Look-Ahead Reasoning on Learning Platforms
Authors: Haiqing Zhu
Tijana Zrnic
Celestine Mendler-Dünner
Venue: NeurIPS 2025 poster ()
Keywords: performative prediction
strategic classification
collective action
performative shift
performativity
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: We analyze how collective strategic behavior influences predictive models, introducing level-k reasoning and showing how user coordination affects equilibrium outcomes.

Abstract: On many learning platforms, the optimization criteria guiding model training reflect the priorities of the designer rather than those of the individuals they affect. Consequently, users may act strategically to obtain more favorable outcomes. While past work has studied strategic user behavior on learning platforms, the focus has largely been on individual strategic responses to a deployed model, without considering the behavior of other users. In contrast, *look-ahead reasoning* takes into account that user actions are coupled, and---at scale---impact future predictions. Within this framework, we first formalize level-k thinking, a concept from behavioral economics, where users aim to outsmart their peers by looking one step ahead. We show that, while convergence to an equilibrium is accelerated, the equilibrium remains the same, providing no benefit of higher-level reasoning for individuals in the long run. Then, we focus on collective reasoning, where users take coordinated actions by optimizing through their joint impact on the model. By contrasting collective with selfish behavior, we characterize the benefits and limits of coordination; a new notion of alignment between the learner's and the users' utilities emerges as a key concept. Look-ahead reasoning can be seen as a generalization of algorithmic collective action; we thus offer the first results characterizing the utility trade-offs of coordination when contesting algorithmic systems.

Look-Ahead Reasoning on Learning Platforms
Haiqing Zhu Australian National University haiging. zhu@anu. edu.auu.dd.aaa. au
Tijana Zrmic
Stanford University
ti jana. zrni crmmiiioaaoordeee
Celestine Mendler-Dünner
ELLIS Institute Tübingen MPI for Intelligent Systems, Tübingen Tübingen Al Center celestine@tue wtw.:/eeeteeeells.eeeeee...
Abstract
On many learning platforms, the optimization criteria guiding model training reflect the priorities of the designer rather than those of the individuals they affect. Consequently, users may act strategically to obtain more favorable outcomes. While past work has studied strategic user behavior on learning platforms, the focus has largely been on individual strategic responses to a deployed model, without considering the behavior of other users. In contrast, look-ahead reasoning takes into account that user actions are coupled, and-at scale-impact future predictions. Within this framework, we first formalize level-k thinking, concept from behavioral economies, where users aim to outsmart their peers by looking one step ahead. We show that, while convergence an equilibrium is accelerated, the equilibrium remains the same, providing no benefit of higher-level reasoning for individuals in the long run. Then, we focus on collective reasoning, where users take coordinated actions by optimizing through their joint impact the model. By contrasting collective with selfish behavior, we characterize the benefits and limits of coordination; a new notion of alignment between the learner's and the users' utilities emerges as a key concept. Look-ahead reasoning can seen as a general- ization of algorithmic collective action; we thus offer the first results characterizing the utility trade-offs of coordination when contesting algorithmic systems.
Introduction
Digital platforms deploy learning algorithms that collect and analyze data about individuals power services, personalize experiences, and allocate resources. As people come to understand how these systems make decisions, they often adapt strategically improve their outcomes. Prior research has largely modeled such strategic behavior as unilateral: each agent responds to the platform's 's decision rule by optimizing their outcome while treating that rule fixed. For example, a job applicant might rephrase their resume to include keywords that align with an automated screening system's preferences. This perspective neglects the fact that many others may be doing the same- thereby collectively shifting the data distribution from which the platform learns in the future.
reality, there ample empirical evidence that users frequently reason about one another's behavior. They may act in solidarity (Tassinari and Maccarrone 2020), anticipate other people's adaptations to gain an advantage [Kneeland] (2015), or coordinate to amplify their collective influ- ence [Chen 2018), in the latter case oftentimes facilitated by labor organizations [e.g., Rideshare Drivers United]. On learning platforms particular, such reasoning involves not only anticipating the behavior of other platform participants but also how those behavioral changes will impact the
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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learning algorithm in the future. We call this look-ahead reasoning. In the resume- screenin example, look-ahead reasoning might surface as choosing to emphasize distinct keywords that others have abandoned, anticipating that popular buzzwords will lose predictive value as they become widespread.
1.1 Our contributions
We characterize how look-ahead reasoning user behavior that anticipates the actions of others in the aopulation-reshapee learning dynamics and equilibria on learning platforms. We begin with selfish agents, who strategically account for the other agents' responses to the deployment of a predictive model and act independently. We then turn to coordinated behavior, where agents strategize together and account for their joint influence on the learner. Finally, contrasting the two settings allows us to characterize the benefits and limitations of coordination when contesting learning platforms.
To capture agents who selfishly aim outsmart their peers, formalize the concept of level-k: thinking [Nagel 1995) from behavioral economics in the context of learning systems. Level-k thinking captures different depths of strategic thought: level-k thinker acts assuming they are "one step ahead" of everyone else, who is level-(k - -) thinker. A level-0 thinker is non-strateicc. Higher levels & are defined recursively. In our setup, actions are determined by hallucinating the data distribution resulting from the assumed behavior of other agents and best-responding to the predictive model induces. We study the dynamics of repeatedly retraining model acting on a population of level-k thinkers. We show that "deeper" thinking achieved for larger l accelerates the learning dynamics, while resulting the same equilibrium solution, no matter the depth of thinking.
Theorem 1 (Informal), For k 2 1, let Ogk E (0, 1) be the fraction of level k-thinkers in the population, Ex=1 Okk = 1. Assume the learner minimizes a loss function that is smooth and strongly convex, and suppose that the agent responses are sufficiently Lipschitz in the model parameters. Then, for some constant e (0, 1), repeated retraining converges to unique stable point at rate
80
Therefore, even if it relies on deeper reasoning, selfish behavior does not improve the agents' utility. Next, we investigate whether agents can move past this obstacle if they coordinate. Look-ahead reasoning with coordination allows steering model updates by systematically taking into account that the learner learns from the data agents report. We show that the gap between coordination and lack thereof in terms of agent utility is governed by notion of alignment between the objectives of the learning platform and the population. Below, use f(2, 0) and u(2, to denote the learner's 's loss and the agent utility for deploying model 0 on instance z. Furthermore, (a, M := a Mb.
Theorem 2 (Informal). Let DD and Då denote the population 's data distributions at equilibrium under selfish reasoning and under collective reasoning, respectively. Then, assuming regularity conditions, the benefit of coordination B, defined as the difference in population utility at the two equilibria, is bounded as
B (BENDD- [Vou"), (V ()
where H' = EEND- [Vgl(z, )*] and H* denotes the equilibrium model under selfish reasoning. We use the short-hund notation Vou" Vau(z, 0*), Vol Vol(z, 0*).
If the average agent utility and the average loss of the learner are orthogonal, there is no benefit to coordination. Similarly, when u zero. However, when there is the right amount of overlap between the objectives that the collective can exploit, coordination can lead to more favorable outcomes than selfish reasoning. For example, think of instance & as a feature-label pair (z. and consider a learner and a collective who both care about predictions fo (z)): the learner cares about making them accurate, and the collective cares about them having certain favorable values. Then, modifying the labels y can be an effective strategy for the collective to steer the predictive model 0, something that individuals reasoning selfishly cannot achieve, enabling a large benefit of coordination. We elaborate on this example later on.
for either >> 0 or < 0, the benefit of coordination is
additional results, we study heterogeneous populations and collectives of varying size. The results explain why larger collectives, despite more leverage, do not always lead to a higher utility for participating agents. They also show how broader participation the collective stabilizes learning dynamics.

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1.2 Background and related work
Strategic classification [Hardt et al. 2016 Brückner and Scheffer 2011] introduces a model to study strategic behavior in learning systems based on assumptions individual rationality. It describes population of agents best-responding to a decision rule by altering their features to achieve positive predictions, given fixed decision rule. Several variations of this basic model have been studied
[e.g.. Dong et al. 2018 Chen et al. 2020 Bechavod et al. 2022 Ghalme et al. 202 Jagadeesam al. 2021 Levanon and Rosenfeld 2022]; see Podimata [2025] for a recent survey of this literature. All these works focus on studying how agents strategize against a fixed decision rule. Our work introduces a new dimension of reasoning to strategic classification, taking into account how individual agents' actions are coupled and how this influences the predictive model the agents strategize against.
Performative prediction [Perdomo et al. 2020) introduces performative stability as an equilibrium notion that characterizes long-term outcomes in the interaction of a population with a learning system. Performative stability is a fixed point of repeated retraining by the learner in a dynamic environment. Prior work in performative prediction [e.g.. Perdomo et al.; 20200] Hendler-Dünner et al. 2020
Drusvyatskiy and Xiao 2023, Brrwnn et al. | Narang et al. [023] has studied the behavior of
retraining and conditions that ensure its convergence stability in different learning settings. We refer to Hardt and Mendler-Dünnet [2025] for an overview of the performative prediction literature. A key concept performative prediction is the "distribution map." " which characterizes how different
model deployments impact the population. In convergence analyses this map is typically treated as a fixed and unknown quantity. We study how different types of strategic reasoning impact the distribution map, thus also impacting the resulting convergence properties and equilibria.
A more recent literature on algorithmic collective action Hardt et al., 2023] studies coordinated agent efforts with the goal of stecring learning systems;
Baumaan and Mendler-Dünne 2024] Ben-Dov& et al., 2024] Gauthier et al., 22025] Sigg et al. [2025] for recent developments in this area, as well as related discussions of data leverage Vincent et al. 2021 and protective optimization technologies Kullnnych et al. 2020]. From the perspective of our work, collective action is a type of look-ahead reasoning: agents plan through model updates under the assumption that they coordinate with other agents. In this work we study the tradeoffs and implications of coordination on the utility of agents participating in the collective. Relatedly, |Hardt et al. [2022] discuss how platforms can reduce risk by actively steering a population. Collective action reverses this perspective and investigates how the population can improve its utility by steering the learner. This perspective is related to [Zmic iet al. 2021], who also deviate from the classical model of strategic classification and instead model the population as the leader in the Stackelberg game against the learning platform. Our framework aims to bridge strategic classification and algorithmic collective action to illuminate the benefits and limits of coordination.
Setup
We consider population of individuals interacting with a learning platform. We assume the platform trains a predictive model on the population's 's data, and individuals strategically alter their data to achieve favorable outcomes. We elaborate below.
Learning platform. Upon observing data about the population, the learner optimizes the parameters 0 E ① of their predictive model fo- We work with the following optimality assumption on the learning algorithm: given loss function f, the learner's 's response A(D) to data distribution D is given by risk minimization, defined as
A(D) arg min EEWD [[(2 BEB
Strategic agents. We assume individuals are described by data points 2 e sampled from a base distribution Do. Typically, z = (I, v) E X' x y are feature-label pairs. Individuals implement a data modification strategy hg : + Z that maps an individual's data point a to a modified data point h2(;;; the strategy can depend on the learning platform's 's currently deployed model 6. We will sometimes omit the subscript 0 if the strategy is independent of the current model, i.e., he = he- for all 8, 8'. We use Dr,, to denote the distribution of ho(2) for 2 ~ Do: in other words, this is the distribution of data points after applying strategy hg to all base data points. Following the terminology of Perdomo et al. [2020). we call Dr, distribution map. Note that different strategies he correspond to different distribution maps. When the strategy clear from the context, we will write Dr,, = D(0). The notation Dr, equally applies to modell-nddpendert strategies h.
3

