1 # 1/2 could hold only if w I 1. There is an upper bound on a beyond which the model provider is incentivized to choose the extreme values of 9, So, if this bound does not hold, after some time, the model provider always benefits from setting 10;1 1/2, even though this prediction is necessarily biased. Additionally, if w < 1, the solution converges 0r + 0% st + S PÊ + Poo in the limit ing us to study the long-term seg-term effects of PRM.
(ar |a + a ||| 73 a))T
P π
2a
3 + aB 7/3(1 - B)
Comparison with Naive Path We have that 0% p.c
1-lal-8 T-o-8 π. The bias of the naive path tends to zero as +
00. The long-term shift is also zero if >> 0. If 0.
allow-
P
1 1-73 |al
< 1.
Ph
1 7/ + 1
We visualize the solution for all cases in Figure (top row). The restriction Vt |0;| * 1/2 does not cover the cases of big positive values a. In such scenarios, the PRM prediction depends on Po discontinuously because the model provider has strong incentive shift the mean extreme values.
The long-term shift of the naive path bigger than that of the PRM path in the negative feedback case.
1, the naive path has smaller bias and shift than the PRM path in the positive feedback case, while the PRM path has smaller shift in the negative feedback case. However, the long-term bias of the naive path is O, even in the negative feedback case.
Similarly
For the rest of this section, we assume that π > 0.
Long-Term Bias
The long-term bias follows
a(1 |al| - 3)
5.1.2. RAPID DEPLOYMENT
0%
Poo 1 2a 3 a3 7/
B)
Theorem 5.2 (Proof in Appendix B.6 Assume that the
PRM path does not take extreme values Vt [0 + 1/2. Then, the solution the problem ( in T rapid case satisfies
Even in the limit t + co, the PRM solution has non- vanishing bias. If > 0 and a is small, the long-term bias positive. Even though the bias increases the error term in Equation (2). the model provider benefits in terms of
2 + (po )k² + 0 Pt (po p + Poo
6

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On the Impact of Performative Risk Minimization for Binary Random Variables
where k= 3
2a T+x' X
1ra(a+0) 01 I 1-788
and
estimates. We visualize the prediction path of the algorithm in Figure ( After some exploration, the predictions and the means of the distribution quickly converge to the paareeticall-prediicee equilibrium values, which validates our theoretical analysis of the perfect information case.
- 713) - |0| -B)Tr
0x = 1- 0 - 3 1 (a a + 3 3(2a + B))
(1 (1-7(a+ + 3) - |a| -9)
Po0 1 - 0 - ra + + ((aa + 3))
6. Discussion and Future Work
Similarly to the slow case, the restriction Vi 10 + 1/2 could hold only if [x] s 1. If |s| < 1. to and Pic represent the long-term values of 02 and pi. respectively.
Our results suggest that the performatively optimal (PRM) path is, in general, biased and introduces non-zero mean shift. These effects are more expressed when the mean re- sponds positively to model predictions or when it responds negatively, but the model is updated rapidly and the perfor- mativity is high. To understand the potential impact of such effects, we now provide two example scenarios and interpret our measures and technical results in a social context.
Figure ( (bottom row) visualizes the solution for all cases. Again, the assumption vt|0ū] * 1/2 does not cover large |a.- If |a| is large, the PRM prediction depends discontinuously on Pa- If a > O, the mean, depending on po. converges to one of two equilibrium values. If < 0, the mean oscillates between two values that correspond to extreme predictions.
Case study: drug efficacy estimation Consider a sce- nario in which a company is trying to estimate the effec- tiveness of a drug they produce against a specific disease. We define our binary random variables as indicators that the drug cures randomly sampled patient. To model the well- placebo effect, under which beliefs about the effectiveness of drug may further increase its positive im- pact, we assume a positive performative response (a > 0). Consider the one-period positive feedback model in Sec- tion 4 Then, a positive/negative bias indicates an exag- gerated/understacee prediction of the average drug efficacy respectively, which may make it harder to find the most ef- fective drug on the market. At the same time, a positive shift indicates a higher drug efficacy due to the placebo effect, which is, of course, desirable for combating the disease.
For the rest of this section, we assume that π. >
Long-Term Bias
We get
70 (1- - |o|| -r
8% - Po0 a- 3 1(a + + (2(2 ()))
The bias is again not zero and behaves similarly the slow case for small |or/.
Long-Term Shift
We get non-zero long-term shift:
(a - la| + a| + ((a) - a)
Po - T
a - 3 7(a + + 3(2x + B))
Comparison with Naive Path Notice that the mean in the naive path case satisfies pf = app + 3pp + (1 |a|- - m. Since a + 8 < 1. the mean converges an equilibrium, which satisfies 0% = p% = I-a-8 π. Again, the long-term bias of the naive path is zero. The shift is zero a > 0. If <
Our results in Section 4 with pa = π suggest that whenever Po > (i.e. the drug effective to begin with), PRM would lead to positive bias, i.e. exaggerated prediction on the drug's 's effectiveness; as well as positive shift and thus increased drug effectiveness due to performativity.
1-lal-8
P' ph π
1+1/al/2-77 -B 78
Case study: traffic prediction Consider a model provider seeking to predict which of the two roads, A or B, is less busy. We model this by defining the binary random variable as an indicator for the event that road A is less busy. Con- sider our infinite horizon negative feedback model. Positive or negative bias of PRM corresponds to the model provider redirecting more traffic to road A or B respectively. At the same time, positive or negative shift indicates an increase in the usage of road B or A respectively. The bias is probably undesirable property of the prediction as it makes some drivers choose sub-optimal road. At the same time, the shift might be benign or adverse, depending on the context.
1.
1+ 1-1lll/2201 7/3 + 1- 5)
21++a DD)
Similarly the slow case, the shift is smaller for 02.
5.2. RL Simulations
Finally, we check whether our results in the perfect infor- mation case transfer to the general performative prediction problem with information restrictions. We consider a usual sequential RL problem. We implement simple heuris- tic algorithm, which learns the performative response by deploying extreme predictions {-1/2, 1/2} at random for the first 4 steps. Then the model provider learns the pa- rameters of the performative response by likelihood maxi- mization and deploys the optimal policy under the resulting
In the slow deployment case, the mean usage of roads be- comes more equalized (Figure 5, top-right part) compared to the case when no performativity present, which is in-
8

