Bayesian Persuasion for Algorithmic Recourse
Anonymous Author(s)
AfﬁliationAddress email
Abstract
When subjected to automated decision-making, decision subjects may strategicallymodify their observable features in ways they believe will maximize their chancesof receiving a favorable decision.
In many practical situations, the underlyingassessment rule is deliberately kept secret to avoid gaming and maintain competitiveadvantage.
The resulting opacity forces the decision subjects to rely on incompleteinformation when making strategic feature modiﬁcations.
We capture such settingsas a game of Bayesian persuasion , in which the decision maker offers a form ofrecourse to the decision subject by providing them with an action recommendation(or signal ) to incentivize them to modify their features in desirable ways.
Weshow that when using persuasion, the decision maker and decision subject arenever worse off in expectation, while the decision maker can be signiﬁcantlybetter off.
While the decision maker’s problem of ﬁnding the optimal Bayesianincentive-compatible (BIC) signaling policy takes the form of optimization overinﬁnitely-many variables, we show that this optimization can be cast as a linearprogram over ﬁnitely-many regions of the space of possible assessment rules.
Whilethis reformulation simpliﬁes the problem dramatically, solving the linear programrequires reasoning about exponentially-many variables, even in relatively simplecases.
Motivated by this observation, we provide a polynomial-time approximationscheme that recovers a near-optimal signaling policy.
Finally, our numericalsimulations on semi-synthetic data empirically demonstrate the beneﬁts of usingpersuasion in the algorithmic recourse setting.
Introduction
High-stakes decision-making systems increasingly utilize data-driven algorithms to assess individualsin such domains as education [31], employment [5, 36] and lending [24].
Individuals subjected tothese assessments (henceforth, decision subjects) may strategically modify their observable featuresin ways they believe maximize their chances of receiving favorable decisions [21, 9].
The decisionsubject often has a set of actions/interventions available to them.
Each of these actions leads tosome measurable effect on their observable features and subsequently, the decision they receive.
From the decision maker’s perspective, some of these actions may be more desirable than others.
Consider credit scoring as an example.
1 Credit scores predict how likely an individual applicantis to pay back a loan on time.
Financial institutions regularly utilize credit scores to decide whetherto offer applicants their ﬁnancial products and determine the terms and conditions of their offers(e.g., by setting the interest rate or credit limit).
Applicants regularly attempt to improve their scoresgiven their (partial) knowledge of credit scoring instruments.
For instance, a business applying for a loan may improve its score by paying off existing debt or cleverly manipulating its ﬁnancial recordsto appear more proﬁtable.
While both of these interventions may improve credit score, the formeris more desirable than the latter from the perspective of the ﬁnancial institution offering the loan.
The question we are interested in answering in this work is: how can the decision maker incentivizedecision subjects to take such beneﬁcial actions while discouraging manipulations?
The strategic interactions between decision-making algorithms and decision subjects has motivated agrowing literature known as strategic learning (see e.g., [18, 11, 40, 29, 19]).
While much of the priorwork in strategic learning operates under the assumption of full transparency (i.e., the assessmentrule is public knowledge), we consider settings where the full disclosure of the assessment rule is notviable.
In many real-world situations, revealing the exact logic of the decision rule is either infeasibleor irresponsible.
For instance, credit scoring formulae are closely guarded trade secrets, in part toprevent the risk of default rates surging if applicants learn how to manipulate them.
In such settings,the decision maker may still have a vested interest in providing some information about the decisionrule to decision subjects in order to provide a certain level of transparency and recourse .
In particular,the decision maker may be legally obliged, or economically motivated, to guide decision subjectsto take actions that improve their underlying qualiﬁcations.
To this end, instead of fully revealing theassessment rule, the decision maker can recommend actions for decision subjects to take.
Of course,such recommendations need to be chosen carefully and credibly; otherwise, self-interested decisionsubjects may not follow them or may utilize the recommendations to ﬁnd pathways for manipulation.
We study a model of strategic learning in which the underlying assessment rule is not revealed todecision subjects.
Our model captures several key aspects of the setting described above: First, eventhough the assessment rule is not revealed to the decision subjects, they often have prior knowledgeabout what the rule may be.
Secondly, when the decision maker provides recommendations todecision subjects on which action to take, the recommendations should be compatible with thesubjects’ incentives to ensure they will follow the recommendation.
Finally, our model assumesthe decision maker discloses how they generate recommendations for recourse—an increasinglyrelevant requirement under recent regulations (e.g., [10]).
Utilizing our model, we aim to design a mechanism for a decision maker to provide recourse to adecision subject who has incomplete information about the underlying assessment rule.
We assumethe assessment rule makes predictions about some future outcome of the decision subject (e.g.,whether they will pay back a loan in time if granted one).
Before the assessment rule is trained(i.e.
, before the model parameters are ﬁt), the decision maker and decision subject have some priorbelief about the realization of the assessment rule.
This prior represents the “common knowledge”about the importance of various observable features for making accurate predictions.
After training,the assessment rule is revealed to the decision maker, who then recommends an action for thedecision subject to take, based on their pre-determined signaling policy .
Upon receiving this actionrecommendation, the decision subject updates their belief about the underlying assessment rule.
Theythen take the action which they believe will maximize their utility (i.e., the beneﬁt from the decisionthey receive, minus the cost of taking their selected action) in expectation.
Finally, the decisionmaker uses the assessment rule to make a prediction about the decision subject.
The interaction described above is an instance of Bayesian persuasion , a game-theoretic model ofinformation revelation originally due to Kamenica and Gentzkow.
The speciﬁc instance of Bayesianpersuasion we consider in this work is summarized below.
Interaction protocol between the decision maker and decision subject
The decision maker and decision subject initially have some prior/belief about the assessment rule that will be trained.
Before training, the decision maker commits to a signaling policy.
After training, the assessment rule is revealed to the decision maker.
The decision maker then uses their signaling policy and knowledge of the assessment rule to recommend an action for the decision subject to take.
The decision subject updates their belief given the recommendation, and chooses an action that they believe maximizes their utility.
The decision subject receives a prediction through the assessment rule.
Our contributions.
Our central conceptual contribution is to cast the problem of offering recourseunder partial transparency as a game of Bayesian persuasion.
Our key technical contributions consistof comparing optimal action recommendation policies in this new setup with two natural alternatives:(1) fully revealing the assessment rule to the decision subjects, or (2) revealing no information atall about the assessment rule.
We provide new insights about the potentially signiﬁcant advantagesof action recommendation over these baselines, and offer efﬁcient formulations to derive the optimalrecommendations.
More speciﬁcally, our analysis offers the following takeaways:
Using tools from Bayesian persuasion,
we
show that it is possible for the decision maker to
provide incentive-compatible action recommendations that encourage rational decision subjects
to modify their features through beneﬁcial interventions.
While the decision maker and decision
subjects are never worse off in expectation from using optimal incentive-compatible recommen-
dations, we show that situations exist in which the decision maker is signiﬁcantly better off in
expectation utilizing the optimal signaling policy (as opposed to the two baselines, Section
We derive the optimal signaling policy for the decision
maker
While the decision maker’s
optimal signaling policy initially appears challenging
to
compute
(as it involves
optimizing
over
continuously many variables)
, we show
that the problem can naturally be cast as a linearprogram deﬁned in terms of a ﬁnite set of variables.
However, solving this linear programmay require reasoning about exponentially-many variables.
Motivated
by this observation, weprovide a
polynomial
-time algorithm to approximate the optimal signaling policy
Section 4)
We empirically evaluate our persuasion mechanism on semi-synthetic data based on the Home
Equity Line of Credit (HELOC) dataset, and ﬁnd that the optimal signaling policy performssigniﬁcantly better than the two natural alternatives across a wide range of instances (Section 5).