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Equilibria and learning dynamics. We study the long-term behavior of the learner repeatedly optimizing their model and the population strategically adapting to it. Formally, we study the learning dynamics of repeated risk minimization:
B1 ADhh,.)
The model 0 is repeatedly updated based on observed data from the previous time step. The natural equilibrium of these dynamics called performative stability (Perdomo et al. 2020). We say a model 8" is performatively stable with respect to a strategy ha if
0" AD
In words, there is no reason for the learner to deviate from the current model, given observations of the strategic response of the population.
Population utility. Different strategies he lead to different equilibria. Rather than just focusing on the learner's ' loss, we evaluate equilibria in terms of the utility they imply for the population. Let 6 denote the equilibrium under strategy hgg i.e., the performatively stable point. Then, Dig- denotes the equilibrium distribution. We denote the population's 's utility after implementing strategy ha by
U(hg) Ez-Pa,- [u(2,
where u(z,0) is the utility of an individual with data point z when the deployed model is 0. Note that the stable point satisfies 0* = A(Dr..) ) and thus the utility is evaluated against the data that determines the predictive model. Again, we will sometimes omit the subscript when denoting the strategy if it is independent of the deployed model.
3 Level-k reasoning
Strategic classification (Hardt et al., 2016) assumes that each individual selfishly best-responds to a deployed model 0. As explained earlier, this model does not account for the agents' awareness that they, as a whole, determine the deployed model. To account for this dimension of reasoning, we build on the cognitive hierarchy framework from behavioral economics [Nagel] 1995] and generalize strategic classification to allow individuals to reason through the other individuals' responses. In particular, we formalize level-k thinking, which categorizes players by the "depth" of their strategic thought. Intuitively, an individual reasoning at level k assumes a level of cognitive reasoning for the rest of the population and tries to "outsmart" them. In other words, they are always one step ahead: a level-k thinker best-responds to the model that would result from a population of level-(A - 1) thinkers. The basic level-k model starts with an explicit assumption about how individuals at level Ó behave. It then defines higher levels of thinking recursively.
Suppose that agents at level 0 are non-strategic and implement hi Then, for every higher level of thinking k 2 1 we define the strategy for level-k thinkers recursively:
(0) (z) = z in response to all 0.
h (z) == argmax./ u(s', A(D)
(1)
where (0) is the distribution obtained by applying the model-dependent strategy hg (k-1) to every 2 ~ Do- At level = 1, we recover the standard microfoundation model of strategic classification [Hardt al. 2016), where individuals best- rest-respond to a fixed model. For larger k, the agents anticipate the actions of other agents and best-respond to the hypothetical model resulting from the shifted distribution. The hypothetical model being 0' A(Dx-1(0))
Different individuals in the population might implement different levels of reasoning. To reflect this we deviate from a homogeneous population and let the population consists of level-k thinkers at different levels k. In particular, we assume that fs.-fracion of the population has cognitive level k, for 1, 2.... and Σ ae = 1. ar = 1 for some k. then all individuals the population have the same level of reasoning. This model results in the distribution map:
D(0) =
a4 (t)).
We characterize the learning dynamics for different levels of thinking. We use the following Lipschitzness assumption on the induced distribution at level = 1:
(O), Di (0)) < 0 0'1l2, v0,0' E O,
where W denotes the Wasserstein-1 I distance. In performative prediction this condition is known as
e-sensitivity (Perdomo et al. 2020].

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Theorem 3 (Retraining with level-k thinkers). Suppose the loss of the learner is y-sstrongly convex in and 8-smooth in 2, and that the distribution map Di(0) (0) e-sensitive. Then, long as e there is a unique stable point 0r such that for any (oxx) k=1 retraining on the mixed population converges as
1 0kk
20
EB
110, 00 S
Σ
1100 0
The core technical step in the proof is to derive how the sensitivity of the distribution map De(0) changes recursively with k. In particular, the distribution map D(0) in (2) has sensitivity --1
Ek=1 on (eB/>)(-1,, e. We refer to Appendix A for the full poof.
For the case where 01 = and thus all agents reason at level k = 1, we recover the retraining result of Perdomo et al. 2020]. There are interesting implications of the generalization in Theorem 3 First, we observe that for populations with higher levels of thinking k; the rate of convergence increases (although the condition for convergence, e < 7/3. remains the same). This can be interpreted as saying that performative distribution shifts are mitigated when the population has deeper level of strategic thought. The second implication is that, as long the agents act selfishly, they cannot benefit from higher levels of reasoning at stability.
Under the assumptions of Theorem 3 holds that U (( = = Uh)))) Vk 2 1. Moreover, the utility at stability remains unaltered for any mixed population consisting of level-k thinkers regardless of (a=)R==
Corollary 1.
This corollary follows from the observation that the stable point 0* the same for any mixed population of level-k thinkers. Another consequence of this fact is that the equilibrium strategies hg- and hence the induced distributions D& (B*), are identical for every We will denote this unique optimal selfish strategy by h" hg- and the implied data distribution by D
4 Collective reasoning
Level-k thinkers anticipate model changes implied by the population's 's actions. We saw that higher levels of selfish reasoning do not improve the agents' utility at equilibrium. The fundamental reason is that individually they cannot steer the trajectory of the learning algorithm: they can merely anticipate it. In the following we show how individuals can achieve more favorable outcomes by joining forces and making decisions collectively; this gives them steering power.
We denote by ha the optimal collective strategy:
htt arg max (h) arg max ELND, [u(2, A(D))]
h
Notice the difference compared ①.. In I the optimization variable 2' does not enter the model training A, while above h directly determines the subsequently deployed model. The optimal collective strategy is Stackelberg equilibrium: the population acts as the Stackelberg leader.
To contrast the optimal collective and selfish strategies, we define the benefit of coordination.
Definition 1 (Benefit of coordination). Let ht" be the optimal selfish strategy and h² the optimal collective strategy. We define the benefit of coordination B Uh²)) Uh²).
Since hF the globally optimal strategy for the population, it holds that B 2 How large B is depends on the goals pursued by the learner and the population, as characterized by & and u, respectively. Through coordinated data modifications, the population can steer the model towards a common target. But to do so, they may have to deviate from the individually optimal strategy. Thus, what governs the benefit of coordination is the tradeoff between the return of steering the model to a better equilibrium and the cost of being evaluated against the modified data.
We start with a simple case where the benefit of coordination is zero. Proposition 4. Suppose u = c l for some + Then, it holds that B
There are two different mechanisms at play, depending on the sign of c. In an adversarial setting where e > O, the game between the learner and the collective is zero-sum game. In this case the
5

