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Privacy and Personalization in a Dynamic Model

  • Yufei Chen ORCID logo EMAIL logo
Veröffentlicht/Copyright: 22. September 2025
The B.E. Journal of Theoretical Economics
Aus der Zeitschrift The B.E. Journal of Theoretical Economics

Abstract

This paper studies a two-period model with a monopolist and a consumer. The firm makes two offers (product-price pairs) in the first period and makes one offer in the second period. The single consumer’s type is unknown to the firm, and she chooses at most one offer each period. Products are horizontally differentiated and the consumer prefers products closer to her type. Two privacy settings are considered. In one setting, the consumer cannot hide her purchase history (i.e., cannot opt out) whereas she can in the other setting. We characterize the firm-optimal equilibria in both settings and show that when the opt-out choice is added, the ex-ante producer surplus and social surplus increase while the ex-ante consumer surplus decreases. How the interim consumer surplus changes depends on her type. Our results suggest that, in some cases, privacy protection tools may inadvertently harm consumers while benefiting firms. Therefore, regulators should account for the dynamic and strategic interactions between these two sides when evaluating the implications of such tools.

Keywords: price discrimination; privacy; personalization
JEL Classification: D42; D82; L11; L12
Corresponding author: Yufei Chen, School of Economics, Nanjing University of Finance and Economics, Nanjing, China, E-mail: 

Acknowledgements

This paper is adapted from the third chapter of my Ph.D. dissertation at the University of Pittsburgh. I am indebted to my advisor Richard Van Weelden for his guidance, support, and encouragement. I also owe Luca Rigotti, Daniele Coen-Pirani, and Alexey Kushnir for their advice and support. I thank the seminar participants at the University of Pittsburgh for their suggestions. I would also like to thank the anonymous referee and the editor for their comments, which significantly improved the article.

  1. Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

A Results from Hidir and Vellodi (2021)

Lemma 1.

(Optimal single offer, Hidir and Vellodi (2021)) When consumer type θ follows a uniform distribution on [ θ ̲ , θ ̄ ] and the firm and the consumer only interact for one period, there exists an optimal offer ( θ ̲ + θ ̄ 2 , p * ) where

  1. If u ̄ ≥ 3 a 4 ( θ ̄ − θ ̲ ) 2 , all consumer types accept the offer, and p * = u ̄ − a 4 ( θ ̄ − θ ̲ ) 2 . The firm’s expected revenue is u ̄ ( θ ̄ − θ ̲ ) − a 4 ( θ ̄ − θ ̲ ) 3 ;

  2. If u ̄ < 3 a 4 ( θ ̄ − θ ̲ ) 2 , not all consumer types accept the offer, and p * = 2 3 u ̄ . The firm’s expected revenue is 4 u ̄ 3 u ̄ 3 a .

Lemma 2.

(Optimal pair, Hidir and Vellodi (2021)) Suppose the firm and the consumer interact for one period,

  1. If u ̄ ≥ 3 a 16 , there exists a unique optimal pair: ( 1 4 , u ̄ − a 16 ) and ( 3 4 , u ̄ − a 16 ) ;

  2. If u ̄ < 3 a 16 , any pair ( v l , 2 3 u ̄ ) and ( v r , 2 3 u ̄ ) (v l  < v r ) is optimal for the firm if it satisfies the following conditions

    v l ≥ w * 2 ,
    1 − v r ≥ w * 2 ,
    v r − v l ≥ w * ,

    where w * = 4 u ̄ 3 a .

The conditions in the second part are that the intervals covered don’t overlap and they don’t extend beyond the boundaries of the market (0 and 1).

Lemma 3.

(Optimal triple, Hidir and Vellodi (2021)) Suppose the firm and the consumer interact for one period,

  1. If u ̄ ≥ a 12 , there exists a unique optimal triple: ( 1 6 , u ̄ − a 36 ) , ( 1 2 , u ̄ − a 36 ) , and ( 5 6 , u ̄ − a 36 ) ;

  2. If u ̄ < a 12 , any triple ( v l , 2 3 u ̄ ) , ( v m , 2 3 u ̄ ) , and ( v r , 2 3 u ̄ ) (v l  < v m  < v r ) is optimal for the firm if it satisfies the following conditions

    v l ≥ w * 2 ,
    1 − v r ≥ w * 2 ,
    v m − v l ≥ w * ,
    v r − v m ≥ w * ,

    where w * = 4 u ̄ 3 a .