1.1 Related work
Strategic responses to unknown predictive models.
To the best of our knowledge, our work is theﬁrst to use tools from persuasion to model the strategic interaction between a decision maker andstrategic decision subjects when the underlying predictive model is not public knowledge.
Severalprior works have addressed similar problems through different models and techniques.
For example,
Akyol et al.
[1] quantify the “price of transparency”, a quantity which compares the decision maker’sutility when the predictive model is fully known with their utility when the model is not revealedto the decision subjects.
Ghalme et al.
[17] compare the prediction error of a classiﬁer when it ispublic knowledge with the error when decision subjects must learn a version of it, and label thisdifference the “price of opacity”.
They show that small errors in decision subjects’ estimates ofthe true underlying model may lead to large errors in the performance of the model.
The authorsargue that their work provides formal incentives for decision makers to adopt full transparency as apolicy.
Our work, in contrast, is based on the observation that even if decision makers are willing toreveal their models, legal requirements, privacy concerns, and intellectual property restrictions mayprohibit full transparency.
So we instead study the consequences of partial transparency—a commoncondition in real-world domains.
Bechavod et al.
[2] study the effects of information discrepancy across different sub-populations ofdecision subjects on their ability to improve their observable features in strategic learning settings.
Like us, they do not assume the predictive model is fully known to the decision subjects.
Instead, theauthors model decision subjects as trying to infer the underlying predictive model by learning fromtheir social circle of family and friends, which naturally causes different groups to form within thepopulation.
In contrast to this line of work, we study a setting in which the decision maker providescustomized feedback to each decision subject individually.
Additionally, while the models proposedby [17, 2] circumvent the assumption of full information about the deployed model, they restrict thedecision subjects’ knowledge to be obtained only through past data.
Algorithmic recourse.
Our work is closely related to recent work on algorithmic recourse [28].
Algorithmic recourse is concerned with providing explanations and recommendations to individualswho have received unfavorable automated decisions.
A line of algorithmic recourse methods including[42, 41, 25] focus on suggesting actionable or realistic changes to underlying qualiﬁcations to decisionsubjects interested in improving their decisions.
Our action recommendations are “actionable” in thesense that they are interventions which promote long-term desirable behaviors while ensuring thatthe decision subject is not worse off in expectation.
Transparency.
Recent legal and regulatory frameworks, such as the General Data Protection
Regulation (GDPR) [10], motivate the development of forms of algorithmic transparency suitable forreal-world deployment.
While this work can be thought of as providing additional transparency intothe decision-making process, it does not naturally fall into the existing organizations of explanationmethods (e.g., as outlined in [7]), as our policy does not simply recommend actions based on thedecision rule.
Rather, our goal is to incentivize actionable interventions on the decision subjects’observable features which are desirable to the decision maker, and we leverage persuasion techniquesto ensure compliance.
Bayesian persuasion.
There has been growing interest in Bayesian persuasion [27] in the computerscience and machine learning communities in recent years.
Dughmi and Xu [12, 13] characterize thecomputational complexity of computing the optimal signaling policy for several popular models ofpersuasion.
Castiglioni et al.
[4] study the problem of learning the receiver’s utilities through repeatedinteractions.
Work in the multi-arm bandit literature [33, 34, 22, 6, 39] leverages techniques from
Bayesian persuasion to incentivize agents to perform bandit exploration.
Finally, linear programmingbased approaches to Bayesian persuasion have been studied in the economics literature [30, 14],although the persuasion setting we study differs considerably.
Other strategic learning settings.
The strategic learning literature [18, 17, 8, 32, 23, 2, 19, 20, 29,16] broadly studies machine learning questions in the presence of strategic decision subjects.
Therehas been a long line of work in strategic learning that focuses on how strategic decision subjects adapttheir input to a machine learning algorithm in order to receive a more desirable prediction, althoughmost prior work in this literature assumes that the underlying assessment rule is fully revealed to thedecision subjects, which is typically not true in reality.
Setting and background
Consider a setting in which a decision maker assigns a predicted label ˆ y ∈ {− 1 , +1 } (e.g., whetheror not someone will repay a loan if granted one) to a decision subject with initial observable featuresx 0 = ( x 0 , 1 , · · · , x 0 ,d − 1 ,
) ∈ R d (e.g., amount of current debt, bank account balance, etc.).
Weassume the decision maker uses a linear decision rule to make predictions, i.e., ˆ y = sign { x (cid:62) 0 θ } ,where the assessment rule θ ∈ Θ
⊆
R d is chosen by the decision maker.
The goal of the decisionsubject is to receive a positive classiﬁcation (e.g., get approved for a loan).
Given this goal, thedecision subject may choose to take some action a from some set of possible actions A to modifytheir observable features (for example, they may decide to pay off a certain amount of existing debt,or redistribute their debt to game the credit score).
We assume that the decision subject has m actions{ a 1 , a 2 , . . .
a m } ∈ A at their disposal in order to improve their outcomes.
For convenience, weadd a ∅ to A to denote taking "no action".
By taking action a , the decision subject incurs some costc ( a ) ∈ R .
This could be an actual monetary cost, but it can also represent non-monetary notions ofcost such as opportunity cost or the time/effort the decision subject may have to exert to take theaction.
We assume taking an action a changes a decision subject’s observable feature values from x 0to x 0 +∆ x ( a ) , where ∆ x ( a ) ∈ R d , and ∆ x j ( a ) speciﬁes the change in the j th observable feature asthe result of taking action a .
For the special case of a ∅ , we have ∆ x ( a ∅ ) = 0 , c ( a ∅ )
= 0 .
As a resultof taking action a , a decision subject, ds , receives utility u ds ( a, θ )
= sign { ( x 0 + ∆ x ( a )) (cid:62) θ } − c ( a ) .
In other words, the decision subject receives some positive (negative) utility for a positive (negative)classiﬁcation, subject to a cost for taking the action.
If the decision subject had exact knowledge of the assessment rule θ used by the decision maker, theycould solve an optimization problem to determine the best action to take in order to maximize theirutility.
However, in many settings it is not realistic for a decision subject to have perfect knowledgeof θ .
Instead, we model the decision subject’s information through a prior Π over θ , which can bethought of as “common knowledge” about the relative importance of various observable featuresin predicting the outcome of interest.
For example, the decision subject may believe that priorpayment history would likely be highly correlated with future default.
We will use π ( · ) to denote theprobability density function of Π (so that π ( θ ) denotes the probability of the deployed assessmentrule being θ ).
We assume the decision subject is rational and risk-neutral.
So at any point during the interaction, if they hold a belief Π (cid:48) about the underlying assessment rule, they would pick anaction a ∗ that maximize their expected utility with respect to that belief.
More precisely, they solvea ∗ ∈ arg max a ∈A E θ ∼ Π (cid:48)
[ u ds ( a, θ )] .
From the decision maker’s perspective, some actions may be more desirable than others.
Forexample, a bank may prefer that an applicant pay off more existing debt than less when applying for a loan.
To formalize this notion of action preference, we say that the decision makerreceives some utility
u dm ( a ) ∈ R when the decision subject takes action a .
In the loan example,u dm ( pay off more debt ) >
u dm ( pay off less debt ) .
2.1 Bayesian persuasion in the algorithmic recourse setting
The decision maker has an information advantage over the decision subject, due to the fact that theyknow the true assessment rule θ , whereas the decision subject does not.
The decision maker may beable to leverage this information advantage to incentivize the decision subject to take a more favorableaction (compared to the one they would have taken according to their prior) by recommending anaction to the decision subject according to a commonly known signaling policy .