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On the Impact of Performative Risk Minimization for Binary Random Variables
tuitively desirable. In the rapid deployment case, if the strength of performativity is small, the usage becomes more equalized (Figure 5, bottom row, third plot). If the perfor- mativity is large, the usage oscillates between roads (Figure 5, bottom row, fourth plot), which may be undesirable.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. Our theoretical analysis contributes to better understanding of the impact of machine learning society. Therefore, we expect that our results can serve positive purpose in increasing the awareness about ML impact and encouraging further research on related topics. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Limitations and future work In this work, we focus on mean estimation of binary variables only and work under the linear response model (3). This makes the analysis of the long-term dynamics driven by T tractable and allows for defining natural metrics of impact and interpreting them in context. Despite its simplicity, we hope that our model can be qualitatively useful in broader settings. First, the linear response naturally arises as a first-order Taylor approxima- tion for any performative response. Thus, our results may (at least qualitatively) transfer to situations of weak performa- tive response. Second, as noted in the discussion of Lemma 3.2, the error-uncertainty decomposition holds for a broad class of distributions. Thus, we can expect PRM to gener- ally prefer distributions with smaller aleatoric uncertainty. For example, in the case of multinomial distribution, the model provider has an additional incentive to concentrate the probability mass on a small subset of outcomes.
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On the Impact of Performative Risk Minimization for Binary Random Variables
Supplementary Material
Appendix A
contains additional results.
Appendix B contains proofs of all results.
Appendix contains details RL experiments.
A. Additional Results
This section presents additional results that were not included in the main text.
A.1. Generalization Impact Metrics
This section discusses possible generalizations the impact metrics defined in Section 3.3
Mean Shift Metric Regarding the mean shift metric, we identify two possible extensions, for parametric and non- parametric distributions, respectively.
Consider setting where the distribution is parametrized by finite number of parameters, D. D(w)), where w, (w!.... .... ) This is, for example, the case for distributions defined via causal graphical models, as well as for many common distributions (e.g., exponential families). Then, one can define the parameter shift metric as shift, = E(w, - w!),, where w are the parameters of the distribution at time if the distribution was not affected by the performativity.
In a non-parametric setting, we can instead define a divergence-based metric shift, = ((,(D,, D.)), where K an arbitrary divergence function (e.g., KL-divergence). The function K can be designed to capture an undesirable shift in the distribution, according the target application.
These metrics can then be studied under different models for the performative response and counterfactual dynamics of the distribution in the absence of performativity, which can be chosen depending on the learning task and application under consideration.
Bias Metric Regarding the bias metric, we provide two extensions suitable for cases where the distribution divided into several groups, which is relevant, e.g., seirees--eensiivee applications.
First, in setting with several groups and multi-laheled data, could calculate matrix of biases with entry for
each group and label defined as follows biag E(E (q (X) - [Y y]|G g)), where qe (X)
(a) (X), ... % (x)) is the vector of model's softmax probabilities at time and G is the group. These biases could be interpreted measure of unfairness among groups
Second, one can use established metrics from the literature of bias amplification see Zhao et al. 2017) y Wang & Russakovsky
(202)): Zhao et al. (2023) Tokas et al. (2024).
A.2. Perfect Information
This section contain additional results in the perfect information setting.
1. ONE- ENE-PERIOD SLOW DEPLOYMENT, PERFECT INFORMATION
Here, we present additional results for Section 1.11
12

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Po- On the Impact of Performative Risk Minimization for Binary Random Variables
In the symmetric case, assume that the equilibrium probability is Comparison with Naive Path in Symmetric Case symmetric, 0. Then,
loss = 1/4 + (1 2a 2/))p
loss = 1/4 1 2a Po
aB
bias = (1 - a -3)po:
bias.
Po,
1 2xx
(a |a| + alal)
shift (a |a| (alA)po:
shifti
1 1-20
If a > 0, we get
bias
biass (1 - (1-
If the performativity, or the inertia, X, is big then the naive prediction is preferable in terms of bias. Otherwise, the optimal prediction is preferable. (The same analysis holds for the shift of estimator.)
S O, we get
biastl
|a|((1 - |a|)A
(1 21 À + |a| + (a|A) <I
[bias]]
Thus, the optimal prediction is preferable in terms of bias. (The same true for the shift.)
Finally, if 2a > the loss penalty equals
( 2xx 2,3 +
g2
2
Po-
1
2a
To analyze it consider two cases: X and À 1. If
get that the following penalty
loss V1 loss = (1 )o
This penalty is bigger than the penalty in the equilibrium case for small If 1, get that po 81. So, get the same answer in the equilibrium case.
A.2.2. TWO-PERIOD SLOW DEPLOYMENT, PERFECT INFORMATION
Here, extend Section 4.1.1 by solving the two-period case and comparing with it.
Proposition A.1 (Proof in Appendix B.7 Assume that 2a: V7lai.. Then, the solution the problem 1 in T slow case satisfies
(1-1a)((1 (a)((1 -2a + 70:32) + 70:3( -)m²)
(1
1 111
= clip
(1 2a)2 2,2,83
2'2
2f - (a(1-la|)ss |5 $ 1 2a (which always holds for a S 0).
We visualize whole solution Figure 6 bottom
Notice that the left part of the picture operate in regime 1 - 2a v In this situation, the optimal prediction depends non-continuously on Po because of the incentive push the mean to the extremes. Additionally notice that the left plot has kink on its right side. This kink corresponds to the transition between the cases 2(1 - lal)) | I 2a and 2(1 - |a])| >
2a.
[0%] < 1/2 the setting of Proposition
we get
a ((1
2nx) 81 + 203 A)T)
p
(1-2a)² 32
For the rest of the subsubsection we assume 0j 1/22.
Bias of 03 We get
0i - pi a(1 (11-)a)((( 2a 21,3 )1 + 7(1 a) A)T)
(1 1-2a)² 722
For equilibrium and symmetric n, the bias of prediction becomes more pronounced because
1 - 2a 7
1
(1 2a)2 722
1 20
13

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On the Impact of Performative Risk Minimization for Binary Random Variables
Shift of 0g
We get
P ((1-2a(aa |a] + a(aa)) + ) + 7a (1- [al))((1 A)T
(1 22aa2 222
Discussion We can see that generally the bias of the optimal prediction is exacerbated in the two-period model. It happens because the motivation of the model provider to skew the distribution becomes stronger due to longer horizon.
Comparison with Naive Path In equilibrium case т = po- given that the impact and bias of the naive path is the same as for the one-period model and our results above, we get that the naive path is even more preferable to the optimal path in terms of bias and shift. For the case of a < 0, we get
bias;
(1 la) 2/a 2/a 7 |a| 2(a
|a]|1
(1 + 11-2/a)9-~/or1
shifti
|a))3
|a|s]
(1+ 2|a)22 - ~
shift?
bias" Jals1
2.
Jals1
By direct calculation, the bias and shift of the naive path is always higher than those of the optimal path.
In the symmetric case 0, if
0, get
bias
a1 - 2a))
bias ((1 - (11-aa) 30 2a))
Notice that ratio /2/(1-2a)) 2 a2/(1- 2a)) is increasing in a. Thus, the ratio biases is increasing in a and X. So, similarly, to the one-period case, the optimal path is preferable the naive path in terms of bias if and À are small enough. (Same analysis holds for the shift.)
If a < O, we get
shiftil
la X(1 2/0| + +
shift?
((1 70033)) |a|
A x [a/2)
This ratio is increasing in À and y. Hence,
shifti shiftt]
2 a
2 8|a + 3n 1.
So, the bias of the optimal path smaller than the bias of the naive path. Similarly, the shift of the optimal path is smaller than the impact of the naive path.
Discussion Similarly to the one-period case, the naive path might be preferable in terms of bias and impact to the optimal path for a 2 0. However, for a < 0, the optimal path is superior to the naive path in terms of bias and shift.
A.2.3. INFINITE HORIZON SLOW DEPLOYMENT, PERFECT INFORMATION
This section contains additional results for Section 5.1
Bias of 0g We get
(1 (1- (1 - lal)
bias
81
2a +
Notice that, if - 2a 2 VRB, then this bias is bigger than in the two-period case. Thus, as previously. the longer time horizon incentivizes the model provider to give more biased predictions.
14