The conditions in the second part are that the intervals covered don’t overlap and they don’t extend beyond the boundaries of the market (0 and 1).

B Omitted Proofs

Proof of Proposition 1

Proof.

Given the consumer’s strategy, it is clear that the firm has no profitable deviation as it’s already providing the optimal pair. Given the firm’s strategy, if the consumer deviates in the first period, the highest surplus she can get in period 2 is zero. She also has no incentive to deviate in t2 as it’s the last period.

Proof of Proposition 2

Proof.

It’s straightforward to see that in any equilibria where all types buy some product in t1, the equilibrium in Proposition 1 indeed gives the firm the highest ex-ante surplus. The reason is as follows. In all such equilibria, consumer types are pooled into two groups in t2: those who bought v l and those who bought v r . In this case, the best the firm can do is to provide the optimal pair in both periods.

On the contrary, in equilibria where some types don’t buy any product in t1, consumer types are pooled into three segments in t2: those who bought v l , those who bought v r , and those who didn’t buy anything. Recall that Θ(v, p) is the set of types served by (v, p). We denote Θ(v l , p l ) ≡ [A, B] and Θ(v r , p r ) ≡ [C, D] for some 0 ≤ A < B < C < D ≤ 1. Notice that if this equilibrium is firm-optimal, then A = 0 and D = 1. This is because, if A > 0 or D < 1, then those types who didn’t buy anything does not constitute an interval, and thus the expected revenue from those types can be increased by shifting [A, B] towards 0 or shifting [C, D] towards 1.

Formally, consider the following kind of equilibrium. In t1, the firm provides (v l , p l ) and (v r , p r ) with Θ(v l , p l ) = [0, B] and Θ(v r , p r ) = [C, 1]. B and C can be expressed in terms of v i (i ∈ {l, r}):

B = 2 v l , C = 2 v r − 1 .

Prices can also expressed in terms of v i (i ∈ {l, r}):

u ̄ − a ( 0 − v l ) 2 − p l = 0 ,
u ̄ − a ( 1 − v r ) 2 − p r = 0 .

Hence p l = u ̄ − a v l 2 and p r = u ̄ − a ( 1 − v r ) 2 . To maximize its expected revenue among all equilibria of this kind, in t2, the firm offers v l , u ̄ − a v l 2 (or ( v r , u ̄ − a ( 1 − v r ) 2 ) ) if the consumer bought v l (or v r ) in t1.

If the consumer didn’t buy anything, the firm offers ( v l + v r − 1 2 , p m ) such that u ̄ − a [ 2 v l − ( v l + v r − 1 2 ) ] 2 − p m = 0 . Therefore, p m = u ̄ − a ( v l − v r + 1 2 ) 2 . In other words, if the firm observes that the consumer didn’t buy any product, it believes that the consumer type is in [2v l , 2v r  − 1], and makes the optimal offer for that interval. It’s easy to confirm that no type wants to mimic any type outside her group.

Thus the firm’s ex-ante surplus is

2 v l u ̄ − a v l 2 ( 1 + δ ) + ( 2 v r − 1 − 2 v l ) u ̄ − a v l − v r + 1 2 2 δ + ( 2 − 2 v r ) u ̄ − a ( 1 − v r ) 2 ( 1 + δ ) ,

which can be rewritten as

(1) w i d l u ̄ − a 4 w i d l 2 ( 1 + δ ) + w i d m u ̄ − a 4 w i d m 2 δ + w i d r u ̄ − a 4 w i d r 2 ( 1 + δ )

where wid l , wid m , and wid r are the width of the left interval, the middle interval, and the right interval, respectively.

We want to find the maximum value of (1) and then compare it to ( 1 + δ ) ( u ̄ − a 16 ) .

To this end, we first prove the following claim.

Claim B.1.

For each given width of the middle interval wid m , the value of (1) is maximized when wid l  = wid r .

Proof.

Denote wid m  = 1 − Δ. Then wid l  + wid r  = Δ. Maximizing (1) is equivalent to maximizing

w i d l u ̄ − a 4 w i d l 2 + ( Δ − w i d l ) u ̄ − a 4 ( Δ − w i d l ) 2 .

The first-order derivative w.r.p. to wid l is − 3 a 4 w i d l 2 + 3 a 4 ( Δ − w i d l ) 2 , which is equal to 0 when wid l  = Δ − wid l . The second-order derivative − 3 a 2 w i d l − 3 a 2 ( Δ − w i d l ) is negative.