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benefit of coordination is zero, as the cost of steering is equal to its return. When c < O, the platform and the agents pursue the same goal and the game becomes a potential game between the two. In this case the benefit of coordination is zero as selfish actions are simultaneously optimal for the collective and there is no benefit to steering the model away from the selfish equilibrium. In general, however, the benefit of coordination can be arbitrarily large. Consider the following example.
Example 1 (Label modifications as an effective collective lever.). Consider a learner who aims to accurately predict labels from features and a collective that prefers these predictions to follow target function 9- We assume that agents can easily manipulate their labels, while their features cannot be changed. The population and the learner have the following utility and loss, respectively:
u(z. 0) -ff(())) -
((2.0) = (fo(x)-))) y)²
In this setting, the collective has a powerful lever to steer the model through label modifications. The optimal collective strategy is clearly (²()) = (I, g(x)). In contrast, selfish agents have no leverage over the learner and would simply report (z) (ær, y). Assuming there exists O such that fo (z) = E[y] this gives a benefit of coordination equal to the suboptimality of the labeling function B- ED [Ely|z] - g(z))², which can be arbitrarily large, depending on Do and g.
To characterize when the benefit of coordination large, we consider linear distribution maps. A generalized version of the result can be found in Appendix [A.6
Assumption 1 (Linearity). ). Let each strategy h(л) be represented by a parameter vector 1J E R. We say that the induced distribution is linear with respect to the parameterization for all a E [0, 1]. (1(an+(1-a))"" = aDDMn)) + (1 o))Pirr)
Intuitively, linearity means that the population's is data distribution is the same whether agents linearly interpolate between two strategies r) and n/', or they split up in two subgroups and each implements one of the two strategies. Under this assumption, the following result provides a bound on the benefit of coordination. The proof can be found in Appendix AA]
Theorem 5 (Benefit of coordination). Let Assumption I hold. Let (h((r)) be 7-strongly concave in and (ah + (1 - a)l(') be difserentiable with respect to a. Then, we have B S L $2 where
24
= (E [Vu((z ",,),,E,D,, [Vol(z, 0)) (H.)-- (=))) and
H- H,ED,
Va [o
This result shows how the benefit of coordination is governed by the alignment between the utility of the population and the loss f of the learner, quantified by the inner product of their gradients at the selfish equilibrium 0", We refer to & as the alignment term.
We have t O in the case where the gradients are orthogonal and the two functions and & are unrelated; in this case agents can optimize their utility independent of the model and there is no benefit to being able steer the model. Further, in line with Proposition 4 when u é for some constant c, we have &- = O because selfish actions are simultaneously collectively optimal and shifting the model away from 0* would not benefit the collective. In particular, 0" is local optimum of the loss under DA-- and thus the first term in the inner product becomes zero
To see cases where 2 can be large, we look at the individual terms in its definition. The first term in the inner product that defines & describes the suboptimality of 0 for the agents under Da. It is non-zero if the collective's 's utility on DD,- can be improved by moving the model away from 0", The second term in the inner product captures the response of the learner to the collective strategy. For this term to be non- non-zero the data modification DA must make 6 suboptimal. Consequently, & is large both terms in the inner product non-zero, and there is overlap in the direction of improvement for the learner and the collective. Moreover, what really matters is not just the raw gradient alignment but one filtered through the local curvature of the loss landscape. The directions that the learner finds "flat" (small Hessian eigenvalues) allow for more influence on the model through small data modifications, and thus they offer more leverage for the collective. We provide example satisfying the assumptions of Theorem 5 with non-trivial closed-form expression for t in Appendix B
Let us revisit Example | Although the example does not satisfy the assumptions in Theorem 5 the alignment term provides the right intuition for why B is large: since the learner aims to accurately predict labels, any systematic label modification induces a model change. The collective can fully control the direction of these changes, i.e., the gradient of the loss, and align it with their objectives. The label modification strategy leverages this to increase the agents' utility, providing non-zero return as long as 8" is suboptimal for the collective under Dr..

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Heterogeneous populations
A perfectly coordinated population where every agent participates, or one that implements the optimal collective strategy, is unlikely to emerge practice. In the following we consider some plausible deviations from the idealized collective studied in the previous section. Unless stated otherwise, we assume the collective implements any fixed strategy h (independent of f), which could be simpler alternative to a potentially hard-to-implement optimal strategy ht.
5.1 Scaling strategies in the presence of non-strategic agents
First, we consider a setting where only a fraction of the population participates in the collective and study conditions under which it is worth scaling up strategy h, meaning larger collective implies higher utility for the collective. To study how the collective's 's utility changes with its size, we consider the following mixture model for the population as proposed in (Hardt et al. 2023]:
D aD + (1-a) Do
where an a- a--raction of the population implements the collective strategy h and the remaining (1 (1- -+)- fraction is non-strategic. Here, the collective strategy h can be any fixed strategy, not necessarily the optimal one. We are interested in the average utility for agents participating in the collective, denoted
as Ue := Eg [u(z; 02)], where 0% is the equilibrium under the mixture model (3).
Proposition (Benefit of scaling up strategy). Consider the mixture model in 3). fix a strat-
egy h, and denote the resulting equilibrium by 02. Then, the benefit of scaling up the strat- egy h for collective of size is positive, i.e., AUa. "Bs > 0, isand only if V < O, where
y = (EewD, [Vou(z; 0-]] BEWDA [Vol(z; (2)1) and H == ViLLaDD [6(2; 05)1
Whether strategy is worth scaling up or not is again linked to a notion of alignment described by V. To provide intuition for the result we consider some special cases. Suppose H is positive semidefinite, which happens when É is convex; then, if u = f, v 2 O, and if -t, V < 0. This means that, under convex losses, when considering the utility of participating agents, scaling up a fixed strategy is always harmful in zero-sum game, and it is always beneficial when the learner and the population optimize the same target. Note that the result holds for any fixed strategy h. We provide additional intuition and empirical insights into how Ua and V/ change with a in Section
Optimal size-aware strategy. Next, we aim to understand what collectives can achieve if they are aware of partial participation and optimize their strategy accordingly. We define the optimal size-aware collective strategy for size as: h arg max, Ez-D, [u(z; A(aD, + a1-a)Do)]:
The global Stackelberg solution h² corresponds to the case where corresponds to the population utility. In the case where a < 1, the collective optimizes the utility of participants, rather than the full population. Agents are informed of their collective size and will choose the best strategy hig accordingly. In the following proposition, we characterize the utility of agents participating in collective that deploys size-ave aware strategy. We use Uü to denote the utility of population of size a implementing the optimal size-aware strategy hif--
1 and the collective utility
Proposition 7 (Benefit of larger collectives). Consider the mixture model (3) with collective of size a implementing the optimal size-aware strategy h = hi,- Then, the average collective utility U aU% achieved by implementing hy, satisfies "a 2 0 if and only if aUa.
-= 2 0.
Note that the derivative takes into account the dependence of the strategy on a. Thus, the result implies that reoptimizing the strategy as function of collective size does not change whether scaling up worth it or not. All that matters the alignment of the pursued goals.
5.2 Learning dynamics in the presence of selfish agents
Finally, we study how partial participation impacts learning dynamics of repeated risk minimization. For a = and fixed strategy h that independent of / the learning dynamics converge a single step. This also holds for a < 1 in the presence of non-strategic agents. However, this changes as soon as agents deviating from the collective strategy act selfishly. To reflect this scenario we consider the following alternative mixture model:
(0) aD, + (1- -D))),
(4)
7

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10.1- 011
+ 08,
-01,
10""
10
20
Donation
Figure I: Convergence of repeated risk minimization on mixture population of level-k thinkers. The curves show how the gap between iterates ||0 Pell2 evolves across iterations for different mixture weights. Error bars indicate one standard deviation over 10 runs.
where an a-fraction of the population implements the collective strategy and an (1 - a) fraction deviates from it. We assume these latter act selfishly and their behavior can be characterized by D(0). In particular, the actions of these agents can depend on the deployed model, such as level-k reasoning discussed in Section
We characterize the rate of convergence of repeated risk minimization under this model and show that larger collectives have the advantage of stabilizing the learning dynamics.
Proposition 8. Consider the heterogeneous population model 4 Suppose É is 7-strongly convex in and 3-smooth in z, and that the distribution map D(0) is e-sensitive. Then, as long as < }. repeated risk minimization is guaranteed to converge to a unique stable point Or at rate
10: - 1- <cB(1 (18(1-a) |10o 0%
7
This result explains how the sensitivity of the non-participating agents to changes in the deployed model, together with the fraction of these agents, determines the rate of convergence stability. For = 0 we recover the result of Perdomo et al. 2020). The smaller a the slower the convergence.
Simulations
We validate our theoretical findings empirically. We adapt the credit-scoring simulator from [eerdoo eet al. 2020) that models how a lending institution classifies loan applicants by creditworthines! We first focus on the results related to level-k reasoning from Section and then offer empirical insights into the trade-offs of collective reasoning from Section 4 and Section
6.1 Retraining dynamics under level-k thinking
Assume the learner fits a logistic regression classifier 9 using cross-entrope loss. The data has 10 features and assume agents can manipulate the subset S 'remaining credit card balance', 'open credit lines', 'number real estate loans"). }. Given some > 0, the utility the agents is given (0,z) 1 ||0 - x x1|2 where Zo is their feature value under Do- The best 2
by ue ue((z, 0)
response of the agents is given by 2 xs cAs, where indexes the strategic features. Note that this corresponds to the strategy for agents reasoning at level-1; indeed, assuming other agents are non-strategic implies strategizing against a fixed model. It is not hard to see that the resulting distribution map D (0) is e-sensitive. Under this model we simulate the repeated retraining dynamics for mixed populations of level-1 and level-2 thinkers of varying proportion.
In Figure I we report the speed of convergence by presenting the iterate gap ||+++1 61l12 against the number of iterations. We choose e = 0.5. First, we can see that under all three mixture weights the gap tends to and the dynamics converge. As the fraction of higher levels of thinking increases, the speed of convergence increases, which confirms our theoretical finding in Theorem 3 We also verified empirically that the dynamics converge unique equilibrium independent of
For the implementation of the simulation, see wwtphh//////......ooo/////////// Ahead Reasoning.
on-rennnnng- LeLeeannng-Platiimann Platforms for the dataset, see http::/wwww..agggeeccomcccceehee