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On the Impact of Performative Risk Minimization for Binary Random Variables
Shift of 0 We get
2a
aa |0|5
shiftj
2a 31.
Similarly to the bias of o. the impact of 0 increases compared to the two-period case if >> 0. However, if < 0, the impact becomes smaller than the two-period impact.
Bias and Shift of % The bias and impact of the naive path in the symmetric case follows
bias" = (1 3)pu.
shift = (a laal))p
The bias the naive path smaller which happens only if > and a is sufficiently big. (The same inequality holds for the shift.)
2a + 3(1 + 7(x + B3(1 - a --)
A.2.4. INFINITE HORIZON RAPID DEPLOYMENT, PERFECT INFORMATION
This section contains additional results for Section 5.1.
Bias of 06
We get
1 X 0 Po
Po-
1 X
Assuming that > O, we get the following classification of the model provider actions. In the case of a >> O, we get that
8% > Po- If < 0 and a + 3 > O, 6 < Po- Finally, if a + < 0, 0 > pu again.
Shift of 0
We get
2x -lalA +
KPO
Xpo
Po-
+ x
Since increases in a and kla=0 X, the shift increases in |arl.
A.2.5. ADDITIONAL VISUALIZATIONS
We visualize the solutions for T' = 1 rapid case, T = 2 rapid case, and T = 2 slow case in Figure 6 As we can see, if a > 0, the prediction and the resulting next-period mean shift to more extreme values. Otherwise, the prediction and mean shift to 0 (the effect is more pronounced for the mean).
A.3. Imperfect Information, = 1 Slow Deployment
Here, we present additional results for Section 4.2
1. BIAS AND MEAN SHIFT THEORETICAL RESULTS
The bias for the naive estimator 0 is given by
bias = po(lal-aj)
For the performative estimator 0z the bias is
bias, = (1 (1-a)B((0) - la|))o
For the naive estimator 0 the mean shift is
shift = Po(a |aa) = -bias
and for the performative estimator 0g. we have
= a E |alpo-
shift =
15

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On the Impact of Performative Risk Minimization for Binary Random Variables
A.3.2. GENERAL VERSION OF THEOREM
Here, we present result that generalizes Theorem 4.3
offering theoretical insights for all possible values of E (-1,
Theorem A.2. For the naive estimator iü the expected loss is
R Diont [(] - =) = p6(2 2a 1) + (2a 1) 4pō 1 + 4'
4m
and for the performative estimator 0g we have
E[(0; =)²]
P-P (1-1a1)
pd
E (-1,0)
10
m
po(1 - (2F) m. + ( -1 + 1-
a 1)
2
Exer((1
22))g(1) 2(1 (()))o((<))p(11))
(po (al) 1-20) F Fm (2-30 m)
+(po(1 -aa)) 1-20 F
2-2a
po
|al)
1-a
a (0,0.5),
where I is the set of integers in
om 2-
(2-3a)m 2-2a
((x) = 1 (m
2). F po (x): E p(x), and
m
1
$/1
m-=
p(x)
+ po
Po
2
Asymptotically, we have that as m + o0
E[(0; - ==)² loss
i.e. as m goes to infinity, 0j approaches the optimal estimator for the risk minimisation problem.
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On the Impact of Performative Risk Minimization for Binary Random Variables
T Rapid, 0.45
T Rapid, -0.45
%
6
Pi
0.4
Pi
81
81
0.2
0.2
0.0
0.0
-0.2
002
-0.4
-0.4
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0
0.2
0.4
= 2, Rapid,
0.45
T = 2, Rapid, a -0.45
%
6
0.4
Pi
0.4
Pi
81
81
0.2
0.2
0.0
0.2
0.2
0.4
-02
0.0
0.4
-0.2
0.0
0.2
0.4
T = 2, Slow, 0.45
T = 2, Slow,
-0.45
%
6
0.4
pi
0.4
Pi
0.0
0.0
-0.4
-0.4
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0
0.2
0.4
Figure 6. The plots depict the dependence of 0g (blue). pi (orange), and si (green) on for X = 0.8, π = 0.2, and = 0.5. Columns
correspond the different values a; the top row corresponds the rapid case; the middle row corresponds T 2 rapid case; the hottom row corresponds to the T = slow case.
17

---

On the Impact of Performative Risk Minimization for Binary Random Variables
B. Proofs
B.1. Proof Lemma 3.2
Proof. Using conditional expectation we have
Bi 2)² 0,, plat]]
El68 -20,2 x² 1 [
=0 -20, E| 0, Pt
+ E[2 Oe, 1
1
0 20,p;" +
4
1
-0² pt
)²
(p(
B.2. Proof of Proposition 4.1
By direct calculation,
8 2000 + (1 - -al))s1) - -22)60 2(1 - lool)s100
If the parabola above opens downwards, 1 2a < 0, it achieves minimum at the extreme point of the domain. By analyzing both extreme points, we get 0i = sign(81) (21)//
the parabola opens upwards,
> achieves the minimum the point our domain closest to the vertex point. 22
Thus, 0- clip (1-la1)s1
B.3. Proof of Proposition 4.2
Going backward, get
0 =P1
which results in the following problem
min 0, - 20.Po xpp s.t. P1 = a00 + (1 - (a))(pp) + (1-A)T), 0, E 1//22 7/2)
00.01.0 .P1
Similarly Appendix B.2 we get
(1+ va5)a + ya(1- - A)TT
0% clip
1- y-7
2
(Notice that 1 ya² > 0, which reduces the number of cases.)
B.4. Proof of Theorems 4.3 and A.2
Before computing the expected loss, we first show the following result regarding the first two moments of the performative estimator.
Lemma B.1 (Moments of the Performative Estimator). For the performative estimator 0ū. we have that the first two moments given by
Dle
a E (-1,0)
1 2a
mm
a (0.5.
1-
px))
EE
I-
F
m.po+
m 2-
22-3 30
F
Fm.
1 2-2a m)
a E (0,0.5)
2
18