Now we want to find the maximum value of:

(2) w i d l u ̄ − a 4 w i d l 2 ( 1 + δ ) + ( 1 − 2 w i d l ) u ̄ − a 4 ( 1 − 2 w i d l ) 2 δ + w i d l u ̄ − a 4 w i d l 2 ( 1 + δ )

The first-order derivative w.r.p. to wid l is 2 u ̄ + 3 a 2 δ ( 1 − 2 w i d l ) 2 − 3 a 2 ( 1 + δ ) w i d l 2 . Since u ̄ ≥ 3 a 4 , 0 < δ ≤ 1, and w i d l ≤ 1 2 , the derivative is no less than 2 × 3 a 4 − 3 a 2 × 2 × 1 4 = 3 a 4 > 0 . The second-order derivative is negative.

Thus for w i d l ∈ ( 0 , 1 2 ) , (2) is bounded above by 1 2 ( u ̄ − a 4 × 1 4 ) ( 1 + δ ) + 1 2 ( u ̄ − a 4 × 1 4 ) ( 1 + δ ) = ( 1 + δ ) ( u ̄ − a 16 ) .

Hence in equilibria where some types don’t buy any product in t1, the firm’s expected revenue is also lower than ( 1 + δ ) ( u ̄ − a 16 ) .

Therefore, the equilibrium in Proposition 1 is firm-optimal.

Proof of Proposition 3

Proof.

In the proposed equilibrium, given the consumer’s strategy, the firm offers the optimal pair in period 1 and the optimal triple in period 2. Thus the firm has no incentive to deviate in either period.

For the consumer, depending on the product purchased and the opt-out choice, her type space can be partitioned into 4 intervals:

  1. When θ ∈ [ 0 , 1 3 ) , the consumer accepts ( 1 4 , u ̄ − a 16 ) in period 1 and accepts ( 1 6 , u ̄ − a 36 ) in period 2;

  2. When θ ∈ [ 1 3 , 1 2 ] , the consumer accepts ( 1 4 , u ̄ − a 16 ) in period 1, chooses to opt out, and accepts ( 1 2 , u ̄ − a 36 ) in period 2;

  3. When θ ∈ ( 1 2 , 2 3 ] , the consumer accepts ( 3 4 , u ̄ − a 16 ) in period 1, chooses to opt out, and accepts ( 1 2 , u ̄ − a 36 ) in period 2;

  4. When θ ∈ ( 2 3 , 1 ] , the consumer accepts ( 3 4 , u ̄ − a 16 ) in period 1 and accepts ( 5 6 , u ̄ − a 36 ) in period 2.

Given the firm’s strategy, if the consumer deviates from her opt-in/out choice, her expected utility in period 2 will decrease. If she deviates from her period-1 purchase decision, her expected utility will decrease in period 1 and weakly decrease in period 2. Moreover, since period 2 is the last period, she has no incentive to deviate from the purchase decision in that period.

Therefore, it is indeed an equilibrium.

Proof of Proposition 4

Proof.

First, consider equilibria where all types buy some product in period 1. In such equilibria, at the beginning of period 2, the firm can segment the market into at most three submarkets – those who bought v l , those who bought v r , and those who chose to opt out. In period 1, the firm can segment the market into at most two submarkets since all types buy some product. Therefore, the expected revenue from period 1 is bounded above by the revenue of offering the optimal pair (which gives u ̄ − a 16 ). The expected revenue from period 2 is bounded above by the revenue of offering the optimal triple (which gives u ̄ − a 36 ). Thus for any equilibria where all types buy some product in period 1, the ex-ante PS is bounded above by u ̄ − a 16 + δ ( u ̄ − a 36 ) .

Next, we consider equilibria where some types don’t buy anything in period 1. For equilibria of this kind, the potential firm-optimal equilibria induce three submarkets in t1 (buy v l , buy v r , and don’t buy anything) and four submarkets in t2 (bought v l , bought v r , didn’t buy anything, and opted out). We first rule out some candidate equilibria by the following claim.

Claim B.2.

Equilibria where a positive measure of types reject both offers in t1 and opt out in t2 are not firm-optimal.

Proof.

