Learning in reverse causal strategic environments with ramifications on two sided markets
Authors: Seamus Somerstep
Yuekai Sun
Yaacov Ritov
Venue: ICLR 2024 poster (2024)
Keywords: Strategic Classification
Performative Prediction
Labor Market
Primary area: societal considerations including fairness, safety, privacy
TL;DR: We develop and study an example of performative prediction that is applicable to economic models of labor markets.

Abstract: Motivated by equilibrium models of labor markets, we develop a formulation of causal strategic classification in which strategic agents can directly manipulate their outcomes. As an application, we consider employers that seek to anticipate the strategic response of a labor force when developing a hiring policy. We show theoretically that employers with performatively optimal hiring policies improve employer reward, labor force skill level, and labor force equity (compared to employers that do not anticipate the strategic labor force response) in the classic Coate-Loury labor market model. Empirically, we show that these desirable properties of performative hiring policies do generalize to our own formulation of a general equilibrium labor market. On the other hand, we also observe that the benefits of performatively optimal hiring policies are brittle in some aspects. We demonstrate that in our formulation a performative employer both harms workers by reducing their aggregate welfare and fails to prevent discrimination when more sophisticated wage and cost structures are introduced.

Published as a conference paper at ICLR 2024
LEARNING IN REVERSE CAUSAL STRATEGIC ENVIRON- MENTS WITH RAMIFICATIONS ON TWO SIDED MARKETS
Seamus Somerstep
Yuckai Sun
la'acov Ritov
Department of Statistics University of Michigan
Department of Statistics
Department of Statistics University of Michigan yr tove umi ch edu
University of Michigan yueka @umi ch. edu
smrstepeume ummi ch. edu
ABSTRACT
Motivated by equilibrium models of labor markets, we develop formulation of causal strategic classification in which strategic agents can directly manipulate compare employers that anticipate the
their outcomes. As an application, strategic response of a labor force with employers that do not. We show through a combination of theory and experiment that employers with performatively optimal hiring policies improve employer reward, labor force skill level, and in some cases labor force equity. On the other hand, we demonstrate that performative employers harm labor force utility and fail to prevent discrimination in other cases.
INTRODUCTION
In many applications of predictive modeling, the model itself may affect the distribution of samples on which it has to make predictions; this problem is known as strategic classification (Hardt et al. 20155 [Bücckne al. 2012) or performative prediction (Perdomo et al., 2020). For example, traffic predictions affect route decisions, which ultimately impact traflic. Such situations can arise in a variety of applications: a common theme that the samples correspond to strategic agents with an incentive to "game the system" and elicit a desired outcome from the model.
In the standard strategic classification setup, the agents are allowed to modify their features, but they do not modify the outcome that the predictive model targets. An example of this is spam classification: spammers craft their messages (e.g. avoiding certain tokens) to sneak them past spam filters. There line of work on causal strategic classification that seeks to generalize this setup by allowing the agents to change both their features and outcomes, usually by incorporating a causal model between
the two (Miller et al. 2020 Kleinberg and Raghavan, 2020 Haghtalab et al.
2023 Horowitz and
Rosenteld 2023).
In this paper, motivated by equilibrium models of labor markets (see Fang and Moro 201 2011a) for a survey), we study a strategic classification setup in which the agents are able to manipulate their attributes via a reverse causal mechanism. This complements prior work causal strategic classification, in which the agents manipulate their attributes via causal mechanisms. As an example of reverse causal strategic classification, we consider the employer's problem in labor market models. In particular, we study the consequences (in terms of employer and labor force welfare) of hiring policies that anticipate reverse causal strategic responses (we will refer such anticipatory policies as strategic and later performative or optimal).
1. In the simple Coate and Loury (1993) labor market model, we show theoretically that such strategic policies lead to higher employer rewards (compared to non-strategic hiring policies). Thus, rational employers should be performative. Further, such hiring policies improve labor force skill level and equity, so performative employers also benefit the labor force.
2. Unfortunately, we also observe that in some aspects, the desirable properties of (reverse causal) strategic hiring policies are brittle. To study their robustness, we developed a more sophisticated general equilibrium labor market model. We show empirically that while our theory generalizes, strategic hiring policies will harm workers by reducing their aggregate welfare and can still lead to disparities amongst labor force participants.

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1.1 RELATED WORK
The study of learning on data distributions that are dependent the learned model (known as performative prediction) was formalized (Perdomo et al. 2020). The works Mendler-Dünner et al. (2020); Drusvyatskiy and Xiao (2020); Wood et al. (2021) extend the problem to a stochastic setting. Perhaps the most prevalent example of performative prediction is strategic classification (Hardt et al. 2015). A work in the strategic classification literature close in spirit to ours is Liu et al. (2022), in which the authors model agents as competing in contests. Our work is also close to the
works that inject element of causality into strategic classification [Alon al. (2020); Kleinberg
and Raghavan (2020); Haghtalab et al. (2023), Millerr et al. (2020); Shavit et al. 2020); Harris et al.
(2022): Horowitz and Rosenfeld (023); Mendler-Dünner al. (2022). Crucially, each of these works assumes that strategic agents manipulate features z, which in turn has a causal effect on outcomes we are interested in the reverse: when strategic agents manipulate y. which in turn has a causal effect
on features J.
Our work is inspired by the economic models of statistical discrimination in labor markets (Arrow 1971 Phelpss 1972 Coate and Loury 1993] Moro and Norman 20033] 2004). See |Fnn and Moro (201 for a survey of this work. The works Liu et al. (2020), Kannan et al. (2019) give a contempo- rary study of Coate and Loury (1993) type models in the context of algorithmic decision-making and affirmative action, respectively.
We provide extended discussion of related works in the appendix.
1.2 PRELIMINARIES
Performative prediction, introduced in Perdomo et al. (2020), seeks to study distribution shifts that dependent model deployment. To be more specific, if the learner picks model parameter 6, then the next samples are drawn from D(0). Thus, a performative learner seeks to minimize
PR(0) & Ez~D(0) {zz: [E)
This problem can be non-convex; most works assume the learner utilizes strategy of repeatedly retraining model. Under this strategy, at round 1 of training, the learner deploys
B+t = arg ming Ez-
The authors of Perdomo al. (2020) show that this strategy (known repeated risk minimization) will converge to stable points, which are defined by:
Saab E arg ming: Ez (((::8))
The second performative solution concept a performatively optimal point, which satisfies
Opp arg ming Ez-
It is worth emphasizing that optimal points generally not stable, and stable points generally not optimal; in particular, stable points are not minima of the performative loss, and optimal points need not be the best response to their induced distribution.
The canonical example of performative prediction is strategic classification.
Example 1.1 (Hardt et al. (0055) Although it predates performative prediction, strategic classifica- tion Hardt et al. (2015) can be viewed as an instance of performative prediction which users game their features. More concretely, there are users with features E X, discrete outcomes y E y ~ [K], x y. The goul of learning to deploy model f X y using training data drawn from P. The cutch is the following: post-training and
and a corresponding base distribution over Z
pre-testing of users will game their features (at some cost c: X x X - Rt )via the update rule:
xz A arg max fx(a'))) c(x,
I x
The goal is to deploy a classifier that is accurate post-distribution shift. The ideal classifier minimizes Ep((((x))))
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2 REVERSE CAUSAL STRATEGIC LEARNING
In the standard strategic classification setting, agents update their features æ, but they do not update their labels y. This is common when the agents wish "game" the model; the standard example is spam classification; spam creators change their emails to try and avoid spam filters (for example, by avoiding certain tokens), but their emails will remain spam.