---

On the Impact of Performative Risk Minimization for Binary Random Variables
and
2/ aa
0.255ppp
+ Pr)
E (-1,0) a 005, 1)
1-2a
m
[[
1-a Exer 1-2a
2
2)x))
m
+ 1 1
m.
2-220
12-3a m)
+ 1
Fm
(2-20 m)
(0.0.5)
where I is the set of integers in
am (2-3a)m
F pm.po+ (x) == -= p(x)). and
2-
2-2a
m
x 1/1 po
m-z
p(x)
+ Po
2
Proof of Lemma B.I Recall that 0g is given by
clip(
[1- la Mi
21 212), 4),
1 2a > 0, - 2a 1 0.
0
sign (2o)
We consider three cases for the value of ac:
ii a E (-1,0]
In this case we have
a
0
1 20 Po
and therefore
- |
1 |a|
1 - |a|
2
pō
E[O]]
20 Po:
[[(())
Elpol
p6
2
1 20
1
20
where we have used that pa,+ ~ Do for i 1...., m, and thus Pa,s + 2 follows a Bernoulli distribution with parameter
Po 1
(ii) E 1)
In this case, have that
Po
0
2 Po 0.
Since Po- q where := 1 -m= % and g == Pa.i, so that qi Bern(po + 2), we know that the events can be written as
(po 2 0) ( 0.5), (0o < = (7 0.5).
Therefore,
1 X11<0055
0 =
2 X
Finally, using the law total expectation, we get that
[
El0i 2 0.5] Pr(g 2 0.5] + E < 0.5] < 0.5)
1
Pr(g 0.5]
Pr(j < 0.5]
2
2
1
-Fm.po+ (0.5m),
2
19

---

On the Impact of Performative Risk Minimization for Binary Random Variables
where we have used that mą Bin(m, + 0.5). Similarly for the second moment
[(
B[ 2 0.5] Pr(g 2 0.5| + [[()) < 0.5) Pr(g < 0.5]
Pr(g 2 0.5] + 4
Pr q < 0.5]
4
(ii) (0, 0.5)
In this case we have
1- Po.
1-a
if
2-2n
1-2a 1-2a
2-2a
= A
06
io >
2-2a =: B
1-2a
2
if
2-2a = C
2
where we have denoted by A, B, C' the random events that we have not clipped the value of the performative estimator, that we have clipped it from above or that we have clipped in from below. Using the law of total expectation, we have
00 = 1 4| Pr
||]
+
E
[6 B
Pr B + E C( Pr[C]
Pr[B]
- Pr[[]
2
1[0ixal +
2
1
Pr
q>
2 3a
1
Pr
q S
a
2 - 20
2
2 2a
The first term can be computed as follows
x
Σ
( x aET 2a m
1 -
2 p(x).
where we have used that mpo + m/2 ~ Bin Po + } and have denoted by p(x) the PMF Bin(m, Po + 3. The last two terms are easily expressed via the CDF of the same distribution, giving us that
B[0G]
Σ 1EI
1 2x
-
a
x
2
1
p(x)
2 2 m.
2 - 3x
2 2a
2
F
m.pot
2
2a
where I is the set of integers in the interval
(2-2a m,
2-3a ml Similarly, for the second moment have that
((()))
[[(())
[r[A]] B[
Pr|
E Bi(0)) (C)
1
[E((0)) XA| + Pr[B] Pr|C]
2 3a
EE((G)) xxa] + Pr
/> 2 20
Pr
q
2 20x
Σ
1 1 2a
a
2 x )((m
1)
2
p(x)
+
4
4
1
F
m.poo
(2 2- 30
2 2a
m
+ F
4
m.po
(a a
( 2- 22220 2a
which finishes the proof.
Now, we are ready to present the full proof.
20

---

On the Impact of Performative Risk Minimization for Binary Random Variables
Proof of Theorems
4.3 and A.2
We begin by rewriting the expected loss follows
2u) = E |E [ 20 + 6011
E [ 20, E[2u [00 + 1
1
08 20-2001110 (06) 4
(1 - 2a) B 2(1 - |a|)po B +
where the expectation is only in terms of the randomness of the observations {po.: ) For the naive estimator, 0g. have that the first moments are
E E po
[ )1 = Po + ( - ( + pa)
which follows since po, ~ Do for I,..., m. Therefore, we get
1
E[(0 2)²1 = (1 2ax) E - 2(1 lal) E(0o] +
4
1 (2a 1)(4p
=2 2a 1) 4
4yn
For the performative estimator, we use the first and second moments of 0 from Lemma B.1 to obtain
E[( =)
(1 (1|aa1)
1-
p
a (-1,0)
20
po - |a))(2F po
1) -
a [0.5,
Exer ((1 22)g(()) - 2(11 - (al)po(()))p(z) + (Po lal))
1-a pmppot
1-20
(2-30 2220
+(po(1 - -aal)= + 2_20) ) po 2-22a) po(1 |al)
2
E 0.5),
where g(x) = (1-2a 51)
Asymptotically, as oo, have that the moments of 0J for a E (-1,0) are given by
(1 - |al)
(1 - lal)
1 2a Pu +
1 1 2a
(1- la))2
00.5 vo
(1 |a: )2
E [[
pn
(1 2xx 22 Pm
(1 ) m
Similarly, for [0, 5, 1), we have
E[0:]
-
F mo 1
m
7
sign(po)
2
2
2
1
B[(OE)
where we have used that the CDF function F F
1
converges to for non-negative Po and to 0 for negative po as
2
m -> DC.
21

---

On the Impact of Performative Risk Minimization for Binary Random Variables
Finally, for (0, 0.5), have that
- lalpo
1
B [
R2
clip
20
2
2
(1
|a| P00
E
X(PDEA)
Pr(po B]
Pr(po E cl
1 2a
(1 lae))pg
1
X(Moεл)
X
2 X/oDEC
2a
2
B a E (0.0.5)l.
where denotes the region (a function of a),, where 0j has been clipped, / represents the region where it has been clipped from above, and C is the region where it has been clipped from below. The third line follows from: (I) the law of large numbers, which ensures that po Po almost surely as oo, and (2) the dominated convergence theorem. The same argument applies for [] Thus, combining this with the other two cases for a, we get the following asymptotic results
lim E[0ū] = 6
lim E((())²] = (0))².
m-020
m=000
Therefore, we can conclude that as m + 00.
BI(0g -z)² - lossø.
B.5. Proof of Theorem 5.1
Consider Lagrangian function
L(w, 9. v. u.
00
Σ
(0 20.pp-1) (a0, 3pc + - (alaaa)(1 A)T pe+1)ur - (1/2 0z) -O0 1/2)m.
KKT conditions for this infinite-horizon problem (see Section 4.5 of Stokey et al. 1989 give
0 27' - P(+1) - 004 + Me
0 22,0,, 0, + -34++1
0 = (1/2 0, 0.) 2 O,
= (0, + 1 1/2) 7 2 O.
Thus, the solution for the case when the restrictions on tg are non-binding satisfies
0,+1 (1-2a
70.32) 2a +
1 1-
3(1 (1-a)a n
0,
s11
7(1 a)33
7(1 - a)
8122 ad0; +B8441 + (1 A)T.
We get that the optimal path satisfies first-order linear recurrence relation for 0, and s,. Its characteristic equation follows
x² 1- 2a + 7
1 = 0.
(11
gives the following eigenvalues
1 - 2x + 7 + /(1 - -1-
[(1 2a. 7,92)
F0.1
27 xx
22

