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The Limits of Manipulation within Personalized Incentive Policies
Andrea Araldo, Alessa Mayer, Ravi Seshadri, Amir Brudner
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
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Summary
This paper analyzes the vulnerability of Personalized Demand Management Policies (PDMP) to strategic manipulation by agents. Using a Stackelberg game framework, the authors demonstrate that unlike previous models assuming instantaneous profile classification, a regulator who learns agent profiles through observed behavior imposes a 'Cost of Manipulation' on agents. This cost deters manipulation, making personalized policies more robust than previously thought. The study establishes an upper bound on manipulation gains and validates these findings using the London Passenger Mode Choice dataset.
Entities (7)
Relation Signals (5)
Stackelberg Game → modelsinteractionbetween → Regulator
confidence 97% · We formalize a Stackelberg game in which an agent may simulate a different profile to extract higher incentives from a regulator.
London Passenger Mode Choice dataset → validates → deterrence effect
confidence 96% · Simulations on the London Passenger Mode Choice dataset illustrate this deterrence effect.
Personalized Demand Management Policies → isvulnerableto → strategic manipulation
confidence 95% · Personalized incentive policies tailor subsidies to agents' needs and constraints, but personalization also exposes the regulator to agents' strategic manipulative behavior.
Regulator → learnsprofilevia → observed past behavior
confidence 94% · we explicitly model the regulator's process of learning the agent profile, based on observed past behavior.
Cost of Manipulation → deters → Manipulation
confidence 93% · By accounting for it, we get a counterintuitive finding: although classification uncertainty reduces personalization accuracy, it may benefit the regulator by deterring manipulation.
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Abstract
Abstract:Demand management policies aim to shift individual behavior toward sustainable choices (e.g., goods, transport modes). Personalized incentive policies tailor subsidies to agents' needs and constraints, but personalization also exposes the regulator to agents' strategic manipulative behavior. The extent to which agents can profitably manipulate such policies remains poorly understood, which limits their applicability. This paper establishes analytically the conditions under which manipulation benefits the agent. We formalize a Stackelberg game in which an agent may simulate a different profile to extract higher incentives from a regulator. Previous work typically assumes that the regulator would instantaneously classify agents into the simulated user profile. In contrast, we explicitly model the regulator's process of learning the agent profile, based on observed past behavior. This makes manipulation harder, since it forces the agent to consistently behave according to a simulated profile, thus renouncing following their true preferences. Manipulative agents thus incur a Cost of Manipulation, which has typically been neglected. By accounting for it, we get a counterintuitive finding: although classification uncertainty reduces personalization accuracy, it may benefit the regulator by deterring manipulation. We establish an upper bound on the agent's gain from manipulation, which becomes stricter as profiles become harder to distinguish for the regulator. Simulations on the London Passenger Mode Choice dataset illustrate this deterrence effect. The framework can therefore be used to stress-test the robustness of personalized incentive policies before deployment.
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- Source: https://arxiv.org/abs/2610.13133v1
- Canonical: https://arxiv.org/abs/2610.13133v1
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The Limits of Manipulation within Personalized Incentive Policies Andrea Araldo Alessa Mayer Ravi Seshadri Amir Brudner Abstract Demand management policies aim to shift individual behavior toward sustainable choices (e.g., goods, transport modes). Personalized incentive policies tailor subsidies to agents’ needs and constraints, but personalization also exposes the regulator to agents’ strategic manipulative behavior. The extent to which agents can profitably manipulate such policies remains poorly understood, which limits their applicability. This paper establishes analytically the conditions under which manipulation benefits the agent. We formalize a Stackelberg game in which an agent may simulate a different profile to extract higher incentives from a regulator. Previous work typically assumes that the regulator would instantaneously classify agents into the simulated user profile. In contrast, we explicitly model the regulator the process of learning the agent profile, based on observed past behavior. This makes manipulation harder, since it forces the agent to consistently behave according to a simulated profile, thus renouncing to follow their true preferences. Manipulative agents thus incur a Cost of Manipulation, which has been typically been neglected. By accounting for it, we get a counterintuitive finding: although classification uncertainty reduce personalization accuracy, it may benefit the regulator by deterring manipulation. We establish an upper bound on the agent’s gain from manipulation, which becomes stricter as profiles become harder to distinguish for the regulator. Simulations on the London Passenger Mode Choice dataset illustrate this deterrence effect. The framework can therefore be used to stress-test the robustness of personalized incentive policies before deployment. 1SAMOVAR, Télécom SudParis, Institut Polytechnique de Paris, 91120 Palaiseau, France 2Institut Polytechnique de Paris 3Technical University of Denmark 4Massachusetts Institute of Technology 5The Hebrew University Business School 1 Introduction Behavioral change remains a major challenge in achieving sustainability goals, especially as technological solutions alone are insufficient. Demand Management Policies (DMP) favor changes in the agents’ behavior toward sustainable choices (IEA 2021). Potential greenhouse gas reduction achievable via DMP is estimated to exceed 40% by 2050, compared to baselines (IPCC 2023; Creutzig et al. 2022). We will consider DMP in which a regulator, e.g., a government or any type of authority, applies a signal. Negative signals (e.g., prices, taxes, bans) generally lack public acceptability (to the point of violent protests (Dabla-Norris et al. 2023)) and are often inadvertently inequitable (Blasco et al. 2023). For instance, increasing taxes on fuel particularly impacts low-income rural households, who are dependent on their cars to perform their everyday activities, and who spend, for transport, already up to 30% of their income (Le Bras 2019). On the other hand, positive signals (e.g., incentives, subsidies) are expensive for the regulator and may not be sufficiently strong to induce relevant behavioral shifts (Gneezy and Rustichini 2000). Personalized DMP (PDMP) can help overcome the above issues (Javaudin et al. 2023; Ascarza and Israeli 2022; Xie et al. 2024) by adapting signals to the condition and the preferences of each agent. Personalized signals can target agents for whom a behavioral shift is less costly, thus mitigating inequitable effects. Thus, PDMP have the potential to be more effective and acceptable than non-personalized DMP. However, PDMP are vulnerable to manipulative agents, who may make dishonest choices in order to conceal their true preferences and manipulate the regulator’s signals in their favor. For instance, it has been shown that some individuals maliciously reduced their labor supply to get a subsidy dedicated to the under-employed (Firpo et al. 2014). In this work, we focus on this critical vulnerability by modeling the interaction between a regulator and a potentially manipulative agent in a Stackelberg game. The regulator (the leader) estimates agent (follower) profiles based on observed choices, while the agent may adopt a simulated (i.e., fake) utility function to secure higher incentives. We investigate how the regulator’s continuous estimation of agent utility impacts manipulation. Extensive work in artificial intelligence on strategic behavior (§2) models agents in ways that are not behaviorally-grounded, and thus not compliant with the way human agents make choices. They are thus not applicable to demand-side mitigation. Our contribution is the first behaviorally-grounded model of repeated interaction between a regulator and an agent within a demand-side mitigation scenario, consisting of a personalized incentive policy. We model agents’ choices, based on Random Utility Theory (RUT) (Strzalecki 2025; McFadden 1981). In contrast with previous work, in which the agent simply reports their profile, we explicitly model the regulator’s preference elicitation (Chapman and Fisher 2025), with which the regulator learns the agent’s profile. Previous results showed that the agent could greatly manipulate the regulator. Our analysis shows instead that such findings are over-pessimistic. Indeed, we theoretically show that realistic preference elicitation deters manipulative behavior and, consequently, personalized incentive policies can effectively reduce social cost. Results are validated in Monte Carlo simulations. The proofs that are not in the main paper are in the Supplementary Material. 2 Related Work Strategic interaction and manipulation are studied in artificial intelligence, game and learning theory. In Adversarial Interaction Attacks (Nguyen et al. 2019), a defender allocates resources to defend targets and an attacker chooses the target to attack. The defender seeks to learn the attacker’s profile to better allocate defense resources. The attacker can manipulate the defender, by implementing certain sequences of attacks that steer the defender’s learning in a way that increases the utility of the attacks. Our case, instead, cannot be simultaneous: first, the regulator must set the incentives for the different available alternatives; then, the agent chooses an alternative. Our setting is thus better described by a Stackelberg game, where the leader (the regulator) takes their action (setting incentives) and then the follower (the agent) takes their action (choosing an alternative). In recent work, when the leader has limited information about the follower, the latter can manipulate the outcome of the interaction. (Gan et al. 2019a; Gan et al. 2019b; Nguyen 2019; Birmpas et al. 2020; Dawkins et al. 2021) show that the follower can implement an optimal manipulative strategy, which gives the follower higher payoff: the leader is thus greatly vulnerable to manipulation, which is simply performed by the follower via strategic reporting. Such previous findings would tend to discourage the application of personalized policies. We however observe that their setting is unrealistic and too pessimistic for the leader. Indeed, manipulation is “too easy”: “To manipulate the [leader], the [follower] reports a fake [profile]”, and the leader just considers the reported profile when deciding their action. The previous quote is from (Gan et al. 2019a), but this assumption is common to all the aforementioned work. Similarly, framing the setting within Bayesian Persuasion (Kamenica and Gentzkow 2011; Jain and Perchet 2024), an agent could manipulate the regulator by choosing the information to report. We consider instead a more realistic setting, in which the regulator (leader) learns the agent (follower) profile, by interacting with them and observing their previous choices, performing preference elicitation (Chapman and Fisher 2025). In the following, we will thus refer to instantaneous and realistic preference elicitation, to contrast, respectively, the setting of the aforementioned work (where the leader immediately considers the profile reported