Deﬁnition 2.1 (Signaling policy) .
A signaling policy S :
Θ
→ A is a (possibly stochastic) mappingfrom assessment rules to actions.
We use σ ∼ S ( θ ) to denote the action recommendation sampled from signaling policy S ,
whereσ ∈
A is the realized recommended action.
The decision maker’s signaling policy is assumed to be ﬁxed and common knowledge.
This is becausein order for the decision subject to perform a Bayesian update based on the observed recommendation,they need to know the signaling policy.
Additionally, the decision maker must have the powerof commitment , i.e., the decision subject must believe that the decision maker will select actionsaccording to their signaling policy In our setting, this will be the case since the decision makercommits to their signaling policy before training the assessment rule.
This can be seen as a form oftransparency, as the decision maker is publicly announcing how they will use their assessment ruleto provide action recommendations/recourse before they train the assessment rule.
For simplicity,we assume that the decision maker shares the same prior beliefs Π as the decision subject over theobservable features before the model is trained.
These assumptions are standard in the Bayesianpersuasion literature (see, e.g., [27, 33, 34]).
In order for the decision subject to be incentivized to follow the actions recommended by the decisionmaker, the signaling policy S needs to be Bayesian incentive-compatible .
Deﬁnition 2.2 (Bayesian incentive-compatibility) .
Consider a decision subject ds with initial observable features x 0 and prior Π .
A signaling policy S is Bayesian incentive-compatible (BIC) for dsif E θ ∼ Π [ u ds ( a, θ ) | σ = a ] ≥ E θ ∼ Π
[ u ds ( a (cid:48) , θ )
| σ = a ] for all actions a, a (cid:48) ∈
A such that S ( θ ) haspositive support on σ = a .
In other words, a signaling policy S is BIC if, given that the decision maker recommends action a , thedecision subject’s expected utility is at least as high as the expected utility of taking any other action a (cid:48) .
We remark that while for the ease of exposition our model focuses the interactions between thedecision maker and a single decision subject, our results can be extended to a heterogeneous population of decision subjects—as long as we assume their interactions with the decision-maker areindependent of one another (e.g., this assumption rules out one subject updating their belief based onthe outcome of another subject’s prior interaction with the decision-maker).
Under such a setting,the decision maker would publicly commit to a method of computing the signaling policy, given adecision subject’s initial observable features as input.
Once a decision subject arrives, their featurevalues are observed and the signaling policy is computed.
Characterizing the utility gains of persuasion
As is generally the case in the persuasion literature [27, 26, 13], the decision maker can achieve higherexpected utility with an optimized signaling policy compared to if they provided no recommendation or fully disclosed the model.
To characterize how much leveraging the decision maker’s informationadvantage may improve their expected utility under our setting, we study the following example.
Consider a simple setting under which a single decision subject has one observable feature x 0(e.g., credit score) and two possible actions: a ∅ = “do nothing” (i.e., ∆ x ( a ∅ ) = 0 , c ( a ∅ )
= 0 ,u dm ( a ∅ ) = 0 ) and a 1 = “pay off existing debt” (i.e., ∆ x ( a 1 ) > 0 , c ( a 1 ) > 0 , u dm ( a 1 ) = 1 ), whichin turn raises their credit score.
For the sake of our illustration, we assume credit-worthiness to be amutually desirable trait, and credit scores to be a good measure of credit-worthiness.
We assume thedecision maker would like to design a signaling policy to maximize the chance of the decision subjecttaking action a 1 , regardless of whether or not the applicant will receive the loan.
In this simplesetting, the decision maker’s decision rule can be characterized by a single threshold parameter θ ,i.e., the decision subject receives a positive classiﬁcation if x + θ ≥ 0 and a negative classiﬁcationotherwise.
Note that while the decision subject does not know the exact value of θ , they insteadhave some prior over it, denoted by Π .
Given the true value of θ , the decision maker recommends an action σ ∈ { a ∅ , a 1 } for the decisionsubject to take.
The decision subject then takes a possibly different action a ∈ { a ∅ , a 1 } , whichchanges their observable feature from x 0 to x = x 0 + ∆ x ( a ) .
Recall that the decision subject’sutility takes the form u ds ( a, θ )
= sign { ( x 0 + ∆ x ( a ))
+ θ } − c ( a ) .
Note that if c ( a 1 ) > 2 , thenu ds ( a ∅ , θ ) > u ds ( a 1 , θ ) holds for any value of θ , meaning that it is impossible to incentivize anyrational decision subject to play action a 1 .
Therefore, in order to enable the decision maker toincentivize action a 1 , we assume c ( a 1 ) < 2 .
We observe that in this simple setting, we can bin values of θ into three different “regions”, basedon the outcome the decision subject would receive if θ were actually in that region.
First, ifx 0 + ∆ x ( a 1 ) + θ < 0 , the decision subject will not receive a positive classiﬁcation, even if they takeaction a 1 .
In this region, the decision subject’s initial feature value x 0 is “too low” for taking thedesired action to make a difference in their classiﬁcation.
We refer to this region as L .
Second, ifx 0 + θ ≥ 0 , the decision subject will receive a positive classiﬁcation no matter what action they take.
In this region, x 0 is “too high” for the action they take to make any difference on their classiﬁcation.
We refer to this region as H .
Third, if x 0 + θ < 0 and x 0 + ∆ x ( a 1 ) + θ ≥ 0 , the decision subjectwill receive a positive classiﬁcation if they take action a 1 and a negative classiﬁcation if they takeaction a ∅ .
We refer to this region as M .
Consider the following signaling policy.
Example signaling policy S ( θ )
Case 1: θ ∈ L .
Recommend action a 1 with probability q and action a ∅ with probability 1 − q
Case 2: θ ∈ M .
Recommend action a 1 with probability
Case 3: θ ∈ H .
Recommend action a 1 with probability q and action a ∅ with probability 1 − q
In Case 2, S recommends the action ( a 1 ) that the decision subject would have taken had they knownthe true θ , with probability 1 .
However, in Case 1 and Case 3, the decision maker recommends, withprobability q , an action ( a 1 ) that the decision subject would not have taken knowing θ , leveragingthe fact that the decision subject does not know exactly which case they are currently in.
If thedecision subject follows the decision maker’s recommendation from S , then the decision makerexpected utility will increase from 0 to q if the realized θ ∈ L or θ ∈ H , and will remain the sameotherwise.
Intuitively, if q is “small enough” (where the precise deﬁnition of “small” depends onthe prior over θ and the cost of taking action a 1 ), then it will be in the decision subject’s best interestto follow the decision maker’s recommendation, even though they know that the decision maker maysometimes recommend taking action a 1 when it is not in their best interest to take that action .
Thatis, the decision maker may recommend that a decision subject pay off existing debt with probabilityq when it is unnecessary for them to do so in order to secure a loan.
We now give a criteria on qwhich ensures the signaling policy S is BIC.
Proposition 3.1.
Signaling policy S is Bayesian incentive-compatible if q = min { π ( M )(2 − c ( a 1 ))
c ( a 1 )(1 − π ( M )) , 1 } ,where π ( M ) :
= P θ ∼ Π ( x 0 + θ < 0 and x 0 + ∆ x ( a 1 ) + θ ≥ 0) .
Proof Sketch.
We show that E θ ∼ Π [ u ds ( a ∅ , θ ) | σ = a ∅ ] ≥ E θ ∼ Π [ u ds ( a 1 , θ ) | σ = a ∅ ] and 279 E θ ∼ Π [ u ds ( a 1 , θ ) | σ = a 1 ] ≥ E θ ∼ Π [ u ds ( a ∅ , θ ) | σ = a 1 ] .