Recently, Horowitz and Rosenfeld (2023) considered a more general problem setting by permitting agents modify their labels (in addition their features) through causal/structural model. Here, the agents still change their features, but changes to the features can propagate through the causal model and (indirectly) lead to changes their labels. In the following, we consider reverse causal setting in which the agents change their labels and the changes propagate through a structural model to their features. Here are two motivating examples for the reverse causal strategic learning setting: one from labor economics and one from game models of content platforms.
(Coate and Loury ). Consider an employer that wishes to hire skilled workers. The worker skill level is represented as Y E (0, 1} with Y ~ Ber(x). The employer implements (noisy) skill level assessment, with the outcome represented as X E [0,,], 1]. The employer receives utility p+ from "qualliied" hire and suffers loss p- from an "unqualified" hire, and seeks to train classifier (xx) [0 10 that optimizes their overall utility:
Example 2.1
m [P+xP((fx)) =1 1) p- p-11)))-((
-00],
Thus far, this a standard classification problem because the workers are non-strategic, i.e. they unaffected by the employer's 's hiring policy. To introduce a strategic component, the workers are allowed to become qualified (at a cost) in response to the employer's policy. Let w > O be the wage paid hired workers and be the (random and drawn from CDF G) cost to a worker of becoming skilled. For an unskilled worker, the expected utility of becoming skilled is
So.n w wf(x) 1) - Uoo. w (f( 0)
if the worker becomes skilled, if the worker remains unskilled,
(f.v)
where D(x: | 3) is the conditional distribution of skill level assessments. So a strategic worker becomes skilled if
w Sho [((()=1)d((x>>> (x = w [o.1]
=)
Example 2.2 (Hron et al.) (2023). Jagadeesan al. (2023)). Consider a social media system. There Pa; RaT from which users arrive to the system sequentially, with
is demand distribution ul representing a vector of charucteristics of the l'th user. There is also a population of content producers which each produce content {si) st E R. Associated with a piece of content is some noisy measurement X E Rd the content generated via CDF Φ(- 8). For example, X could consist of the sentiment of "comment" receives from users, as well as the amount of likes and views s generates. In general, we interpret E||X]] 1/ the "amount" of the content that creator with signal X produces. The learner wishes to train recommendation system R(E, x. u) that recommends content s the user us with probability:
P(u is assigned content $)) R(E,
ΣØ exp( (at)T
The learner wishes to maximize the true enjoyment user receives from content, which is given by
r(E*, u' = (st""""""
Content producers are not static, however, and will adapt content production. Since exposure to users directly connected the learned recommendation system, a content creator with original content will tend towards producing content s' that optimizes
Eu-PPaar R(E, u) 1B-001111
2.1 THE REVERSE CAUSAL STRATEGIC LEARNING PROBLEM
In reverse causal strategic learning, the samples (X, Y) are agents, and the learner wishes to learn a (possibly randomized) policy/rule X Y predict the agents' responses Y from their features
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X. The agents are fully aware of the learners policy f and such, we assume that the response of the agents to is to change their outcomes strategically:
Y Y - arg max, arg. -(f.
(2.1)
where W is welfare function that measures the agent's welfare and c is (possibly random) cost function that encodes the cost (to the agent) of changing their outcome from to y'. Example 2.3 Exxample2.I]ceee In the labor market example, the agents are the workers, and their welfare is their expected wage
w(f.v) Soo. (f())d(( l/)a E (0,1),
while the cost is
dg',y) cf': G, 3' 10.
Example 2.4 (Example 2.2 cont). In the content creation example, the agents are the content producers. Their welfare function is
W(E, Paar RE,,,,
while the cost function is simply the effort () 2 required to produce a certain "amoun!" of content.
The key ingredient that makes this setting reverse causal is that strategic change in the outcome propagates to cause change in the generation of features X via:
X ( Y) - X+ ~ 2( (+(F))),
where ( Y =3) y) is the conditional distribution of the features X given the outcome Y in the base distribution of the agents. In other words, the agents are unable to directly change their features in the reverse causal setting; they only change their features by changing their outcomes. This distinguishes the reverse causal strategic learning setting from the problem settings in other works on performative prediction and strategic classification (Hardt al. 2015 Horowitz and Rosenfeld 2023).
Going back the learner, their welfare depends on the post-response agents, they must account for the strategic response of the agents. A learner which does this is performative and minimizes the post-strategic response loss:
(2.2) In the rest of the paper, we will return to the labor market, demonstrating that a learner with proactive strategy is often mutually beneficial for both the learner and strategic agents. A discussion on the minimization of Objective2222 is reserved for appendix A in which we provide the a simple algorithm for the learner to minimize the performative loss in a stochastic setting.
ser E (())((
STRATEGIC HIRING IN THE COATE-LOURY MODEL
In this section, we focus on example 2 and study the impact of performatively optimal hiring policies on employer and labor force welfare in the Coate-Loury model. Despite the strategic nature of the labor force, prior studies of labor market dynamics generally assume employers merely reactive (instead of proactive) to the strategic responses of the labor force (Fang and Moro 2011a). We show that in variety of market settings, performatively optimal hiring policies improve employer welfare, labor force welfare, and labor market equity.
3.1 PERFORMATIVE PREDICTION IN THE COATE-LOURY MODEL
Recall the setup from example 2.1 We impose some standard assumptions on the problem:
1. P(X==ly=1) dzly=1)
is monotonically increasing in s;
P(X=rb==) ((=)=0)
Φ( y) is continuously differentiable for y E {0, 1}.
3. G()) G is continuously differentiable, and G is almost surely bounded above by Mc-
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The second two assumptions are technical, but the first is more substantive. It resembles the monotone likelihood ratio assumption in statistical hypothesis testing that ensures consistency of hypothesis tests. In this case, it ensures the optimal hiring policy is a threshold policy of the form 1 1 0) for some 0, this is the operating assumption in the original Coate and Loury (993) paper, and we will proceed with it as well.
Armed with the assumption of a threshold hiring policy, we discuss performative prediction in the Coate and Loury (1993) model. Recall the strategic response of the labor force participants in example 2.1 In aggregate, the proportion of skilled labor force participants becomes
π (0) G[w 0 y 1)
P(x 0 y 0j])-
Given this, the employer's ' strategic hiring problem really an instance of performative prediction with the performative employer utility given by
(0) (0) I(X
> 0 0lv- y- ()) P-P y 0) +
As in employer that deploys an optimal policy anticipates and accounts for the labor force participants
other performative settings, the employer may opt to deploy an optimal or stable policy. An
responses to their decisions and thus deploys policy that solves the performative problem
Copt E arg max, Upar(0).
(3.1)
An employer that deploys stable policy is reactive rather than anticipatory towards labor force strategic responses, i.e. they deploy a stable policy, which is any policy that maximizes employer utility on its own induced distribution:
saab E argmax P+ (0saat) > 0 = p-(1-p1111 (1 - n
= 0)-
The concept of performative stable solutions in the context of a micro-cconomic model has a game theoretic interpretation. A performative solution is the Stackleberg equilibrium to a 2-player game between the firm and the labor force, with the firm as the leader and the labor force the follower. On the other hand, the stable solution is the Nash equilibrium between the firm and the labor force there is no leader-follower dynamic.