by the agent when setting incentives) and the more realistic setting of our work (where the leader considers instead the profile they learn by observing the agent’s behavior). Related are screening contracts, with which a regulator could design a menu of options, specifically to distinguish among agent profiles (Einav et al. 2022). Our setting differs because the options (alternatives) are exogenous and agent choices are stochastic and thus do not perfectly reveal their profile. In (Sessa et al. 2020) the agent can adversarially choose one profile at every timeslot. However, their setting is too pessimistic (and thus inappropriate) for studying personalized policies, for different reasons. First, we assume that agents behave according to Random Utility Theory (RUT) (Strzalecki 2025; McFadden 1981), which is more realistic in the economic settings we are tackling. This assumption constrains the agent’s alternatives to be probabilistically chosen, at every timeslot, according to a single profile. The manipulation of the agent thus consists in committing to a “fake” profile, instead of adversarially selecting a new profile at every timeslot as in (Sessa et al. 2020). Moreover, we assume agents aim to maximize their utility, and not necessarily act adversarially (e.g., aim to maximize the loss of the regulator). Second, (Sessa et al. 2020) assume the leader adopts probabilistic policies, while in our case incentive policies are constrained to be deterministic. Indeed, it would not be acceptable that a regulator (e.g., a government) gives randomized incentives to individuals. Ionescu et al. (2023) study strategic behavior in matching algorithms, while we consider demand-side management policies. Moreover, their manipulation is implemented by “returning agents”, which take the role of our regulator. We instead study user-side manipulation. A different, but related, problem is the personalized reserved price setting in a repeated auction, considered by (Nedelec et al. 2020). In their work, however, the set of leaders (corresponding to the buyers) are those who can potentially manipulate the follower (the seller). As a consequence, their findings do not hold in our case, in which agents (followers) can manipulate the regulator (leader). In Performative Prediction (P) (Perdomo et al. 2020), predictions produced by a supervised learning model alter the data-generating process itself, thereby changing the distribution on which future predictions are evaluated. However, in P the action of the leader is to change classification parameters, whereas the action of our leader (the regulator) is to propose incentives. Moreover, different from P, the goal of our leader is not to predict, but to propose incentives. In P, the leader aims to reduce classification loss, whereas our leader aims to reduce generalized social cost. In P, the agent influences the classifier parameters, which is not our case. Also, P is not applicable to demand-side policies: in (Perdomo et al. 2020, §5.1), the regulator is allowed to change the classifier over time. However, for transparency constraints, a regulator, such as a public authority, must fix and announce the rules in advance; they would otherwise be unacceptable and perceived as black-box or arbitrary by citizens. This is why we fix the agent’s profile classification is performed via the Maximum Likelihood Estimator, according to real preference elicitation cases (Chapman and Fisher 2025). Our setting has some similarity with the Vickrey–Clarke–Groves (VCG) mechanism. The latter computes prices so as to maximize social welfare, while ensuring truthful behavior of the agents. One could be tempted to use a VCG-like mechanism in our setting. This is however not possible, because in VCG, the social welfare is the sum of the agent’s utilities, while in our case the social cost that the regulator aims to minimize is not such a sum, but is rather related to sustainability goals (e.g., emissions). Despite the rich literature in the analysis of manipulative behavior, the settings considered are either too simplistic - the agent can manipulate by simply misreporting - or too pessimistic - the manipulative behavior is not appropriately constrained to be economically and behaviorally grounded. This paper aims to fill this gap. 3 Model Notation is entirely defined in the paper. For ease of reference, it is summarized in Table 1 of the the Suppl. Material. We formulate a repeated Stackelberg game (as (Birmpas et al. 2020)) with a leader (regulator) and a follower (agent). At each iteration, the agent chooses an alternative i∈ℐi , which generates externality ci≥0c_i≥ 0. Let alternatives ℐ=1,…,II=\1,…,I\ be sorted in the increasing order of externalities, i.e., from the most to the less sustainable. To encourage the choice of the most sustainable alternative, at the beginning of each iteration the regulator proposes incentive y≥0y≥ 0. To give our model some context, we briefly mention some real-world applications. For instance, consumers (agents) can choose to buy a short-lasting, unethically produced clothing item, which would correspond to alternative 22, with c2c_2 representing the environmental and social externality caused by producing this good. On the other hand, alternative 11 would be to buy this clothing item second-hand, which has low externality c1c_1. Incentive y would be a subsidy to encourage buying second-hand. Another application consists in the choice of a traveler (agent) of public transport (alternative 11) or taking the car (alternative 22) for a trip. Here, externality c2c_2 would represent the car-related externality, e.g. pollution, induced congestion. The government could provide an incentive y to encourage taking public transport, which produces lesser externality c1c_1. A last example is the decision of a firm (agent) to prefer electricity generated from renewable energies (alternative 11) or traditional, fossil-fuel-based electricity (alternative 22), which imposes a higher externality c2c_2 in the form of CO2 emissions. The regulator can set a subsidy y to encourage the firm to use renewable sources. 3.1 Random Utility Modeling We model agent behavior according to random utility theory (RUT) (Strzalecki 2025; McFadden 1981). The agent has a utility function v(i,y)v(i,y), associating a certain systematic utility (or, briefly, utility (Cascetta 2009, §3.2)) to alternative i when the regulator proposes incentive y. When the agent chooses alternative i∈ℐi , they get random utility: systematic utility v(i,y)v(i,y) plus a random variable ϵi _i, called random term. Following RUT, we assume ϵi _i are identically and independently drawn from a Gumbel distribution with 00-mean and scale parameter θ≥0θ≥ 0. The random term accounts for some inherent randomness in agent’s behavior (Cascetta 2009, p100), which makes the agent’s choice not easily predictable for the regulator. This randomness may be due to the inability of the agent to estimate the exact alternatives’ utility or to an additional utility disturbance, which is completely observable only after the choice is made. The agent adopts in general a mixed strategy, i.e., they make choices according to a probability distribution over the available alternatives. We focus here on a class of mixed strategies, well-known in RUT, namely the logit model (McKelvey and Palfrey 1995, §3): if the agent has a systematic utility function v(⋅,⋅)v(·,·) and the regulator proposes incentive y, they will choose alternative i with probability: ℙiv(y)=ev(i,y)/θ∑i′∈ℐev(i′,y)/θ.P_i^v(y)= e^v(i,y)/θ _i e^v(i ,y)/θ. (1) The probabilistic nature of the choice accounts for the bounded rationality of the agent (Wheeler 2018), who does not always make the theoretically optimal decisions, due to limited information. In our case, the agent does not know in advance the actual outcome v(i,y)+ϵiv(i,y)+ _i of their choice, which is revealed only after the choice is made. This is because the actual realization of ϵi _i is not known before alternative i is chosen. Note that, the larger the scale parameter, the greater the randomness in the agent’s choice. The behavior of the agent is determined by their profile, capturing inherent agent’s preferences or needs impacting their choices. A profile is mapped to a certain systematic utility function. The set of profiles is =0,1,..,K−1K=\0,1,..,K-1\ and their respective utility functions are uk(⋅,⋅)u_k(·,·). We allow any finite number K of agent profiles. According to standard RUT ((Cascetta 2009, §3.2.5a)), each alternative i is described with a (possibly) agent-dependent attribute vector i,kx_i,k. Considering the case in which alternatives represent transport modes, the components of i,kx_i,k may represent monetary cost, walking travel time, in-vehicle time, comfort, etc. If alternatives represent different products, i,kx_i,k may instead represent characteristics such as the price, the reputation of the brand, the weight, etc. Some components are agent-dependent: e.g., the in-vehicle time to reach a certain destination depends on the starting location, e.g., the residence, which varies from one agent to another. This is why i,kx_i,k is indexed by k. Each profile is associated with a coefficient vector k α_k, which captures how individuals with profile k value the various attributes. For instance, older individuals may assign greater importance to walking time compared to younger individuals. The systematic utility is thus k⋅i,k α_k·x_i,k in the absence of any incentive policy. Lemma 1. Assuming utility is quasi-linear with respect to income11 1 This is a standard assumption in econometrics ((Javaudin et al. 2023)), which means that both k⋅i,k α_k·x_i,k and the incentive y are expressed in the same unit as income and that y has an additive effect on utility. utility function uk(⋅,⋅)u_k(·,·) can be written as uk(i,y)=ak+yi=1bki≠1;ak:=k⋅1,k;bk:=θln∑i>0ek⋅i,kθ u_k(i,y)= casesa_k+y&i=1\\ b_k&i≠ 1 cases; array[]la_k:= α_k·x_1,k;\\ b_k:=θ Σ _i>0e α_k·x_i,kθ array Also, pk(y):=ℙ1uk(y)=σ(ak−bk+yθ) , p_k(y):=P_1^u_k(y)=σ ( a_k-b_k+yθ ) (4) where σ(⋅) is the sigmoid σ(·) is the sigmoid This equation summarizes the fact that, depending on their profiles, some agents might have higher disutility than other when choosing alternative 11, and we compensate them for this loss, adapting incentive y to their case. This personalized compensation introduces vertical equity (Black et al. 2022, §3.2), since it distributes incentives to those who need it the most to change their behavior. Quantity ak−bka_k-b_k interestingly captures how much profile k is inherently responsible, i.e., how much they prefer the more sustainable alternative i=1i=1 over the others, independent of the proposed incentive. We call ak−bka_k-b_k the responsibility intensity of that profile. With no lack of generality, we order all profiles accordingly: ak−1−bk−1≥ak−bk, for k=1,..,K−1 a_k-1-b_k-1≥ a_k-b_k, for k=1,..,K-1 (5) Lemma 2. Let us consider an agent following a certain profile k∈k . Then, the expectation of the externalities caused by them and of the incentive collected are, respectively: extk(y) ^k(y) =pk(y)⋅c1+(1−pk(y))⋅c¬1; =p_k(y)· c_1+ (1-p_k(y) )· c_ 1; inck(y) ^k(y) =pk(y)⋅y =p_k(y)· y with c¬1 c_ 1 :=∑i>1eu0(i,y)θ∑i′>1eu0(i′,y)θci>c1 := _i>1 e u_0(i,y)θ _i >1e u_0(i ,y)θc_i>c_1 (6) The generalized social cost when the agent chooses i consists of social cost cic_i and the cost of incentive yiy_i provided by the regulator, the latter term weighted by a parameter β>0β>0, which expresses the relative importance the regulator gives to incentive-related expenditures versus externalities and also normalizes the units of measure. The expected social cost is f~k(y)=extk(y)+β⋅inck f^k(y)=ext^k(y)+β·inc^k (7) Note that computing the expected value above requires knowing the utility v^(⋅,⋅) v(·,·) that the agent is adopting. This is information is not available to the regulator, which can only thus only estimate this expectation, based on estimate v^(⋅,⋅) v(·,·), inferred via preference elicitation. 