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On the Impact of Performative Risk Minimization for Binary Random Variables
Notice that the product of these eigenvalues is 1/7. Thus, of the eigenvalues is necessarily bigger than in absolute value. Due the restrictions on we the homogeneous solution corresponding this eigenvalue should be zero.
Consider the case of - 2a 2 VT3, then, the smallest eigenvalue, w, satisfies
(1 20 7/32) 2y - a
v(1 (1 2a.² -1:32 3
2
2a + g
Corresponding eigenvector gives the following homogeneous solution
6+1 sw", 0 2(1 2(1-lax) rw.
2a
One of inhomogeneous solutions satisfies
(1 -1-20 2a - 7,311 -)) 1)TT
(1 (1- - A)TT
0 = - 2a - a3 7 3)
1 2a - B 3+ aBB
- -33(1 aBB
3)
Using the initial conditions, we get the desired solution.
B.6. Proof of Theorem 5.2
Consider Lagrangian function
L(w, v, M. =
y (0? - -00pp) - (aB, + Spc + (1 - (a-la))(1 - A)T - P(+1) - (1/2 0e)su - (0c + +122)n
KKT conditions for this infinite-horizon problem (see Section 4.5 of Stokey et al. 1989 give
0 = 27 (0, pc) al/: + Mt Th.
0 = 2e + 14-1 - 3yr.
0 = (1/2 - 0r) us 2 0,
0 (0: + 1/2) Mc: Th 2 O.
Thus, the solution for the case when the restrictions on 0g are non non-binding satisfies
0.11 1+ 7a.
7(a + 3) 0, 7(a + B) Pt + 3(1 la)) A) n
a + 3
P+1 a0, + Bp: + (1 (1- (al \)=.
We get that the optimal path satisfies first-order linear recurrence relation for 0g and Its characteristic equation follows
x² 1 1+ 3) 1(a + 3)
1 0. x +
7
gives the following eigenvalues
30.1 = 1+ 7+73 + 3) 1 /(1 - 719)(( l 72a + B)2)
27(a + 33)
Notice that the product of these eigenvalues is 1/7. Thus, one of the eigenvalues is necessarily bigger than in absolute value. Due the restrictions on w, the homogeneous solution corresponding this eigenvalue should be zero.
The smallest eigenvalue, satisfies
1 1+7 + B) -V(1 --00)(1 (2xx + 3)
2a
K
27(a +
1 1 X
23

---

On the Impact of Performative Risk Minimization for Binary Random Variables
Thus, homogeneous solution follows
2
q: = = 0%
gr.
1+1 x
One of inhomogeneous solutions satisfies
(1 - 7a + (1))(())) - A)T
- 11-)8)(1 [a])(1 \)TT
d
0
1 - a -
- 8(2a + 5))
1
-7(a B -3(2a + B))
B.7. Proof of Proposition A.1
Using the results of Proposition we get
clip
[1-1a - 2'2/
1 2a >
0 =
2m
sigu(sa)
2a - 0,
which results in the following loss in the second period:
2(1 [2)
< -1-20 2a,
(01)² - -2
20
1-20
(1 (1-la|)]s2l
2(1 |a]) 2a.
Notice that
(1 1al)2s3
(07)2 p0 2
- 2a
Thus,
Σ 7 (0) 20.p(+1) 2 0 -200p1 -20pp1 7(1 1al)2s3
1 -2a
t=0
So, the minimizer the right hand side satisfies 2(1 - |a |s/
|hr S 1 - 2a, it will minimize the left-hand side.
Similarly Appendix B.2
we have that the minimizer of the right-hand side satisfies
hs.
clip (ا-اا)] [111 20 27 (aa 2)
1 2a > V
(1- 2012 72
sign
[1 20 + +7a 1-)a
1 1 < /
Thus, when 1 - 2a Viaaß and 2(1 |a])|sš| < 1 2a, get the desired solution problem.
24

---

On the Impact of Performative Risk Minimization for Binary Random Variables
C. Details of RL Simulations
This section gives additional details about RL-like simulations in Sections 4.2.2 and 5.2
C.1. One-period Episodic Simulations
We consider episodic exploration of = slow model, where we additionally assume that = and known to the provider. In this setting, we assume that each episode has the following structure.
1. Nature samples qu-
2. The provider observes {= 1m-1 D
3. The provider deploys Pb.
4. The provider observes (- DT.
We implement Algorithm|| 1 adopted version of Algorithm 1 of et al. (2022), where we denote the episode number by т, with hyperparameter / = 2- find the optimal prediction. (Notice that the first period observations are non- non-informative for the log-likelihood maximization because À 0.)
Algorithm Optimistic Maximum Likelihood Estimation
Initialize: B = {aa, 7) E (-1,1), E D = 0
T = to do
Deploy 0 arg
mill 1oss(0 a,
Observe Si (DT)
Add S1) to D
Update BT+1
(a, T) E B": (o,))e log Pr(S a, n,,) 2 max (o, log Pr(S) a, 0) -3
end for
C.2. Infinite Horizon Simulations
We consider episodic exploration of = co slow model, where the provider know the value of y. In this setting, we assume that each step has the following structure.
1. The provider observes
m-1 ~ D
2. The provider deploys dz.
For this case, we implement heuristic Algorithm where we denoted the value function as
V(po, a, = min
'looss(0, 00-1..... Go Po, a, A).
(6, (2 t=0
This algorithm learns the performative response by deploying extreme predictions {-1/2, 1/2} at random for the first 4 steps. Then the model provider learns the parameters of the performative response by likelihood maximization and deploys the optimal policy under their estimates.
25

---

On the Impact of Performative Risk Minimization for Binary Random Variables
Algorithm 2 Greedy Exploration
Observe So D
for f = 0 to do
Deploy 0g at random from 1-1/
Observe S.+1 D
end for
for t- 4 to T do
Estimate (ax, n, A, po) = arg m1xx 118 A.po Ei=o log Pr(S.
Deploy 0₂ arg ming, (0, Pt+1 (0 oo Po. a, nI A)) +
Observe S D
end for
26