We will study both low-wage and high- wage markets. Low-wage markets possess the desirable property that any employer that follows "greedy" strategy, iee. they sequentially deploy
8+41 - arg max P((X
= (0,) - p P_X > y = 0) - xx()))
(3.2)
will eventually converge on stable policy. This is because 3.2 is an instance of RRM, and low worker wages (in addition to some regularity conditions) will ensure that the requirements for RRM convergence provided in Perdomo et al. (2020) are satisfied (see Appendix D).
High-wage markets are independently interesting, as in such markets the benefits of optimal policies for firm will be substantial. On the other hand, in high-wage markets, the convergence of RRM may not be universal. This is not necessary for any of our results (stable policies will always exist and any greedy employer that does stabilize will do also demonstrate that, in practice, convergence of reactive firms is not sensitive to market conditions.
stable policy). Additionally, empirical results
Finally, we remark that each theorem will need additional assumptions market (for example, worker wage and firm reward). The social situations that the Coate and Loury (1993) model applies to are broad Fang and Moro (b) and thus the applicability of each theorem will depend on the social environment at hand. For any specific social situation, a full justification of these assumptions would require a study of the model on real labor market data (or the alternative social situation). Such studies are relatively rare in the literature, but two examples are Arcidiacono al. (20IT) (studies admissions into Duke), Duke),(Altonj) and Pierret (200I) (studies education data).
the structure of the labor
3.2
EFFECTS OF STRATEGIC HIRING ON EMPLOYER AND LABOR FORCE WELFARE
The main result of this subsection compares employer welfare (in terms of the employer's expected utility) and labor force welfare (in terms of the fraction of skilled labor force participants) resulting from optimal and stable hiring policies. We impose two additional assumptions on the market.
1. there exists 0 [0, 1] such that P(X > Ø y = 1) - P(X 1) 0.
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2. d(a 1) ((
82 for any z E [0,
Together, these ensure that some distinction between skilled and unskilled workers is possible, but no policy perfectly distinguish between the two. These assumptions ensure that stable and optimal policies will be distinct.
We study markets in both the waw- and high-wage regimes. In the large-wage regime, firms receive substantial benefits from being optimal.
Theorem 3.1. If > Ma and > p-pp (0)10)) then the following holds for all stable parameters stab and all optimal parameters opps
1. (00pp) TI (0
Uper((stat)) S Uppr((opt) (O10 + 61).
On the other hand, in the low-wage regime, the firm's benefits from an optimal strategy may be small.
Theorem 3.2. If w O. p+ > max(1, (1-=(0)) P-) and 8162 > p-, then the following holds for all stable parameters subb and all optimal parameters opp:
x(
1. (Bupp) T (6suub))
Uper((stab)) S Uper((oopt)
To see the intuition behind Theorems 3.1 and 3.2
consider an extreme case where p- 0. Here, since for any > O, the best reactive employer response is 0 0, any stable policy will satisfy (6smm) (and thus spar(( 0), while clearly an optimal employer can do better than this and will in general deploy model that results in (00pp) > 0. Theorems 3.1 and 3.2 are generalization of these simple dynamics.
The alignment between the social welfare of the workers and the reward to the learner is in contrast with other areas of strategic learning. For example, Milli et al. (2018) show that there a direct trade-off between the social burden strategic agents and learner accuracy in strategic classification.
3.3 EFFECTS OF STRATEGIC HIRING ON LABOR FORCE BQUITY
Besides labor force skill level, another pressing issue in labor markets equity. In fact, the Coate- Loury model was developed to show how inequities may arise in labor markets despite the lack of explicit discrimination in the market. First, we recall the original two-group version of the Coate-Loury model.
Employers now seek to hire workers from large population of labor force participants consisting of two identifiable groups denoted as Maj or Min, with X denoting the fraction of workers with Maj group membership. Crucially, employer profits, worker costs, wages, and worker signals are agnostic with respect to group membership. Denoting the proportion of qualified workers in group Maj and Min the (non-performative) employer utility is:
U (OMi, Min) XU Main (1 (OMn)
The population level response of labor force participants to deployed policies is largely similar in the group case: (BMaa) gMin) G(w[P(x
Maj y 1) P(x Maii 0)]),
(3.3)
Min (MMa), gMin) G(w[P(x MMn y y= 1) P(x Min 0)))
Note that under these assumptions, both the employer's non-performative hiring problem and the performative one are seperable; i.e. the employer can simply solve two hiring problems for each
group separately. As such, define stable pair B (MMaj. @Min) pair of policies that each stable and optimal pairs 6m = (eM (MMin) as a pair of policies that are each (performatively) optimal.

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In Coate and Loury (193) policy pair 0 is discriminatory if i Mi) + <MMi(0), this will also be our metric for discrimination in strategic hiring. It is worth emphasizing that our definition of discriminatory stable pair is exactly the definition of a discriminatory market equilibrium in Coate and Loury 1993).
We will that, in certain sense, employer with a performatively optimal strategy enforces fairness amongst the groups, while a reactive employer provides no such guarantees. Some technical assumptions are needed the conditions of the market; as the employer's hiring problem is separable with respect to group, we state these assumptions in the context of single group market, with the implication that the requirements hold when the market is constrained either group.
We will study markets with the following assumptions:
1. There exists such that P(x > B = 1) - P(x fly= = 0) > 1-
2. p+ p-
The second assumption is primarily to simplify the problem. On the other hand, the first assumption is crucial. It implies that the firm find hiring policy that provides excellent separation between qualified and unqualified workers (generally can be thought of as small). Under such an assumption, a performative firm will have both ability and motive to steer the market towards equitable equilibrium.
Theorem 3.3. Assume 10
( (0) is Lipschitz on [0, 0). where 0" (IT) is the best employer response if Ply = 1) = π and that w > Ma/(1 (1 - - Then the following hold simultaneously:
1. There exists a stable pair Foob such that
n Min (maa)))
Mall (Fopt) Miiin < ffr all optimal pairs app-
The first assumption requires sufficient smoothness on the inverse of the firm's best response; in general, is easy to construct markets where both e and are small. Such an example is presented in Appendix B, and some sufficient market conditions for these conditions to hold are provided in Appendix C.
We also study the problem in low-wage markets, a strong concavity assumption on the firms non- performative utility (0) needed. This is generally not too difficult to meet, and a discussion supplied in Appendix D.
Theorem 3.4. Assume that the non-per non-oerformatiie firm utility U()) is 7 strongly concave for all Also, assume s(-), and Φ(-() are bounded above by K1, and they are disferentiable and g'(.), o' d'(-) o1010)
are bounded above by K2. If ( (1-0)XX0111000000) < w < 7/(2 K2) then the the following hold simultaneously:
1. There exists a stable pair Funb such that (T Min Minn(( (Feaa))
0.
2. Mo (Opp)))) Minn (ōpp)) = 0 for all optimal pairs Japa-
For this to truly be a "low-wage market" we need G G (TT-))000 +)))0)) example markets that satisfy this and the other assumptions the appendix.
o(0
PJ O, we will see an
The intuition behind optimal policies enforcing fairness in Theorems 3.3 and 3.4[is relatively simple; a performative employer that can distinguish skill level well posseses both direct incentive and ability to ensure that both majority and minority group worker qualification levels will be high, and such the discriminatory gap will be small. On the other hand, a reactive employer has no direct incentive to increase the fraction of skilled workers; i.e. they are too myopic to maximize the fraction of skilled workers. Thus, there is no invisible hand steering the market to more equitable equilibria.