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cilletive
dependaits
Allgmma 0- Itesiiy of
20-
deppendents
9.710
9.700
0.695
10
0.600
9.685
colleative
Figure 3: Collective utility decreases with col- lective size in the zero-sum case. The collective implements the optimal size-aware strategy for 0 in a mixed population with non-strategic agents. Small collectives can realize large gains, but the response by the learner impedes gains at larger sizes
Figure 2: Aligument serves as a good prexy for the benefit of coordination. We consider the utility instantiation in (5) and evaluate alignment & and the benefit of coordination for different values of A. We show them for two strategies that modify the feature "age', and #dependents', respectively.
6.2 offde-offs in collective reasoning
With the same credit-scoring data, we study the benefit of coordination and illustrate agent utility for different strategies and collective size. We again consider a learner that trains a logistic regres- sion classifier using cross-entropy loss. For the population, we consider strategies that consist of misreporting single feature. We choose this to be either 'age' or 'number of dependents'; "age" is the most important feature for the classification problem, and 'number of dependents' the least important one (see Figure the Appendix). Contrasting the two strategies allows us to vary the impact of a strategy on the learner an isolated and systematic way.
Alignment proxy for the benefit of coordination. We investigate empirically how the alignment metric in Theorem 5 relates to the benefit of coordination. We instantiate the agents' utility as
u((z, 0) l((x, 0) A 11012,
where é is the loss of the learner and the regularization term À E [o, 1] controls the alignment between the learner's and the agents' objectives. For À we get zero-sum game. The larger À the more different the learner's 's and the agents' objectives are.
Note that, this setting, the linearity and the strong concavity our theory are not satisfied. We are still interested in investigating what extent the alignment metric serves useful proxy for the benefit of coordination. of coordination B. We find that B Φ which is in line with our theory. Furthermore, alignment correlates with the benefit of coordination and accurately predicts that one strategy is more effective than the other. Recall that we restrict the strategy space to the modification of single feature; the solid and the dashed line compare two different settings.
Figure 2 use À to vary alignment and plot & against the benefit
Cost of steering and why larger collectives can be worse off. Next, we illustrate the challenges that come with large collectives in a misaligned setup. For this purpose we consider the zero-sum and sixxed-oopulation composed of a collective and non- non-strategic agents. The
setting with collective implements the optimal size-aware strategy hi for modifying each of the two features. We approximate this optimal strategy using gradient descent with learning rate 0.01 and 250 epochs.
In Figure we visualize the collective utility function of the collective size a. We that the collective utility is maximized as a + 0 and decreases with size. The gap at a = 0 shows the benefit of strategic data reporting against a fixed model. Small is advantage in the zero-sum case as agents have diminishing influence on the learner and they can move almost independently of 0. For larger collectives it becomes increasingly hard realize benefit because the model responds the agents' actions and an equivalent change to the feature has a larger effect on the learner's loss. This counter-force in the case of conflicting utilities the reason why the collective utility may decrease with the collective size. Scaling up a fixed strategy. Finally, we verify Proposition 6 emmirically. For simplicity, we assume the learner performs binary classification to predict whether person experienced 90 days past due

---

0.-
Algiment Metic Mesic
1.0
-02
Thilng
-04
-0.5
-1.
0.2
0.4
0.5 1.0
0.2
0.4
0.6
0.5
1.0
Figure 4: Change in alignment metric (left) and utility (right) with collective size, for three fixed strategies. The utility non-monotonic size, and the sign of the alignment metric v accurately predicts whether it is worth scaling up strategy or not. We consider the setting in (6 and evaluate three different strategies, corresponding to the optimal size-aware strategy hig at a (0.3, 0.5, 0.8).
delinquency or worse. The collective is uniformly sampled from the data with label and aims to maximize their utility
ue (1r, mz), - 110 uurges 11
where Ourget a fixed target model that the collective would like to achieve (see Appendix C.2 for details). We again assume the collective modify individual features and consider three strategies h =000 he =0.5- and A - where each strategy is optimal for a given collective size a {0.3, 0.5, 0.8). We refer to Appendix C.2 for details on how the strategies are computed.
In Figure illustrate the collective utility and the alignment metric V as defined in Proposition for different collective sizes We observe that, in accordance with our theory, when the alignment metric is positive, the average utilities of the collective decrease as the sizes of the collective increase. Analogously, when the alignment is negative, the average utilities of the collective increase with collective size. Notably, a approaches the assumed sizes 0.3, 0.5 and 0.8, the utility converges toward O, indicating that the model closely matches the target marger. Beyond this point, however, the collective "over-pushes" the model, leading to decrease in utility and positive V/
7
Conclusion
We study look-ahead reasoning new aspect of strategic reasoning learning platforms. While traditional analyses of strategic classification treat users as reacting independently to a fixed model, look-ahead reasoning highlights that users' incentives and actions inherently interdependent each agent's 's actions influence future model deployments, and thus the utility of other agents in the population. Within this broad theme, find that higher-order reasoning accelerates convergence toward equilibrium but does not improve individuals' long-run outcomes, suggesting that attempts "outsmart" others may offer only transient advantages. In contrast, collective reasoning-wwe where users coordinate their behavior through their shared impact on the loodel-allowe the agents to steer the model towards a desirable state. Our results show that this can be very effective lever for the collective when the loss of the learner and the utility of the population are appropriately aligned. However, for conflicting objectives, we find that the excessive steering power that comes with larger collectives can prevent large utility gains.
central goal of our work was to provide a unifying framework to contrast selfish reasoning with collective reasoning when contesting machine learning predictions. There several natural extensions of our work. One is investigating look-ahead reasoning under imperfect information. As common in economic models, assumed agents determine their actions under perfect information. In our case, this concerns knowledge about the learner's loss function and the population's is data distribution. However, in practice this information needs to be estimated from finite data which raises questions of statistical complexity. A related question would be to study how model misspecificationn estimation errors, or imperfect coordination impact outcomes in look-ahead reasoning.
More broadly, we hope our work can support recent discussions concerning data poisoning and [e.g.|Nestaas |estaas al.) 2025), and open up new pathways to transfer insights from strategic classification to algorithmic collective action, and vice versa.
multi-party adversarial attacks [e.g..
10

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Acknowledgements
Celestine Mendler-Dümn acknowledges the financial support of the Hector Foundation. The work was conducted during an internship of Haiqing Zhu the Max Planck Institute for Intelligent Systems. Tübingen.
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12

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A Proofs
A.1 Auxiliary results
Lemma 9. Let denote the Wasserstein-I distance, and Ei=1 Og 2 with a; 2 0, then
n
n
W
Σ DgD Σ Da
DD, Dj).
==1
Proof. By definition, the Wasserstein- distance can be written as
min (x(X, [||X - -Y x(X,))
where j is the joint distribution of X, and X ~ Ei= a; Ei= o,DY Consider the
measures μ4 defined
min Bx(X.)) [||; Yill] ((X;.Vi)
14
where X, Dis Y, D. Then, with Ei 0:/4s notice that
w Ea ag Lo S Ep III YI1
Σ au -||X-III] Σ (D,,
where the first inequality follows from the minimization property of the Wasserstein- distance.
Below we state bound on the benefit of coordination that does not assume linearity (Assump- tion
Theorem
Let each strategy h()) be represented by a parameter vector 1) E IRd, Suppose (h(r)) y-strongly concave 1). Then, have
5
0 B 27
Vo [E(2, (()2 (H') (H''"''EDD.'" Vou(z, 00]2,
with H* = EEEDA strategy h"
(2,0((, and 0* denoting the stable point corresponding to the selfish
A.2 Proof of Theorem
The key step is to prove Lemma 10 below. The claim in the theorem follows by combining Lemma 10
with Theorrm 3.5 in Pereoooo et al. [2020].
Lemma 10. Let ag be the portion of the population with cognitive level k. Suppose Di is e-sensitive and the loss is y-strongly convex and 3-smooth in 2, Then the distribution map D(0) ;= => k=1
(0) is E-sensitive with
EB 7
k-1
k=1
Proof. Denote 8 = (Ok-(1)))) and 80 A. Similarly, &² = (ot-(1) and 60 á. From Equation can that (D D (0h-1). Consider the map D4; then, we have the recursion:
(O),
W (0h-1) Di ( s s - 04-2
By Perdomo al. 2020), Theorem 3.5, also have
E k-1
| 10---66-11 11 <
110 Φ112
Finally, we notice that
[G3 k-1
W Σ (O), X (..Dil,,DDDDttt
Σ
S Σd ak
110
k=1
k=1
k=
where the first inequality follows from Lemma
g
13

---

A.3 Proof of Corollary
We start from Theorem
Define the contraction factor
Pa
a&
can be seen that Pr. is positive for some a, holds that the dynamies will converge to the equilibrium for any with _k=1 OK 1. Similarly, if it is zero, this holds so for any a. Thus, simple contraction argument shows that the trajectory converges the same stable point independent of a. The same holds for the special case 0*k = 1. At this point, no agent is moving and thus the equilibrium strategies h h are identical, and so are the utilities:
(K') hig
U(hTr) (8) = U(hg-))
A.4 Proof of Proposition 4
Since (h", (*) is the performatively stable point, by definition, it is clear that
Ez~D,- [ol(2, 0² 0.
Therefore, since C-C, we can conclude that IE,, [Vuu(z [r)[ = c- C--EE [ol((z,0)] 0. Then, by Theorem we have B = 0.
A.5 Proof of Theorem
We the parameterization in Assumption 1 The optimal strategy is defined as ha == h(r²) where nf == arg max, U(h(r)). The performatively stable point (h", 0*)) = (h(v)"), 0²) is the point such that
1)" = arg max Daon [u(z, 0"),
0" = arg min EN-Da [((2,
We consider a hypothetical mixture population which is represented as aD,; + a)D (M(o)+((11))7))) For this mixture population,
indicates the equilibrium of selfish actions.
This means that, if we fix the learner's action to be 0", the action ha = fu(an² + +1 (1-a))n) will maximize the population's expected utility only when 0. Formally, consider the function [u(2, 0*)]. By Assumption we must
((a) = EgDD ant +(1o)h have
[u(2, 8*)] = EgwD BewDPtont-(--o
d
da a= BEWD,, [m(2, 0²)] BEND,, (u(200") = 0.
At the stable point (h*,0"), we also have that BE.DD [oL(2,
Consider the function
e(a) = U (ah# + +11 - a)hh*) (h(an + - )) we have
de
= =ESDD, [u(z, 0¹)] - EEND,. [u(z, 0*)
da a=0
- -EENDD [ 0*)] LEND, V 0*)] - EL-D, Vol(z,0-)]] (H-)-1 (E.ND,- Vou(2, 0*) EEND, [ol(2, EEND,, Vol(2, (H-)-1
(E.-D [Vou(2 0*) EEwD, [ol(z, 0*)) (
Finally, the result follows from the PL-inequality.
A.6 Proof of Theorem 57
For notational simplicity, we set f(n, 0) = EENDaGn [u(2;0)] and g(n, 0) = EewDaun [{(2:0)] Recall that
U(h(n)) f(n. A(D()())))) EaNDaon [u(2, A A(Dnm)))))
14