3.2 Interaction between Regulator and Agent The interaction between the regulator and the agent is a sequential game. In each timeslot t: 1. The regulator infers profile k^t=ψ(t−1) k^t=ψ(h^t-1) of the agent, via a deterministic inference rule ψ⋅ψ· applied on previous observations t−1=y1,i1,…,yt−1,it−1h^t-1=y^1,i^1,…,y^t-1,i^t-1, where ys,isy^s,i^s are the incentives proposed by the regulator and the alternative selected by the agent at iteration s=1,…,t−1s=1,…,t-1. In the numerical results, inference rule ψ(⋅)ψ(·) is the maximum likelihood estimator, details in (80) of the supplemental material. 2. The regulator takes its action. An action is the incentive yty^t that the regulator proposes to the agent at timeslot t for choosing alternative 11. The regulator proposes incentive yt=ϕ(k^t)y^t=φ( k^t), which is associated with inferred profile k^t k^t via deterministic function ϕ(⋅)φ(·). We assume that this function returns the incentive so as to minimize the estimated expectation of the generalized social cost (7). 3. The agent chooses alternative it∈ℐi^t , according to ℙiv(yt)P^v_i(y^t) formulated in (1). 4. The regulator transfers incentive yty^t to the agent, if it=1i^t=1, or no incentive if it>1i^t>1. 5. The agent gets utility v(it,yt)+eitv(i^t,y^t)+e_i^t, where eite_i^t is a realization of random noise ϵit _i^t. The regulator may interact with multiple agents. We focus on one regulator-agent interaction, assuming cross-agent effects do not materially alter individual choices. This standard assumption (Javaudin et al. 2023) is appropriate because realistic incentive policies, even when producing non-negligible effects, rarely induce shifts massive enough to substantially change the surrounding choice environment, e.g., affecting congestion. Since in this case aggregate social cost is additive: the system then decomposes into separate regulator-agent interactions, and our analysis applies to each agent. Regulator budget constraints could be added in step 2, which we do not do as it would not add conceptual added value. Moreover, the regulator could solve complex optimization problems when calculating yty^t, e.g., to maximize some long-term cumulative objective. However, such complex optimizations would not be implementable by real decision makers: for political acceptance and transparency to citizens, policies must be based on simple, understandable rules, to prevent citizens from wrongly perceiving that incentives are calculated by an arbitrary black box. Our personalized incentives are instead easy to understand by the agents, since they are based on the simple concept that an agent with profile k deserves a certain incentive yk⋆y_k , whose value can be transparently announced by the regulator a priori. Therefore, yt=yk^t⋆y^t=y_ k^t . Prop. 3 will detail the way in which such a value is calculated. 4 Analysis 4.1 Optimal Incentives There are unique incentives that minimize the estimated expectation of the generalized social cost f~v^t(y) f v^t(y): Proposition 3. There is a unique optimal incentive for inferred profile uk(⋅,⋅)u_k(·,·): yuk⋆=max(c¬1−c1β−θ(1+W(eak−bkθ+c¬1−c1θβ−1)),0)y _u_k= ( c_ 1-c_1β-θ (1+W(e a_k-b_kθ+ c_ 1-c_1θβ-1) ),0 ) where W(.) is the Lambert’s W-function. Moreover, Corollary 4. Let yk∗y^*_k be the optimal incentives as defined in Proposition 3. Then yk∗y^*_k is non-increasing with responsibility intensity ak−bka_k-b_k. This proposition tells us which profiles expect to have higher incentives. Thanks to ordering (5) and Cor. 4 we can thus determine the order of proposed incentives. The following proposition prevents optimal incentives from being infinitely large, which is crucial when evaluating the economic feasibility of the regulator’s policy. Proposition 5. The optimal incentive is upper-bounded, i.e., 0≤yv^t⋆<c¬1−c1β.0≤ y _ v^t< c_ 1-c_1β. As expected, the previous bound grows with the externality c¬1c_ 1 generated when not choosing alternative 11 and decreases with the social cost c1c_1 of alternative 11 since the regulator can prevent more social cost by encouraging alternative 11 if there is larger difference between the costs caused by alternatives 11 and the others. 4.2 Agent’s Strategic Behavior We focus here on an agent that is inherently responsible, i.e., their true profile is the responsible profile v(⋅,⋅)=u0(⋅,⋅)v(·,·)=u_0(·,·) (see ordering (5)) and who has a natural preference toward sustainable alternatives, i.e., a0>b0a_0>b_0. We study in which case such an agent may be better off manipulating, by adopting (i.e., simulating) another profile. We assume that the agent has bounded rationality, i.e., they seek to maximize their utility subject to cognitive and computational limitations, which prevents them from adopting arbitrarily complex optimization. This is compliant with human behavior (Wheeler 2018) and rules out sophisticated Markovian or RL planning strategies that require accounting for long-term rewards. We thus assume that, if the agent decides to be manipulative, they will imitate another profile r∈r , via adopting utility ur(⋅,⋅)u_r(·,·), i.e., consistently choosing alternative it=1i^t=1 or it>1i^t>1 according to probability pr(⋅)p^r(·) (see Lemma 1) in all the timeslots. In the following analysis, among profiles K, three deserve particular attention: • k=0k=0 is the most responsible profile. We focus on a user whose true profile is 00 and study under which conditions they are better off manipulating. • r is the profile adopted by the agent, which determines probability pr(⋅)p^r(·) to choose the sustainable alternative. If the agent behaves honestly, r=0r=0, otherwise, r>0r>0. • k^t k^t is the regulator’s estimate at time t of the agent’s profile. The true utility collected by the agent when the regulator proposes incentive y is u0(y)=a0+yif choosing i=1b0if choosing i>1, u_0(y)= casesa_0+y&if choosing i=1\\ b_0&if choosing i>1, cases (8) no matter whether the agent is honest (r=0r=0) or not (r>0r>0). When a user is manipulative, it incurs a loss of true utility, due to the fact that they do not choose alternatives in accordance with their true profile. On the other end, they have a gain, obtained via shifting the probability distribution of the proposed incentive YtY_t toward a more favorable one. We capture this loss and gain in the definitions of Gain from Manipulation and Cost of Manipulation, which will be given in Lemma 7. We need first to introduce some notation. Definition 6. [Notation of the random variables] Let K^t=ψ(t−1) K^t=ψ(H^t-1) be the random variable (r.v.) denoting the profile inferred by the regulator at time t. It is the result of applying a deterministic function ψ(⋅)ψ(·) (see (80)) to t−1H^t-1, which is the r.v. representing the history. t−1H^t-1 is stochastic since the agent’s choices are stochastic (Lemma 1). Let Yt=ϕ(K^t)Y^t=φ( K^t) denote the incentive proposed at time t. It is a r.v., whose possible values are yk⋆,k∈y_k ,k . Let U0(y)U_0(y) denote the random variable representing such a utility. It takes values a0+ya_0+y and b0b_0 with probability pk(y)p_k(y) and 1−pk(y)1-p_k(y), respectively (Lemma 1). The true utility collected by the agent at time t is thus random variable U0(Yt)U_0(Y^t). Its stochasticity comes from the stochasticity of YtY^t, as well as the stochasticity of U0(y)U_0(y) for any value of y. For notational compactness, we introduce the following conditional expectation and probability operators k[⋅]=[⋅|agent adopts profile k], _k[·]=E[·|agent adopts profile k], (9) ℙk[⋅]=ℙ[⋅|agent adopts profile k]. _k[·]=P[·|agent adopts profile k]. (10) Let r.v. YktY_k^t and K^kt K_k^t represent the incentive proposed at time t and the profile inferred by the regulator, when the agent adopts profile k. Observe that all r.v. K^kt,∀k∈,∀t K_k^t,∀ k ,∀ t have the same domain K. They only differ in how the probability mass, i.e., Pk[A],∀⊆P_k[A], (10), is distributed in the domain. Similarly, all r.v. Ykt,∀k∈,∀tY_k^t,∀ k ,∀ t have the same domain yk⋆k∈\y_k \_k . This notation makes it clear that, by adopting a certain profile k∈k , the agent induces a certain probability distribution over the inferred profile, and thus over the incentive proposed. Lemma 7. Let CMr(y)CM_r(y) be the Cost of Manipulation incurred by an agent simulating profile r when the proposed incentive is y∈yk⋆k∈y∈\y_k \_k : CMr(y):=0[U0(y)]−r[U0(y)]. _r(y):=E_0[U_0(y)]-E_r[U_0(y)]. (11) Let CMrtCM_r^t be its mean: CMrt:=0[CMr(Yt)]=∑k∈ℙ0(Yt=yk⋆)⋅CMr(yk⋆) _r^t:=E_0[CM_r(Y^t)]= _k array[]rP_0(Y^t=y_k )\\ ·CM_r(y_k ) array Let GMrtGM_r^t be the Gain from Manipulation when simulating profile r, defined as GMrt:=∑k∈(ℙr[Yt=yk⋆]−ℙ0[Yt=yk⋆])r[U0(yk⋆)]. _r^t:= _k ( array[]rP_r[Y^t=y_k ]\\ -P_0[Y^t=y_k ] array )E_r[U_0(y_k )]. Let ΔUrt U_r^t be the manipulation advantage, defined as the additional expected true utility collected by the agent at time t when simulating profile r over acting honestly. Such an advantage can be decomposed in ΔUrt:=r[U0(Yt)]−0[U0(Yt)]=GMrt−CMrt. U_r^t:=E_r[U_0(Y^t)]-E_0[U_0(Y^t)]=GM_r^t-CM_r^t. (16) Proof. By the law of total expectation, for each j∈0,rj∈\0,r\, j[U0(Yt)]=∑k∈ℙj(Yt=yk⋆)j[U0(yk⋆)]E_j[U_0(Y^t)]= _k P_j(Y^t=y_k )\,E_j[U_0(y_k )]. Replacing these expressions in (16) and adding and subtracting ∑k∈ℙ0(Yt=yk⋆)r[U0(yk⋆)] _k P_0(Y^t=y_k )E_r[U_0(y_k )] gives the Lemma. ∎ We now study how the agent can manipulate the probability distribution of the incentives. Let IErtIE_r^t denote the intensity of elicitation of profile r at time t, i.e., how much the agent can change the probability distribution of the profile inferred by the regulator, by simulating profile r>0r>0: IErt:=DKL(K^0t∥K^rt), _r^t:=D_KL( K_0^t\| K_r^t), (17) which is the Kullback-Leibler (KL) divergence between the profile inferred by the regulator when the agent adopts true profile 00 or simulates profile r, respectively. Similarly, let IPrtIP_r^t denote the intensity of personalization of profile r at time t, i.e., how much the agent can change the probability distribution of the incentives proposed by the regulator, by simulating profile r>0r>0: IPrt:=DKL(Y0t∥Yrt)≤IErt, _r^t:=D_KL(Y_0^t\|Y_r^t) _r^t, (18) where the inequality is obtained thanks to the contraction of divergence induced by function Y^kt=ϕ(K^kt) Y_k^t=φ( K_k^t) (Polyanskiy and Wu 2025, Theor. 7.4). Theorem 8. The gain from manipulation is bounded by 0 0 ≤GMrt≤RrIPrt/2≤RrIErt/2, _r^t≤ R_r IP_r^t/2≤ R_r IE_r^t/2, (19) where Rr R_r :=r[U0(yK−1⋆)]−r[U0(y0⋆)] :=E_r[U_0(y_K-1 )]-E_r[U_0(y_0 )] (20) is the range of expected true utility. Proof. Thanks to Prop. 3, YrtY_r^t First-Order-Stochastically Dominates (FOSD) Y0tY_0^t, which intuitively means that by simulating r>0r>0 the agent shifts the probability mass toward higher values of incentives. Observe that y↦[U0(y)]y [U_0(y)] is increasing. Looking at the sum defining GMrtGM_r^t (7), the terms with high values of [U0(y)]E[U_0(y)] are multiplied by positive numbers (ℙr[⋅]>ℙ0[⋅]P_r[·]>P_0[·], due to FOSD) and only low values of [U0(y)]E[U_0(y)] are multiplied by negative numbers. Simple calculation thus yields a positive sum and thus GMrt≥0GM_r^t≥ 0. As for the upper bound, we will first bound GMrtGM_r^t in terms of total variation δ(Y0t,Yrt)δ(Y_0^t,Y_r^t) between the probability distributions of Y0tY_0^t and YrtY_r^t (def. in (Polyanskiy and Wu 2025, (7.3))). Since ∑k∈(ℙr[Yt=yk⋆]−ℙ0[Yt=yk⋆])=0 _k (P_r[Y^t=y_k ]-P_0[Y^t=y_k ] )=0, we can introduce any constant A in the expression of GMrtGM_r^t (Lemma 7) without changing its value, as follows: GMrt=∑k∈(ℙr[Yt=yk⋆]−ℙ0[Yt=yk⋆])⋅(r[U0(yk⋆)]−A) _r^t= _k ( array[]lP_r[Y^t=y_k ]\\ -P_0[Y^t=y_k ] array )·(E_r[U_0(y_k )]-A) =Rr2∑k∈(ℙr[Yt=yk⋆]−ℙ0[Yt=yk⋆])⋅r[U0(yk⋆)]−ARr/2 = R_r2 _k ( array[]lP_r[Y^t=y_k ]\\ -P_0[Y^t=y_k ] array )· E_r[U_0(y_k )]-AR_r/2 =Rr2∑k∈(ℙr[Yt=yk⋆]−ℙ0[Yt=yk⋆])⋅f(yk⋆), = R_r2 _k ( array[]lP_r[Y^t=y_k ]\\ -P_0[Y^t=y_k ] array )· f(y_k ), where we define function f(yk⋆)=r[U0(yk⋆)]−ARr/2f(y_k )= E_r[U_0(y_k )]-AR_r/2. The previous expression is equivalent to GMrt=Rr2(r[f(Yt)]−0[f(Yt)]) _r^t= R_r2 (E_r[f(Y^t)]-E_0[f(Y^t)] ) (26) Let us choose in particular A A =(r[U0(yK−1⋆)]+r[U0(y0⋆)])/2 = (E_r[U_0(y_K-1 )]+E_r[U_0(y_0 )] )/2 Since y↦r[U0(y)]y _r[U_0(y)] is an increasing function. Also, thanks to Prop. 3, yk⋆y_k is non decreasing with k. Since A is the midpoint between the maximum and minimum expected value, |A−r[U0(yk⋆)]|≤Rr/2,∀k∈. |A-E_r[U_0(y_k )]|≤ R_r/2,∀ k . (27) and thus |f(yk⋆)|≤1,∀k∈|f(y_k )|≤ 1,∀ k . We can thus apply the sup-representation of total variation from Polyanskiy and Wu (2025, (7.18)) and obtain GMrt≤Rr⋅δ(Y0t,Yrt). _r^t≤ R_r·δ(Y_0^t,Y_r^t). (28) Thanks to the Pinsker inequality first (Polyanskiy and Wu 2025, (7.27)) δ δ (Y0t,Yrt)≤12DKL(Y0t∥Yrt), (Y_0^t,Y_r^t)≤ 12D_KL(Y_0^t\|Y_r^t), (29) Combining (28), (29) and (18) concludes the proof. ∎ Theor. 