EXPERIMENTS ON MARKETS WITH CONTINUOUS SKILL LEVELS
In this section, study the sensitivity of the promising labor force skill and equity results in the preceding section to the underlying Coate-Loury model. A key limitation of the two-group Coate- Loury model is that the labor markets for the groups operate independently of each other. Here
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we study the problem using a more sophisticated general equilibrium model (introduced in Appendix B) which allows for such cross-grouds group effects. In our formulation of labor market, we empirically observe the following economic takeaways:
1. As in the Coure-Loury model, the employer is always incentivized to be performative. Also, performative employer benefits labor force participants because it increases labor force skill levels.
2. Unfortunately, performative employer harms workers by reducing their aggregate welfare. Additionally, the fairness benefits of performative employer are brittle with respect to the assumptions of the underlying market.
Due to space limitations, we only briefly describe our modifications to the Coate-Loury model here and defer a detailed description of the market to Appendix B. The first is that worker skill level Y has Lebesgue density p(y). implying that qualification is not binary, but rather a continuum of possible productivity levels. The second modification to the reverse causal updates of the workers; while they remain strategic and update their outcomes according to the reverse causal mechanism 2.1 the wage structure and cost are different. The cost is now a fixed function: o(.,)) y y + RT. The wage and utility structure will be more complex, again we refer to Appendix B for detailed description of changes.
4.1 EXPERIMENTS WITH LINEAR UTILITY AND FLAT WAGES
In the Coate and Loury (193) model, it is assumed that production functions are linear and that the wage structure is flat. In our context, this is the case where w(x, f) w and u(y, (y)) = H (yy - 1). Finally, the cost is specified to be quadratic: c(y', y) = 5 (v' y)
To study worker welfare and employer utility, we assume p(y) is known, so the employer's policy optimization is non-stochastic. Figure [ diagrams the worker side. In general, an optimal policy results in larger portion of qualified workers, while stable policy leads to greater aggregate worker welfare. The right-hand side of Figure plots the firm side; unsurprisingly, the employer prefers an optimal policy: additionally, the left-hand side of Figure demonttrates algorithm I (appendix A) in a labor market.
75 Qui ifted Workers
0.52 0.54 0.56 0.58
3.8
Worker Welfere
Stable Policy Optimal Polty
optimal milyy
0.5
Stable Poliyy
Stable Palley
Optimal Policy
Optimell Policy
Firm Utility Coeft. (a)
Eduation Cost Param. (c)
Firm Ltility Coeff. (a)
Eduation Cost Param, (c)
(a) Left: w = 1, c 8 // Right: = 1,a= 2
(b) Left: w = 1, = 8 Right: w = 1, = 2
Figure 1: Worker Welfare and Proportion of Qualified Workers for Optimal and Stable Policies
farameter Value
0.4
1.00
Stable Paiccy
0OR 0.1 014
Stable Palicy Optimal
Fimm Utiltyy
Firm Utiliy
0.3
0.10
0.2
6.08
21
Optraal Farmmster
10
30
2.0
2.5
Round
Round
Firm Ltility Coeff. (a)
Eduation Cost Param, (c)
Figure 2: Left: RSGD and Alg 1 (a = 1 5/ Right: employer Utility Under Policies (Left: 1, c = 5/ Right: 1,a 2)
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4.2 EXPERIMENTS WITH NON-LINEAR UTILITY AND NON-FLAT WAGES
Next, we present experiments on markets that include identifiable groups and move beyond the assumptions of linear utility and flat wages. In the subsequent experiments, wages are non-linear and are determined by a Nash equilibrium among competing employers (inspired by (Moro and Norman )004)). Additionally, we assume u(y. Hy) y ())) the important difference being that M is non-linear in hiring probability, which removes the separability property present in the Coure-Loury model. Finally, we no longer assume that cost is agnostic with respect to group; cost
now specified by c(y', y, _ (v' y)²; E MMaj. Min).
Figure
diagrams diagramss the welfare of each group under different policies and parameters; the top row, group costs and proportions are adjusted (costs are adjusted so total cost remains constant), while in the bottom row, the utility parameter a is adjusted.
76 Qualified Workers
056
Werker Utility
622
0.22 0:54
38
0.5
3.6
8.7
0.
6.
65
0.5
D.T
on
0.
05
0.6
0.7
on
0.
0.5
86
9.7
0.8
0.9
Majority Group Proportion
Majority Group Propartion
Majarity Group Propertion
Majority Group Proportion
(a) Left: a 2// Right: a 0.5
(b) Left: a 2 Right: a
T6 Quuallifeed workers
0.38
Worker Utility
666
4.50
0:56
42 644
235
4.00
0.5
100
7.0
05
1.0
1.5
2.0
05
3.0
1.5
2.0
0.5
1.0
1.5
2.0
Utility Power (alpha)
Utility Power (alpha)
Utility Power (alpha)
Utility Power (alpha)
(c) Left: OMin OMay 5 // Right: CMtnn
OMay
20
(d) Left: OMin
CMta
5 Right:
OMin
CMaa
Figure 3: Worker Welfare and Proportion of Qualified Workers for Optimal and Stable Policies
similar pattern as before emerges in Figure 3 Workers tend to be more qualified under performative policies (though this is broken in Figure 3 when the utility function is concave in the proportion of qualified workers), but they tend to have a higher average welfare under stable policies. Clearly, optimal policies no longer enforce fairness, there is appreciable difference in discrimination between the two.
5 CONCLUSION
We have introduced a problem setting that extends strategic classification by allowing agents to directly manipulate y, and in turn, cause shift in æ; additionally, we provided an algorithm for risk minimization in this setting. As an application of this new framework, we studied the effects of stable and optimal strategies on strategic agent welfare and equity in labor market models. We have demonstrated that strategic/perffommaiiv employer can help or harm strategic workers depending on measurement of agent welfare used. In general, proactive learner assists the strategic workers in becoming more skilled but harms the strategic agents by reducing their overall aggregate welfare. Additionally, in some but not all cases, a performative learner can assist in preventing discriminatory equilibrium in labor markets. Finally, have that employer will always benefit from deploying a performative//strateiii hiring policy.
ACKNOWLEDGMENTTS
This paper is based upon work supported by the National Science Foundation (NSF) under grants no. 2027737 and 21 13373.

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A STOCHASTIC OPTIMIZATION FOR REVERSE-CAUSAN STRATEGIC LEARNING
In this section, we develop stochastic optimization algorithm to minimize the performative risk (.. Previously, the work Izzo et al. (222T) gave algorithm for performative optimization under the assumption that the performative map is of the form D()) = p(z; f(0)), with only f uunknww.. Our work is in distinct environment, where the learner has an apriori model for the map D but the base distribution is unknown. This is similar to the assumptions in Levanon and Rosenfeld (2021).
which tackles the optimization for non-causal strategic classification.
A.1 LEARNING SET UP
In practice, learning in reverse causal strategic environments is done sequentially (this is similar to the assumed in other performative prediction works). At each round of learning the learner publishes decision I and all data received in that round drawn via D()). Succinetly, in stochastic (or finite data) setting the order of learning at time is as follows:
1. The firm deploys decision 0,
2. Firm receives data (and corresponding loss/reward) drawn from ((0,). The firm may use this to either deploy a reactive decision or in some algorithm that converges towards an optimal policy (for example in the methodology illustrated below).
Our methodology will be applicable in this stochastic setting, the two strongest assumptions we need are follows:
1. As in previous works on optimization for strategic classification [evvanon and Rosenteld (022) 1 the learner is fully cognisant of the reverse causal strategic map (+(, 0).