Since these conditions are satisﬁed, Sis BIC.
The full proof may be found in Appendix C.
Next, we show that the decision maker’s expected utility when recommending actions according tothe optimal signaling policy can be arbitrarily higher than their expected utility from revealing fullinformation or no information.
We prove the following result in Appendix D.
Proposition 3.2.
For any (cid:15) > 0 , there exists a problem instance such that the expected decisionmaker utility from recommending actions according to the optimal signaling policy is 1 − (cid:15) and theexpected decision maker utility for revealing full information or revealing no information is at most (cid:15) .
Computing the optimal signaling policy
In Section 3, we show a one-dimensional example, where a signaling policy can obtain arbitrarily 289 better utilities compared to revealing full information and revealing no information.
We now derivethe decision maker’s optimal signaling policy for the general setting with arbitrary numbers of 291 observable features and actions described in Section 2.
Under the general setting, the decisionmaker’s optimal signaling policy can be described by the following optimization:
[Equation]
1) where we omit the valid probability constraints over p ( σ = a | θ ) , a ∈ A for brevity.
In words, thedecision maker wants to design a signaling policy S in order to maximize their expected utility,subject to the constraint that the signaling policy is BIC.
At ﬁrst glance, the optimization may initiallyseem hopeless as there are inﬁnitely many values of p ( σ = a | θ ) (one for every possible θ ∈ Θ ) thatthe decision maker’s optimal policy must optimize over.
However, we will show that the decisionmaker’s optimal policy can actually be recovered by optimizing over ﬁnitely many variables.
By rewriting the BIC constraints as integrals over Θ and applying Bayes’ rule, our optimization over 300 p ( σ = a | θ ) , a ∈ A takes the following form
[Equation]
Note that if u ds ( a, θ ) − u ds ( a (cid:48) , θ ) is the same for some “equivalence region” R ⊆ Θ (which weformally deﬁne below), we can pull u ds ( a, θ ) − u ds ( a (cid:48) , θ ) out of the integral and instead sum overthe different equivalence regions.
Intuitively, an equivalence region can be thought of as the set ofall θ ∈ Θ pairs that are indistinguishable from a decision subject’s perspective because they lead tothe exact same utility for any possible action the decision subject could take.
Based on this idea, weformally deﬁne a region of Θ as follows.
Deﬁnition 4.1 (Equivalence Region) .
Two assessments θ , θ (cid:48) are equivalent (w.r.t.
u ds ) if u ds ( a, θ ) −u ds ( a (cid:48) , θ ) =
u ds ( a, θ (cid:48) )
− u ds ( a (cid:48) , θ (cid:48) ) , ∀ a, a (cid:
) ∈ A .
An equivalence region R is a subset of Θ such thatfor any θ ∈ R , all θ (cid:48) equivalent to θ are also in R .
We denote the set of all equivalence regions by R .
For more intuition about the deﬁnition of an equivalence region, see Figure 3 in
Appendix E. Afterpulling the decision subject utility function out of the integral, we can integrate p ( σ = a | θ ) π ( θ ) overeach equivalence region R .
We denote p ( R ) as the probability that the true θ ∈ R according to theprior.
Finally, since it is possible to write the constraints in terms of p ( σ = a | R ) , ∀ a ∈ A , R ∈ R ,it sufﬁces to optimize directly over these quantities.
For completeness, we include the constraintswhich make each { p ( σ = a | R ) } a ∈A , ∀ R a valid probability distribution.
Theorem 4.2 (Optimal signaling policy) .
The decision maker’s optimal signaling policy can be 317 characterized by the following linear program OPT-LP:
[Equation]
where p ( σ = a | R ) denotes the probability of sending recommendation σ
= a if θ ∈ R .
Note that thelinear program OPT-LP is always feasible, as the decision maker can always recommend the actionthe decision subject would play according to the prior, which is BIC.
Similarly, always recommendingthe action the decision subject would take had they known the assessment rule is also feasible.
While the problem of determining the decision maker’s optimal signaling policy can be transformedfrom an optimization over inﬁnitely many variables into an optimization over the set of ﬁnitelymany equivalence regions R , |R| may be exponential in the number of observable features d (see
Appendix F for more details).
This is perhaps unsurprising as without any assumptions on Θ , therepresentation of the prior Π can scale exponentially with d .
In this case, we expect the runningtime of an algorithm which takes the entire prior as input to be exponential in the number of featuresas well.
This motivates the need for a computationally efﬁcient algorithm to approximate (OPT-LP),which does not require the full prior Π as input.
We adapt the sampling-based approximation algorithm of Dughmi and Xu to our setting in order tocompute an (cid:15) -optimal
and (cid:15) -approximate signaling policy in polynomial time, as shown in Algorithm1 in Appendix G.
At a high level, Algorithm 1 samples polynomially-many times from the priordistribution over the space of assessment rules, and solves an empirical analogue of (OPT-LP).
In Appendix G, we show that the resulting signaling policy is (cid:15) -BIC, and is (cid:15) -optimal with highprobability, for any (cid:15) > 0 .
Formally, we prove the following statement.
Theorem 4.3.
Algorithm 1 runs in poly( m, 1 (cid:15) ) time (where m = |A| ), and implements an (cid:15)
-BICsignaling policy that is (cid:15) -optimal with probability at least 1 − δ .
We leave open the question of whether there are classes of succinctly represented priors that permitefﬁcient algorithms for computing the exact optimal policy in time polynomial in d and m .
It is alsoplausible to design efﬁcient algorithms that only require some form of query access to the prior distribution.
However, information-theoretic lower bounds of [13] rule out query access through sampling.
Experiments
In this section, we provide experimental results using a semi-synthetic setting where decision subjectsare based on individuals in the Home Equity Line of Credit (HELOC) dataset [15].
The HELOCdataset contains information about 9,282 customers who received a Home Equity Line of Credit.
Each individual in the dataset has 23 observable features related to an applicant’s ﬁnancial history(e.g.
, percentage of previous payments that were delinquent) and a label which characterizes theirloan repayment status (repaid/defaulted).
We compare the decision maker utility for different modelsof information revelation: our optimal signaling policy, revealing full information, revealing noinformation.
As our theory predicts, the expected decision maker utility when recommending actionsaccording to the optimal signaling policy either matches or exceeds the expected utility from revealingfull information or no information about the assessment rule across all problem instances.
Moreover,the expected decision maker utility from signaling is signiﬁcantly higher on average.
Next, we explorehow the decision maker’s expected utility changes when action costs and changes in observablefeatures are varied jointly.
Our results are summarized in Figures 1 and 2.
In order to adapt the HELOC dataset to our strategic setting, we select four features and deﬁne ﬁvehypothetical actions
A = { a ∅ , a 1 , a 2 , a 3 , a 4 } that decision subjects may take in order to improvetheir observable features.
Actions { a 1 , a 2 , a 3 , a 4 } result in changes to each of the decision subject’sfour observable features, whereas action a ∅ does not.
For simplicity, we view actions { a 1 , a 2 , a 3 , a 4 }as equally desirable to the decision maker, and assume they are all more desirable than a ∅ .
Usingthese four features, we train a logistic regression model that predicts whether an individual is likelyto pay back a loan if given one, which will serve as the decision maker’s realized assessment rule.
For more information on how we constructed our experiments, see Appendix I.
Given a { ( c ( a i ) , ∆ x ( a i )) } 4 i =1 instance and information revelation scheme, we calculate the decisionmaker’s total expected utility by summing their expected utility for each applicant.
Figure 1 shows theaverage total expected decision maker utility across different ∆ x ( a ) and cost conﬁgurations for priorswith varying amounts of uncertainty.