9 formalizes that the more the intensity of personalization, the more the agent can, at each timeslot, gain from manipulation. Theor. 9 result adds that, complying with intuition, the more the interactions between the regulator and the agent, the more the latter can manipulate the former. However, Theor. 9 limits the rate of growth of the potential for manipulation: such a growth can be at most linear. Theorem 9. Manipulation potential IErtIE_r^t is bounded by IErt _r^t ≤(t−1)IEr ≤(t-1)IE_r (30) where IEr:= IE_r:= maxk∈DKL(Ber(p0(yk⋆))∥Ber(pr(yk⋆))), _k D_KL (Ber(p_0(y_k ))\|Ber(p_r(y_k )) ), (31) and Ber(p)Ber(p) is a Bernoulli distribution with probability p. Moreover, the gain from manipulation is bounded by 0 0 ≤GMrt≤Rr12(t−1)IEr. _r^t≤ R_r 12(t-1)IE_r. (32) Proof. The history of proposed incentives and chosen alternatives is t−1=y1,i1,..,yt−1,it−1h^t-1=y^1,i^1,..,y^t-1,i^t-1. Therefore, the realization of K^t K^t is obtained as a deterministic function ψ(t−1)ψ(h^t-1). Let kt−1H_k^t-1 be the random variables representing the possible histories when the agent adopts profile k. Then IErt=DKL(K^0t∥K^rt)≤DKL(0t−1∥rt−1) _r^t=D_KL( K_0^t\| K_r^t)≤ D_KL(H_0^t-1\|H_r^t-1) (33) ≤[∑s=1t−1∑∈ℋs−1ℙ0[s−1=]⋅DKL(ℙ0(Is|s−1=)∥ℙr(Is|s−1=))] ≤ [ _s=1^t-1 _h ^s-1 array[]lP_0[H^s-1=h]·\\ D_KL ( array[]lP_0(I^s|H^s-1=h)\\ \|P_r(I^s|H^s-1=h) array ) array ] =[∑s=1t−1∑∈ℋs−1ℙ0[s−1=]⋅DKL(Ber(p0(yψ(s−1)⋆))∥Ber(pr(yψ(s−1)⋆)))] = [ _s=1^t-1 _h ^s-1 array[]lP_0[H^s-1=h]·\\ D_KL ( array[]lBer(p_0(y_ψ(h^s-1) ))\\ \|Ber(p_r(y_ψ(h^s-1) )) array ) array ] ≤(31)(t−1)IEr, ≤ eq:gammar(t-1)IE_r, (40) Inequality (33) simply exploits contraction of divergence induced by applying a function (Polyanskiy and Wu 2025, Theor. 7.4). Eq. (4.2) is the chain rule obtained as in (Okoroafor et al. 2023, (9)) and ℋs−1H^s-1 is the set of all possible histories up to time s−1s-1. Finally, Eq (32) is the combination of Theor. 8 and (30). ∎ Is it thus possible to understand up to which timeslot manipulation is deterred, as formalized in the following corollary. Corollary 10. The manipulation of the regulator via imitating profile r is deterred in any timeslot t s.t. t t <1+2IEr(CMrminRr)2 <1+ 2IE_r ( CM_r^minR_r )^2 (41) where CMrmin=mink∈CMr(yk⋆). _r^min= _k CM_r(y_k ). (42) Proof. Lemma 7 implies that manipulation is beneficial only if GMrt−CMrt>0GM_r^t-CM_r^t>0: 0<GMrt−CMrt≤Rr12(t−1)IEr,−CMrmin 0<GM_r^t-CM_r^t≤ R_r 12(t-1)IE_r,-CM_r^min (43) The last inequality is obtained by applying (32) and (42). Therefore, (41) ensures manipulation is deterred. ∎ 5 Impact of Preference Elicitation We now examine how preference elicitation limits the agent’s ability to manipulate the regulator. It not only makes it harder for the agent to convince the regulator that their adopted profile is r>0r>0, but also substantially reduces the expected utility gained from manipulation. We compare two preference-elicitation models. The first, Instantaneous Preference Elicitation (InstPE), is implicitly assumed in previous artificial intelligence works (Dawkins et al. 2021; Birmpas et al. 2020; Gan et al. 2019a; Gan et al. 2019b; Nguyen 2019). Under InstPE, the regulator immediately accepts the agent’s simulated utility profile, so that k^t=r k^t=r for all t. This corresponds to strategic misreporting and allows an agent behaving according to profile r to immediately maximize their expected utility. We instead introduce Realistic Preference Elicitation (RealPE). Here, the regulator observes the agent’s choices over time and updates its belief ψ about the agent’s utility function at each timeslot t. The agent must therefore establish the simulated profile through repeated interaction and may meanwhile be classified under another profile. This repeated simulation is costly because it may require choosing alternatives that the agent does not prefer. To distinguish the two settings, we denote expectations by InstPEE^InstPE and RealPEE^RealPE. Under InstPE, the agent can directly compare the expected utilities of honest and manipulative behavior. Observation 11. In InstPE (i) the regulator proposes incentive yk⋆y_k if the agent declares profile k∈k and (i) the agent always behaves according to their true profile 00, no matter the declared profile k. Therefore, the operators (10)-(9) are such that ℙk[⋅]=ℙ0[⋅]P_k[·]=P_0[·] and k[⋅]=0[⋅]E_k[·]=E_0[·]. The following Lemma shows that the cost of manipulation is null under InstPE and the gain from manipulation is directly the difference between the expected true utilities, under incentive yr∗y_r^* of simulated profile r and under incentive y0⋆y_0 associated to the true profile 00. Lemma 12. GMrt,InstPE=0[U0(yr⋆)]−0[U0(y0⋆)], _r^t,InstPE=E_0[U_0(y_r )]-E_0[U_0(y_0 )], CMrt=0, _r^t=0, ∀t ∀ t 6 Case Study: Travel-Mode Incentives We here validate the theoretical findings and show the deterring impact of Realistic Preference Elicitation (RealPE) on manipulative users, compared to Instantaneous Preference Elicitation (InstPE). We run a comprehensive Monte Carlo simulation campaign calibrated on transport mode choices in London (open dataset of Hillel et al. (2018)), which reports choices of travelers (agents), with i=1i=1 (bike), i=2i=2 (public transport (PT)), i=3i=3 (car). We calibrated the parameters related to 33 profiles k=0,1,2k=0,1,2 and the respective behavioral parameters, and ordered them according to the responsibility intensity. Without imposing it, the calibration returned profiles with preference toward biking, PT, and car. We then run Monte Carlo simulation over multiple seeds, to implement the stochastic choice of the agent, the profile inference (preference elicitation) of the regulator and the calculation of the proposed incentive. The important parameters assigned based on previous experimented incentive policies and CO2 emission evaluations, as well as values obtained via calibration and (3) are θ=3.7θ=3.7, akk=6.1,−6.8,−6.6\a_k\_k=\6.1,-6.8,-6.6\, bkk=5.8,−11.7,−11.9\b_k\_k=\5.8,-11.7,-11.9\, cneg1,kk=2.8,2.2,2.6\c_\-neg1,k\_k=\2.8,2.2,2.6\, β=0.6β=0.6, yk⋆k=0,0.1,0.9(£)\y_k \_k=\0,0.1,0.9\( ) . A detailed explanation of the dataset, the hyperparameters, the calibration method and the parameters resulting from calibration §D1 of the of the Suppl.Material. A Python notebook that reproduces dataset downloading and preprocessing, calibration and simulation is provided in the Suppl.Material. It runs entirely in less than 2 minutes in Google Colab. Figure 1: Time-averaged true agent utility. Averages across all the random realizations (corresponding each to a random seed) and 95% confidence intervals are represented (a) (b) Figure 2: (a) Cost of and gain from manipulation (CM,GMCM,GM); (b) change in the percentage of cycling trips Fig. 2(a) shows that Cost of Manipulation is higher than the Gain under RealPE. Due to (16), time-averaged true utility (1/t)∑s=0tU0(Yt)(1/t) _s=0^tU_0(Y^t) is higher when acting honestly under RealPE (Fig. 1). Manipulation is thus deterred. Fig. 1-2(a) also shows that the situation would be inverted under the over-pessimistic InstPE setting. Therefore, under realistic conditions, personalized incentive policies can be effective in fostering sustainable behavior, as shown in Fig. 2(b) where we computed the induced shift in bike usage. 7 Conclusion By explicitly modeling the regulator’s learning process in personalized incentive policies, we establish an upper bound on the gain achievable from manipulative agents. 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Sect. 3, before Eq. (1) ℙiv(y)P_i^v(y) Probability that an agent with systematic utility v chooses alternative i under incentive y. Eq. (1) =0,…,K−1K=\0,…,K-1\, k Set of profiles, its cardinality, and a profile index. Sect. 3, before Lemma 1 uk(i,y)u_k(i,y) Systematic utility associated with profile k. Sect. 3, before Lemma 1 i,kx_i,k, k α_k Attribute vector of alternative i for profile k, and the corresponding coefficient vector. Sect. 3, before Lemma 1 aka_k, bkb_k Systematic utility of alternative 11 without incentives, and aggregate utility of the remaining alternatives, for profile k. Lemma 1 pk(y)p_k(y), σ(⋅)σ(·) Probability that profile k chooses alternative 11, and the sigmoid function. Eq. (4) ak−bka_k-b_k Responsibility intensity of profile k. Immediately before Eq. (5) extk(y)ext^k(y), inck(y)inc^k(y) Expected externality and expected collected incentive when the agent adopts profile k. Lemma 2 c¬1c_ 1 Expected externality conditional on not choosing alternative 11. Lemma 2 β Weight assigned to incentive expenditures relative to externalities. Eq. (7) f~k(y) f^k(y) Expected generalized social cost under profile k and incentive y. Eq. (7) v^(⋅,⋅) v(·,·), v^t(⋅,⋅) v^t(·,·) Regulator’s estimate of the utility profile adopted by the agent. After Eq. (7) Sequential interaction and optimal incentives t, s Current timeslot and a generic preceding-timeslot index. Step 1 of Sect. 3.2 t−1h^t-1 Realized history of incentives and choices observed before timeslot t. Step 1 of Sect. 3.2 k^t k^t Profile inferred by the regulator at timeslot t. Step 1 of Sect. 3.2 ψ(⋅)ψ(·) Deterministic profile-inference rule. Step 1 of Sect. 3.2 yty^t, iti^t Incentive proposed and alternative chosen at timeslot t. Steps 2–3 of Sect. 3.2 ϕ(⋅)φ(·) Deterministic mapping from an inferred profile to an incentive. Step 2 of Sect. 3.2 yk⋆y_k Optimal incentive associated with inferred profile k. Prop. 3 Table 2: Notation used throughout the paper (continued). Symbol Meaning First introduced Manipulation analysis 00, r Agent’s true profile and the profile imitated by a manipulative agent. Sect. 4.2 u0(y)u_0(y) True realized utility under incentive y, expressed according to the selected alternative. (8) t−1H^t-1, K^t K^t, YtY^t Random history, inferred profile, and proposed incentive at timeslot t. Def. 6 U0(y)U_0(y), U0(Yt)U_0(Y^t) Random true utility under a fixed incentive y, and under the random incentive proposed at timeslot t. Def. 6 k[⋅]E_k[·], ℙk[⋅]P_k[·] Expectation and probability conditional on the agent adopting profile k. Eqs. (9)–(10) YktY_k^t, K^kt K_k^t Random incentive and inferred profile conditional on the agent adopting profile k. After Eqs.(9)-(10) CMr(y)CM_r(y), CMrtCM_r^t Cost of Manipulation under incentive y, and its mean at timeslot t. Eqs. (11)–(7) GMrtGM_r^t Gain from Manipulation at timeslot t when imitating profile r. Eq. (7) ΔUrt U_r^t Manipulation advantage: additional expected true utility obtained by imitating profile r. Eq. (16) DKL(⋅∥⋅)D_KL(·\|·) Kullback–Leibler divergence. Eq. (17) IErtIE_r^t Intensity of elicitation: distinguishability between the inferred-profile distributions induced by profiles 00 and r. Eq. (17) IPrtIP_r^t Intensity of personalization: distinguishability between the corresponding incentive distributions. Eq. (18) RrR_r Range of expected true utility over the incentives the regulator may propose. Theor. 8, Eq. (20) δ(⋅,⋅)δ(·,·) Total-variation distance between probability distributions. Proof of Theor. 8 IErIE_r Maximum one-step KL divergence between the choices induced by profiles 00 and r. Eq. (31) Ber(p)Ber(p) Bernoulli distribution with success probability p. Theor. 9, Eq. (31) kt−1H_k^t-1, ℋs−1H^s-1, IsI^s Random history conditional on profile k, set of possible histories, and random alternative at timeslot s. Proof of Theor. 