2. The learner has knowledge of the feature generating mechanism d(<ry)) The justification here is that in many reverse causal strategic settings ((xly) is based on some standards that the learner sets (eg a hiring firm giving an interview) and thus the learner should have knowledge of this mechanism
Performative learning in a scenario where Y+ 0) is unknown is an open and interesting problem. If Φ(x v) is unknown, it can be estimated from data pairs (x, y).
A.2 METHODOLOGY
We let d(xly) denote the conditional density of x given y. We also assume that we deploy some parametric model f0(x), and write Y+ (Je. v) A Ga(y). In this notation, the cost function in the anti-causal objective can be rewritten as (recall Z X ))
L(0)= Jz ((fa (e(
We derive the gradient in the following lemma.
Lemma A.1. The gradient of T((0) is VI.(0) =
where
(IL = 2 0 (dfncchll((6dchhcccc]]]44666.6d
V2L(0) = 12 30GGw)/b4(((occhccccc4444444444
(A.1)
VaL(0) = (GGGG)))((((()),,,)))))))444666
and (g)) is the solution to the following:
fx 0g ([](())])[[[Goo)))) + (xx fo -4
0. (A.2)
We propose that the learner utilize Monte-Carlo techniques to numerically approximate the integrals
of A.2 and A. One applicable option is to use REINFORCE approximation (see Izzo et al. (0021)
or Williams (1992)). This technique allows us to re-write the integrals of A.2
fx fo())))))(x(())))))))))))))))))) Goo))) bcte
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Since the learner knows these expressions be computed via drawing samples from ( (y)). The other integrals in A.2 and A.1 are readily computable via this method in their vanilla forms. We summarize the proposed algorithm for the performative prediction problem in Algorithm
Algorithm 1 Reverse Causal Strategic SGD
Input: u, 77, n, Pg
while not converged do
Draw {ye 3 ~ p and observe Go(vi)
1n Draw {xr) fijml from o(x Go(U:)
Compute (0) {x)]Ø==) with REINFORCE
0r t- 0 E, (I(()()), (1)))=))
end while
Figure gives an example of this algorithm utilized in the hiring of strategic workers.
A.3 PROOF OF LEMMA A.I
The only tricky term to evaluate is (g) To deal with problem we utilize the implicit function theorem. We have
OgGo(y) 0g argmax, / folo)olols)de .)y.v))
Via the implicit function theorem dyGo(y) must solve
0o / fo(=))(rl/ddd c(v',))) -0].
Which implies that (y)o(y) is given by:
J 0o Bo(fo(e))od(c//oo)))))) + 1 (cl.8//occccc//e------6000000000 =0. (A.3)
B MARKET FORMULATION AND EXPERIMENTS
B.1 MARKET DETAILS
This market is conceptually similar to that Co and Loury (1993). with the twist being that worker skill y is now continuous, with y E y IR generated from ply). The employer still makes hires based on noisy skill assessment X generated from & (xxly) and group membership
Ber[A]. Employer
production specified by utility function which depends on the skill level of worker and the probability that worker is hired: u with (w) A fx (((x)d the employer's 's (non performative) policy maximization objective given by
x [0, 1] - Letting Hy(v) = > (v) + (1 - A) (v)
fax E[u(y. Hy()))
hired worker is given wage w(x, f), dependent on the observed skill level and policy choice and they can improve skills at cost c(.,)) + R+; thus workers strategically choose outcomes by selecting
(v, f) arg max / w(x, (=)l)) c(y', .y)
Using the same notation as in section 2, the employer's post strategic response problem given by jax Blu((++ (v, f), HY (v)))
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In comparing worker well-being under stable and optimal policies, we first must quantify the well- being of the workers. As before, one metric we use is the proportion of qualified workers, which in this case is the probability worker drawn from distribution induced by policy is skilled enough to provide positive employer utility.
We also introduce second metric, the cost adjusted worker welfare. Towards this, define the welfare of a worker with a given natural skill y and group membership as their net benefit in the market post policy deployment and strategic update. Specifically, this is given by
Wi v) = Ix f( (1) d (x)+((u, fi)) - c(Y+ (v. ft).
We will define the aggregate worker welfare as the expectation of (. / y) with respect to p(y).
To study the techniques discussed in appendix A, we must focus on the case when the optimal policy is parametric policy class, i.e. faz 0); 0 E R. In the appendix, we provide sufficient conditions for this to hold.
Proposition B.1. Assume the ratio o(x / (( is increasing in x for all y1 > yo. and the utility u(y) is increasing in y. Then the optimal policy choice for the employer is some threshold policy f(x) = 1=20-
Proof. We denote the marginal density of the signal x as v(x) and subscript any distribution that depends on the policy with f; also WLOG assume y = [0. 1]. We can write the employers (non- performative) utility as the following:
u(f) utf) ( /
u(y)p(ul=)dyda
Thus the optimal policy is of the form:
(() (w)=111) 11 u(y)p(lo)dd 0}
Thus the lemma statement is equivalent to u(y)py/x)dd being monotone in x. Using integration by parts re- re-write this as the following:
f (uy)p(ulz)dd = P(y
/ 1'''v1 ((())(( & x d)dy
0 - / u'()) S y z)dy
=
Since u' vz) > by assumption; we need to show that P(Y S y x) decreasing in x for all y. We have the following:
P(Y g/X x x)) S P.Y v/X = x)dy Std (alv))(()d
So (cll))(()dd
S (o
= =1((1
I ((
By assumption 3o Y1 the ratio ((zl)))/((x(x) is decreasing in x. Thus for all y and any distribution p the ratio 1 ((zlg)p(y)dd// [, ((x)))p(y)dd is monotonically decreasing in x.
B.2 NON-LINEAR UTILITY AND WAGES
In several alternative labor market models (Arrow 1971 Moro and Norman 2004), wage structures not flat; instead they determined by competition between competing employers. Additionally, the authors of Moro and Norman (2004) find that moving beyond linear utility functions introduces inter-group interaction the labor market, which is key to revealing sources of discrimination.
Throughout this section we will assume that u(y. Hz)) = ((H+()))((E for strictly increasing functions 7 [0, 1] > [0, 1]: R - R; and additionally that [2;;; 0) Tzz0-
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Limitations of Linear Utility:
We briefly discuss the limitations of the model if the production function is linear, i.e. 7(:) is the identity function. In such case, via the linearity of expectation, write the vanilla employer utility objective as
max, AEH
())H(())u(y)
Thus, in this case, the employer can simply pick an optimal policy и; for each group separately. Performative stable policies with which क (z) * (o(z) may exist; but one group does benefit from discrimination of another group. This lack of interaction is why we go beyond the linear utility case.
Determination of Wage Structure:
We adhere to the principle (which is shown rigorously in Moro and Norman (2004)) that a Nash equilibrium among two more competing employers, the net profits of each employer should be zero. Equivalently: H (V)
[x[((;; ['] 0. Thus, the wage offered to a worker with signal and group membership should be the expected utility of such worker conditioned on these traits. We can write this for an individual in group as
= xi = ((r((f(())) = = (x)((x))) ((Woo(ala)dp()) Ja d(sls)dpt))
As such, for this section we will assume the wages proffered by the employer are
=(f(==))((((()) Sa u(vlo(olu)dp(()) Sa d(wl))dp(y)
(B.1)
Example B.2. As an example of this assumed wage structure, consider a market specified by ((y) y, p(y) = NO0, o)), f(rlv) = N(y og)) In this market, a employer (with hiring threshold
8) offers wage
wlx, 0) = (306/03) 1sz 2...