See Figure 7 in Appendix I.3 for plots of all instances whichwere used to generate Figure 1.
Across all instances, the optimal signaling policy (red) achieveshigher average total utility compared to the other information revelation schemes (blue and green).
The difference is further ampliﬁed whenever the decision subjects are less certain about the trueassessment rule (i.e., when σ is large).
Intuitively, this is because the decision maker leverages the
[Figure]
The optimal signaling policy (red) consistently yields higher utility compared to the two baselines: revealing full information (blue) and no information (green).
This gap increases when the decision subject is less certain about the model being used (higher σ 2 ).
[Figure]
decision subjects’ uncertainty about the true assessment rule in order to incentivize them to takedesirable actions, and as the uncertainty increases, so does their ability of persuasion.
To better understand how the decision maker’s expected utility changes as a function of c ( a ) and∆ x ( a ) , we sweep through multiple { ( c ( a i ) , ∆ x ( a i )) } 4 i =1 tuples on a grid of ( c ( a i ) , ∆ x ( a i )) ∈{ 0 , 0 .
5 }×{ 0 , 0 . 5 , 1 . 0 } for i ∈ { 1 , 2 , 3 , 4 } and measure the effectiveness of the three informationrevelation schemes.
Figure 2 shows the surface of the decision maker utility as a function of( c ( a i ) , ∆ x ( a i )) for the optimal signaling policy (red), revealing full information (blue), and revealingno information (green).
When c ( a i ) is high and ∆ x ( a i ) is low, the total expected decision makerutility is low as there is less incentive for the decision subject to take actions (although even underthis setting, the optimal signaling policy outperforms the other two baselines).
As c ( a i ) decreasesand ∆ x ( a i ) increases, the total expected decision maker utility increases.
Conclusion
We investigate the problem of offering algorithmic recourse without requiring full transparency (i.e.,revealing the assessment rule).
We cast this problem as a game of Bayesian persuasion, and offerseveral new insights regarding how a decision maker can leverage their information advantage overdecision subjects to incentivize mutually beneﬁcial actions.
Our stylized model relies on severalsimplifying assumptions, which suggest important directions for future work:
Public persuasion.
We assume that the recommendations received by each decision subject areprivate .
However, if a decision subject is given access to recommendations for multiple individuals,it may be possible for them to reconstruct the underlying model.
While out of the scope of this work,it would be interesting to study models of public persuasion in the algorithmic recourse setting.
Beyond linear decision rules.
We focus on settings with linear decision rules and assume alldecision subject parameters (e.g., cost function, initial observable features, etc.) are known to thedecision maker.
We leave it for future work to extend our ﬁndings to non-linear decision rules, orsettings in which some of the decision subjects’ parameters are unknown to the decision maker.
Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s
contributions and scope?
[Yes]
(b) Did you describe the limitations of your work?
[Yes] See Section 6.
(c) Did you discuss any potential negative societal impacts of your work?
[Yes] SeeAppendix A.(d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?
[Yes]
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes] See Sections3 and 4.
(b) Did you include complete proofs of all theoretical results?
[Yes] See Appendices C, D,F, G.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main ex-
perimental results (either in the supplemental material or as a URL)?
[Yes] See
supplementary material.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
[Yes] See Appendix H and I.
(c) Did you report error bars (e
g., with respect to the random seed after running experiments multiple times)?
[N/A](d)
Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] See Appendix B
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using/curating?
[N/A]
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information or offensive content?
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If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
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spent on participant compensation?
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A Societal implications
Persuasion in the algorithmic recourse setting is usually socially beneﬁcial, as the quantities thedecision maker wants to incentivize are usually aligned with socially desirable actions.
Our workemphasizes these potential social beneﬁts through our running example with credit lending whichillustrates how persuasion can be used to incentivize individuals to pay off more existing debt.
However, we recognize that there may be decision makers whose interests are negatively correlatedwith the interests of society who may use persuasion as a tool to incentivize socially undesirablebehavior.
Our work illustrates the ways in which persuasion may play a role in the deployment ofalgorithmic decision-making, capturing important dynamics between the decision maker and decisionsubject in the study of the long-term impact of these systems.
B Computational Resources
We ran our experiments on an Intel Core i7-8700 3.2 GHz 6-Core Processor with 16GB of RAM anda GTX1070 GPU, although the GPU was not used.
C Proof of Proposition 3.1
Proof.
Based on the decision subject’s prior over θ , they can calculate(1) π ( L ) =
P Π ( x 0 + ∆ x ( a 1 ) + θ < 0) , i.e., the probability the decision subject is in region Laccording to the prior(2) π ( M ) =
P Π ( x 0 + θ < 0 and x 0 + ∆ x ( a 1 ) + θ ≥ 0) , i.e., the probability the decisionsubject is in region M according to the prior(3) π ( H ) =
P Π ( x 0 + θ ≥ 0) , i.e., the probability the decision subject is in region H accordingto the prior
Case 1 : σ = a 0 .
Given the signal σ = a ∅ , the decision subject’s posterior probability densityfunction π ( ·| σ = a ∅ ) over L , M , and H will take the form
[Equation]
[Equation] [Equation]
If the decision subject receives signal σ = a 0 , they know with probability 1 that they are not in region
M with probability 1 .
Therefore, they know that taking action a 1 will not change their classiﬁcation,so they will follow the decision maker’s recommendation and take action a ∅ .
Case 2 : σ = a 1 .
Given the signal σ = a 1 , the decision subject’s posterior density over L , M , and Hwill take the form
[Equation] [Equation]
[Equation]
The decision subject’s expected utility of taking actions a ∅ and a 1 under the posterior induced byσ = a 1 are
E
[Equation]
[Equation] [Equation]
) and
E
[Equation]
∼
[Equation] [Equation]
+
[Equation]
a 1 ))
In order for S to be BIC,
[Equation]
( a ∅ , θ ) | σ = a 1 ] .
Plugging in our expressions for E θ ∼ Π [ u ds ( a 1 , θ ) | σ = a 1 ] and E θ ∼ Π [ u ds ( a ∅ , θ ) | σ = a 1 ] , we see thatπ ( H | σ = a 1 ) · (1 − c ( a 1 ))
+
π ( M | σ = a 1 ) · (1 − c ( a 1 ))
+ π ( L | σ = a 1 ) · ( − 1 − c ( a 1 ))
≥ π ( H | σ = a 1 ) − π ( M | σ = a 1 ) − π ( L | σ = a 1 )
After canceling terms and simplifying, we see that−
[Equation]
π
[Equation]
L |
[Equation]
=
a
[Equation]
+
[Equation]
|
[Equation]
a 1 )
[Equation]
=
[Equation] [Equation]
− c ( a 1 )) ≥
Next, we plug in for π ( L | σ = a 1 ) , π ( M | σ = a 1 ) , and π ( H | σ = a 1 ) .
Note that the denominators ofπ ( L | σ = a 1 ) , π ( M | σ = a 1 ) , and π ( H | σ = a 1 ) cancel out.
[Equation]
Solving for q , we see that
[Equation]
Note that q ≥ 0 always.
Finally, in order for q to be a valid probability, we restrict q such that
[Equation]
This completes the proof.
D Proof of Proposition 3.
Proof.
Consider the example in Section 3.
Expected utility from revealing no information.
If the decision subject acts exclusively accordingto the prior, they will select action a 1 with probability 1 if E θ ∼ Π [ u ds ( a 1 , θ )] ≥ E θ ∼ Π [ u ds ( a ∅ , θ )] andwith probability 0 otherwise.