---

where the first argument in f only applies in distribution that is taken against and the second argument corresponding to the second argument of u. Then, by the implicit function theorem we have
A(Dr(m)) (1,20(1, where V, and V2 denote the gradient operator on the first/second argument of the function, For the equilibrium of the selfish strategy (h(v)), ), 0*), we must have (1 = O. Let vy arg max, ((h(n)) by the Pl-inneuuliiy, we obtain that
v,U(((()) - (1, - v.
(2,
U (h(F)) (h(n"))))
1 2
VV229'n' (V220l A(Dn(r)))] V2f(n" A(P
A.7 Proof of Proposition 6
Consider the derivative of Un with respect to variable
aU
DEgDDa [u(z; 02)]
DE [u(2; 02)
08%
da
da
00 da
Next, we notice that
Vo Di [[( = =0,
where we further write the left-hand side
Vo +(1-) [((2; 0)] = a EDND, [ol(2; [)] + (1 - a)) EgND, [ol(2; [)] = 0.
Therefore, EZND, (Vl((2; 1-a EzDD, (ol(2; 0). Then, we consider the term 00_
By
Br
the implicit function theorem, with
have
0r:
(((=;)))) (EENDA Vel(2;0)] - EEND. [ol((;))))
da
1
-1 (EzwD, [V)((;))))
1
Therefore, finally write
OUa 1
da 1 a (BenDa Vou(2;00))) (elc)))"(Vlllle (Vo, [((2;)))) -1 (EEgDD [v)((=;0)).
A.8 Proof of Proposition 7
The proof is the same the proof of Proposition 6 up use of the envelope theorem. For completeness, we restate the proof here and highlight the use of the envelope theorem. For notational simplicity, abbreviate hü as h. Consider the derivative of U% with respect o,
OU UE.-D, [u(2: 01)]
ÖE Dr, [u(2; 01)
08
da
On
00
Oo.
where the second equality follows from the implicit function theorem and the envelope theorem. Next, we notice that
V Vo +(1-a)Do [6(2; 0)) O, where we can further write the left-hand side as
VoLesaDD++(-a)22 ((1-a) [<(2; 0)) = a EZnDy [ol 0)] + (1 - ax) EZ [Vo((z; 0)] = 0.
Therefore, EZNDe Vol(z;0) - - EZND, V [o[(2; Consider the term function theorem, with 0, we have
00
By the implicit da
00"
0. g Beaaa +1 ()) (E.ND, [o[(2; - ELND, [vl(2;01)
da
1
(V2 BraaDaD (0 (()= (RELD, [v){(2;0)))
We finally write
OU
1 (EzND, (ou(z;0))))" (Tin ( (((=; [[(2: -1 (E.wD, [vo((2;)))
1
da
15

---

A.9
Proof of Proposition 8
By Lemma 10 the sensitivity of the mixture distribution can be computed as
k-1 c= c1
Σ
where 01 = a and og a. Combining this with Theorem 3.5 in Perdomo et al. 2020 yields the result.
B Example to illustrate Theorem 5
Consider a toy setting to derive closed-form expression for the benefit coordination. Assume learner estimates the centroid 0 of a distribution supported on three anchor points P1, P2, Pa-the corners of an equilateral triangle. The collective applies a strategy h that moves their data point to of the three anchors with probability (W1, W2, Wa). respectively, Σ, wy = 1 and wy 2 0. This implics distribution D Ei wpdp, where the strategy determines w. The learner minimizes the squared loss; the solution is A(D;) E, wy The collective, the other hand, prefers the centroid lie between P1 and P2, and maximizes
P3
0*
P1
8
P2
u(2, 0) = - ||Pı 012 1| - 01 | 012.
When the collective distributes mass uniformly, i.e., h* = (1/3, 1/3, 1/3), the centroid 0* Ei=1 P; is performatively stable point. There is no incentive for either party to change their strategy since they are both respornsg to the current state. However, a look-ahead collective would prefer A = (1/2, 1/2, 0), as they are aware that they could collectively shift the model away from this stable state further down in the triangle. This leads to look-ahead optimal point OF = + P2), which deviates from 0*
In this example, Assumption I is satisfied. Moreover, 2 0 since t 22 where ||P 2- The benefit of coordination B (hF) - U(h)) , is strictly positive long the anchor points are appropriately spaced. One can check that U is 2tr-stry concave, and Theorem 5 can be verified since < 2y concavity assumption.
&² 2² which is tight up to factor 1/4 coming from the slack in the strong
C Simulations
C.1
Feature importance
In Figure we show the importance of different features in the credit-scoring simulator. We simulate modifications to feature by replacing 2 with a value z/. which we sample independently from a standard normal distribution. Subsequently data. The values of the bars indicate the drop of test accuracy compared to the baseline classifier trained without misreporting. The error bars indicate standard deviations over 10 different train-test splits.
train logistic regression classifier the modified
C.2 Binary prediction setup
Consider the case in which the collective wants to implement fixed strategy, which modifies their strategic features "age" and "number of dependents" as
h (zx 91 = (xi + Ho,s vt).
where fo.s is the mean of the strategic features when the label equals 0. Since the features are all
centered across the whole dataset, this transformation can be interpreted geometrically as translating the samples with label along the direction of μoo moving them toward the center of the distribution with label 0 in feature space.
16

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Effid of each Jeature
Acumncy to buscilns
0.04
0.02
0.01
0.00
RevelsigUUl
PartDuc00 59
Age
DeldRatic
Incete
Opeal iavs
PasDhesgg
RealEstate
PanDacsd.35
Depandants
Figure 5: Accuracy drop against modifying individual features.
We run simulations with three strategies h =0.3 h 0055 and h =008 where each strategy opti- mally chooses for given collective size a E {0.3, 0.5, 0.8). The target model Ptarget we used in the simulation For each a E {0.3, 0.5, 0.8). we can solve for an optimal 7e such that the resulting model coincides with barget In other words, for all three collective sizes, there exists manipulation strength 7)a that enables the collective to attain maximum utility (equal to 0). This ensures that comparisons across different values of well-defined.
fixed to be the model produced by the strategy with parameters (17 = 0.5, 0.3).
17