Wages scale linearly with perceived skill of a worker; additionally, they increase as the profitability- noise ratio increases (a decrease in o o
Proof. The wage structure is given by:
Jn ye- (a- via vi2 1202 -2 120 by
Sn ye 0000-1/000111223000=0000000000
dy dy
1820
Laz0
S (e-(=-))/203 e - // 2 dy
Sn -
We multiply the top and bottom row by the needed normalization constants, then the top row is the expectation a normal random variable with mean xz(1 + /²/o?)) and the bottom row integrates one.
For Figure 3 we stick with this example of market; additionally assuming that the cost is again quadratic cly', v) = (3' - ² and that ((Hy(y)) (z))
B.3 ADDITIONAL EXPERIMENTS
In figure 4 we present an example of a market that demonstrates the phenomena discussed in Theorems 3.3 and The left hand side plots two curves: the dotted one represents proportion of qualified
workers resulting from a hiring policy, while the solid one employers best response as a function of the proportion of qualified workers; intersections of these correspond to stable policies. There two stable policies with an appreciable gap worker qualification; simultancously the performative utility has unique maximum and thus all pairs of optimal policies are fair.
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0.55
0.30
Warkars
0.25
0.
Frm Ubitty
-0 30
02
Peapor
Optival Parmmater
Deployed Paramater
Deployed
Figure 4: A market with fair optimal pairs and discriminatory stable pairs
Here is another demonstration of Algorithm in the context of labor markets.
1.08
0.15
Fi Utilty
1.06
Parametor
0.10
Optimal Parametat
0.05
RoL
Round
Figure 5: Algorithm 1 and RSGD with a = 4, w = 1, =
B.4 EXPERIMENTAL DETAILS
Section 4.2
1. Figure 2 (left): Experiment is run on market parameterized by y: Unif(0, 11, 6(x|y) =
(y 1 Lo<=<11 u(v) ay 1, c(y, v/) ( y²² = 2, population of 1000 were drawn and at each round 100 are sampled, learning was run for 20 rounds. 10 separate trials were run, with darker lines showing the mean and the shade indicating 90 percent confidence interval. Step size (0.1/(t 1)) is used for both RSGD and Algorithm 1 iterations, with an initial seed of for each.
1, 5). A total
Figure 1 and figure 2 (right): The same market configuration is used (specific parameters indicated in captions). Optimal policies were located using sci-py (Virtanen et al. (2020) mini- mization packages. To locate stable points RRM was run until iterations were within a distance of 0.001 of one another.
Figure 4
For simplicity, a simpler market parametrization is used with u(v) = (w2205) - (v50.5). <(xly) yy) 2.rl + 1 * [ysu.s), p(y) U(0 0.5), c(y, v') cly - y')+
Appendix
50' y For the top 1. Figure 4 u(y) 15y, p(v) = No 1). ((x(y) .N 1), D),c(// 5
row group costs are set as CMin = 25/(1 A); = (2)( In the bottom row group costs are at CMin 25, CMaj 20 CMin = 40, CMai 20, while 0.8. Optimal and stable policies are located in a similar manner, with RRM being terminated after iterations are within 0.00001 of one another for stable polices.
2. Figure 5 Experiment is
market parameterized by ys Unif0O 1), d(zly) (v 1)x 1occac1 u(y) = ay - 1, c(y. f y)² (a = 4, w = 1, c 4). total population of 1000 drawn and each round 100 sampled, learning run for 20 rounds. 10
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separate trials were run, with darker lines showing the mean and the shade indicating 90 percent confidence interval. Step size (0.1/(t + 1)) is used for both RSGD and Algorithm I iterations, with initial seed of 0.1 for each.
C SECTION 3 PROOFS
C.I PROOF OF THEOREM 3.1
Proof. For notational convenience we refer to the performative employer utility as U. We also denote TPR(#) (0) P(X > fyy 1) and FPR(Ø) P(X > fly 0) as the true postive rate (resp. false positive rate) of the deployed classifier. The (decoupled) performative leamner utility is the following:
U(01, 02) - p+ (T[[ FPR (22))) - - (TP((22) - FPR 0g FPR(()
Consider differentiating this wrt the first argument:
0.U 02) 0o ((2) { d0(x [1] p- - ((2)) l d4 d=(0)]
81
FIC P+ (04 11) 1) p- (1-x(12)) (((22))(()))) 10)
By the definition of stability (stable points are optimal on their induced distribution, and end points will not be optimal) any stable point O. must satisfy (0,, 02) 101 =0==0 0. Thus must have:
- P -0. = + +- (0.))) = 0) =
Or cquivalently:
p- - T(.))) ((t,11)
(0.) 6(0,10)
From here we apply the assumption that (((11)//(())) > 6z:
p- (1-(())))
82 - p- (0.)) > ((.))
(0,)
p p+ + ग T(a)).
p-p (0,) + pp
P 6ptp The intuition from here is straightforward, if p+ is large then (0.)) must be small but the optimal labor force skill level for learner should not be small. To formalize this, consider plugging in the upper bound for π (0,) the learners objective. Any which satisfies the bound will have:
So any stable point 9 satisfies x (00)) <
p-
(0))= < P+P- (PR(0) S
P- 82
P4 +p-
On the other hand consider U0) (with Ō defined as assumption 1.) Assume that w is large enough so that wōz > My. then have the following:
U()) p+ O p- 0 FPR(O) > p+ (81 FPR(O))
Now clearly if p+ > P- (0/0((8)))) it holds that (61 + FPR(Ø)) > 182 In this case, since (0)) > (()) for all such that 7 (f) p-p/ pp/0pp02 + p-- it can not hold that x (Pop)) ) < p-/(p448 82 + p-): this completes the proof of the first part. To prove the second part note that U ((0pp) 2 (0) 2 01- Thus:
(0pp))
(faabb) - 81 81 +p)
Since by assumption 645,82 > p_ we have U (8 < p- ((Op))/ + 81) which completes the proof.
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C.2 PROOF OF THEOREM 3.2
Proof. An identical argument as before will show that any stable point O, satisfies T(,)) p- which in turn implies that for all stable points U(0,) < p-/02. Now consider U(O). We have the following:
P462+p-
((0) = P+ (0) - p- (1 x (0))
By the assumptions of the theorem, we have that p. ( > p- - (0)), and thus U()) > d1. Since 81 > p2-62, for all 0,, (()) (0.),, and in particular no optimal point Oppt can have T (0op)) P사b--pー
p-
The statement that Uper((ssall I Uper(B (fops) follows directly from the definition of optimal points.
C.3 PROOF OF THEOREM 3.3
Proof. We first prove the statement on stable points. Let 0* (T) denote the optimal policy for the vanilla utility if the proportion of qualified workers is т. Note that 6 (T) is a monotonically decreasing function of æ. The existence of the stable set daaa (B @Min) such that T. (fxubh) ( is equivalent to the existence of multiple intersections of the following functions from e = [0, 1 to [0, 1]:
f1(0) (0),
f2(0) = G(w(((zw > fly = 1)) - P(x yy= 0))
Let =(0)= consider 7()). By the assumptions on w, (2) = 1, and since [i) is strictly decreasing in (Ō) < 1 so (() < O. By the assumptions Z is continuous and thus by IVT Z has at least one additional zero on [0, 1J. WLOG let 0Min = 1 that (0 0, and let MMai be the other stable point whose existence we have just proved. Note that May < 0 so by the c-Lipsehitz property of fi (0) we have the following:
- (1 - (2)). Note that ((0) = 1 - > and that 2(1) = 0 0. Additionally,
I Ma)))
f1(0)) 1 (B
S ct Mai c..
stab
Since fi (B))) stab = (M)))
stab we have T Mai) - cd 1- - c which completes part of the proof.