Consider an equilibrium where a positive measure of types reject both offers in t1 and opt out in t2. Denote the set of types who don’t buy anything in t1 as Θ1∅. In t2, Θ1∅ can be partitioned into two sets – Θ 1 ∅ i n (possibly empty) and Θ 1 ∅ out . The two sets correspond to types in Θ1∅ who choose opt-in and opt-out, respectively.

If Θ 1 ∅ i n = ∅ , then consumer types are pooled into three submarkets in t2 – those who bought v l , those who bought v r , and those who opted out. Thus the firm’s expected revenue from t2 is bounded above by δ ( u ̄ − a 36 ) . In t1, the market is pooled into three submarkets, but only types in two submarkets buy some products, thus the firm’s expected revenue in this period is bounded above by u ̄ − a 16 . Therefore, the candidate equilibrium is not firm-optimal.

If Θ 1 ∅ i n ≠ ∅ , the consumer types are pooled into four submarkets in t2 – those who bought v l , those who bought v r , those who bought nothing, and those who opted out. We can construct another equilibrium with higher expected revenue in which no type rejects both offers in t1 and opts out in t2. The new equilibrium is constructed from the candidate equilibrium in the following way: keep the firm’s period-2 offers, the consumer’s period-2 purchase decision, and her opt-in/out choice unchanged; construct a new pair of period-1 offers so that those who opt out find it optimal to choose one contract (instead of rejecting both contracts) from the pair. Therefore, the candidate equilibrium is not firm-optimal.

Now we consider equilibria where every type who didn’t buy anything in t1 chooses to opt in. Θ1∅ is an interval and we denote its width as wid∅. We first fix wid∅ and calculate the maximum ex-ante PS. Then we show that those maximum values are bounded above by u ̄ − a 16 + δ ( u ̄ − a 36 ) . Note that when wid∅ is fixed, the [0, 1]\Θ1∅ can be treated as a market where all types buy some product in t1, and the width of this market is 1 − wid∅. As discussed at the beginning of this proof, in this submarket, the maximum ex-ante PS is attained when the firm makes the optimal pair of offers in t1 and makes the optimal triple in t2. Therefore, for the original market [0,1], the maximum ex-ante PS with given wid∅ is

u ̄ ( 1 − w i d ∅ ) − a 16 ( 1 − w i d ∅ ) 3 + δ u ̄ ( 1 − w i d ∅ ) − a 36 ( 1 − w i d ∅ ) 3 + δ u ̄ ⋅ w i d ∅ − a 4 ⋅ w i d ∅ 3 .

The above expression is a function of wid∅. Taking derivative we get

− u ̄ + 3 a 16 ( 1 − w i d ∅ ) 2 + δ a 12 ( 1 − w i d ∅ ) 2 − δ 3 a 4 w i d ∅ 2 .

Since u ̄ ≥ 3 a 4 , we have

− u ̄ + 3 a 16 ( 1 − w i d ∅ ) 2 + δ a 12 ( 1 − w i d ∅ ) 2 − δ 3 a 4 w i d ∅ 2 ≤ − 3 a 4 + 3 a 16 ( 1 − w i d ∅ ) 2 + δ a 12 ( 1 − w i d ∅ ) 2 − δ 3 a 4 w i d ∅ 2 < − 3 a 4 + 3 a 16 + a 12 = − 23 48 a < 0 .

The second-order derivative w.r.t. wid∅ is − 3 a 8 ( 1 − w i d ∅ ) − δ a 6 ( 1 − w i d ∅ ) − δ 3 a 2 w i d ∅ , which is negative.

Thus as wid∅ increases, the maximum value decreases. Those maximum values are bounded above by u ̄ − a 16 + δ ( u ̄ − a 36 ) (attained at wid∅ = 0). But this upper bound corresponds to the ex-ante PS in the equilibrium in Proposition 3. Therefore this equilibrium is firm-optimal.

Proof of Proposition 5

Proof.

In the equilibrium without opt-out, the type space is divided into two intervals.

For θ ∈ [ 0 , 1 2 ] , the expected utility is

( 1 + δ ) u ̄ − a θ − 1 4 2 − u ̄ − a 16 = a ( 1 + δ ) − θ 2 + θ 2 .

For θ ∈ ( 1 2 , 1 ] , the expected utility is

( 1 + δ ) u ̄ − a θ − 3 4 2 − u ̄ − a 16 = a ( 1 + δ ) − θ 2 + 3 2 θ − 1 2 .