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Reviews:
[{"review_id": "u8iPbE7DUG", "reviewer": "Reviewer_vhC6", "summary": "This paper studies strategic user behavior in the context of performative prediction, where the data distribution depends on the deployed predictive model. Prior work has largely assumed independent, myopic user behavior. This paper generalizes the strategic behavior of agents by introducing two models from behavioral economics and game theory: (1) Level-k reasoning, where agents reason recursively about other agents’ strategies. (2) Collective reasoning, where agents treat themselves as part of a statistically significant population capable of steering the learning system through coordinated actions.\n\nThe paper characterizes learning dynamics and equilibrium outcomes under these models and introduces the concept of the price of selfish reasoning to compare selfish and coordinated strategies. It also analyzes the role of mixed populations—part selfish, part collective—and studies how alignment between population utility and model loss impacts convergence, utility, and incentives to participate in collective strategies. Simulations are provided using a credit scoring dataset to support theoretical claims.", "questions": "1. Can the authors formally model the Stackelberg game between the learner and the population under collective reasoning? What are the individual agents’ incentives, and do they form an equilibrium? Do they receive benefit from each model update or only in the PS state?\n\n2. Can the authors characterize which subset of agents are more willing to collaborate and which subsets are more willing to free-ride? How does this depend on the transition mapping, the initial data distribution and the choice of model hypothesis space?\n\n3. Can the authors add simple examples (even synthetic) to show concrete strategies under level-1 vs. level-k vs. collective reasoning? How do individual utilities compare?\n\n4. Are the results robust to violations of the linearity and strong concavity assumptions? Would the conclusions still hold with more realistic agent behavior or stochasticity?\n\n5. The simulation section would benefit from baseline comparisons, error bars, and quantitative metrics (e.g., average regret, time to convergence, etc.), as well as more real-world datasets in previous works. Can the authors extend this?", "limitations": "not beyond weaknesses and questions", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. Novel conceptual framing: Introduces collective reasoning into performative prediction, bridging the gap between strategic classification and algorithmic collective action.\n\n2. Technically sound derivations: Provides multiple formal results on convergence, utility trade-offs, and price of selfish reasoning under various assumptions.\n\n3. Insightful alignment analysis: Shows how gradient alignment between utility and loss determines the benefits or costs of collective action.\n\n4. Simulation studies: Includes a few simulations that qualitatively support theoretical insights, especially regarding convergence speed and utility curves in mixed populations.\n\nWeaknesses\n1. Limited Formalization of Game-Theoretic Structures:\nAlthough the problem involves Stackelberg-like interactions, the game-theoretic formalism is weakly developed.\n\nThe notion of agents “best-responding” collectively is used, but no explicit utility functions or equilibria for individual agents are presented beyond collective averages. Whether the agents receive utility in a single round or repeatedly in a repeated game can significantly change the analysis.\n\nThe Stackelberg structure is only implicitly referenced and lacks formal modeling of individual incentives in multi-agent dynamics.\n\n2. Absence of Agent-Level Utility Analysis:\n\nThere is no detailed discussion of agent best responses, deviation incentives, or whether the proposed collective strategies are individually rational.\n\nThe conditions under which agents would prefer to improve or manipulate features are not clearly contrasted, as is standard in the strategic learning literature.\n\n3. Simulation Limitations:\n\nThe simulations are qualitative and small-scale, with limited quantitative rigor (e.g., no standard deviations, statistical tests, or multiple runs reported).\n\nOnly a single real-world dataset is used, and no baselines (e.g., existing performative prediction or strategic classification methods) are included for comparison.\n\nClaims about convergence speed and utility alignment are not empirically validated across diverse models or settings.\n\n4. Missing Empirical Demonstration of Key Ideas:\n\nCore concepts such as the price of selfish reasoning, collective shift bounds, and benefit of participation are not clearly illustrated with examples or real decision-making scenarios.\n\nThe paper lacks simple examples or visualizations to build intuition about higher-order reasoning or the effects of collective action.\n\n5. Some Assumptions Are Strong or Unverified:\n\nKey results rely on linearity assumptions of the distribution, strong concavity of utility, and smoothness of losses—all of which may be hard to verify or hold in practice.\n\nThese assumptions are presented without discussion of their practical implications or limitations.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback. In the following we address your comments in sequence. Please let us know if anything requires further clarification.\n\nQ1. **Stackelberg structure**   \nIn the context of collective reasoning (Section 4), the actions of the learner and the population are defined as follows: \n\nThe learner is assumed to be risk minimizing (recall Assumption 1): observing $\\mathcal D$, they choose $\\mathcal{A}(\\mathcal{D}) = \\arg\\min\\_{\\theta \\in \\Theta} \\mathbb{E}\\_{z \\sim \\mathcal{D}}[\\ell(z; \\theta)]$. In words, the learner best-responds to the population; their utility corresponds to the negative expected loss, and their action to a choice of parameters. \nThe collective in turn finds a strategy that maximizes their *collective* utility $U$ by planning through the learner’s response. This optimal strategy is defined in L176 as $ h^\\sharp = \\arg\\max_h \\mathbb{E}_{z \\sim \\mathcal{D}_h}[u(z, \\mathcal{A}(\\mathcal{D}_h) ]$. \n\nTogether, this structure corresponds to a two-player Stackelberg game where the collective acts as the leader and the learner as the follower. Note that the agents in the population are not necessarily optimizing their *individual* utilities. Like in traditional cases of collective action, participating in a collective is not always individually rational, but individuals may opt in for altruistic reasons.\n\nIn later sections we consider deviations from this model where the strategy space of the collective only permits the manipulation of a subset of the population ($\\alpha<1$). This corresponds to some agents prioritizing their individual utilities, and others acting altruistically in the interest of the collective. We hope this clarifies your question about the game-theoretic formalization of collective reasoning.\n \n**Do individuals benefit from model updates?** Agents’ utilities are not guaranteed to be monotone over time and even at stability a benefit is not guaranteed at the individual level. As explained above, our goal is to study agent behavior that is driven by *collective* goals rather than individual incentives. We demonstrate how a group of agents can improve their aggregate utility through coordination, more so than if all of them perform individually rational actions. The reason is that collectives can ‘steer’ the learner, rather than just their own data. \n\n\nQ2. **Participation dynamics**   \nThis question relates to an important conceptual departure between our work and strategic classification. We study agents that purposefully optimize the collective utility. This behavior does not arise from behavior that is rational according to an individual utility. Taking this collective perspective, our results show when the collective is better off coordinating, rather than acting myopically. In the present work, we do not study the dynamics by which collectives form, or mechanisms to incentivize participation. Understanding the role of heterogeneous incentives in the context of algorithmic collective action is an interesting and, to the best of our knowledge, an unexplored question.\n \nQ3. **Example**   \nOne concrete example of level-k strategies is demonstrated in the simulation of Section 6.1. It is an extension of the classical strategic classification setting with a quadratic cost and linear utility. Here, agents at level $k$ modify their strategic features $x_S$ assuming the other agents act according to level $(k-1)$. The actions of agents can be written in closed form as $x\\_S^k = x_S - \\epsilon \\cdot \\theta^{k-1}\\_S$, where $\\theta^k = \\arg\\min_\\theta \\sum_{i} \\ell(x^k_i, y_i, \\theta)$. In the following we use this example to numerically compare the utilities of level-1 and level-2 agents with the same initial features $x = (0, \\ldots, 0)$. We report averages over 10 runs.\n\n|       Timestep      |        1       |        2       |        3       |         4        |        5        |      >= 6     |\n|:-------------------:|:--------------:|:--------------:|:--------------:|:----------------:|:---------------:|:-------------:|\n| $u^{k=2} - u^{k=1}$ | $7.2*10^{-1}$ | $4.5*10^{-5}$ | $2.1*10^{-9}$ | $-2.8*10^{-9}$ | $2.1*10^{-9}$ | <= $10^{-9}$ |\n\nWe observe that the utility gap decreases across time steps, as the procedure converges towards stability. However, there is no evidence on the monotonicity of utilities against their cognitive levels. We will include an extended version of this numerical analysis in the paper. Thank you for the suggestion!\n\nQ4. **Assumptions**:   \n\nWe discuss the three assumptions individually:\n\n- *Strong concavity*. The simulation in Section 6.2 is an example when the utility is not strongly concave, but the results still qualitatively align with our theoretical findings.\n- *Linearity*. The condition is only used in Theorem 4 and 6 in the last step of the proof to permit a cleaner and more interpretable bound. A “less clean version” of the result does not require this assumption and preserves the qualitative take-aways. For Theorem 4, for example, $\\Phi$ would need to be replaced by $\\Vert\\nabla\\_h\\nabla\\_\\theta\\mathbb{E}\\_{z\\sim \\mathcal{D}\\_{h\\^*}}\\[u(z, \\theta)\\](\\mathbf{H}\\^\\star)\\^{-1}\\mathbb{E}_{z \\sim \\mathcal{D}\\_{h\\^\\ast}}\\[\\ell(z, \\theta)\\]\\Vert$. In general, for simple densities the assumption can always be made possible by reparameterising the density functions which allows to differentiate through the individual terms.  \n- *Agent behavior* The analysis of the mixture distribution (Equation 3) is our attempt to understand heterogeneous populations and deviate from the typical model of a homogeneous population with a fixed level of reasoning. One insight we have is that convergence to stability is achieved for varying proportions of agents at different levels of cognitive hierarchy. We find that convergence is robust to these deviations.\n\nQ5. **Experiments**   \nThank you for the suggestions, we are happy to extend our experimental section along the dimensions you mentioned. We added error bars (as one standard deviation) for all plots in the paper. See below for the numbers.