9 CMrminCM_r Minimum Cost of Manipulation over all incentives that may be proposed. Eq. (42) Preference-elicitation models InstPE, RealPE Instantaneous and Realistic Preference Elicitation. Sect. 5 InstPE[⋅]E^InstPE[·], RealPE[⋅]E^RealPE[·] Expectation under Instantaneous and Realistic Preference Elicitation. Sect. 5 Zt:=It=1Z^t:=1\I^t=1\, ztz^t Binary random variable and its realization indicating whether the sustainable alternative 11 is chosen. Appendix C Lkt(t−1)L_k^t(h^t-1) Likelihood of observing history t−1h^t-1 under profile k. Eq. (77) lnLkt(t−1) L_k^t(h^t-1) Log-likelihood of observing history t−1h^t-1 under profile k. Eq. (78) Appendix B Proofs Proof of Lemma 1 Lemma. Assuming utility is quasi-linear with respect to income,22 2 This is a standard assumption in econometrics ((Javaudin et al. 2023)), which means that both k⋅i α_k·x_i and the incentive y are expressed in the same unit as income and that y has an additive effect on utility. utility function uk(⋅,⋅)u_k(·,·) can be written as uk(i,y)=ak+yi=1bki≠1;ak:=k⋅1,k;bk:=θln∑i>1ek⋅iθ u_k(i,y)= casesa_k+y&i=1\\ b_k&i≠ 1 cases; array[]la_k:= α_k·x_1,k;\\ b_k:=θ Σ _i>1e α_k·x_iθ array Also, pk(y):=ℙ1uk(y)=σ(ak−bk+yθ) , p_k(y):=P_1^u_k(y)=σ ( a_k-b_k+yθ ) (46) where σ(⋅) is the sigmoid σ(·) is the sigmoid Proof. The quasi-linearity of the utility with respect to income implies that the systematic utility associated with alternative 11 is uk(1,y)=k⋅i+yu_k(1,y)= α_k·x_i+y, and thus uk(1,y)=ak+yu_k(1,y)=a_k+y. If the agent does not choose alternative 11, they will choose the best among the remaining alternatives i>1i>1. As explained in §3, the random utility of the agent associated with alternative i is uk(i,y)+ϵiu_k(i,y)+ _i, where ϵi _i is a Gumbel distribution with scale parameter33 3 The scale parameter is also called “noise parameter”. θ. As suggested by (Strzalecki 2025, §1.8), it is reasonable to assume that, when choosing among alternatives i>1i>1, the agent chooses the best among them. The expected utility in this case is bk b_k :=[uk(i,y)|i>1]=[maxi>1[uk(i,y)+ϵi]] :=E [u_k(i,y)|i>1 ]=E [ _i>1[u_k(i,y)+ _i] ] (47) =θln∑i>1ev(i,y)θ =θ _i>1e v(i,y)θ (48) where the last equality is obtained via (Strzalecki 2025, Prop. 3.6). Observe that we do not need to carry the specification of y in symbol bkb_k, because no incentive is given to any alternative i>1i>1. By simple calculus ebkθ=∑i>1ev(i,y)θe b_kθ= _i>1e v(i,y)θ. Therefore, probability (1) can be written as: ℙ1uk _1^u_k =eak+yθeak+yθ+∑i>1ev(i,y)θ=11+ebkθ/eak+yθ = e a_k+yθe a_k+yθ+ _i>1e v(i,y)θ= 11+e b_kθ/e a_k+yθ (49) =11+e−ak+y−bkθ = 11+e^- a_k+y-b_kθ (50) which is the sigmoid calculated in ak+y−bkθ a_k+y-b_kθ. ∎ Proof of Lemma 2 Lemma. Let us consider an agent following a certain profile k∈k . Then, the expectation of the externalities caused by them and of the incentive collected are, respectively: extk(y) ^k(y) =pk(y)⋅c1+(1−pk(y))⋅c¬1,k; =p_k(y)· c_1+ (1-p_k(y) )· c_ 1,k; inck(y) ^k(y) =pk(y)⋅y =p_k(y)· y with c¬1,k c_ 1,k :=∑i>1euk(i,y)θ∑i′>1euk(i′,y)θci>c1 := _i>1 e u_k(i,y)θ _i >1e u_k(i ,y)θc_i>c_1 (51) Proof. The result is obtained by applying the law of total expectation: extk(y) ^k(y) =ℙ1vk(y)(ext(y)| agent adopts k,i=1) =P_1^v_k(y)E (ext(y)| agent adopts k,i=1 ) +(1−ℙ1uk(y))(ext(y)| agent adopts k,i≠1) + (1-P_1^u_k(y) )E (ext(y)| agent adopts k,i≠ 1 ) =pk(y)⋅c1 =p_k(y)· c_1 +(1−pk(y))⋅∑i>1euk(i,y)θ∑i′>1euk(i′,y)θci + (1-p_k(y) )· _i>1 e u_k(i,y)θ _i >1e u_k(i ,y)θc_i (52) The conditional distribution among the alternatives i>1i>1 does not depend on y, because the incentive modifies only the utility of alternative 11. The expected incentive collected is computed similarly. ∎ Lemma. The estimated expectation of the generalized social cost f~k(y):[0,+∞[→ℝ f^k(y):[0,+∞[ has a unique minimum. Let y^k y_k be the root of gk(y) g^k(y) :=y+θ+θe(y+ak−bk)/θ−c¬1,k−c1β :=y+θ+θ e^(y+a_k-b_k)/θ- c_ 1,k-c_1β (53) Then, if c¬1,k−c1β>θ(1+e(ak−bk)/θ) c_ 1,k-c_1β>θ(1+e^(a_k-b_k)/θ) (54) the min of f~k:[0,+∞[→ℝ is at y^k>0 min of f^k:[0,+∞[ is at y_k>0 (55) else such a minimum is at y=0 such a minimum is at y=0 (56) Proof. The estimated expectation of the generalized social cost, which the regulator aims to minimize, is given by (7). By Lemmas 2 and 1 this is f~k(y) f^k(y) =e(ak+y)/θe(ak+y)/θ+ebk/θ(c1+βy) = e^(a_k+y)/θe^(a_k+y)/θ+e^b_k/θ (c_1+β y ) (57) +ebk/θe(ak+y)/θ+ebk/θc¬1,k. + e^b_k/θe^(a_k+y)/θ+e^b_k/θc_ 1,k. (58) By taking its derivative, we get (f~k(y))′ ( f^k(y)) =βye(y+ak+bk)/θ+βθe(y+ak+bk)/θ+βθe2(y+ak)/θ(e(ak+y)/θ+ebk/θ)2 = β ye^(y+a_k+b_k)/θ+βθ e^(y+a_k+b_k)/θ+βθ e^2(y+a_k)/θ(e^(a_k+y)/θ+e^b_k/θ)^2 +c1e(y+ak+bk)/θ−c¬1,ke(y+ak+bk)/θ(e(ak+y)/θ+ebk/θ)2 + c_1e^(y+a_k+b_k)/θ-c_ 1,ke^(y+a_k+b_k)/θ(e^(a_k+y)/θ+e^b_k/θ)^2 (59) To find the extrema we take (f~k(y))′=0( f^k(y)) =0 and, thus, we obtain βye(y+ak+bk)/θ+βθe(y+ak+bk)/θ+βθe2(y+ak−bk)/θ β ye^(y+a_k+b_k)/θ+βθ e^(y+a_k+b_k)/θ+βθ e^2(y+a_k-b_k)/θ −(c¬1,k−c1)e(y+ak−bk)/θ=0 -(c_ 1,k-c_1)e^(y+a_k-b_k)/θ=0 Dividing by βe(y+ak+bk)/θβ e^(y+a_k+b_k)/θ, we obtain ⇔gk(y):=y+θ+θe(y+ak−bk)/θ−c¬1,k−c1β=0 g^k(y):=y+θ+θ e^(y+a_k-b_k)/θ- c_ 1,k-c_1β=0 (60) So the extremum of f~k(y):ℝ→ℝ f^k(y):R is at the root of gk(y)g^k(y), which we will refer to as y^k y_k. Note that gk(y)g^k(y) is an increasing function, which implies that its root y^k y_k is unique. Moreover, observe that (f~k)′( f^k) is equal to gkg^k multiplied by a positive quantity, namely (βe(y+ak+bk)/θ)/(θ(e(ak+y)/θ+ebk/θ)2)(β e^(y+a_k+b_k)/θ)/(θ(e^(a_k+y)/θ+e^b_k/θ)^2). Therefore, gk(y)>0g^k(y)>0 implies (f~k(y))′>0( f^k(y)) >0, i.e., f~k(y) f^k(y) increasing, and gk(y)<0g^k(y)<0 implies (f~k(y))′<0( f^k(y)) <0, i.e., f~k(y) f^k(y) decreasing. For y<y^ky< y_k, function f~k(y) f^k(y) is decreasing, since y<y^ky< y_k implies gk(y)<gk(y^k)g^k(y)<g^k( y_k), i.e. gk(y)<0g^k(y)<0. For y>y^ky> y_k, function f~k(y) f^k(y) is increasing, since y>y^ky> y_k implies gk(y)>gk(y^k)g^k(y)>g^k( y_k), i.e. gk(y)>0g^k(y)>0. The root of gkg^k is, thus, the minimum of f~k(y) f^k(y) on ℝ→ℝR . Suppose that the root y^k y_k of gkg^k is non-positive. In this case, in the interval [0,+∞[[0,+∞[, function f~k(y) f^k(y) is always increasing. Its minimum is thus at 00. Otherwise, the minimum of f~k(y):[0,+∞[→ℝ f^k(y):[0,+∞[ is the root y^k y_k of gkg^k. Condition (54) determines the sign of the root y^k y_k. Indeed If c¬1,k−c1β>θ(1+e(ak−bk)/θ), c_ 1,k-c_1β>θ(1+e^(a_k-b_k)/θ), (61) then 0>θ(1+e(ak−bk)/θ)−c¬1,k−c1β then 0>θ(1+e^(a_k-b_k)/θ)- c_ 1,k-c_1β (62) ⇒gk(y^k)>gk(0) g^k( y_k)>g^k(0) (63) To obtain the last implication, we simply use the definition of gk(y)g^k(y) (60) (replacing y=0y=0) and the fact that it is zero at its root, i.e., gk(y^k)=0g^k( y_k)=0. Since gk(y)g^k(y) is increasing, equation (63) implies that y^k≥0 y_k≥ 0, and so the minimum of f~k:[0,+∞[→ℝ f^k:[0,+∞[ corresponds to y^k y_k. On the contrary, if condition (54) does not hold, the root y^k<0 y_k<0 and the minimum of f~k:[0,+∞[→ℝ f^k:[0,+∞[ is 00. ∎ Proofs of Proposition 3 and Corollary 4 Proposition. 3. There is a unique optimal incentive for profile k∈k : yk⋆=max(c¬1,k−c1β−θ(1+W(eak−bkθ+c¬1,k−c1θβ−1)),0)y_k = ( c_ 1,k-c_1β-θ (1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) ),0 ) (64) where W(.) is the Lambert’s W-function. Proof. The estimated expectation of the generalized social cost, that the regulator aims to minimize, is given by (7). By (4) this is f~k(y) f^k(y) =e(ak+y)/θe(ak+y)/θ+ebk/θ(c1+βy) = e^(a_k+y)/θe^(a_k+y)/θ+e^b_k/θ(c_1+β y) +ebk/θe(ak+y)/θ+ebk/θc¬1,k + e^b_k/θe^(a_k+y)/θ+e^b_k/θc_ 1,k Let yk⋆=0y_k =0 be point at which f~k(y):[0,+∞[→ℝ f^k(y):[0,+∞[ has its minimum (which is unique - see (53)). Let us start by assuming that (54) holds. In this case, yk⋆y_k is the the root of gk(y)g^k(y) and is positive (see (53)). Therefore yk⋆+θ+θe(ak−bk+yk⋆)/θ−c¬1,k−c1β=0 y_k +θ+θ e^(a_k-b_k+y_k )/θ- c_ 1,k-c_1β=0 ⇒yk⋆θ+1+e(ak−bk+yk⋆)/θ−c¬1,k−c1θβ=0 y_k θ+1+e^(a_k-b_k+y_k )/θ- c_ 1,k-c_1θβ=0 ⇒e(ak−bk+yk⋆)/θ=c¬1,k−c1θβ−yk⋆θ−1 e^(a_k-b_k+y_k )/θ= c_ 1,k-c_1θβ- y_k θ-1 We substitute u=c¬1,k−c1θβ−yk⋆θ−1u= c_ 1,k-c_1θβ- y_k θ-1 and, by inserting u and yk⋆θ=c¬1,k−c1θβ−1−u y_k θ= c_ 1,k-c_1θβ-1-u into the last equation, we get eak−bkθ+c¬1,k−c1θβ−1−u=u e a_k-b_kθ+ c_ 1,k-c_1θβ-1-u=u ⇒eak−bkθ+c¬1,k−c1θβ−1=ueu e a_k-b_kθ+ c_ 1,k-c_1θβ-1=ue^u Now we can apply the Lambert W-function on both sides W(eak−bkθ+c¬1,k−c1θβ−1)=W(ueu)=u W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1)=W(ue^u)=u By reinserting u=c¬1,k−c1θβ−yk⋆θ−1u= c_ 1,k-c_1θβ- y_k θ-1, we obtain W(eak−bkθ+c¬1,k−c1θβ−1)=c¬1,k−c1θβ−yk⋆θ−1 W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1)= c_ 1,k-c_1θβ- y_k θ-1 ⇒yk⋆=c¬1,k−c1β−θ−θ⋅W(eak−bkθ+c¬1,k−c1θβ−1) y_k = c_ 1,k-c_1β-θ-θ· W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) Summarizing, we obtain that, if (54) holds, then yk⋆=c¬1,k−c1β−θ⋅(1+W(eak−bkθ+c¬1,k−c1θβ−1))>0y_k = c_ 1,k-c_1β-θ· (1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) )>0 (65) If, instead, (54) does not hold, then yk⋆=0y_k =0 (see (56)). Therefore, yk⋆=max(c¬1,k−c1β−θ⋅(1+W(eak−bkθ+c¬1,k−c1θβ−1)),0)y_k = ( c_ 1,k-c_1β-θ· (1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) ),0 ) (66) ∎ Corollary. 4. Let yk∗y^*_k be the optimal incentives as defined in Proposition 3. Then yk∗y^*_k is non-increasing in ak−bka_k-b_k. Proof. By Proposition 3, yk⋆=max(c¬1,k−c1β−θ(1+W(eak−bkθ+c¬1,k−c1θβ−1)),0)y_k = ( c_ 1,k-c_1β-θ (1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) ),0 ) First, note that eak−bkθ+c¬1,k−c1θβ−1e a_k-b_kθ+ c_ 1,k-c_1θβ-1 is increasing with ak−bka_k-b_k, because of the term ak−bkθ a_k-b_kθ inside the exponential. Following this, W(eak−bkθ+c¬1,k−c1θβ−1)W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) is increasing with ak−bka_k-b_k, since if the argument of the Lambert W function is positive and increasing, the Lambert W function is increasing as well. It follows that −θ(1+W(eak−bkθ+c¬1,k−c1θβ−1))-θ (1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) ) is decreasing with ak−bka_k-b_k and so is the whole expression c¬1,k−c1β−θ(1+W(eak−bkθ+c¬1,k−c1θβ−1)) c_ 1,k-c_1β-θ (1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1) ) Due to the fact that yk⋆=0y_k =0 if ak−bka_k-b_k is sufficiently large (see (54)), it follows that yk⋆y_k is non-increasing with ak−bka_k-b_k, which concludes the proof. ∎ Proof of Proposition 5 Proposition. 5. The optimal incentive is upper-bounded, i.e., 0≤yk⋆<c¬1,k−c1β.0≤ y_k < c_ 1,k-c_1β. Proof. By Proposition 3, it suffices to show that c¬1,k−c1β−θ(1+W(eak−bkθ+c¬1,k−c1θβ−1)<c¬1,k−c1βCLOSE c_ 1,k-c_1β-θ(1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1)< c_ 1,k-c_1β (67) i.e., −θ(1+W(eak−bkθ+c¬1,k−c1θβ−1)<0CLOSE-θ(1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1)<0 (68) ⇔1+W(eak−bkθ+c¬1,k−c1θβ−1)>0 1+W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1)>0 (69) ⇔W(eak−bkθ+c¬1,k−c1θβ−1)>−1 W(e a_k-b_kθ+ c_ 1,k-c_1θβ-1)>-1 (70) This is true, since if the argument of the Lambert W function is positive, the Lambert W function is positive as well. ∎ Proof of Lemma 12 Lemma. GMrt,InstPE=0[U0(yr⋆)]−0[U0(y0⋆)], _r^t,InstPE=E_0[U_0(y_r )]-E_0[U_0(y_0 )], CMrt=0, _r^t=0, ∀t ∀ t Proof. The following equalities follow from a straightforward application of Obs. 11: GMrt,InstPE _r^t,InstPE =∑k∈ℙrInstPE[Yt=yk⋆]r[U0(yk⋆)] = _k P_r^InstPE[Y_t=y_k ]E_r[U_0(y_k )] (71) −∑k∈ℙ0InstPE[Yt=yk⋆]r[U0(yk⋆)] - _k P_0^InstPE[Y_t=y_k ]E_r[U_0(y_k )] (72) =r[U0(yr⋆)]−r[U0(y0⋆)] =E_r[U_0(y_r )]-E_r[U_0(y_0 )] (73) =0[U0(yr⋆)]−0[U0(y0⋆)] =E_0[U_0(y_r )]-E_0[U_0(y_0 )] (74) and CMrInstPE(y)=0[U0(y)]−0[U0(y)]=0⟹CMrt=0. _r^InstPE(y)=E_0[U_0(y)]-E_0[U_0(y)]=0 _r^t=0. ∎ Appendix C Preference Elicitation via a Maximum-Likelihood Estimator We now specify the deterministic profile-inference rule ψ(⋅)ψ(·) used in the numerical experiments. At every timeslot s, the agent chooses among all alternatives is∈ℐi^s according to the multinomial-logit model in Eq. (1). For online preference elicitation, however, the regulator uses only whether the incentivized alternative 11 was selected. Define the binary random variable and its realization as Zs Z^s :=Is=1, :=1\I^s=1\, zs z^s :=is=1. :=1\i^s=1\. (75) Thus, zs=1z^s=1 when the agent chooses alternative 11, whereas zs=0z^s=0 