Thus the ex-ante CS is

∫ 0 1 2 a ( 1 + δ ) − θ 2 + θ 2 d θ + ∫ 1 2 1 a ( 1 + δ ) − θ 2 + 3 2 θ − 1 2 d θ = a ( 1 + δ ) − 1 3 θ 3 | 0 1 + 1 4 θ 2 | 0 1 2 + 3 4 θ 2 − 1 2 θ | 1 2 1 = a 24 ( 1 + δ ) .

In the equilibrium with opt-out, as shown in the proof of Proposition 3, the type space is divided into four intervals.

For θ ∈ [ 0 , 1 3 ) , the expected utility is

u ̄ − a θ − 1 4 2 − u ̄ − a 16 + δ u ̄ − a θ − 1 6 2 − u ̄ − a 36 = a − θ 2 + θ 2 + a δ − θ 2 + θ 3 .

For θ ∈ [ 1 3 , 1 2 ] , the expected utility is

u ̄ − a θ − 1 4 2 − u ̄ − a 16 + δ u ̄ − a θ − 1 2 2 − u ̄ − a 36 = a − θ 2 + θ 2 + a δ − θ 2 + θ − 2 9 .

For θ ∈ ( 1 2 , 2 3 ] , the expected utility is

u ̄ − a θ − 3 4 2 − u ̄ − a 16 + δ u ̄ − a θ − 1 2 2 − u ̄ − a 36 = a − θ 2 + 3 2 θ − 1 2 + a δ − θ 2 + θ − 2 9 .

For θ ∈ ( 2 3 , 1 ] , the expected utility is

u ̄ − a θ − 3 4 2 − u ̄ − a 16 + δ u ̄ − a θ − 5 6 2 − u ̄ − a 36 = a − θ 2 + 3 2 θ − 1 2 + a δ − θ 2 + 5 3 θ − 2 3 .

Thus the ex-ante CS is

∫ 0 1 3 a − θ 2 + θ 2 + a δ − θ 2 + θ 3 d θ + ∫ 1 3 1 2 a − θ 2 + θ 2 + a δ − θ 2 + θ − 2 9 d θ + ∫ 1 2 2 3 a − θ 2 + 3 2 θ − 1 2 + a δ − θ 2 + θ − 2 9 d θ + ∫ 2 3 1 a − θ 2 + 3 2 θ − 1 2 + a δ − θ 2 + 5 3 θ − 2 3 d θ = a 24 + a δ 54 < a 24 ( 1 + δ ) .

In the equilibrium without opt-out, the social surplus is

T S = P S + C S = ( 1 + δ ) u ̄ − a 16 + ( 1 + δ ) a 24 = ( 1 + δ ) u ̄ − a 48 .

In the equilibrium with opt-out, the social surplus is

T S = P S + C S = u ̄ − a 16 + δ u ̄ − a 36 + a 24 + δ a 54 = u ̄ − a 48 + δ u ̄ − a 108 > ( 1 + δ ) u ̄ − a 48 .

□

Proof of Proposition 6

Proof.

For all types, the difference is in the second period. We calculate the difference in expected utilities below (with – without).

For θ ∈ [ 0 , 1 3 ) , the difference is

a δ − θ 2 + θ 3 − a δ − θ 2 + θ 2 = − θ 6 a δ .

For θ ∈ [ 1 3 , 1 2 ] ,the difference is

a δ − θ 2 + θ − 2 9 − a δ − θ 2 + θ 2 = θ 2 − 2 9 a δ .

For θ ∈ ( 1 2 , 2 3 ] , the difference is

a δ − θ 2 + θ − 2 9 − a δ − θ 2 + 3 2 θ − 1 2 = − θ 2 + 5 18 a δ .

For θ ∈ ( 2 3 , 1 ] , the difference is

a δ − θ 2 + 5 3 θ − 2 3 − a δ − θ 2 + 3 2 θ − 1 2 = θ 6 − 1 6 a δ .

Clearly, the difference is positive for θ ∈ ( 4 9 , 5 9 ) , negative for θ ∈ ( 0 , 4 9 ) ∪ ( 5 9 , 1 ) , and zero for θ ∈ { 0 , 4 9 , 5 9 , 1 } .

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Received: 2024-08-08
Accepted: 2025-08-17
Published Online: 2025-09-22

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https://doi.org/10.1515/bejte-2024-0085
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