Who Leads and Who Follows in Strategic Classification?

1 Introduction . Individuals interacting with a decision-making algorithm often adapt strategically to the decision rule in order to achieve a desirable outcome . While such strategic adaptation might increase the individuals ’ utility , it also breaks the statistical patterns that justify the decision rule ’ s deployment . This widespread phenomenon , often known as Goodhart ’ s law , can be summarized as : “ When a measure becomes a target , it ceases to be a good measure ” [ 49 ] . A growing body of work known as strategic classification [ 14 , 19 , 28 ] models this phenomenon as a two-player game in which a decision-maker “ leads ” and strategic agents subsequently “ follow. ” Specifically , the decision-maker first deploys a decision rule , and the agents then take a strategic action so as to optimize their outcome according to the deployed rule , subject to natural manipulation costs . For example , a bank might make lending decisions using applicants ’ credit scores . Knowing this mechanism , loan applicants might sign up for a large number of credit cards in an effort to strategically increase their credit score at little effort . One of the main goals in the literature is to develop strategy-robust decision rules ; that is , rules that remain meaningful even after the agents have adapted to them . Recent work has studied strategies for ∗Equal contribution . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . finding such rules through repeated interactions between the decision-maker and the agents [ 6 , 20 , 46 ] . In particular , the decision-maker sequentially deploys different rules , and for each they observe the population ’ s response . Under certain regularity conditions , over time the decision-maker can find the optimal solution , defined as the rule that minimizes the decision-maker ’ s loss after the agents have responded to the rule . With the emergence of online platforms such as social media and e-commerce sites , repeated interactions between decision-makers and the population have become ever more prevalent . Online platforms continuously monitor user behavior and update pricing algorithms , recommendation systems , and popularity rankings accordingly . Users , on the other hand , take actions to ensure favorable outcomes in the face of these updates . A distinctive feature of online platforms is the decision-maker ’ s dominant computational power and abundant data resources , allowing the platform to react to any change in the agents ’ behavior virtually instantaneously . For example , if fake news content changes over time , automated algorithms can quickly detect this and retrain the classifier to incorporate the shift . It has been observed [ see , e.g. , 16 , 18 , 45 ] that , when faced with such “ reactive ” algorithms , strategic agents tend to take actions that anticipate the algorithm ’ s response . That is , through repeated interactions , agents aim to find actions that maximize the agents ’ utility after the decision-maker has responded to these actions . This suggests that the order of play in strategic interactions can in fact be reversed , such that the agents “ lead ” while the decision-maker “ follows. ” To give an example of such a reversed strategic interaction , consider ride-sharing platforms that deploy algorithms for determining travel fare as a function of trip length and relevant traffic conditions . These pricing mechanisms are frequently updated based on the current supply and demand , and in particular a dip in the supply of drivers triggers a surge pricing algorithm . Möhlmann and Zalmanson [ 45 ] observed that drivers occasionally coordinate a massive deactivation of drivers from the system , artificially lowering driver supply , only to get back on the platform after some time has passed and the prices have surged . In this interaction , the drivers essentially make the first move , while the platform ’ s pricing algorithm reacts to their action . Other examples of users aiming to exert control over algorithms can be found in the context of social network analyses [ 16 , 18 ] . In this work , we argue that the order of play in strategic classification is fundamentally tied to the relative update frequencies at which the decision-maker and the strategic agents adapt to each other ’ s actions . In particular , we show that , by tuning their update frequency appropriately , the decision-maker can select the order of play in the underlying game . Furthermore , in natural settings we show that allowing the strategic agents to play first in the game can actually be preferable for both the decision-maker and the agents . This is contrary to the order of play previously studied in the literature , whereby the decision-maker is always assumed to make the first move . 