aggregates all choices is>1i^s>1. This binary observation model is plausible in applications where the regulator need not collect the agent’s complete choice. In particular, the data-minimization principle of the General Data Protection Regulation (GDPR) supports limiting the collected information to what is necessary for the specified purpose. Accordingly, an implementation may reveal only whether the incentivized alternative was selected—the information needed to determine eligibility for the incentive—rather than which non-incentivized alternative was chosen. Let t−1=(y1,i1,…,yt−1,it−1)h^t-1=(y^1,i^1,…,y^t-1,i^t-1) be the realized history before timeslot t. Although the complete alternatives isi^s appear in the history, the inference rule depends on them only through the binary observations zsz^s. Conditional on candidate profile k and incentive ysy^s, the binary observation follows Zs|s−1=s−1∼Ber(pk(ys)), Z^s ^s-1=h^s-1 \! (p_k(y^s) ), (76) where pk(ys)p_k(y^s) is the probability of choosing alternative 11 defined in Lemma 1. The likelihood of the binary observations induced by t−1h^t-1 under candidate profile k is therefore Lkt(t−1) L_k^t(h^t-1) :=∏s=1t−1[pk(ys)]zs[1−pk(ys)]1−zs := _s=1^t-1 [p_k(y^s) ]^z^s [1-p_k(y^s) ]^1-z^s =Lkt−1(t−2)[pk(yt−1)]zt−1[1−pk(yt−1)]1−zt−1, =L_k^t-1(h^t-2) [p_k(y^t-1) ]^z^t-1 [1-p_k(y^t-1) ]^1-z^t-1, (77) with Lk1=1L_k^1=1. The proposed incentives do not introduce an additional factor in the likelihood because each ysy^s is a deterministic function of the preceding observed history. To avoid numerical underflow, the implementation maintains the log-likelihood lnLkt(t−1) L_k^t(h^t-1) =∑s=1t−1[zslnpk(ys)+(1−zs)ln(1−pk(ys))] = _s=1^t-1 [z^s p_k(y^s)+(1-z^s) \! (1-p_k(y^s) ) ] =lnLkt−1(t−2)+zt−1lnpk(yt−1) = L_k^t-1(h^t-2)+z^t-1 p_k(y^t-1) +(1−zt−1)ln(1−pk(yt−1)), +(1-z^t-1) \! (1-p_k(y^t-1) ), (78) with lnLk1=0 L_k^1=0. Let the set of maximum-likelihood profiles be ℳt(t−1):=argmaxk∈lnLkt(t−1). ^t(h^t-1):= _k L_k^t(h^t-1). (79) The regulator selects, among the profiles in ℳt(t−1)M^t(h^t-1), the profile associated with the smallest optimal incentive. If several such profiles have the same optimal incentive, the smallest profile index is selected. Consequently, the deterministic inference rule used in the numerical experiments is k^t=ψ(t−1):=mink∈ℳt(t−1):yk⋆=minj∈ℳt(t−1)yj⋆. k^t=ψ(h^t-1):= \k ^t(h^t-1):y_k = _j ^t(h^t-1)y_j \. (80) This online Bernoulli likelihood must not be confused with the multinomial likelihood used for the offline calibration in Sect. D1. Accordingly, the Bernoulli distributions in Eq. (4.2) concern the binary variable ZsZ^s, rather than the complete categorical alternative IsI^s. Appendix D Appendix to the Case Study D1 Dataset and Calibration Procedure D1.1 Data The London Passenger Mode Choice (LPMC) dataset combines trip diaries from the London Travel Demand Survey with the routes, travel times, public-transport fares, driving fuel costs, and congestion charges associated with walking, bike, public transport (PT), and car.44 4 Tim Hillel, Mohammed Z. E. B. Elshafie, and Ying Jin. 2018. “Recreating Passenger Mode Choice-Sets for Transport Simulation: A Case Study of London, UK.” Proceedings of the Institution of Civil Engineers–Smart Infrastructure and Construction 171(1):29–42. See Sec. 2.1, p. 30–33. https://doi.org/10.1680/jsmic.17.00018. The public release contains 81,086 trips collected between April 2012 and March 2015.55 5 Tim Hillel. 2019. London Passenger Mode Choice. Transport and Mobility Laboratory, EPFL. See Table 1, p. 3. https://transp-or.epfl.ch/documents/technicalReports/CS˙LPMC.pdf. We retain travelers aged at least 12, since they are assumed to be responsive to monetary incentives, and trips between home and work, between home and educational activities, or between home and another activity. These are the home-based work, home-based education, and home-based other purposes in the LPMC documentation5. Only trips made by bike, PT, or car are retained. For each trip, the following information concerning the trip or the traveler is reported: the travel time associated with each mode; out-of-pocket costs for PT and car; the number of PT interchanges; possession of a driving licence; car ownership, measured as the number of cars in the household divided by the number of people in the household; trip distance; and the bus and rail portions of PT in-vehicle time. Total PT time is the sum of access, bus, rail, and interchange time. Trips missing a quantity required by the utility specification are discarded. Let calD_cal denote the calibration set and simD_sim the simulation set. Among the retained bike trips, 70% are assigned to calD_cal and 30% to simD_sim; the same split is performed independently among PT trips and among car trips. Hence, the proportions of bike, PT, and car trips are preserved in both sets. The resulting set calD_cal contains 34,198 trips made by 20,577 travelers, whereas simD_sim contains 14,656 trips made by 11,709 travelers. For a trip set S∈cal,simS∈\D_cal,D_sim\, let SN_S denote its travelers, d¯S d_S its mean trip distance, and ωbus,S _bus,S the share of bus time in total bus-plus-rail in-vehicle time. Table 3 separates quantities computed directly from the trip records from quantities estimated by the calibration procedure. Table 3: Quantities computed directly from the calibration and simulation sets. A dash indicates that the quantity is not used for that set: d¯S d_S and ωbus,S _bus,S define the representative simulation context and are therefore computed from simD_sim. Quantity Symbol calD_cal simD_sim Trips |S||S| 34,198 14,656 Travelers |S||N_S| 20,577 11,709 Mean trip distance (km/trip) d¯S d_S – 5.41 Bus-time share ωbus,S _bus,S – 0.65 D1.2 Profile-based choice model Let =0,1,2K=\0,1,2\ be the set of possible true profiles. After calibration, the profiles are labeled so that k=0k=0 is bike-oriented, k=1k=1 is PT-oriented, and k=2k=2 is car-oriented. For every k∈k , the systematic utility uk(i,y)u_k(i,y) introduced in the main paper is instantiated for trip τ as uk,τ(1,y) u_k,τ(1,y) =ξkbike−νtτbike+y, = _k^bike-ν t_τ^bike+y, (81) uk,τ(2,y) u_k,τ(2,y) =ξkPT−νtτPT−CτPT−ιkmτ, = _k^PT-ν t_τ^PT-C_τ^PT- _km_τ, uk,τ(3,y) u_k,τ(3,y) =−νtτcar−Cτcar+ζklicχτlic+ζkownχτown. =-ν t_τ^car-C_τ^car+ _k^lic _τ^lic+ _k^own _τ^own. Symbol Definition τ Index of a recorded trip. i∈1,2,3i∈\1,2,3\ Travel alternative: i=1i=1 denotes bike, i=2i=2 public transport (PT), and i=3i=3 car. tτit_τ^i Travel time of alternative i for trip τ, in hours. CτPTC_τ^PT Out-of-pocket PT cost for trip τ. CτcarC_τ^car Out-of-pocket car cost for trip τ. mτm_τ Number of PT interchanges for trip τ. χτlic _τ^lic Indicator equal to 11 if the traveler making trip τ holds a driving licence, and 00 otherwise. χτown _τ^own Car ownership of the traveler making trip τ, measured as the number of cars in the household divided by the number of people in the household. ν Common value of travel time, in pounds per hour. ιk _k Profile-k utility penalty for each additional PT interchange. ζklic _k^lic Profile-k increment in car utility associated with holding a driving licence. ζkown _k^own Profile-k increment in car utility per unit of χτown _τ^own. ξkbike _k^bike Intrinsic profile-k preference for bike not represented by the other trip or traveler characteristics. ξkPT _k^PT Intrinsic profile-k preference for PT not represented by the other trip or traveler characteristics. Table 4: Notation used in the trip-specific systematic utilities. The notation used in the trip-specific utility functions is summarized in Table 4. For any alternative i, trip τ, profile k, and incentive y, let ℙiuk,τ(y)P_i^u_k,τ(y) denote the probability in Eq. (1) evaluated with the trip-specific systematic utility function uk,τu_k,τ in Eq. (81). In particular, ℙiτuk,τ(0)P_i_τ^u_k,τ(0) is the probability that profile k chooses the mode iτi_τ recorded for trip τ when no bike incentive is offered. The argument 00 is used during calibration because the trips in the dataset were recorded without the simulated incentive policy. D1.3 Calibration procedure The offline calibration described in this subsection is distinct from the regulator’s online learning process in Appendix C. The offline calibration is used only to obtain behavioral parameters for the numerical experiment. Its log-likelihood ℓnkcal _nk^cal, defined below from the LPMC trips, must therefore not be confused with the regulator’s likelihood Lkt(t−1)L_k^t(h^t-1) in Eqs. (77)–(78). Let calN_cal be the travelers represented in calD_cal, let ncalT_n^cal be the set of calibration trips made by traveler n∈caln _cal, and let iτi_τ be the mode chosen on trip τ. For parameter vector Θ , the offline calibration log-likelihood of traveler n under candidate true profile k is ℓnkcal(Θ)=∑τ∈ncallogℙiτuk,τ(0). _nk^cal( )= _τ _n^cal _i_τ^u_k,τ(0). (82) The trip and traveler quantities in Eq. (81) are fixed inputs. The parameters to be calibrated are Θ=(ν,θ,ξkbike,ξkPT,ιk,ζklic,ζkownk∈), = (ν,θ, \ _k^bike, _k^PT, _k, _k^lic, _k^own \_k ), (83) where θ is the Gumbel scale in Eq. (1). The initial profile-membership probabilities are based on each traveler’s mode shares. Let μnk _nk be the fraction of trips in ncalT_n^cal made using the mode associated with profile k: bike for k=0k=0, PT for k=1k=1, and car for k=2k=2. Before enforcing equal total membership across profiles, the initial profile-membership probabilities are q~nk(0)=exp(ηinitμnk)∑j∈exp(ηinitμnj),ηinit=2. q_nk^(0)= ( _init _nk) _j ( _init _nj), _init=2. (84) The mode-share scaling parameter ηinit _init Controls how strongly observed mode shares influence the initial profile probabilities. A larger value gives more probability to profiles associated with frequently used modes; 0 would give equal probability to every profile.. We heuristically set ηinit=2 _init=2. At calibration step d≥1d≥ 1, the parameters Θ(d) ^(d) determine the pre-adjustment profile-membership probabilities q~nk(d)=exp(ℓnkcal(Θ(d)))∑j∈exp(ℓnjcal(Θ(d))). q_nk^(d)= \! ( _nk^cal( ^(d)) ) _j \! ( _nj^cal( ^(d)) ). (85) The superscript (0)(0) in Eq. (84) denotes initialization from empirical mode shares, whereas the superscript (d)(d) in Eq. (85) denotes calibration step d. The tilde identifies probabilities before imposing the profile-size condition in Eq. (87). For every d∈0,…,Nstepd∈\0,…,N_step\, adjustment parameters λk(d)∈ℝ _k^(d) are selected so that qnk(d) q_nk^(d) =q~nk(d)exp(λk(d))∑j∈q~nj(d)exp(λj(d)), = q_nk^(d) ( _k^(d)) _j q_nj^(d) ( _j^(d)), (86) ∑n∈calqnk(d) _n _calq_nk^(d) =|cal|||,k∈. = |N_cal||K|, k . (87) The quantity qnk(d)q_nk^(d) is the membership probability assigned to traveler n and profile k at calibration step d. Therefore, ∑n∈calqnk(d) _n _calq_nk^(d) is the fraction of agents associated to profile k within the calibration set, and Eq. (87) sets this quantity to one third of |cal||N_cal| for every k∈k . We do so to prevent calibration to favor one profile over the others.66 6 We also run calibration imposing that the fraction of trips (and not of agents) be 1/3 per each profile. Final result did not change significantly and are thus omitted. Observe that the adjustment parameters λk(d) _k^(d) shift the log-membership scores in Eq. (86). Adding the same constant to every λk(d) _k^(d) leaves the probabilities unchanged; hence, we set λ0(d)=0 _0^(d)=0 and solve the two equations in Eq. (87) corresponding to k=1k=1 and k=2k=2 for λ1(d) _1^(d) and λ2(d) _2^(d). For fixed profile-membership probabilities qnk(d−1)q_nk^(d-1), we find the parameters Θ that minimize (Θ,q(d−1)) ( ;q^(d-1)) =NLL(Θ,q(d−1))+ρℛ(Θ), =J_NLL( ;q^(d-1))+ρ\,R( ), (88) NLL(Θ,q(d−1)) _NLL( ;q^(d-1)) =−1|cal|∑n∈cal∑τ∈ncal =- 1|D_cal| _n _cal _τ _n^cal ∑k∈qnk(d−1)logℙiτuk,τ(0). 