We now complete the second part of the proof. Plugging in 0 to the performative utility and using the assumptions on G, c, w we have
U()) > + - -6 ppp +
p+ (1 -)
Since any optimal point Opn satisfies (Dapp) 2 (0) we know that for all optimal points:
p4 > Oopely = 1) - p_ - π (fop))) P(x > 0pply = 0) > p+ - -))
T(0mp) > Oppt y 1) + P(x > Oppt y 0)] - P(x Oop y 0) > p+ (1-
1 -
(00pp)) > P(x Oaps y 1) + P(x: Opp y 0) +
P(x > 0ep y 0)
P(r > 0opt 1) + P(x > 0op 0)
Since 0 < P(x > Oppe y = 1) < and 0 < p(x > Osp y = 0). simple calculation will show
1-6 6
P(x > Oopt y = 1) + P(x > 0ept 1/ = 0)
P(æ > 0ep
0)
>1-c.
P(x > 0opt 1) + P(x > 0pp 1 0)
This in turn implies T (00p)) must satisfy Mac (Dpp)) Min (1Tp))
e for any optimal point. Thus all optimal pairs Begi (aMa)) AMinn
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Published as a conference paper at ICLR 2024
Remark C.1. It is generally possible construct markets in which both and are small. In fact, the derivative of 0 (() given by
0 (0)] 0 (0) (z) (011) (100-'00)0000000)
Φ + x(((0)
Thus, for example any market for which ø' (æ 1) for x 1 and x(x10) can satisfy both. See Figure iin in appendix B for an example.
C.4 PROOF OF THEOREM 3.4
Proof. We first prove the statement on stable points. Let 8* (T)) denote the optimal policy for the vanilla utility the proportion of qualified workers т. The existence of the stable set (6 stab Min)
such that x-Majj (f * (i is equivalent to the existence of multiple intersections of the following functions from = [0, to [0,1]:
f1) (0),
f2(0) = G(w(((wP > fly 1)) P(x y-==0)) =0)
Let Z(0)= f1(0) f2))). Note that (1) convexity assumptions, we can directly calculate that:
0. Additionally, consider Z(0). By the strong
(0) x(0 + 6(0
Thus, by assumption, (1) f2(0) and since these functions continuous, there must be an additional 0' such that (0') (2))', thus the pair (0', 0) will be discriminatory.
Next prove the statement on optimal policies. We will show that if the vanilla utility U(0 л) is Y strongly concave and that w is small enough, the performative utility will be concave and thus Mai = e Minn since the optimal policy will be unique.
Consider the decoupled performative utility:
(0, 02) = (0) + + - (02)))
The second derivative of Uprr(() (0) is given by the following:
82
8²
(0) 307 Uper(O1,
(0,
00?
We wish to show that (0) < - form some o > 0. Note that by assumption we have the following:
02
003
-n7
Letting A()) (PR(0) FPR(O)
8²
Uper (01, 02) = ((wg 11 6(0 0)) + g(u(A(0))( 6'(0 (O))
002
Because of the boundedness assumptions 9() and <() we have that
82
503 (0, 02) < 2w RwKKK
Thus if w < < the performative firm utility will be strongly concave and thus optimal points will be unique and non-discnin aon-dscriminatory.
Remark C.2. Consider the family of markets with 6(x 0) -ax and oxx 11 (n 1)x" for some a small and large. Note that if p+ p- large enough, this market is strongly concave. Additionally, it is easy see that there will be O that provides good seperation and additionally that 6 0) 23 1 while o(0 1) O, and thus the lower bound on in this market be small well.
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Published as a conference paper at ICLR 2024
D DISCUSSION ON CONVERGENCE OF RRM (THE MYOPIC FIRMS PROCESS)
For the purpose of this discussion assume that X E [0, 1]. From Perdomo et al. 2020 a myopic firm is sure eventually stabilize if the following conditions met the market:
1. (0) must be c- Lipschitz continuous, for some not large. Note that:
[0)( S w suprecI(c) jo. 1) o(x [0]
In general, as long as 9() and <(ly) are bounded functions, then this condition will hold for w K. Additionally, if wages low, e will be small.
2. The firms non-performative utility (0) and the agents aggregate response T(0) must be suf- ficiently smooth. This will follow as long as the functions G(), ((-ly) are smooth, which is a technical condition and does not impact the dynamics of the market.
3. The firms vanilla utility (() p+ TPR(() p- (1 FPP(()) should be y- strongly concave for x. Note that:
U p+ To' (0]1) p- (1 10--))p T))
For example, if d'(11) > r/p+ and if (010) < -7/p- then 7 strong concavity will be guaranteed.
The most crucial ingredient for convergence is the sensitivity of π (0). In general, if the function T(0) = 0 (λ) x (0)
8" (x) arg ma (T)
has a derivative with upper bound less then one then convergence of RRM will be ensured. This will occur, for example, if the market is regular enough so that 0" (T) has bounded derivative and w is small.
E EXTENDED RELATED WORKS
Performative Prediction: Performative prediction, introduced in Perdomo al. (2020), seeks to study distribution shifts that are dependent on model deployment. In particular, the reverse causal distribution map considered in this paper leads form of subpopulation shift [Matty et al. [202T] 2022b). This line of research has been extended to the stochastic optimization setting in the works (Mendler-Dünnes et al., 2020 Drusvyatskiy and Xiao 2020 Wood et al. 2021 Maity al. 2022a).
Stateful performative prediction. introduced in Brown et al. (2022). allows the map D to depend on both and the current data distribution. The authors Izzo et al. (2021) and Izzo et al. (2022) propose
methods for minimizing the performative risk under parametric assumption (D(O) p(2; (())). The work Miller et al. (202T) establishes the necessary conditions for the performative risk to be convex. The authors of Jagadeesan al. (2022) introduce zero't 'th order algorithm for minimizing regret in a performative setting. Our algorithm for anti-causal strategic learning is similar in spirit to the work of Izzo al. (202T). The main difference is that in our case the performative map is known apriori, which simplifies the procedure and allows for use beyond the parametric setting.
Strategic Classification:
Although it predates performative prediction, strategic classification Hardt et al. (015)) can be viewed as an instance of performative prediction in which users game their features. The authors Levanon and Rosenfeld (2021] 2022) give efficient algorithms for learning in general strategic settings. The work et al. (2020) introduces methods for minimizing Stackelberg regret in online strategic classification. The authors of Yu et al. (2022) establish algorithms for strategic offline reinforcement learning problem. One line of work seeks to study the case where users and the learner not share all information; the authors of Dagadeesan
al. (2021) assume users view noisy version of the model; the work Dong et al. (2018) assumes the learner is blind to the strategic agents utility function; the works Ghalme et al. (2021): Bechavod et al. (2022): Barsotti et al.) (2022) study the case where the strategic agents must also learn the deployed classifier. The authors of Zrnic et al. (20T)) study an extension of strategic classification where the roles are reversed; the agents make strategic decisions before the learner deploys model. In order to introduce interaction among the strategic agents, the authors of Liu et al. (2022) study a problem where agents compete in contests.
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Published as a conference paper at ICLR 2024
A line of work similar in vein to work and strategic classification is causal recourse Karimi jet al. (2021), König et al. (2023) which seeks to provide actionable intervents for strategic agents to improve their features.