Plugging in our expressions for E θ ∼ Π [ u ds ( a 1 , θ )] and E θ ∼ Π [ u ds ( a ∅ , θ )] ,we see that the decision subject will select action a 1 only ifπ
[Equation]
1 )) ≥ π ( L )( − 1 − 0)+ π ( M )( − 1 − 0)+ π ( H )(1 − 0)
Canceling terms and simplifying, we see that
[Equation]
must hold for the decision subject to select action a 1 .
Finally, substituting π ( L ) + π ( H )
= 1 − π ( M )gives us the condition 2 π ( M ) − c ( a 1 ) ≥ 0 .
Alternatively, if π ( M ) c ( a 1 ) < 12 , the decision subject willselect action a ∅ with probability 1 .
Intuitively, this means that a rational decision subject would takeaction a 1 if the ratio of π ( M ) (the probability according to the prior that taking action a 1 is in thedecision subject’s best interest) to c ( a 1 ) (the cost of taking action a 1 ) is high, and would take actiona ∅ otherwise.
Expected utility from revealing full information.
If the decision maker reveals the assessment ruleto the decision subject, they will select action a 1 when θ ∈ M and action a ∅ otherwise.
Thereforesince u dm ( a 1 ) =
1 and u dm ( a ∅ ) = 0 , the decision maker’s expected utility if they reveal fullinformation is π ( M ) .
Expected utility from S .
Recall that the decision maker’s signaling policy S from Section 3 setsq =
min { π ( M )(2 − c ( a 1 )) c ( a 1 )(1 − π ( M )) , 1 } .
Under this setting, the decision maker’s expected utility is min { 1 ·π ( M ) + q · (1 − π ( M )) , 1 } .
Substituting in our expression for q and simplifying, we see that thedecision maker’s expected utility for recommending actions via S is min { 2 π ( M ) c ( a 1 ) , 1 } .
Suppose that 2 π ( M )
=
c ( a 1 )(1 − (cid:15) ) and c ( a 1 ) = 2 (cid:15) , for some small (cid:15) > 0 .
The decision maker’sexpected utility will always be 0 from revealing no information because 2 π ( M ) c ( a 1 ) = 1 − (cid:15) < 1 .
Thedecision maker’s expected utility from recommending actions via S will be 2 π ( M ) c ( a 1 ) = 1 − (cid:15) .
Sinceπ ( M ) =
(cid:15) (1 − (cid:15) ) < (cid:15) , the decision maker’s expected utility from revealing full information will beless than (cid:15) .
Therefore, as (cid:15) approaches 0 , the decision maker’s expected utility from revealing fullinformation approaches 0 (the smallest value possible), and the decision maker’s expected utilityfrom S approaches 1 (the highest value possible).
This completes the proof.
[Figure]
[Table]
[Figure]
The decision maker’s expected utility as a function of their possible strategies is summarized in
Table 1.
Note that when 1 { π ( M ) ≥ c ( a 1 ) 2 } = 1 , q = 1 .
Therefore, the decision maker’s expectedutility is always as least as good as the two natural alternatives of revealing no information about theassessment rule, or revealing full information about the rule.
E Region Illustration
In Figure 3, we show an example of how different equivalence regions might partition the space ofpossible assessment rules Θ .
In this example, there are two actions and two observable features, andthe space of Θ is partitioned into three different equivalence regions.
Note that as long as the set ofactions
A is ﬁnite, |R| < ∞ .
F Computational Barriers
In this section, we show that even in the setting where each action only affects one observable feature(e.g., as shown in Figure 4), the number of equivalence regions in (OPT-LP) is still exponential in thesize of the input.
While somewhat simplistic, we believe this action scheme reasonably reﬂects realworld settings in which the decision subjects are under time or resource constraints when decidingwhich action to take.
For example, the decision subject may need to choose between paying off someamount of debt and opening a new credit card when strategically modifying their observable featuresbefore applying for a loan.
Under this setting, (OPT-LP) optimizes over Θ( m |R| ) variables, where m is the number of actionsavailable to each agent and |R| is the number of equivalence regions.
In order to determine the size of
R , we note that an equivalence region can be alternatively characterized by observing that assessmentrules θ and θ (cid:48) belong to the same equivalence region if the difference in their predictions for any twoactions a and a (cid:48) is the same.
(This follows from straightforward algebraic manipulation of Deﬁnition4.1.)
As such, an equivalence region R can essentially be characterized by the set of actions A R ⊆
Awhich receive a positive classiﬁcation when θ ∈ R .
5 a i,a d,a ∅ a 1 ,... a 1 ,m... a i,m i ... a d,m d ...
Armed with this new characterization of an equivalence region, we are now ready to show the scaleof |R| for the setting described in Figure 4.
Proposition F.1.
For the setting described in Figure 4, there are |R| = Π di =
m i − 1 equivalenceregions, where d is the number of observable features of the decision subject and m i ( ∀
i ∈
[ d ] ) is thenumber of actions the decision subject has at their disposal to improve observable feature i .
Proof.
In order to characterize the number of equivalence regions |R| , we deﬁne the notion of adominated action a , where an action a is dominated by some other action a (cid:48) if ∆ x ( a ) (cid:22) ∆ x ( a (cid:48) ) ,with strict inequality holding for at least one index.
Using this notion of dominated actions andour reﬁned characterization of an equivalence region, it is straightforward to see that if action a isdominated by action a (cid:48) , then a
(cid:48) ∈
A R for any equivalence region R where a ∈ A R .
Proposition F.1then follows directly from the fact that each action only affects one observable feature.
Proposition F.1 shows that the computation of (OPT-LP) quickly becomes intractable as the numberof observable features grows large, even in this relatively simple setting.
This motivates the need foran approximation algorithm for (OPT-LP), which we present in Section ?? .
G Proof of Theorem 4.3
Proof.
First, since the approximation algorithm solves an approximation LP (APPROX-LP) ofpolynomial size, it runs in polynomial time.
Lemma G.1.
Algorithm 1 runs in poly( m, 1 (cid:15) ) time.
By bounding the approximation error in the BIC constraints of (APPROX-LP), we show that theresulting policy satisﬁes approximate BIC.
Lemma G.2.
Algorithm 1 implements an (cid:15) -BIC signaling policy.
Next, we show that a feasible solution to (APPROX-LP) exists which achieves expected decisionmaker utility at least OPT - (cid:15) with probability at least 1 − δ .
In order to do so, we ﬁrst show thatthere exists an approximately optimal solution (cid:101) S to (OPT-LP) such that each signal is either (i) large(i.e.
, output with probability above a certain threshold), or (ii) honest (i.e., the signal recommendsthe action the decision subject would take, had they known the true assessment rule θ ).
Next, weshow that (cid:101) S is a feasible solution to (APPROX-LP) with high probability by applying McDiarmid’sinequality [35] and a union bound.
Input: θ ∈ Θ , (cid:15) > 0 , δ >
Output: Signaling policy (cid:98) S :
= { p ( σ = a | R θ ) } ∀ a ∈A (where region R θ contains θ )
Set K = (cid:108)
[Equation]
log (cid:16)
[Equation]
(cid:17)(cid:109)
Pick (cid:96) ∈ { 1 , . .
, K } uniformly at random.
Set θ (cid:96) =
θ .
Sample (cid:101)
Θ = { θ 1 , . . .
, θ (cid:96) − 1 , θ (cid:96) +1 , . .
, θ K } ∼ π ( θ ) .
Let (cid:101) R denote the set of observed regions.
Compute ˜ p
( R ) , ∀ R ∈ (cid:101) R , where ˜ p ( R ) is the empirical probability of θ (cid:48) ∈ R .
Solve
[Equation] [Equation]
Return signaling policy (cid:98) S :
= { p ( σ = a | R θ ) } ∀ a ∈A .
Lemma G.3.