Reviews:
[{"review_id": "RJAVR20pa8", "reviewer": "Reviewer_5pBv", "summary": "This paper studied the long-term impact of performative prediction on the predictor quality (as measured by bias) and population distribution (as measured by shift). It examined separately the settings of slow and rapid classifier updates with or without perfect information. The paper analyzed the long-term bias and shift in the context of binary mean estimation and compare and contrast them with the naive setting. The generalizability is implied with RL simulations.\n\n## update after rebuttal\nThank the authors for providing detailed responses to my questions. I think the strength of this paper generally outweighs its weakness, so  I would consider raising the recommendation.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thoughtful and constructive review! We address your concerns and questions below and remain available for further discussion.\n\n**Setting $\\lambda = 0$ seems too restrictive, no? ...**\n\nThank you for your comment. Indeed, in Section 4.2.2 we had to make assumptions in order to ensure that we have one optimal value of the mean in the second period, $p_1$. First, we need to assume that the mean in the first period is also fixed, $p_0 = const$. Secondly, to make our setting estimable, we need to have some restrictions on either $\\pi$ or $\\lambda$, such as $\\pi = 0$, $\\pi=p_0$, or $\\lambda = 0$ (otherwise, $p_0$ would not be distinguishable from $\\pi$). Given that the purpose of experiments was to see how exploration interferes with performative risk minimization, we chose $\\lambda = 0$ since this case is harder in terms of exploration compared to $\\pi=0$ or $\\pi=p_0$.\n\nNotice, however, that one can observe the effects of self-evolution in our experiments in Section 5.2. There we use $\\lambda \\neq 0$. When $T \\to \\infty$, the mean evolves over a long period of time and, for small values of $|\\alpha|$, converges to an equilibrium value.\n\nSome trends we observe there as $T \\to \\infty$ are as follows:\n\n- The equilibrium mean, $p^*$, goes closer to the extreme values with an increase in $\\alpha$, when $\\alpha > 0$.\n- For the slow deployment case and $\\alpha < 0$, $p^*$ goes closer to zero with an increase in $|\\alpha|$.\n- For the rapid deployment case and $\\alpha < 0$, for small values of $|\\alpha|$ the mean also goes closer to zero. However, for big values of $|\\alpha|$ it starts to oscillate.\n\n**Can you provide more details on how to obtain the optimal $\\theta_t$ ...**\n\nThank you for raising this point, we will clarify this in the next version of the manuscript. We find the value function, $V$, using the value iteration algorithm. Due to the Blackwell's contraction mapping theorem and the boundedness of the loss function, this process converges. Then, we solve the optimization problem for $\\theta_t$ directly.\n\n**To complement the findings, would the case where the classifier is myopic ...**\n\nThe naive path exactly serves this purpose. The naive path corresponds to a situation when the classifier is myopic, and the performativity is present, while the bias and mean shift compare the PRM-induced distribution path when the performativity is present with the distribution path when the performativity is absent.\n\n**If so, in what way the current model is more advantageous ...**\n\nOur results generally indicate that the naive (myopic) path induces less bias and shift in the positive feedback case. Thus, if one is concerned about the impact of performativity (see Section 6) and could sacrifice accuracy, the naive (myopic) classifier could be preferable to the PRM one. The negative feedback case is more complicated. When the performativity is weak ($|\\alpha|$ is small), the naive path introduces a bigger mean shift. However, when the performativity is big, in the infinite horizon case, the naive path is more stable (e.g., it will not have such behavior as depicted by Figure 5, bottom row, fourth plot).\n\n**Typos**\n\nThank you! We will fix these typos in the next version of our manuscript.", "claims_and_evidence": "All claims are supported by either theoretical or experimental evidences.", "methods_and_evaluation_criteria": "The method makes sense and the theoretical results are instructional.\nEvaluatoin metrics: bias and shifts make sense and the benchmark being the naive path (where the performativity is absent) is a reasonable baseline to study the impact of performativity.", "theoretical_claims": "I checked correctness of proofs in section 4 and Theorem 5.1, where no issues were spotted.", "experimental_designs_or_analyses": "RL simulations validate the theoretical findings with information restriction. The simulated curves converge to the theoretical prediction.", "relation_to_broader_scientific_literature": "The work builds on performative prediction and long-term fairness, the combination of which is an emerging study in the literature. It contributes a quantifiable framework for PRM’s societal impact. The limitation lies in that focusing on binary variables restricts connections to broader performativity literature.", "essential_references_not_discussed": "Not recognized.", "other_strengths_and_weaknesses": "This paper focuses on a novel problem and is the few work looking at the long-term societal impact of performative prediction. Although the methodology may be too simplified to capture practical situations, the findings on the classifier bias and distribution shift in the long run are insightful.", "other_comments_or_suggestions": "1. Right column on page 5: the second to last line contains \"Theorem 4.1\", does it mean \"Proposition 4.1\"?\n2. Last sentence of the caption of Figure 5: it seems both top and bottom rows correspond to the slow case.", "questions_for_authors": "To complement the findings, would the case where the classifier is myopic, i.e., optimize instantaneous objectives at every step instead of solving for a long-term policy, be different from the model considered in the paper? If so, in what way the current model is more advantageous; and if not, why would you or the society prefer one instead of the other?", "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": "", "extra_scores": {}}, {"review_id": "EDc8ThrltB", "reviewer": "Reviewer_5zXv", "summary": "The paper studies settings where predictions actively shape the data distribution, a formalism known as performative prediction. Beyond just predictive accuracy, their main focus is on understanding how different update procedures and learning algorithms influence the data distribution and understanding to what extent do performatively optimal predictors bias the distributions towards extreme values. \n\nThey study these questions in an outcome performative (predictions only influence outcomes not features) binary prediction setting. They further impose a linear model of how predictions shape the conditional distribution over outcomes. Their main results exactly characterize essentially all aspects of this binary problem: what is the performative optimal solution, what is the optimal loss, what is the bias between predictions and outcomes in the performatively optimal case, and how do predictions look like if one performs naive updates. They do this for both a stateless and stateful versions of their model.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thoughtful and constructive review! We address your concerns and questions below and remain available for further discussion.\n\n**I think that the paper would benefit by including a discussion to the following papers.**\n\nThank you for the suggested references! We agree that, while orthogonal to our work, they are certainly relevant. We will cite and discuss them in the next version of our manuscript.\n\nIndeed, Kim and Perdomo (2023) investigate how to steer distributions towards a\nmore desirable outcome by using omnipredictors. While steering is indeed\ndifferent from accuracy maximization, the focus of this and our work are different. We\nanalyze the secondary effects of performative risk minimization on the\ndistribution and outcomes, which were not intended by the model provider. In contrast, Kim and Perdomo (2023) assume that the model provider\nintentionally changes the distribution according to their metric, which can be\nexpressed in the omniprediction framework. Hence, the works could be seen a\ncomplementary. Our work allows the model provider to discover and analyze the\nunintended distribution changes, while the omniprediction could potentially\nallow the model provider to encode penalties for these changes, if they could be\nexpressed in the omniprediction framework.