\n\nFigure 1:\n\n|                   Timestep                  |       1       |        2       |        3       |        4       |        5        |        6        |      >=7      |\n|:-------------------------------------------:|:-------------:|:--------------:|:--------------:|:--------------:|:---------------:|:---------------:|:-------------:|\n|   standard deviation: $\\alpha = \\{0,1,0}$   | $2.9*10^{-1}$ | $2.48*10^{-5}$ | $2.18*10^{-7}$ | $1.95*10^{-9}$ | $1.76*10^{-11}$ | $1.53*10^{-13}$ | <= $10^{-14}$ |\n| standard deviation: $\\alpha = \\{0,0.5,0.5}$ | $1.9*10^{-1}$ |  $1.6*10^{-5}$ |  $2.2*10^{-7}$ |  $2.0*10^{-9}$ |  $1.8*10^{-11}$ | $1.61*10^{-13}$ | <= $10^{-14}$ |\n|   standard deviation: $\\alpha = \\{0,0,1}$   | $1.7*10^{-1}$ |  $1.5*10^{-7}$ |  $1.4*10^{-9}$ | $1.2*10^{-11}$ |  $1.0*10^{-13}$ |  <= $10^{-14}$  | <= $10^{-14}$ |\n\nFigure 2:\n|                $\\alpha$                |      0.02     |      0.13     |      0.23     |      0.34     |      0.45     |      0.55     |      0.66     |      0.77     |      0.87     |      0.98     |\n|:--------------------------------------:|:-------------:|:-------------:|:-------------:|:-------------:|:-------------:|:-------------:|:-------------:|:-------------:|:-------------:|:-------------:|\n|           utility std - #Age           | $4.6*10^{-3}$ | $2.0*10^{-3}$ | $1.2*10^{-3}$ | $8.2*10^{-4}$ | $4.6*10^{-4}$ | $4.9*10^{-4}$ | $2.8*10^{-4}$ | $1.8*10^{-4}$ | $7.6*10^{-5}$ | $6.0*10^{-6}$ |\n|        utility std - #Dependents       | $7.2*10^{-3}$ | $1.6*10^{-3}$ | $7.4*10^{-4}$ | $6.8*10^{-4}$ | $5.0*10^{-4}$ | $3.0*10^{-4}$ | $1.9*10^{-4}$ | $1.5*10^{-4}$ | $8.0*10^{-5}$ | $6.0*10^{-6}$ |\n|     Benefit of Participation - #Age    | $4.8*10^{-3}$ | $2.6*10^{-3}$ | $1.9*10^{-3}$ | $1.6*10^{-3}$ | $1.1*10^{-3}$ | $1.5*10^{-3}$ | $1.1*10^{-3}$ | $1.1*10^{-3}$ | $9.5*10^{-4}$ | $3.9*10^{-4}$ |\n| Benefit of Participation - #Dependents | $7.4*10^{-3}$ | $2.1*10^{-3}$ | $1.2*10^{-3}$ | $1.3*10^{-3}$ | $1.2*10^{-3}$ | $9.4*10^{-4}$ | $7.9*10^{-4}$ | $9.1*10^{-4}$ | $1.0*10^{-3}$ | $3.6*10^{-4}$ |\n\nIn terms of convergence: note that Figure 1 (in the paper) illustrates convergence for the experiment of Section 6.1. For the experiment conducted in Section 6.2, for the inference of the collective strategy, convergence is achieved after 200 steps for all cases. We will give a more comprehensive report on these quantitative metrics in the revised version.  \n\nRegarding real-world datasets, we would like to emphasize that the credit scoring simulator is based on *real* data (data collected from 250,000 borrowers including historical information about the individual and information on whether they defaulted on a loan). More generally, in related literature, there are few experiments on real-world data since that would require obtaining samples under different model deployments. If the reviewer has a suggested dataset we could look at, we would be happy to incorporate it."}, {"review_id": "HRkatiu7jE", "reviewer": "Reviewer_DVWb", "summary": "This work studies the impacts of collective reasoning in predictive systems. They formalize level-k-thinking and and analyze how these more sophisticated levels of thinking impact outcomes within the context of performative prediction. They show that individuals making use of higher levels of strategic thinking influence the rate of convergence to an equilibrium, whilst not impacting the stable point that is converged to. They then move to consider the impact of collective reasoning in trying to achieve better outcomes.  In this setup, the authors consider situations in which populations are mixed between agents engaging in collective optimization and agents who engage in level-k reasoning. They first establish a benefit of 'free-riding' by considering the case when k = 1. They then move to establish results indicating the price of partial participation and the implications for convergence to a stable point. They then move on to consider the case of having an unstrategic population mixed with a strategic population and prove a set of results on the cost of steering the learner and the benefit of participating in and the benefit of implementation.", "questions": "In the collective reasoning section, it seems as though the work only considers k=1 for selfish participants. What would be the implications if we were to extend this to higher levels of k reasoning for the selfish participants?", "limitations": "Authors adequately addressed limitations", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Clearly presented and well structured: This paper is well written, and the structure makes it easy to follow the key contributions the authors present. \n- Formalizing and instantiating key behavioural economics concepts: I found the incorporation of level k reasoning within the framework of performative prediction to be particularly interesting. I think this is particularly valuable as these ideas are natural, particularly within the context of performative prediction\n- Wide range of formal settings explored: Another key strength of the paper is that it provides a wide range of results spanning many settings from results on equilibria convergence given differences in k-level reasoning of sub-populations to and quality of stable points under different mixtures of agents\n\nWeaknesses:\n- The justification for considering an unstrategic segment of the population is not made particularly clear in the broader context of the paper, wherein strategic agents were the focus before. This transition seemed somewhat arbitrary to get to the results they had?", "was_revised": true, "final_justification": "I retain my positive assessment of this work.", "rebuttal": "Thank you for the positive assessment of our work. We elaborate on your questions and comments below:\n\n**Justifying unstrategic segments in the population**  \nThroughout, our goal is to study heterogeneous populations of agents and to contrast different reasoning strategies, including the lack of strategic thought. In the context of collective action there is ample evidence that in practice typically not all agents in a population participate in collective strategies. For example, Burrell et al. report that a strategic control action was found in 17% of the tweets they coded for their study [1], still the strategy was effective. Our goal is to model this level of heterogeneity by incorporating non-strategic users. \n\n[1] J. Burrell, Z. Kahn, A. Jonas, D. Griffin.”When Users Control the Algorithms: Values Expressed in Practices on the Twitter Platform” ACM CHI, 2019.\n\n**What would be the implications if we were to extend this to higher levels of k reasoning for the selfish participants?**\n\nIn our results we characterize the stable point in scenarios where we have heterogeneous populations, meaning both agents who act selfishly (as studied in Section 3) and agents who reason collectively. In the following we explain why considering the mixture with only a single component at $k=1$ as a representative case for selfish agents is sufficient.\n\nAt the performatively stable point the behavior of agents at different levels $k \\geq 1$ is the same as the behavior of a homogeneous population with only $k=1$ agents, since the second argument in the utility in Eq. (2) is fixed. Further, by Theorem 3 we know that the interplay between the population and the model converges to a unique stable point for *any* instantiation of the mixture weights in Eq. (3). As a result, the stable point across varying mixtures of selfish agents, no matter what their $k$ is. We will clarify this and explain that our results apply to selfish agents more broadly."}, {"review_id": "c6PoTxvqoZ", "reviewer": "Reviewer_KUfC", "summary": "* This paper investigates performative retraining dynamics under different models of strategic response.\n* In the setting, the population is initially distributed according to a base distribution $D_0$ over feature-label pairs. A population may alter its distribution to $D$ to improve utility, where each individual moves from its initial point $z\\\\sim D_0$ to a point $h_\\\\theta(z)$, where $\\\\theta$ represents the currently deployed model. The learner performs repeated risk minimization to optimize $\\\\theta$ at each step. \n* In Section 3, user response is inspired by $K$-level thinking: level-$0$ response is the identity function $h^{(0)}_\\\\theta(z)=z$, and the response of level-$k$ thinkers is a best response to the distribution induced by level-$(k-1)$ thinkers. The main theoretical result shows that retraining converges to a performatively stable point at an exponential rate depending on the composition of different-level thinkers in the population, that the convergence rate is higher when there are higher-order thinkers in the population, and that the performatively stable does not depend on the composition of different-order thinkers in the population. \n* In Section 4, analysis presents a bound on the gap between performatively optimal and “utility maximizing points” based on Hessian-weighted gradient alignment. Section 5 presents convergence bounds for mixture populations (collective action and level-$k$ selfish agents in Section 5.1), a lower bound for the amount of “collective action” to reach a target state (Section 5.2), a characterization of utilitarian agents’ utility at equilibrium as a function of their collective size (Section 5.3), and a similar characterization for fixed strategies (Section 5.4). Finally, Section 6 presents experiments on empirical data with simulated response, showing alignment with theoretical analysis.", "questions": "* What are the information/computational requirements of level-$k$ thinkers? What do they need to know, and what do they need to be able to compute?\n* The setting presented in Section 5.2 seems very similar to the setting analyzed by Hardt et al. (“Algorithmic Collective Action in Machine Learning”, ICML 2023). How does Proposition 8 relate to the results they present?\n* Small question about notation: In L127, the agent utility is denoted by $u(z,\\\\theta)$ (i.e. only depends on a single feature-label pair $z$), however in the classic Strategic Classification model of Hardt et al., the utility of of each agent depends on the “distance” between the original point $z\\\\sim D_0$ and the strategic response $\\\\Delta(z)$. Does the utility $U(h)$ as defined in L126 capture such dynamics?\n* For Theorem 6, it seems that the bound is vacuous for $\\\\alpha=1$. However, if I understand correctly then for this case convergence is “immediate” in some sense, as $D_{h^\\\\sharp}$ does not depend on the currently deployed model. Maybe it is possible to derive a “best of both worlds” bound based on this observation?", "limitations": "Limitations and broader impact are not discussed in a dedicated section.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* Interesting bridge between performative prediction and behavioral dynamics.\n* Results further widen the applicability of repeated risk minimization.\n* Work has substantial breadth.\n\nWeaknesses\n* Attached code does not seem to generate the graphs presented in the paper, has missing modules and dataset, and most of its document is not in English, making it hard to verify results and build upon them.\n* Theoretical results in sections 4,5 characterize alignment as a function of an inverse-Hessian weighted gradient inner product of the performative response. However, it is unclear how the gradient alignment conditions can be computed or verified in practice.\n* There doesn't seem to be a direct discussion of limitations and broader impact.", "was_revised": true, "final_justification": "The authors have addressed my concerns in the rebuttal. I now find that the paper offers new and interesting perspectives on performative prediction, which I believe will be of interest to the community.", "rebuttal": "Thank you for your thorough review. We incorporated your feedback and we will use the additional page available for the revision to include a dedicated section on limitations, discussing the points that came up in the reviews. Below we discuss your comments in order. \n\nQ1. **Computation requirement** \nThe level-k thinkers assume all other agents are level-$(k-1)$ thinkers. Hence, they need to be able to compute the moves of level-$\\{k-1,k-2, \\ldots, 1\\}$ thinkers (in order to compute $\\mathcal{D}_{k-1}(\\theta)$). Further, they need to be able to compute the output of the learning algorithm. The former assumption that agents at higher levels are able to emulate agents at lower levels is standard in economic models of cognitive hierarchies, see Nagel 1995. The latter is specific to our learning setting and requires that the agents know the specifics to the learner’s objective (which is common in strategic classification).