There exists an (cid:15) 2 -optimal signaling policy (cid:101) S that is large or honest.
Lemma G.4.
With probability at least 1 − δ , (cid:101) S is a feasible solution to (APPROX-LP) and theexpected decision maker utility from playing (cid:101) S is at least OPT - (cid:15) .
By Lemmas G.3 and G.4, the decision maker’s expected utility will be at least OPT - (cid:15) with probabilityat least 1 − δ .
Proof of Lemma G.1.
Proof.
Lines 1-3 trivially run in poly( m, 1 (cid:15) ) time.
˜ p
( R ) , ∀ R ∈ (cid:101) R can be computed in poly( m, 1 (cid:15) )time in an online manner as follows: for each k ∈ { 1 , . .
, K } , check if θ k belongs to an existingregion.
(Note that this can be done in O ( m ) time for each region.)
If θ k belongs to an existing region,update the existing empirical probabilities.
Otherwise, create a new region.
Finally, note that LP(APPROX-LP) has poly( m, 1 (cid:15) ) variables and constraints, and can therefore be solved in poly( m, 1 (cid:15) )time using, e.g., the Ellipsoid Algorithm.
Proof of Lemma G.2.
Proof.
By the principle of deferred decisions, θ ∼ Π (cid:48) , where Π (cid:48) is the uniform distribution over (cid:101) Θ .(APPROX-LP) implements an (cid:15) -BIC signaling policy for Π (cid:48) by deﬁnition, so
[Equation]
Finally, apply the law of iterated expectation with respect to (cid:101) Θ to obtain the desired result.
Proof of Lemma G.3.
In order to prove Lemma G.3, we make use of the following deﬁnitions.
Deﬁnition G.5 (Large signal) .
A signal σ = a is large if p ( σ = a ) =
(cid:80) R ∈R p ( σ = a | R ) p ( R ) >(cid:15) 2 m .
Deﬁnition G.6 (Honest signal) .
A signal σ = a is honest if a ∈ arg max a (cid:48) ∈A u ds ( a (cid:48) , R ) .
Proof.
We proceed via proof by construction.
Let S ∗ be the optimal BIC signaling policy.
Deﬁne thesignaling policy (cid:101) S as follows: for a given θ , it ﬁrst samples a signal a ∼ S ∗ ( θ ) .
If the signal is large ,output signal σ = a .
Otherwise, output signal σ = a ∈ arg max a (cid:48) ∈A u ds ( a (cid:48) , R θ ) .
Every signal of(cid:101) S is trivially large or honest.
(cid:101) S is BIC since S ∗ is BIC and (cid:101) S only replaces recommendations of
S ∗ with honest recommendations.
Finally, since the total probability of signals that are not large isat most (cid:15) 2 , and the decision maker’s utilities are in [0 , 1] , their expected utility is no worse than (cid:15) 2smaller than their expected utility from S ∗ .
Proof of
Lemma
G.4.
The following claim will be useful when proving Lemma G.4.
Claim G.7.
The expected decision maker utility from playing (cid:98) S is (cid:80) a ∈A (cid:80) R ∈R p ( R ) p ( σ =a | R ) u dm ( a ) .
Proof.
The expected decision maker utility from playing (cid:98) S is E θ ∼ Π [ (cid:80) a ∈A (cid:80) R ∈ (cid:101)
R ˜ p
( R ) p ( σ =a | R )
u dm ( a )] =
E θ ∼ Π
[ (cid:80) a ∈A (cid:80) R ∈R ˜ p
( R ) p ( σ = a | R )
u dm ( a )] , by the principle of deferred decisions.
Apply the law of iterated expectation with respect to (cid:101) Θ to obtain the desired result.
Additionally, we will make use of McDiarmid’s inequality [35], stated for completeness below.
Lemma G.8 (McDiarmid’s Inequality [35]) .
Let X 1 , . . .
, X n be independent random variables, with
X k taking values in a set A k for each k .
Suppose that the (measurable) function f
: Π A k
→ Rsatisﬁes
[Equation]
whenever the vectors x and x (cid:48) differ only in the kth coordinate.
Let Y be the random variablef ( X 1 , . . .
, X n ) .
Then for any t > 0 .
[Equation]
We are now ready to prove Lemma G.4.
Proof.
First, note that the (cid:15) -BIC constraints can be rewritten using the observed decision rules as
[Equation]
where θ k ∈ R k .
Note that this is a bounded function of θ 1 , . .
, θ K .
Let Y ( a, a (cid:48) )
= 1 K (cid:80)
Kk =1 p ( σ =a | R k )( u ds ( a, R k ) − u ds ( a (cid:48) , R k )) .
Note that E [ Y ( a, a (cid:48) )]
= (cid:80) R ∈R p ( σ = a | R ) p ( R )( u ds ( a, R )
−u ds ( a (cid:48) , R )) .
Applying Lemma G.8, we see that ∀ a,
a (cid:48) ∈ A ,
[Equation]
Similarly, let Z = (cid:80) a ∈A (cid:80) R ∈ (cid:101)
R ˜ p
( R ) p ( σ = a | R )
u dm ( a ) =
1 K (cid:80)
Kk =1 (cid:80) a ∈A p ( σ = a | R k ) u dm ( a )(where R k contains θ (cid:48) k ).
By Claim G.7, E
[ Z ] = (cid:80) a ∈A (cid:80) R ∈R p ( R ) p ( σ = a | R ) u dm ( a ) .
Applying
Lemma G.8,
[Equation]
Applying the union bound, we see that the probability that all m 2 + 1 above inequalities hold is atleast 2( m 2 + 1) exp (cid:0) − K(cid:15) 2 / 2 (cid:1) .
By inverting the tail bound and picking K = 2 (cid:15) 2 log (cid:16) 2( m 2 +1) δ (cid:17) , weget that | Z − E
[ Z ] | ≤ (cid:15)/ 2 and | Y ( a, a (cid:48) ) − E
[ Y ( a, a (cid:48) )] | ≤ (cid:15) , ∀ a, a (cid:48) ∈ A , with probability at least1 − δ .
Therefore, with probability at least 1 − δ , (cid:101) S is a feasible solution for LP (APPROX-LP) andthe objective value is at most OPT − (cid:15) 2 − (cid:15) 2 =
OPT − (cid:15) .
[Figure]
=
D e c
i s i o n - m a k e r ' s
T o t a l E x p e c t e d
U t ili t y
[Figure]
None
[Figure]
2 c ( a 1 )
D e c
i s i o n - m a k e r ' s
T o t a l E x p e c t e d
U t ili t y = 670, x ( a 1 )=
BIC Full
None     50 x ( a 1 )
D e c
i s i o n
[Figure]
m a k e r ' s
T o t a
[Figure]
x p e c t e d
U
[Figure]
t
[Figure] [Figure]
H Instantiating 1-Dimensional Scenario
In this section we instantiate the example introduced in Section 3 and demonstrate the decisionmaker’s gain in utility from the optimal signaling policy over other baselines.
To contextualize thissimple synthetic setup, consider a banking institution deciding whether approve a loan applicationfrom an applicant based on credit score x 0 ∈ [300 , 850] with a simple threshold classiﬁer.
The bankapproves the application ( ˆ y = 1 ) if x 0 + θ > 0 and rejects ( ˆ y = − 1 ) otherwise.
Here, we assume theground-truth threshold value used by the decision maker to be 670 (i.e. θ = − 670 ), which is typicallyconsidered as a decent credit score.
Recall that a ∅ = “do nothing” and a 1 = “pay off existing debt”and set the utility of the decision maker to be u dm ( a 1 ) = 1 , u dm ( a ∅ ) = 0 , as, for the sake of ourillustration, we assume credit score to be a good measure of credit-worthiness.