1.1 Our contribution . To give an overview of our results , we recall some relevant game-theoretic concepts . In the existing literature strategic classification is modeled as a Stackelberg game . A Stackelberg game is a twoperson game where one player , called the leader , moves first , and the other player , called the follower , moves second , with the possibility of adapting to the move of the leader . Previous work assumes that the decision-maker acts as the leader and the agents act as the follower . This means that the decision-maker first deploys a model , and the agents then modify their features at some cost in order to obtain a favorable outcome according to the model . The decision-maker ’ s goal is to find the Stackelberg equilibrium—the model that minimizes the decision-maker ’ s loss after the agents have optimally adapted to it . This optimal reaction by the agents is called the best response to the model . An important parameter that has been largely overlooked in existing work is the rate at which the agents re-evaluate and potentially modify their features . Most works studying the interaction between a decision-maker and strategic agents implicitly assume that , as soon as the model is updated , the data collected from strategic agents follows the best response to the currently deployed model . In the current work we do not assume that the agents react instantaneously to model updates . Instead , we assume that there is a natural timescale according to which the agents adapt their features to models . Allowing agents to adapt gradually to deployed models gives the decision-maker a new dimension upon which to act strategically . Faced with agents that adapt gradually , the decision-maker can choose the timescale at which they update the deployed model . In particular , they can choose a rate of updates that is faster than the agents ’ rate , or they can choose a rate that is slower than the agents ’ rate . We call decision-makers that follow a faster clock than the agents reactive , and if they follow a slower clock we call them proactive . Given that existing work on strategic classification relies on instantaneous agent responses , the previously studied decision-makers are all implicitly proactive . Our first main result states that the decision-maker ’ s choice of whether to be proactive or reactive fundamentally determines the order of play in strategic classification . Perhaps counterintuitively , by choosing to be reactive it is possible for the decision-maker to let the agents become the leader in the underlying Stackelberg game . Since changing the order of play changes the game ’ s natural equilibrium concept , this choice can have a potentially important impact on the solution that the decision-maker and agents find . Throughout , we refer to the Stackelberg equilibrium when the decision-maker leads as the decision-maker ’ s equilibrium and the Stackelberg equilibrium when the agents lead as the strategic agents ’ equilibrium . Theorem 1.1 ( Informal ) . If the decision-maker is proactive , the natural dynamics of strategic classification converge to the decision-maker ’ s equilibrium . If the decision-maker is reactive , the natural dynamics of strategic classification converge to the strategic agents ’ equilibrium . To provide some intuition for Theorem 1.1 , imagine that one player makes updates with far greater frequency than the other player . This allows the faster player to essentially converge to their best response between any two updates of the slower player . The slower player is then faced with a Stackelberg problem : they have to choose an action , expecting that the faster player will react optimally after their update . As a result , the optimal choice for the slower player is to drive the dynamics toward the Stackelberg equilibrium where they act as the leader . It is well known ( see , e.g. , Section 4.5 in [ 5 ] ) that under general losses , either player can prefer to lead or follow in a Stackelberg game , meaning that a player achieves lower loss at the corresponding equilibrium . Our second main takeaway is that in classic learning problems it can be preferable for both the decision-maker and the agents if the agents lead in the game and the decision-maker follows . One setting where this phenomenon arises is logistic regression with static labels and manipulable features . Theorem 1.2 ( Informal ) . Suppose that the decision-maker implements a logistic regression model and the strategic agents aim to maximize their predicted outcome . Then , both the decision-maker and the strategic agents prefer the strategic agents ’ equilibrium to the decision-maker ’ s equilibrium . Theorem 1.2 suggests that there are other meaningful equilibria than those previously studied in the literature . Moreover , Theorem 1.1 proves that such equilibria can naturally be achieved if the decision-maker is reactive and agents are no-regret . Seeing that the decision-maker ’ s equilibrium has also been shown to imply a cost to social welfare [ 30 , 44 ] , our results pave the way for studying new , potentially more desirable solutions in strategic settings . 1.2 Related work . Our work builds on the growing literature on strategic classification [ see , e.g. , 2 , 11 , 14 , 19 , 20 , 25 , 26 , 28 , 30 , 35 , 38 , 42 , 44 , 51 , and the references therein ] . In these works , a decision-maker seeks to deploy a predictive model in an environment where strategic agents attempt to respond in a post hoc manner to maximize their utility given the model . Given this framework , a number of recent works have studied natural learning dynamics for learning models that are robust to strategic manipulation of the data [ 6 , 17 , 20 , 30 , 38 , 48 ] . Such problems have also been studied in the more general setting of performative prediction [ 13 , 31 , 41 , 43 , 46 ] . Notably , all of these works model the interaction between the decision-maker and the agents as a repeated Stackelberg game [ 50 ] in which the decision-maker leads and the agents follow , and these roles are immutable . This follows in a long and continuing line of work in game theory on playing in games with hierarchies of play [ 3–5 ] . Learning in Stackelberg games is itself a growing area in game-theoretic machine learning . Recent work has analyzed