Our work is most aligned with recent efforts to inject causality into strategic classification. Much of the focus has been on improvement (incentivizing agents to improve features in way that causes responses to improve), beginning with the works Alon et al. (2020); Kleinberg and Raghavan (2020); Haghtalab et al. (2023). The authors of Miller al. (2020) show that learning good models for improvement is equivalent to solving causal inference problems. A follow-up line of work considers this problem in specific scenarios; with Shavit al. (2020) focused on regression, and Harris et al.) (2022) on instrumental variables. The work Mendler- Dinler et al. (2022) reframes performative predictions as causal interventions. A closely related work is Horowitz and Rosenfeld 2023) which gives method for minimizing the causal strategic empirical risk.
Economic models labor markets: Studies of statistical theories discrimination began with the two lines of work (Arrow) 1971; Phelps 1972). In Phelps (1972), discrimination in labor markets due to two groups of workers being exogenous, while Arrow (197)) shows that even groups are endogenous, discrimination can occur in equilibrium. These ideas were developed by Coate and Loury into a labor market model (Coate and Loury 1993). More recently, the authors of Moro and Norman (2004) and Moro and Norman (2003) extended this model by including interactions among groups of workers and a wage structure set by inter-employer competition. The line of work Fryar al. (2008); Frraar and Loury (2005) (013); Craig and Fryer (2017) studies the impacts of different afemployerative action policies (i.e. color blinded vs color sighted) in variety of market settings including those similar in spirit to (Coate and Loury] 1993). The more contemporary works
Liu et al. (2020);[Somerss) [Somerstep et al. (2023) utilize the Coate and Loury model to study discrimination algorithmic decision making. For a survey on statistical discrimination in labor market, see (Fang
and Moro 2011a).
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Reviews:
[{"review_id": "mU1MXri6tE", "reviewer": "Reviewer_jt3C", "summary": "This paper proposes a novel formulation of the causal strategic classification problem. Here, there is a set of agents, each described by x, the values for a set of features and with a true label y. A classifier wishes to classify agents based on their features x to minimize some loss function defined w.r.t. the true labels y of the agents. \n\nHowever, the agents are strategic, and given a classifier with parameters \\theta, can change their label y by changing their features at a certain cost, in order to obtain a utility from the classification. The problem is motivated through the running example of the labor market, where the employer has a hiring policy based on the skill of the worker. Workers can change their skill in order to increase their chances of being hired and obtain the wage from being hired. \n\nThe model assumes that there is a (reverse) causal model that determines the relationship between the features and the label (or vice-versa respectively). The paper focuses on the setting with a reverse causal model. Here, agents are allowed only to determine whether they want a change in their label, and the reverse causal model determines how the features must be changed. \n\nThe technical results of the paper deal with analyzing the solutions of the game that results from the interactions between the classifier and agents under such a reverse causal model, with interesting results and consequences for a well-studied model of the labor market. The paper provides interesting results on the effect of a reactive classifier (which iteratively updates the parameters \\theta after the agents respond to the classifier used in the previous iteration) and a strategic classifier which anticipates the best response of the agents on the welfare of the classifier and agents.", "questions": "None", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- First, I found the model very interesting, and the main technical contribution of the reverse causal setting for performative prediction to be interesting and significant. The topic is clearly relevant to ICLR, and similar models may be relevant several other applied fields.\n- The main technical results appear sound, and I appreciate the nice discussions following the theorems discussing how to interpret the results and the implications for the running example of the labor market.\n- The findings that the presence of a performative classifier can hurt agent welfare and that its fairness properties are brittle are also interesting.", "weaknesses": "- No major weakness. The authors may consider doing a more thorough pass to fix some minor typos.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KasoiY2scC", "reviewer": "Reviewer_L1Mv", "summary": "* The paper introduces a reverse-causal strategic classification setting, and analyzes it primarily within the context of the Coate-Loury labor market model.\n* In the Coate-Loury labor market model, an employer trains a screening function $f(x)$ to detect high-skill workers ($y=1$), and workers respond strategically by possibly increasing their skill level at cost $c$. Worker features $x\\\\in[0,1]$ are a noisy function of skill ($x\\\\sim\\\\Phi(\\\\cdot | y)$), making the strategic response reverse-causal.\n* The analysis investigates the gap between the associated repeated risk minimization (RRM/”stable”) and performatively optimal (PO/”performative”) policies (eq. (2.2) and eq. (2.3), respectively).\n* For a Coate-Loury model with a uniform decision rule (same decision threshold $\\\\theta$ for all population), Theorem 3.1 gives conditions under which a PO policy results in a higher proportion of skilled workers and increased employer’s utility compared to RRM.\n* For a model with non-uniform decision policies (two groups {Maj,Min} and a separate decision threshold for each), Theorem 3.2 gives conditions under which RRM leads to a discriminatory population composition, and PO leads to population composition which is approximately balanced.\n* Finally, a generalization of the Coate-Loury model with inter-group interactions is investigated by numerical simulation. Results indicate that a PO policy leads to higher employer utility, but can generally reduce the welfare of the workers.", "questions": "* RRM convergence: How does the claim about convergence to performative stability relate to the formal guarantees given by Perdomo et al.?\n* Additional related results in Perdomo et al.: In the paragraph below the statement of Theorem 3.1, it is claimed that “Theorem 3.1 gives conditions for there to be an appreciable gap. This complements prior results (for example, in Perdomo et al. (2020)) that provide conditions under which the gap is small.”. In contrast, Theorem 4.3 in Perdomo et al. 2020 predicts that the gap between the PO and RRM policies is expected to be small. What is the relation between the gaps presented in this paper and Theorem 4.3 in Perdomo et al.? If some required Theorem 4.3 are not met, which ones? And how does it relate to the RRM convergence guarantees discussed in the question above?\n* Learning setting: At what stage data is available to the employer, and how do they learn from it? How do the main results extend to scenarios where predictors are learned from finite datasets?\n* Do similar results hold for the content creation scenario described in Example 2.2? What would be required in order to apply the results in other scenarios?\n* Small question about notations: What is the difference between $w\\\\int_X 1_{\\\\{f(x)=1\\\\}} d\\\\Phi(x|Y=1)$ and $\\\\int_{[0,1]} w f(x) d \\\\Phi(x|1)$ in Example 2.1?\n* Small typo in Section 2: examplse. Appendix has undefined references: (??).", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* Problem is well-motivated. Theoretical framework is interesting.\n* Performative response is derived directly from an established economic model.\n* Running examples in the introduction aid understanding, and strengthen applicability.\n* Model assumptions are presented clearly, and relaxed gradually.", "weaknesses": "* Soundness concern: Repeated risk minimization (RRM) plays a central role in the theoretical analysis (eq. (2.2) ,eq. (3.1)), and convergence is claimed to be due to “repeated risk minimization, which is known to converge to performatively stable policies (Perdomo et al., 2020)”. However, if I understand correctly, the convergence guarantees in Perdomo et al. 2020 rely on strong regularity assumptions (e.g., $\\\\beta$-joint smoothness, $\\\\gamma$-strong convexity, $\\\\varepsilon$-sensitivity) and RRM can fail to converge in their absence (see Theorem 3.5 and Proposition 3.6 in Perdomo et al. 2020). I was unable to find a discussion of these assumptions and their applicability in the paper, and therefore it is not clear why RRM is guaranteed to converge in this context.\n* The learning setting is unclear (see questions below).\n* Code is not provided, making it hard to validate and reproduce the results of Section 4, which rely on numerical evaluation.\n* Interaction model assumes one dimensional features and strict monotone likelihood ratio. It is not clear how results extend to higher-dimensional features and more complex distribution structures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)