\n\nSimilarly, Golowich et al. (2024) focuses on distribution steering.\nHowever, their results do not inform the model provider about the unintended\ndistribution changes due to the performative risk minimizaiton.\n\nAs for Miller et al. (2021), we will make sure to add this paper in the existing\ndiscussion of learning algorithms in the performative setting.\n\n**The main weakness is that their setup is perhaps overly simplistic.**\n\nOur work indeed analyzes the specific learning problem of binary mean estimation (as clearly acknowledged in the title). Already on this classic learning problem, our work points towards potential shortcomings/side-effects of performative risk\nminimization. Therefore, while do not make specific claims for other settings, we hope that our work will encourage further exploration of the impact of performative methods.\n\nWhile the technical results are specific to the considered setup, we hope that our analysis can be a useful starting point for future exploration of other learning tasks for several reasons.\n\nFirst, while we only analyze the linear response model, we note that the\nlinear response naturally arises as a first-order Taylor approximation for any\nperformative response. Thus, we believe that our results will (at least\nqualitatively) transfer to **situations of weak performative response** (i.e.,\nsmall $|\\alpha|$).\n\nSecond, as we noted in the discussion of Lemma 3.2, **the error-uncertainty\ndecomposition holds for a broad class of distributions**, not only the\nBernoulli distribution. Thus, we can expect PRM to prefer distributions with\nless aleatoric uncertainty, which are usually more degenerate. For example, in\nthe case of the multinomial distribution, the model provider would try to\nconcentrate the probability mass on a small subset of outcomes.\n\nThird, our results can easily be extended to the following **more general group\nsetting**. Imagine that clients consist of several independent groups, and each\ngroup reacts to the predictions of the model in the same way as the whole\ndistribution in our paper. Also, assume that the model provider additionally\ngets some covariates that determine group membership before making a\nprediction. This modification makes our problem much closer to the usual\nsupervised learning tasks where the model provider needs to simultaneously\nlearn a model for membership prediction and outcomes for each group. At the\nsame time, our results in the perfect information setting can be directly\napplied to this setup by independently applying the previous analysis to each\ngroup. The main limitation of such an extension is the assumption that groups\nevolve independently. This assumption could hold in the setting of drug\nefficacy prediction, but it will probably not hold in\ntraffic prediction.\n\n**Typos**\n\nThank you! We will fix these typos in the next version of our manuscript.", "claims_and_evidence": "Yes, all the claims in the submissions are supported by formal results.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I did not check the proofs in detail.", "experimental_designs_or_analyses": "The experimental designs are just visualizations of their theoretical results. These are valid and sound.", "relation_to_broader_scientific_literature": "The paper contributes to the emerging area of performative prediction. Relative to other papers in this area, they focus on a very specific setup and aim to understand the long term impacts of predictions on the distribution instead of designing algorithms to exactly solve for optimal solutions. \n\nSome of their main conceptual takeaways related to their finding that the performatively optimal solution is biased were known, at least in limited regimes. See for instance the biased coin toss example in the initial paper in the area (Perdomo et al 2020). However, they illustrate these ideas in a much richer setting.", "essential_references_not_discussed": "I think that the paper would benefit by including a discussion to the following papers.\n\nKim & Perdomo (ITCS 2023). This paper studies a very similar setup where a learner optimizes an outcome performative binary outcome and designs learning algorithms that can both forecast outcomes accurately as well as steer them towards particular targets. (e.g. optimize the likelihood that y=1). Given the similarities in setup and focus on the broader impacts of prediction on the data beyond just accuracy, I think the paper would be improved by including it in the related work. Similarly, Miller et al ICML 2021 was one of the first papers to think about algorithms for performative optimality. \n\nOutside of performative prediction, their dynamics model is nearly identical to the dynamics considered in Golowich et al Neurips 2024 in their paper on online control of population dynamics. \n\nThe focus of all of these papers is different to that considered here, but very much complementary. The reader would benefit from knowing how this work relates to the broader literature.", "other_strengths_and_weaknesses": "The strengths of the paper is that it provides a lot of insight and a comprehensive analysis of the problem they study. The main weakness is that their setup is perhaps overly simplistic. They think about a scalar optimization problem over binary outcomes where the dynamics are highly constrained. It is not clear how these insights apply to other settings.", "other_comments_or_suggestions": "The estimators paragraph at the bottom of pg 5 has a typo “extend” ---> “extent”. Also it references Theorem 4.1 when it should be Proposition 4.1?", "questions_for_authors": "NA", "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": "", "extra_scores": {}}, {"review_id": "jdxUsAAZ57", "reviewer": "Reviewer_y6Ux", "summary": "The paper devises two metrics to study the path taken for models in a performative setting and applies them in the study of binary random variables (shifted Bernoulli).", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the detailed and constructive review! We address your concerns and questions below and remain available for further discussion.\n\n**The paper never defines what they understand by PRM … They cite three papers but those solve different optimization problems**\n\nThank you for your comment. The optimization problem (1), whose solution we analyze, in the special case of T = 1 and slow deployment, is exactly the classic problem of finding a performatively-optimal model, introduced in Perdomo et al. The cases of T > 1 and rapid deployment cover extensions discussed in Brown et al. and Ray et al..\n\nBy Performative Risk Minimization we simply mean the solution (sequence of models) which minimizes the performative risk. This is exactly the solution to problem (1), which we introduce at the start of Section 3. It is also equivalent to finding a performatively-optimal path. \n\nWhile many works develop methods for finding performatively-stable models instead, this is arguably because of the hardness of finding performatively-optimal points in general settings. Already Perdomo et al. argue (see Section 2 in their ICML version) that minimizing the performative risk is a natural objective in performative prediction. They also show that, under some assumptions, stable and optimal points are “nearby” (their Section 4). \n\n**The usefulness of the metrics cannot be fully supported with just the example of binary random variables … The paper seems like an extension of examples used before in papers like Perdomo et al. (2020) or Izzo. et al (2022).”**\n\nThe studied learning problem is indeed less general than those usually considered in the performative prediction literature. We refer to our response to reviewer 5zXv for an explanation of how our work can serve as a foundation for analyzing more general learning tasks.