\n\n*Additional experiment in response to the comment* To investigate the sensitivity to violations of this assumption we have performed an additional empirical ablation where we allow agents to only approximately solve the learning problem. In particular, we assume all agents can only perform one stochastic gradient descent step on the deployed model to evaluate the model update with learning rate of 0.01. We set up the experiment with 2 levels of agents (same as the experiment in Section 6.1). The following table reports the convergence of the model parameters as the differences between iterations $\\Vert \\theta_{t+1} - \\theta_t \\Vert$ (same as Figure 1):\n\n|         Iterations        |   1   |       2       |        3       |       4       |        5       |        6       |      >=7     |\n|:-------------------------:|:-----:|:-------------:|:--------------:|:-------------:|:--------------:|:--------------:|:------------:|\n|       Level-1 agents      |  0.37 | $3.4*10^{-5}$ |  $4.3*10^{-7}$ | $4.5*10^{-9}$ | $5.3*10^{-11}$ | $5.7*10^{-13}$ | <= $10^{-13}$ |\n| 50% Level-1 + 50% Level-2 |  0.47 | $2.6*10^{-5}$ | $3.16*10^{-7}$ | $3.0*10^{-9}$ | $3.7*10^{-11}$ | $4.4*10^{-13}$ | <= $10^{-13}$|\n|       Level-2 agents      | 0.009 | $2.9*10^{-5}$ |  $9.1*10^{-8}$ | $2.1*10^{-9}$ | $5.8*10^{-12}$ | $2.9*10^{-13}$ | <= $10^{-13}$ |\n\nWe observe that convergence is preserved, and a larger proportion of higher-level agents still accelerates convergence. \n\n\nQ2. **How does Proposition 8 relate to Hardt et al.?**\nWe assume you refer to Section 4.1 in Hardt et al. They analyze a fixed strategy whereas our bound holds for *any* possible strategy that could potentially shift the model to the target state. We demonstrated that the collective’s effort to implement such a strategy (modeled by the shift needed) is inversely proportional to the size of the collective. This is not a quantity studied in prior work. However, what is similar is that the norm of the gradient at the target solution naturally emerges as a notion of suboptimality (under the base distribution) for what the collective aims to achieve.\n\nQ3. **Connection to strategic classification utility model** \nThanks for pointing this out. For simplicity we removed the cost term from our problem statement to be more consistent with the performative prediction formalism. To fully recover the strategic classification’s utility model of Hardt et al., one would need to extend the notation as $U(h) = \\mathbb{E}\\_{z\\sim \\mathcal{D\\_0}}[u(z, h(z), \\mathcal A (D_h))]$. However, this does not change the results in our paper as it acts as a regularizer that equally applies to any level of reasoning for any fixed agent. For example, in Theorem 3, the convergence still holds as long as the sensitivity of the distribution map satisfies $\\epsilon \\leq \\frac{\\gamma}{\\beta}$. And for later results still hold as the envelope theorem still applies. We will clarify this point in the revised version.\n\nQ4. **Clarification on Theorem 6**   \nYes, the model indeed converges immediately since $h^\\sharp$ is not dependent on the current model parameter. Our goal is to characterize the suboptimality due to deviating from the case $\\alpha=1$. We are not sure we understand what you are suggesting - could you please clarify what you mean by a best-of-both-worlds bound?\n\n**Code**   \nThe code requires additional modules that can be found in the github repo of the paper \"Performative prediction.\" ICML 2020. The code is runnable if the dataset (Kaggle contest: Give Me Some Credit, `cs-training.csv`) and the modules (`data_prep.py`, `optimization.py`, `strategic.py`) are placed in the root folder of the notebook. We will work on the documentation and release our polished code publicly alongside the revised version.\n\n**Alignment condition**  \nAs long as the loss function and the agents’ utility functions are available, the gradient alignment expressions are in principle computable. The Hessian of utilities in differential games is indeed an important indicator of the “alignment” of the player’s objectives (see Section 2,  Balduzzi et al., “The Mechanics of n-Player Differentiable Games”, ICML’18). It is inevitable to utilise the second-order information in characterising the exact alignment of the players’ objectives. In practice, it is possible for practitioners to infer the adversarialness of the learner and the collective through qualitative analysis without direct computations. For example, in the simulations, we measure the alignments using the importance of features, and the empirical results also align with our theoretical observations. However in general we see the main value of our analysis in highlighting the role of steering in collective reasoning (which is absent from strategic classification) and offering the first formal characterization of the tradeoff involved in strategizing collectively against a risk minimizing learner."}, {"review_id": "NxgoncOlFl", "reviewer": "Reviewer_cpZT", "summary": "This paper investigates the effects of sophisticated user strategies in the context of performative prediction, where a machine learning model's predictions influence the data distribution it is trained on. The authors introduce and analyze two main forms of strategic user behavior:\n\nLevel-k Thinking: Drawing from behavioral economics, the paper formalizes a hierarchy of strategic depth where \"level-k\" agents best-respond to a model they anticipate will be trained on the actions of \"level-(k-1)\" agents.\n\nCollective Reasoning: This models a scenario where a group of users coordinates their actions to \"steer\" the learning algorithm toward a more favorable outcome for the population, effectively acting as a leader in a Stackelberg game.\n\nThe paper provides theoretical characterizations for both scenarios. It finds that higher levels of individual strategic thinking (level-k) accelerate convergence to a stable equilibrium but do not change the equilibrium point itself. In contrast, collective action can lead to a qualitatively different and more beneficial equilibrium for the users. The paper quantifies this benefit, termed the \"price of selfish reasoning,\" and shows it is governed by the alignment between the population's utility and the learner's loss function. Finally, it analyzes mixed populations containing both collective and selfish agents, exploring the dynamics of free-riding, the benefits of participation, and the costs of steering the model. These theoretical findings are supported by simulations on a credit scoring task.", "questions": "1. The analysis relies on the loss function being smooth and strongly convex. While common for theoretical analysis, many complex machine learning models, such as neural networks, operate in non-convex landscapes. The paper does not discuss how the results might change in such settings.\n\n2. Assumption 2, which posits that the data distribution is a linear function of the population's strategy, is a significant simplification. While the authors note it can be seen as an approximation, the implications of violating this assumption on the derived bounds are not explored.\n\n3. The paper models the collective as a unified actor capable of selecting and implementing a globally optimal strategy (h‡). It successfully analyzes the consequences of this optimal collective action but abstracts away the significant practical challenge of how a large, decentralized population would coordinate to find and execute such a strategy.\n\n4. The model of strategic manipulation, where an agent can transform their data from z to h(z), is highly abstract. In the simulations, this is instantiated as a simple additive change to feature values. Real-world strategic behavior often involves complex, feature-dependent costs and hard constraints (e.g., it is costly to change income but impossible to change age). A more nuanced cost model could lead to different strategic equilibria.", "limitations": "See question section.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength: The paper's primary strength is its successful integration of concepts from behavioral economics (level-k thinking) and game theory (Stackelberg games, collective action) into the machine learning framework of performative prediction. The paper delivers clear and significant theoretical contributions.\n\nWeakness: The theoretical results depend on several strong, standard assumptions that may not hold in all practical settings.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough review and the valuable feedback. Please see below for our point-by-point response to your questions.\n\n\nQ1. **Convexity assumption**  \nIt is inevitable to establish most theoretical results with the standard convexity and smoothness assumptions. However, we note that the simulation we conducted in Section 6.2 violated the strong convexity assumption of the collective’s utility. As indicated by the experimental results, there is no qualitative inconsistency with our theoretical results. The insights gained in convex settings are often indicative of the phenomena happening under a more complex landscape. \n\nQ2. **Linearity assumption**  \nWe note that only Theorems 4 and 6 require the linearity assumption in the last step of the proof to permit a cleaner and more interpretable expression. We will rewrite the proof to expose the expression before applying the linearity assumption. For Theorem 4, for example, $\\Phi$ would need to be replaced by $\\Vert\\nabla\\_h\\nabla\\_\\theta\\mathbb{E}\\_{z\\sim \\mathcal{D}\\_{h\\^*}}\\[u(z, \\theta)\\](\\mathbf{H}\\^\\star)\\^{-1}\\mathbb{E}_{z \\sim \\mathcal{D}\\_{h\\^\\ast}}\\[\\ell(z, \\theta)\\]\\Vert$. In this form the bound still holds without the assumption.\n\nQ3. **Practical challenges of coordinating populations**  \nWe agree that coordination is an important aspect to implement collective action strategies in practice. Reported cases have successfully used forums and social media groups to coordinate such efforts; see, for example, the #DeclineNow movement on Doordash. There is also ongoing research dedicated to developing digital tools for facilitating information exchange among platform participants. See, for example, [1, 2] who develop tools for gig workers on digital platforms. All of this is to say that empirically there is some evidence of large, decentralized populations unifying to execute collective strategies.\n\n[1] Carlos Toxtli and Saiph Savage. Designing AI Tools to Address Power Imbalances in Digital Labor Platforms, 2023.\n[2] Kashif Imteyaz,Claudia Flores-Saviaga, Saiph Savage. GigSense: An LLM-Infused Tool for Workers' Collective Intelligence. 2024\n\nQ4. **Complexity of strategies $h$**  \nIn the context of collective reasoning, our goal is to understand the effectiveness of a given strategy $h$ that a collective might deploy, rather than model organic human behavior based on incentives and costs. As you pointed out in your previous question, implementing and coordinating large-scale efforts can be challenging in practice. This is why simple strategies, such as simple feature manipulations, play an important role (especially since they are often implemented by people who do not have algorithmic expertise). For example, in the \\#DeclineNow movement on Doordash drivers on a food delivery platform coordinate to decline orders above a threshold of $7 to raise the overall pay level. The numerical threshold they picked is not necessarily optimal but very easy to communicate and implement. It is an example of a simple strategy that was effective in practice."}]

Author rebuttal:

Meta-review: This paper studies a performative learning setting in which user behavior is modeled as k-level reasoning and as collective reasoning. All reviewers appreciate the novelty of the setup and agree that the paper makes a nice link between retraining and micro behavioral modeling through the dependencies that follow from collective action. However, while the authors claim that most works have so far considered individual responses, there are several recent works that do consider interdependent user behavior (e.g., see [1-4] and references within), which should be addressed. The reviews were also helpful in identifying areas in which the paper can be improved, such as: a more formal game-theoretic description of the setup, the need for making strong assumptions, and the breadth and robustness of the experiments. The author's rebuttal was helpful in addressing some of the concerns, but they are nonetheless encourage to incorporate the reviewers' suggestions into the final version. 

[1] Strategic Classification with Graph Neural Networks. ICLR 2023. 
[2] Performative Prediction on Games and Mechanism Design. AISTATS 2025. 
[3] Strategic classification with externalities. ICLR 2025.  
[4] Learning Classifiers That Induce Markets. ICML 2025.
Decision: Accept (poster)