Finally, we assumethe prior to be π ( θ ) ∼ N ( µ θ , σ 2 θ ) .
In Figure 5, we verify that our optimal signaling policy ( BIC , red) yields higher decision maker utilitycompared to the two baselines: revealing full information ( Full , blue) and revealing no information( None , green) 6 .
To measure the total amount of decision maker’s expected utility yielded by eachpolicy, we assume a uniform distribution of the decision subjects’ credit scores x 0 in the populationand take the sum of expected decision maker utility values across different scores.
We plot these totalutility values in Figure 5a, and as expected, the larger the σ θ is, the more comparative advantage ourmethod has over the baselines.
As the decision subjects’ uncertainty about the true θ increases (i.e.,the standard deviation of the prior distribution increases from 10 to 50), the decision maker beneﬁtsfrom our optimal signaling policy even more.
When action a 1 becomes more cost-prohibitive (or less effective), as there is less incentive for thedecision subjects to take the action, we expect the decision maker’s utility to decrease 7 .
As shown in
Figure 5b, we indeed observe such a trend as c ( a 1 ) increases (top) and ∆ x ( a 1 ) decreases (bottom).
Nevertheless, our optimal signaling policy yields consistently higher total decision maker utilitycompared to the baselines across all conditions.
In Figure 6, we show 2-D slices of Figure 2 along the c ( a ) axis (left) and ∆ x ( a ) axis (right).
As is expected, with small cost and sufﬁciently large ∆ x ( a ) (top row, right), the two baselines
Pair
[Table]
become as effective as the optimal signaling policy.
Interestingly, we note that changes in different( c ( a i ) , ∆ x ( a i )) result in signiﬁcantly different rates of change in decision maker utility.
For example,the optimal signaling policy (red) and revealing full information (blue) are more resistant to theincrease in c ( a 4 ) in range [0 , 0 .
25] than they are for the increase in other c ( a i ) , i (cid:54) = 4 , showing aconcave drop in utility rather than a convex one (bottom row, left).
Such behavior can be attributed tothe relative weight of each feature on the assessment rule, where | θ 4 | > | θ 3 | >
| θ 1 | > | θ 2 | .
Because the fourth feature has the largest weight, taking action a 4 will have the largest effect on anindividual’s prediction.
As a result, the maker is the least sensitive to increases in the
I Experiment Details and Additional Results
Common prior.
We assume the common prior over the realized assessment rule θ takes the formof a multivariate Gaussian N ( θ , σ 2 I 4 × 4 ) before training.
This captures the setting in which boththe decision maker and decision subjects have a good estimate of what the true model will be, butare somewhat uncertain about their estimate.
We note that our methods extend to more complicatedpriors beyond the isotropic Gaussian prior we consider in this setting.
Changes in observable features.
In order to examine the effects that different ∆ x ( a i )( i ∈{ 1 , 2 , 3 , 4 } ) have on the decision maker’s expected utility, we consider settings in which each∆ x ( a i ) takes a value in { 0 , 0 . 25 , 0 . 5 , 0 .
Utilities and costs of actions.
As the decision maker views actions { a 1 , a 2 , a 3 , a 4 } as equallydesirable, we deﬁne u dm ( a i )
= 1 , i ∈ { 1 , 2 , 3 , 4 } and u dm ( a ∅ ) = 0 .
Since there are 1,320individuals in our test dataset, the maximum utility the decision maker can obtain is 1,320.
Asproposed in [37], we use the Bradley-Terry model [3] to generate the decision subject’s cost c ( a i ) oftaking action a i , for i = 1 , 2 , 3 , 4 .
See Appendix I.2 for details on our exact generation methods.
I.1 Remark on the decision maker’s assessment rule for HELOC dataset
To simulate a setting in which the decision maker employs a machine learning model for making decisions about the decision subjects, we train a simple logistic regression model on the subset of HELOCdataset.
We speciﬁcally work on four features selected in Table 2, and split the dataset into train/testset (7425, 1857 data points respectively).
The test accuracy of the model was 71.08 percent, and thecorresponding model coefﬁcients were θ =
[ − 0 .
52023116 , − 0 .
61600619]with the bias term − 0 .
Note that each coefﬁcient term has the sign that is aligned with howthe desired action was deﬁned in Table 2 (i.e., for features where increasing the value is desirable, thesign is positive and vice-versa).
To further make sure that the deﬁned actions correctly align with themodel, we select the test samples that the trained model made no mistakes on.
This resulted in a totalof 1,320 samples from the test set on which each policy was optimized.
[Figure]
1250 c ( a 1 )=0.
0.5 1.0 x ( a 2 ) c ( a 2 )=0.
0.5 1.0 x ( a 3 ) c ( a 3 )=0.
0.5 1.0 x ( a 4 ) c ( a 4 )=0.
I.2 Computing different costs for HELOC dataset using Bradley-Terry model
While exact action costs may be unknown, it is often reasonable for the decision maker to know anordering over possible actions in terms of their cost for decision subjects.
For example, it may becommon knowledge that opening a new credit card is easier than paying off some existing amountof debt, but exactly how much easier may be unclear.
The Bradley-Terry model uses exponentialscore functions to model the probability that feature x i is more costly for a decision subject to takecompared to feature x j .
Speciﬁcally, it assumes
[Equation]
Given pairwise cost comparisons (generated from common knowledge or gathered from experts)we can estimate P ( a i (cid:31) a j ) and solve for the parameters { c ( a i ) } 4 i =1 using maximum likelihoodestimation.
In order to gain more insight into how different action cost orderings affect the decisionmaker utility, we consider several different ground-truth cost orderings over actions and simulateexpert advice in order to estimate P ( a i (cid:31) a j ) , ∀ a
i , a j ∈ A .
While the expert advice is purelysynthetic in our setting, this method provides a principled way to estimate action costs wheneverinput from domain experts (e.g., ﬁnancial advisors) is available to the decision maker.
We use the following set of comparison inputs (manually generated) in Table 3a-3d to generatecost values with the relative ordering presented in Section 5.
While these comparison inputs aregenerated arbitrarily for the simulations, these can be obtained by querying several domain expertsand aggregating their answers regarding which feature is more difﬁcult to change.
The resulting costvalues are shown in Table 3e.
I.3 Additional results for different cost and ∆ x ( a ) conﬁgurations
Figure 7 shows more exhaustive results on different cost conﬁgurations (i)-(iv) as deﬁned in Table 3eand ∆ x ( a i ) ∈ 0 , 0 . 25 , 0 . 5 , 0 .
for i = 1 , 2 , 3 , 4 on HELOC datset.
For all conﬁgurationsconsidered, our optimal signaling policy (red) consistently yields utility no less than both baselines:revealing full information about the assessment rule (blue), and revealing no information (green).
Previously we viewed actions { a 1 , a 2 , a 3 , a 4 } as equally desirable to the decision maker ( u dm ( a i ) = 1 for i = 1 , 2 , 3 , 4 ).
Figure 8 and 9 show the results with a different utility proﬁle, u dm ( a 1 )
=0 .
25 , u dm ( a 2 ) = 0 .
5 , u dm ( a 3 ) = 0 .
75 , u dm ( a 4 ) = 1 , where each action gives different utility for
[Table] [Figure] [Figure] [Figure]
the decision maker.
Similar to what we previously observed before, the optimal signaling policy (red)effectively upper-bounds the two baselines, revealing everything (blue) and revealing nothing (green)in all settings.
As expected, when certain action has higher utility (e.g. a 4 compared to a 1 ), the totalutility becomes more sensitive to changes in c ( a ) and ∆ x ( a ) .