\n\nHowever, we also study a different mathematical problem, compared to that prior work. \n\nWorks in performative prediction focus on finding performatively optimal/stable points. In contrast, we study the properties (e.g. bias and shift) of models that minimize the performative risk (assuming that such are already found). This requires a different set of mathematical tools, as existing assumptions (e.g.  convexity, smoothness, Lipschitzness w.r.t the Wasserstein distance) from prior work are suited for optimizers searching for optimal/stable points and do not necessarily facilitate an analysis of our problem. \n\nWe restrict to the analysis of a mean estimation problem for two reasons:\n\n- First, to understand the impact of choosing performatively optimal models on the distribution, we need structural assumptions on how the distribution changes as a function of the deployed model. To our awareness, such assumptions have previously only been proposed in specific examples (e.g. those referred to by the reviewer, location families etc). Creating plausible models of how general distributions change in performative settings is indeed an interesting research direction, but orthogonal to our work. \n\n- Second, obtaining theoretically-precise results about the impact of PRM was already challenging for binary mean estimation, as is evident by the need to resort to simulations in more advanced cases. We certainly agree that exploring other learning setups in the future is an important research direction, but we expect that this will have to be limited to simulations, due to likely mathematical intractability. While we hope that our work can serve as a useful foundation for such exploration, this is beyond the scope of our theoretical paper. This is also clearly reflected in the paper title and the contributions part of the introduction.\n\n**Related work**\n\nThank you for the suggested papers. We will cite and discuss them. \n\nWe agree that minimizing the performative loss does not always translate to accuracy. We indeed intended to say that other works focus on loss minimization. We will edit the text as suggested and cite the works in the related work section. \n\nWe will also cite Jaagadeesan et al. 2022 and Mandal et al. 2022, alongside with Brown et al. (2022) and Ray et al. (2022a). All these works indeed motivate our loss (1), whose minimizers we study throughout the paper. \n\nWe do not see a link between our mean shift metric and the regret of Jaagadeesan et al. Could the reviewer clarify?\n\n**Would it be possible to apply the metrics to other problems?**\n\nWe again refer to our response to reviewer 5zXv for a thorough discussion. \n\n**Isn’t this conclusion expected, as there is a distribution shift? Why are all mathematical derivations relevant?**\n\nOur theory not only proves the presence of, but also quantifies the shift and bias, for a wide range of problem parameters. The reason we put such a general comment as a summary was that conclusions about the direction and amount of shift and bias are, in general, case-dependent. We have described the more detailed, case-specific conclusions throughout the technical sections.", "claims_and_evidence": "-   They claim that they analyze the impact of Performative Risk Minimization. However, as far as I can tell, there is no such thing in the previous literature (especially not in Perdomo et al. (2020)) and the paper never defines what they understand by PRM. They cite three papers but those solve different optimization problems: Perdomo et al. (2020) and Brown et al. (2022) optimize for the stable point, while Ray et al. (2020) optimizes for the optimal point. It is unclear in the beginning of the paper what they are trying to do and what is their objective. In addition, they claim that Performative Prediction tries to minimize the objective function presented in (1). However, the standard definition of Performative Prediction in Perdomo et al. (2020) does not fall into this equation. It is unclear how the other papers relate to equation (1) too.\n\n-   The usefulness of the metrics cannot be fully supported with just the example of binary random variables. I would suggest extending the analysis for other problems (like what other papers in the field – like Izzo et al. (2022) – do). Furthermore, there is lack of motivation of the binary problem; why is it relevant for Performative Prediction? A slight justification only appears in the last section with the use cases. The case studies are too simplistic to be considered realistic and, therefore, it is hard to understand why this is an interesting problem for Performative Prediction.", "methods_and_evaluation_criteria": "-   The claims on the binary random variables are supported mainly by theoretical evidence, which make sense in this case.", "theoretical_claims": "-   I have not checked the correctness of proofs. The assumptions (loss and problem) of the problem should be better explained (see **Claims and Evidence**)", "experimental_designs_or_analyses": "-   I did not reproduce the results of experimental experiments", "relation_to_broader_scientific_literature": "-   This paper adds a couple of metrics to study the bias of estimators in Performative Prediction.\n\n-   The link of the optimization algorithm they use “performative risk minimization” and the loss is not very clear\n\n-   The paper seems like an extension of examples used before in papers like Perdomo et al. (2020) or Izzo. et al (2022). Nevertheless, in these papers they used them to exemplify their claims, not as their main results.", "essential_references_not_discussed": "- The paper doesn't discuss Lin & Zrinic (2024), Izzo et al. (2022)  and *Performative Prediction with Neural Networks* (Mofakhami et al. 2023) when saying  “To our awareness, the only work that studies properties beyond accuracy is that of Jin et al. (2024)”. Works in Performative Prediction study the Performative Loss generally, which sometimes might not transfer into accuracy (see *Performative Prediction in NN*). Nevertheless, others like Izzo et al. (2022) or Lin & Zrinic (2024) study how the model converges to the closed-formed solution. This could be fixed if accuracy was changed by “performance”.\n- The paper doesn't discuss how the regret proposed in *Regret minimization with Performative Feedback* (Jaagadeesan et al. 2022) is similar to the mean shift that you propose in your paper. Its citation is also missing.\n- The paper lacks a discussion of how your equation (1) is similar to the discounted loss that they consider in *Performative Reinforcement Learning* (Mandal et al. 2022). Its citation is also missing.", "other_strengths_and_weaknesses": "-   Strengths\n\n    -   The paper identifies and explores the impacts of Performative Prediction with a broader scope that just performance of the models, which is an very interesting direction for the Performative Prediction field.\n\n-   Weaknesses\n\n    -   The problem studied is too narrow. Would it be possible to apply the metrics to other problems? How would the conclusions be different? Why is the binary random variables problem key for Performative Prediction?\n\n    -   Conclusions are too shallow for the number of theoretical derivations, experiments and plots. (“Our results suggest that the performatively optimal (PRM) path is, in general, biased and introduces a non-zero mean shift”). Isn’t this conclusion expected, as there is a distribution shift? Why are all mathematical derivations relevant?\n\n    -   Relation to the previous literature is not clear enough (See claims and evidence, first point)", "other_comments_or_suggestions": "none", "questions_for_authors": "See weaknesses.\n\nI would be willing to change my evaluation of the paper if the metrics were applied to other problems or if it is better justified why the binary random variables problem is key for Performative Prediction. Furthermore, authors would need to show if these metrics can be used to obtain deeper conclusions. Another direction to improve my evaluation can be to study more realistic case studies where the effects of biased paths can be shown.", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper studies how performative risk minimization affects quality of prediction and distribution shift in a particular setting of interest. The paper establishes several theoretical results for several study cases and complements these with simulations. Reviewers were generally positive about the paper's contribution, and appreciated the authors' efforts to study the long-term impacts of performative learning beyond the narrow goal of risk minimization. At the same time, reviewers felt that the particular setting studied was perhaps too simplified – albeit natural for the questions considered, this made it difficult to see how results would generalize to more realistic and practical settings. Some reviewers also found the related work and the connections currently made in the paper require some additions and a revision. The authors' rebuttal was helpful in addressing some of the concerns and clarifying several points. The authors should take all point made in the reviews into account and integrate them into the paper.
Decision: Accept (poster)