the asymptotic convergence of gradient-based learning algorithms to local notions of Stackelberg equilibria [ 20–22 ] assuming a fixed order of play . The emphasis has largely been on zero-sum Stackelberg games , due to their structure and relevance for min-max optimization and adversarial learning [ 21 , 33 ] . Such results often build upon work on two-timescale stochastic approximations [ 10 , 36 ] in which tools from dynamical systems theory are used to analyze the limiting behavior of coupled stochastically perturbed dynamical systems evolving on different timescales . In this paper we depart from this prior work in both our problem formulation and our analysis of learning algorithms . To begin , the problem we consider is asymmetric : one player , namely the strategic agents , makes updates at a fixed frequency , while the opposing player , the decision-maker , can strategically choose their update frequency as a function of the equilibrium to which they wish to converge . Thus , unlike prior work , the choice of timescale becomes a strategic choice on the part of the decision-maker and consequently the order of play in the Stackelberg game is not predetermined . Our analysis of learning algorithms is also more involved than previous works since both the leader and follower make use of learning algorithms . Indeed , throughout our paper we assume that the population of agents is using no-regret learning algorithms , common in economics and decision theory [ 8 , 9 , 24 , 27 , 29 , 39 , 47 ] . This captures the reality that agents gradually adapt to the decisionmaker ’ s actions on some natural timescale . In contrast , existing literature on strategic classification and learning in Stackelberg games assume that the strategic agents or followers are always playing their best response to the leader ’ s action . One exception is recent work [ 32 ] that replaces agents ’ best responses with noisy responses , which are best responses to a perturbed version of the deployed model . While this model does capture imperfect agent behavior , it does not address the effect of learning dynamics and relative update frequencies on the agents ’ actions . Given this assumption on the agents ’ learning rules , we then show how decision-makers who reason strategically about their relative update frequency can use simple learning algorithms and still be guaranteed to converge to game-theoretically meaningful equilibria . Our analysis complements a line of recent work on understanding gradient-based learning in continuous games , but crucially does not assume that the two players play simultaneously [ see , e.g. , 12 , 40 ] . Instead , we analyze cases where both the agents and decision-maker learn over time , and play asynchronously . This represents a departure from work in economics on understanding equilibrium strategies in games and the benefits of different orders of play [ see , e.g. , 3 , 4 ] in Stackelberg games . In our problem formulation , players do not have fixed strategies but must learn a strategy by learning about both their loss and their opponent ’ s through repeated interaction . Some of our analyses touch on ideas from online convex optimization [ 52 ] , specifically derivative-free optimization [ 1 , 15 , 23 , 37 ] . Several works [ 20 , 43 ] within strategic classification and performative prediction apply similar zeroth-order tools to find the decision-maker ’ s equilibrium , but once again assuming immediate best responses to deployed models . We show that such algorithms are versatile enough to be used without such strong assumptions while still having strong convergence guarantees .

Summary: This paper studies a variant of strategic classification where there is not instantaneous interaction between the learner and the agents, but rather, there is a “slower” and a “faster” player (i.e., the whole interaction is governed by the frequencies according to which the learner and the agents update their decisions). This variant is closer to real-life settings, where for example the learner may have much more computational power compared to the agents. In their model, the agents are assumed to update their decisions at a fixed rate, and the learner gets to decide whether to be proactive (i.e., update his decisions slower than the agents) or reactive (i.e., update his decisions faster). After the learner announces the deployed model, the agents are broadly “rational”; to be more specific, they decide their reports based on no-regret algorithms (rather than the standard instantaneous best-response in prior work). The difference in timescales results in fact in a change of who the “leader” and who the “follower” is in the setting. The question that the paper addresses is what changes in terms of loss for the learner and the agents if there exists this timescale difference and whether there are settings where it is best for both the agents and the learner to always have one of the two.  The paper has two main results. First, it shows that if the learner is proactive (resp. reactive), then the learning dynamics of the system converge to the learner’s equilibrium (resp. agents’ equilibrium). Second, it shows that if the learner runs linear or logistic regression and the agents aim to maximize their predicted score, it is preferred by both the learner and the agents that the agents “lead” (i.e., that the learner is reactive). In terms of techniques, the paper uses variants of standard tools in Bandit Convex Optimization (BCO), assuming that everything is smooth enough (i.e., convex, smooth, Lipschitz).