22.76219pt _k q_nk^(d-1) _i_τ^u_k,τ(0). (89) Here, |cal||D_cal| is the number of calibration trips and NLLJ_NLL in Eq. (89) is the average negative log-likelihood of the modes chosen in each recorded trip, weighted by the profile-membership probabilities qnk(d−1)q_nk^(d-1). The second term in Eq. (88) is the regularization term, with ℛ(Θ)=∑k∈[ ( )= _k [ (ξkbike−ξ¯bike)2+(ξkPT−ξ¯PT)2 ( _k^bike- ξ^bike)^2+( _k^PT- ξ^PT)^2 +(ιk−ι¯)2+(ζklic−ζ¯lic)2 +( _k- )^2+( _k^lic- ζ^lic)^2 +(ζkown−ζ¯own)2], +( _k^own- ζ^own)^2 ], (90) where each barred quantity is the mean of the corresponding coefficient over k∈k . Because both the traveler memberships qnkq_nk and the profile-specific coefficients in Eq. (83) are estimated, markedly different coefficient vectors can produce similar values of the weighted likelihood in Eq. (89). The regularization term in Eq. (90) favors the less extreme of such parameterizations by penalizing cross-profile dispersion. The coefficient ρ=0.01ρ=0.01 gives this term a small weight relative to the average negative log-likelihood. The common value of time ν and common Gumbel scale θ are calibrated but not regularized because each has a single value shared by all profiles; neither contributes to cross-profile dispersion. The calibration has Nstep=4N_step=4 steps. At calibration step d, we first initialize Θ at Θ(d−1) ^(d-1); for d=1d=1, the initial values Θ(0) ^(0) in Table 6 are used. We then find Θ(d) ^(d) that minimizes Eq. (88), holding qnk(d−1)q_nk^(d-1) fixed and enforcing the bounds in Table 6. To do so, we use the analytic gradient of loss (88) with respect to parameters Θ and the limited-memory Broyden--Fletcher--Goldfarb--Shanno algorithm for bound-constrained optimization (L-BFGS-B).77 7 Richard H. Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu. 1995. “A Limited Memory Algorithm for Bound Constrained Optimization.” SIAM Journal on Scientific Computing 16(5):1190–1208. See Sec. 2, p. 1193–1198. https://doi.org/10.1137/0916069. The resulting Θ(d) ^(d) is substituted in Eqs. (82) and (85); the adjustment in Eqs. (86)–(87) then gives qnk(d)q_nk^(d). Profile 0 is required to satisfy a0−b0≥£0.15.a_0-b_0≥ 0.15. (91) This condition represents a profile for which the zero-incentive bike utility is slightly larger than the inclusive utility of the non-bike alternatives in the representative travel context. The value £0.15 is a small separation margin rather than an empirical estimate. At the calibrated scale θ^=£3.68 θ= 3.68, the boundary in Eq. (91) implies, by Eq. (4), p0(0)≥σ(0.153.68)=51.02%.p_0(0)≥σ\! ( 0.153.68 )=51.02\%. (92) Thus, the condition requires only a 1.02-percentage-point preference above indifference between bike and the aggregate non-bike option. It identifies a strictly bike-oriented profile without imposing a large behavioral separation. The calibrated value is a^0−b^0=£0.36 a_0- b_0= 0.36, which is above the boundary in Eq. (91). The quantities a0a_0 and b0b_0 are defined as in Eq. (4), using the profile-weighted trip attributes at the current calibration step. After estimating Θ(d) ^(d) and again after obtaining qnk(d)q_nk^(d), the bike constant is updated according to ξ0bike←ξ0bike+max0,£0.15−(a0−b0), _0^bike← _0^bike+ \0, 0.15-(a_0-b_0)\, (93) which restores Eq. (91) whenever that condition is violated. Table 5: Settings of the offline calibration procedure. Quantity Symbol Value Number of profiles |||K| 3 Mode-share scaling in Eq. (84) ηinit _init 2 Calibration steps NstepN_step 4 Maximum L-BFGS-B iterations per step NoptN_opt 80 Function tolerance εfun _fun 10−1010^-10 Gradient tolerance εgrad _grad 10−510^-5 Maximum line-search steps NlineN_line 30 Regularization coefficient ρ 0.01 Profile-0 margin in Eq. (91) a0−b0a_0-b_0 at least £0.15 Table 6: Initialization, admissible ranges, and calibrated values. Profile-indexed values are ordered as k=0,1,2k=0,1,2. The calibrated profile-specific coefficients enter the simulations through the constructed utilities aka_k and bkb_k reported in Table 7. Parameter Θ(0) ^(0) Range Calibrated ξkbike _k^bike: intrinsic bike preference (£) (5,1,−1)(5,1,-1) [−15,15][-15,15] through aka_k ξkPT _k^PT: intrinsic PT preference (£) (0,2,−1)(0,2,-1) [−15,15][-15,15] through bkb_k ν: value of time (£/h) 10 [3,30][3,30] 10.48 ιk _k: PT-interchange penalty (£/interchange) (1,1,1)(1,1,1) [0,8][0,8] through bkb_k ζklic _k^lic: driving-licence utility increment (£) (2,2,4)(2,2,4) [0,10][0,10] through bkb_k ζkown _k^own: car-ownership utility increment (£/unit) (2,2,4)(2,2,4) [0,10][0,10] through bkb_k θ: Gumbel scale (£) 3.50 [1,12][1,12] 3.68 aka_k: constructed bike utility (£) – – (6.12,−6.78,−6.59)(6.12,-6.78,-6.59) bkb_k: constructed non-bike utility (£) – – (5.76,4.94,5.35)(5.76,4.94,5.35) ak−bka_k-b_k: constructed utility difference (£) – Eq. (91) (0.36,−11.73,−11.93)(0.36,-11.73,-11.93) The mixture negative log-likelihood used to evaluate descriptive fit is mix(Θ)=−1|cal|∑n∈callog[∑k∈1||exp(ℓnkcal(Θ))].J_mix( )=- 1|D_cal| _n _cal \! [ _k 1|K| \! ( _nk^cal( ) ) ]. (94) A nat is the unit of log loss obtained when the natural logarithm is used. The fitted value of mixJ_mix in Eq. (94) is 0.67 nat/trip. Uniform choice gives 1.10 nat/trip, and assigning each mode its aggregate frequency in calD_cal gives 0.83 nat/trip. For a model M with average negative log-likelihood MJ_M, define the geometric mean probability assigned to the modes chosen in the recorded trips as p¯geomM=exp(−M). p_geom^M= (-J_M). (95) Using the unrounded losses gives (p¯geomfit,p¯geomshares,p¯geomuniform)=(51.12%,43.74%,33.33%). ( p_geom^fit, p_geom^shares, p_geom^uniform )=(51.12\%,43.74\%,33.33\%). (96) The fitted profile model therefore assigns 7.38 percentage points more geometric-mean probability to the realized choices than the aggregate-share baseline and 17.79 percentage points more than uniform choice. Hence, the profile-specific utilities capture systematic variation in mode choice beyond the aggregate frequencies of the three modes. D1.4 Calibrated parameter values, profile construction and context parameters Let simN_sim be the travelers represented in simD_sim, and let nsimT_n^sim be the trips made by traveler n∈simn _sim in that set. The calibrated parameters Θ give ℓnksim(Θ^)=∑τ∈nsimlogℙiτuk,τ(0). _nk^sim( )= _τ _n^sim _i_τ^u_k,τ(0). (97) The simulation-set membership probabilities qnksimq_nk^sim are obtained from Eq. (85) by replacing ℓnkcal(Θ(d)) _nk^cal( ^(d)) with ℓnksim(Θ^) _nk^sim( ). The profile-size adjustment in Eqs. (86)–(87) is not applied to simD_sim. The share of profile k in the simulation set simD_sim is πk=1|sim|∑n∈simqnksim. _k= 1|N_sim| _n _simq_nk^sim. (98) For each trip τ∈simτ _sim and alternative i, define τi=(CLOSE _τ i= ( i=1,i=2,tτi,Cτi,mτi=2, _\i=1\,I_\i=2\,t_τ^i,C_τ^i,m_τI_\i=2\, (99) χτlici=3,χτowni=3)⊤, _τ^licI_\i=3\, _τ^ownI_\i=3\ )^\! , where Cτ1=0C_τ^1=0, Cτ2=CτPTC_τ^2=C_τ^PT, and Cτ3=CτcarC_τ^3=C_τ^car. Thus, τix_τ i contains the alternative indicators, alternative-specific travel time, out-of-pocket cost, number of PT interchanges, driving-licence indicator, and household car ownership entering Eq. (81). Let n(τ)n(τ) denote the traveler who made trip τ. For every profile–alternative pair (k,i)(k,i), the representative attribute vector is i,k=∑τ∈simqn(τ)ksimτi∑τ∈simqn(τ)ksim.x_i,k= _τ _simq_n(τ)k^simx_τ i _τ _simq_n(τ)k^sim. (100) The weight qn(τ)ksimq_n(τ)k^sim gives greater influence to trips made by travelers whose profile-membership probability qn(τ)ksimq_n(τ)k^sim is higher. This ensures that the representative attribute vector i,kx_i,k is more strongly affected by profile memberships inferred with higher confidence. Substituting i,kx_i,k and Θ into Eq. (81) determines the profile-level systematic utilities uk(i,0)u_k(i,0). Profiles are relabeled so that a0−b0≥a1−b1≥a2−b2a_0-b_0≥ a_1-b_1≥ a_2-b_2, consistently with the ordering in the main paper. Table 7 reports the common calibrated parameters ν ν and θ θ, the profile shares πk _k, and the profile-level quantities obtained from Eqs. (98)–(100). The zero-incentive mode probabilities ℙiuk(0)P_i^u_k(0), i∈1,2,3i∈\1,2,3\, are evaluated using Eq. (1) with the utilities uk(i,0)u_k(i,0). Table 7: Calibrated parameter values and profile-level context quantities. Common quantity Value Unit Value of time ν ν 10.48 £/h Gumbel scale θ θ 3.68 £ k πk _k ℙ1uk(0)P_1^u_k(0) ℙ2uk(0)P_2^u_k(0) ℙ3uk(0)P_3^u_k(0) aka_k bkb_k (%) (%) (%) (%) (£) (£) 0 21.26 52.44 19.12 28.44 6.12 5.76 1 39.36 3.96 52.08 43.96 -6.78 4.94 2 39.38 3.75 43.10 53.15 -6.59 5.35 Figure 3: Zero-incentive mode probabilities ℙiuk(0)P_i^u_k(0) for the three calibrated profiles k∈k . Alternatives i=1,2,3i=1,2,3 denote bike, PT, and car, respectively. For clarity, before rounding, the representative profile-level quantities used throughout the case study are akk=02=(6.1178,−6.7834,−6.5886),\a_k\_k=0^2=(6.1178,-6.7834,-6.5886), bkk=02=(5.7590,4.9421,5.3451),\b_k\_k=0^2=(5.7590,4.9421,5.3451), and therefore ak−bkk=02=(0.3588,−11.7255,−11.9337).\a_k-b_k\_k=0^2=(0.3588,-11.7255,-11.9337). D1.5 Externalities and calibration of the policy weight β Table 8 reports the externally specified quantities. Greenhouse-gas (GHG) factors are expressed as carbon-dioxide equivalent (CO2e), because the official conversion factors aggregate the climate effects of the covered greenhouse gases into an equivalent mass of CO2.88 8 UK Department for Energy Security and Net Zero. 2026. Greenhouse Gas Reporting: Conversion Factors 2026. See methodology Secs. 1 and 5 and the flat-file worksheet “Factors by Category.” https://w.gov.uk/government/publications/greenhouse-gas-reporting-conversion-factors-2026. Bike is assigned zero direct operational emissions. The monetary carbon value VCO2eV_ CO_2e follows UK government appraisal guidance.99 9 UK Department for Energy Security and Net Zero and UK Department for Business, Energy and Industrial Strategy. 2021. Valuation of Greenhouse Gas Emissions: For Policy Appraisal and Evaluation. See Annex 1. https://w.gov.uk/government/publications/valuing-greenhouse-gas-emissions-in-policy-appraisal/valuation-of-greenhouse-gas-emissions-for-policy-appraisal-and-evaluation. The road externalities CcongC_ cong and CotherC_ other are taken from the Institute for Fiscal Studies assessment of motoring external costs.1010 10 Stuart Adam and Rebekah Stroud. 2019. “A Road Map for Motoring Taxation.” In The IFS Green Budget: October 2019. Institute for Fiscal Studies, London. See Sec. 9.2 and Tables 9.1–9.2. https://ifs.org.uk/books/road-map-motoring-taxation. The cycling payment rate RCicR_ Cic is taken from a field experiment.1111 11 Alice Ciccone, Aslak Fyhri, and Hanne B. Sundfør. 2021. “Using Behavioral Insights to Incentivize Cycling: Results from a Field Experiment.” Journal of Economic Behavior & Organization 188:1035–1058. See Sec. 3.1, p. 1038–1039. https://doi.org/10.1016/j.jebo.2021.06.011. The exchange rate XNOK/GBPX_ NOK/GBP is the 2021 annual average reported by DNB Markets.1212 12 DNB Markets. 2021. Monthly and Yearly Averages for Major Currencies 2021. Annual-average GBP/NOK mid-rate; the table states that the reported values average daily mid-rates. https://w.dnb.no/en/markets/fx-and-ir/fx-rates/monthly-yearly-averages/average-rates-2021?la=EN. Table 8: Externally specified inputs. The reference column identifies the precise source location. Input Symbol Value Reference London local-bus emissions EbusE_ bus 63.60 gCO2e/passenger-km Category “Local bus, London” and methodology in [Footnote 8 - Sec. 5, p. 74–76.] National-rail emissions ErailE_ rail 30.92 gCO2e/passenger-km Category “National rail” and methodology in [Footnote 8 - Sec. 5, p. 78.] Average-car emissions EcarE_ car 165.91 gCO2e/vehicle-km Category “Average car” and methodology in [Footnote 8 - Sec. 5, p. 46–69]. Central carbon value VCO2eV_ CO_2e £264/tCO2e Central 2026 value in [Footnote 9 - Annex 1] Congestion externality CcongC_ cong £0.81/vehicle-km Marginal congestion cost of one additional vehicle-kilometre on principal roads in London, estimated in [Footnote 10 Sec. 9.2, Table 9.2]. Other car externalities CotherC_ other £0.03/vehicle-km Sum of accident, local-air-pollution, noise, and infrastructure costs, estimated in [Footnote 10 Sec. 9.2, Table 9.2]. Reference biking incentive RCicR_ Cic 2.00 NOK/cycled-km Pay-per-kilometre treatment, tested in a real world experiment - see [Footnote 11 Sec. 3.1, p. 1038–1039]. We use this value only to set the value of β, as discussed in Eqs. (106)-(109). We do not use it directly as the incentives proposed, which we instead calculate via Prop. 3. NOK/GBP exchange rate XNOK/GBPX_ NOK/GBP 11.83 NOK/£ 2021 annual-average GBP rate [Footnote 12]. Using the direct simulation-set quantities d¯sim d_D_ sim and ωbus,sim _bus,D_ sim in Table 3, the operational emissions of a representative PT trip and a representative car trip are EPTtrip E_ PT trip =d¯sim[ωbus,simEbus+(1−ωbus,sim)Erail], = d_D_ sim [ _bus,D_ simE_ bus+(1- _bus,D_ sim)E_ rail ], (101) Ecartrip E_ car trip =d¯simEcar. = d_D_ simE_ car. (102) Thus, emissions from car trips EcartripE_ car trip are obtained by multiplying the official average-car vehicle-kilometre factor EcarE_ car by the average kilometres traveled d¯sim d_D_ sim computed from the simulation set simD_ sim. PT emissions EPTtripE_ PT trip use the same distance and a bus/rail factor weighted by the bus-time share ωbus,sim _bus,D_ sim. The per-trip externalities c1c_1, cPTc_ PT, and ccarc_ car are c1 c_1 =0, =0, cPT c_ PT =EPTtripVCO2e106, = E_ PT tripV_ CO_2e10^6, (103) ccar c_ car =EcartripVCO2e106+d¯sim(Ccong+Cother). = E_ car tripV_ CO_2e10^6+ d_D_ sim(C_ cong+C_ other). (104) Here, VCO2eV_ CO_2e is the monetary value assigned to one tonne of CO2e, and 10610^6 converts grams to tonnes. The car externality ccarc_ car adds the carbon cost, the marginal congestion cost CcongC_ cong of an additional vehicle-kilometre on principal London roads, and the remaining road costs CotherC_ other. For profile k, the expected externality conditional on not choosing bike, c¬1,kc_ 1,k, is c¬1,k=ℙk(2∣i≠1)cPT+ℙk(3∣i≠1)ccar.c_ 1,k=P_k(2 i≠ 1)c_ PT+P_k(3 i≠ 1)c_ car. (105) The generalized social-cost objective f~v~(y) f v(y) in Eq. (7) weights expected incentive expenditure by β. To set the policy weight β, we anchor the optimal incentive of the car-oriented profile k=2k=2 to an incentive implemented in a cycling field experiment. Ciccone, Fyhri, and Sundfør paid participants RCic=2.00R_ Cic=2.00 Norwegian kroner (NOK) for each kilometre cycled in their pay-per-kilometre treatment11. Applying this rate to the mean trip distance d¯sim=5.41 d_D_ sim=5.41 km/trip gives yCicNOK=RCicd¯sim=10.81NOK/biketrip.y_ Cic NOK=R_ Cic d_D_ sim=10.81\ NOK/bike\ trip. (106) The annual-average exchange rate XNOK/GBP=11.83X_ NOK/GBP=11.83 NOK/£converts this amount to yCic=yCicNOKXNOK/GBP=£0.91perbiketrip.y_ Cic= y_ Cic NOKX_ NOK/GBP= 0.91\ per\ bike\ trip. (107) Let p2(y):=ℙ1u2(y)p_2(y):=P_1^u_2(y) and Δc2:=c¬1,2−c1 c_2:=c_ 1,2-c_1. For profile k=2k=2, differentiating the objective f~u2(y) f^u_2(y) in Eq. (7), and using p2′(y)=p2(y)[1−p2(y)]/θp_2 (y)=p_2(y)[1-p_2(y)]/θ from Eq. (4), gives 0 0 =p2′(y)(c1−c¬1,2+βy)+βp2(y) =p_2 (y) (c_1-c_ 1,2+β y )+β p_2(y) =p2(y)θ−[1−p2(y)]Δc2+β[θ+(1−p2(y))y]. = p_2(y)θ \- [1-p_2(y) ] c_2+β [θ+ (1-p_2(y) )y ] \. (108) We impose y2⋆=yCicy_2 =y_ Cic in Eq. (108); solving for β yields β=[1−p2(yCic)](c¬1,2−c1)θ+[1−p2(yCic)]yCic.β= [1-p_2(y_ Cic)](c_ 1,2-c_1)θ+[1-p_2(y_ Cic)]y_ Cic. (109) The values issued from the calibration are the common scale θ^=£3.68 θ= 3.68 and the profile-2 parameters entering u2(i,y)u_2(i,y) in Eq. (81). The payment yCicy_ Cic is instead constructed in Eqs. (106)–(107). Substituting Θ , i,2x_i,2, and yCicy_ Cic into Eq. (81), and then evaluating Eq. (1), gives p2(yCic)=4.76%p_2(y_ Cic)=4.76\%. Thus, p2(yCic)p_2(y_ Cic) is a post-calibration model evaluation, not a calibrated parameter. The context construction in Eqs. (103)– (105) gives c¬1,2=£2.67c_ 1,2= 2.67 and c1=0c_1=0. Equation (109) therefore gives β=0.56β=0.56. D1.6 Optimal incentives and induced mode choices After calibration, the profile-specific behavioral parameters ξkbike,ξkPT,ιk,ζklic,ζkownk∈\ _k^bike, _k^PT, _k, _k^lic, _k^own\_k , the common parameters ν ν and θ θ, and the representative attributes i,kx_i,k determine the systematic utilities uk(i,0)u_k(i,0) via Eq. (81). The externalities c1c_1 and c¬1,kc_ 1,k and the policy weight β then determine the optimal incentive yk⋆y_k through Prop. 3. For each profile k∈k and mode i∈1,2,3i∈\1,2,3\, define Pik0:=ℙiuk(0),Pik⋆:=ℙiuk(yk⋆).P_ik^0:=P_i^u_k(0), P_ik :=P_i^u_k(y_k ). (110) The probabilities Pik⋆P_ik represent the choices generated by the calibrated profile k if the regulator offers yk⋆y_k . Table 9 reports Pik0P_ik^0, Pik⋆P_ik , and the resulting change in bike probability ΔP1k:=P1k⋆−P1k0 P_1k:=P_1k -P_1k^0. Table 9: Profile-specific optimal incentives and induced mode choices. For i∈1,2,3i∈\1,2,3\, denoting bike, PT, and car, Pik0=ℙiuk(0)P_ik^0=P_i^u_k(0) and Pik⋆=ℙiuk(yk⋆)P_ik =P_i^u_k(y_k ). The change ΔP1k=P1k⋆−P1k0 P_1k=P_1k -P_1k^0 is reported in percentage points. k πk _k P1k0P_1k^0 P2k0P_2k^0 P3k0P_3k^0 ak−bka_k-b_k c¬1,kc_ 1,k yk⋆y_k P1k⋆P_1k ΔP1k P_1k (%) (%) (%) (%) (£) (£/trip) (£/bike trip) (%) (p) 0 21.26 52.44 19.12 28.44 0.36 2.88 0 52.44 0 1 39.36 3.96 52.08 43.96 -11.73 2.22 0.15 4.11 0.15 2 39.38 3.75 43.10 53.15 -11.93 2.67 0.91 4.76 1.01 D1.7 Monte Carlo simulation The experiment consists of NMC=100N_MC=100 independent Monte Carlo replications, each with horizon T=18T=18 timeslots. We select T=18T=18 to examine the initial learning phase, in which the regulator has fewer than twenty choices from which to infer the adopted profile. For replication m∈1,…,NMCm∈\1,…,N_MC\ and timeslot t∈1,…,Tt∈\1,…,T\, a common random number Um,t∼Unif(0,1)U_m,t (0,1) is generated. The honest and manipulative mode choices are obtained by applying the inverse cumulative distribution of their respective choice probabilities ℙiuk(yt)P_i^u_k(y^t) in Eq. (1) to the same Um,tU_m,t. Pairing the same Um,tU_m,t removes variation caused solely by using different random numbers when the two trajectories are compared. Table 10: Settings of the Monte Carlo experiment. Quantity Symbol Value Simulation horizon T 18 timeslots Monte Carlo replications NMCN_MC 100 Confidence level pCIp_CI 0.95 Common random numbers Um,tU_m,t paired trajectories Under InstPE, declaring profile r immediately determines the offered incentive yr⋆y_r , while choices continue to follow the true profile-00 distribution. Under RealPE, an agent adopting profile r chooses among bike, public transport, and car according to the multinomial-logit probabilities of profile r. Recall that, for online profile inference, the regulator uses only the binary observation zs=is=1,z^s=1\i^s=1\, so that a bike choice is recorded as zs=1z^s=1, while both public transport and car choices are recorded as zs=0z^s=0. Before timeslot t, the regulator evaluates the Bernoulli log-likelihoods lnLkt(t−1) L_k^t(h^t-1) defined in Eq. (78) and selects the inferred profile k^t k^t according to Eq. (80). If several profiles attain the same maximum log-likelihood, the profile with the smallest optimal incentive yk⋆y_k is selected; if a tie remains, the smallest profile index is selected. Let ΔUr,ms,PE U_r,m^s,PE denote the realized manipulation advantage in timeslot s of replication m under PE∈InstPE,RealPEPE∈\InstPE,RealPE\. The estimated average manipulation advantage through timeslot t is ΔU¯^r,tPE=1NMC∑m=1NMC1t∑s=1tΔUr,ms,PE. U_r,t^PE= 1N_MC _m=1^N_MC 1t _s=1^t U_r,m^s,PE. (111) The 95% confidence interval for ΔU¯^r,tPE U_r,t^PE is computed from the NMCN_MC replication-specific time averages using a Student-t critical value. Figure 4 shows ΔU¯^r,tPE U_r,t^PE over the complete horizon. Figure 4: Estimated time-averaged manipulation advantage ΔU¯^r,tPE U_r,t^PE from Eq. (111). Vertical intervals are 95% Student-t confidence intervals across the Monte Carlo replications. The time-averaged gain from manipulation and cost of manipulation are obtained by replacing ΔUr,ms,PE U_r,m^s,PE in Eq. (111) with GMr,ms,PEGM_r,m^s,PE and CMr,ms,PECM_r,m^s,PE, respectively. Their difference equals the manipulation advantage by Eq. (16). Figure 2(a) and Table 11 report the resulting decomposition at T=18T=18. Table 11: Manipulation results at T=18T=18. The estimate ΔU¯^r,TPE U_r,T^PE and its 95% interval are in pounds per timeslot; cumulative quantities are in pounds. PE r ΔU¯^r,TPE U_r,T^PE 95% interval InstPE 1 0.08 [0.08,0.09][0.08,0.09] InstPE 2 0.55 [0.53,0.57][0.53,0.57] RealPE 1 -0.16 [−0.18,−0.15][-0.18,-0.15] RealPE 2 -0.16 [−0.17,−0.15][-0.17,-0.15] PE r ∑s=1TGMrs _s=1^TGM_r^s ∑s=1TCMrs _s=1^TCM_r^s ∑s=1TΔUrs _s=1^T U_r^s InstPE 1 1.47 0 1.47 InstPE 2 10.04 0 10.04 RealPE 1 0.36 3.34 -2.99 RealPE 2 0.49 3.36 -2.87 For each preference-elicitation model PEPE and simulated profile r, define the analytic long-run average manipulation advantage and its finite- horizon discrepancy as ΔU¯r,∞PE U_r,∞^PE :=limt→∞1t∑s=1tΔUrs,PE, := _t→∞ 1t _s=1^t U_r^s,PE, (112) Γr,TPE _r,T^PE :=|ΔU¯^r,TPE−ΔU¯r,∞PE|. := | U_r,T^PE- U_r,∞^PE |. (113) For InstPE with r=1,2r=1,2, the values of ΔU¯r,∞InstPE U_r,∞^InstPE are 0.08 and 0.56 £/timeslot; for RealPE with r=1,2r=1,2, the values of ΔU¯r,∞RealPE U_r,∞^RealPE are −0.17-0.17 and −0.13-0.13 £/timeslot. The corresponding discrepancies Γr,TPE _r,T^PE at T=18T=18 are 0.00, 0.00, 0.00, and 0.03 £/timeslot. Thus, the finite-horizon estimates have the same sign as all four analytic long-run averages; the largest remaining finite-horizon discrepancy is the RealPE case with r=2r=2. D2 Social-Impact Relevance The proposed framework is intended as a pre-deployment stress test for personalized incentive policies. Before implementing such a policy, a regulator can calibrate behavioral profiles from observed choices, compute the incentives associated with these profiles, and evaluate whether the additional incentive that an agent may obtain through manipulation compensates for the resulting Cost of Manipulation. The theoretical bounds provide an analytical robustness certificate, while the simulation procedure evaluates the same mechanism under calibrated behavioral heterogeneity and stochastic choices. The London case study provides empirical grounding for this evaluation. The choice model and behavioral profiles are calibrated from observed travel-mode choices, after which the interaction between preference elicitation, personalized incentives, and strategic behavior is simulated under controlled and reproducible conditions. This design isolates the mechanism studied in the paper: whether repeatedly imitating another behavioral profile remains advantageous once the regulator learns from the agent’s choices. The results show that the conclusion obtained under instantaneous preference elicitation can be reversed when this learning process is explicitly represented. In the calibrated instance, realistic preference elicitation makes the Cost of Manipulation exceed the Gain from Manipulation while the incentive policy continues to increase the share of sustainable choices. This use of calibrated simulation is appropriate for policy screening: it allows candidate policies to be rejected or revised before exposing individuals or public budgets to an insufficiently robust intervention. It also makes the assumptions governing the policy explicit and auditable. In particular, the regulator applies a deterministic and pre-announced mapping from inferred profiles to incentives, and the preference-elicitation procedure can operate on the binary information required to determine whether the incentivized alternative was selected, rather than requiring the complete observation of every non-incentivized choice. The analysis therefore supports a concrete deployment workflow: calibrate the behavioral model for the population of interest, evaluate manipulation incentives under realistic preference elicitation, and deploy the policy only when the resulting manipulation advantage remains non-positive over the relevant interaction horizon. When this condition is not satisfied, the same framework identifies the profiles and time horizons for which the policy should be redesigned. The contribution is thus not tied to a single transport intervention: it provides a reproducible procedure for assessing the strategic robustness of personalized sustainability policies before implementation.