Screening with Privacy on (Im)persistency
-
Alice Peng-Ju Su
Abstract
I study how private information on the evolution of preference interacts with the dynamic screening contracts in a principal-agent framework with short-term commitment. Privacy on the evolution of preference preserves the agent’s future information advantage, even following truthful revelation of preference. This relaxes the ratchet effects if the consumer’s initial preference is skewedly distributed and the evolution of preference is distributed sufficiently evenly, while it strengthens the ratchet effects if otherwise. Through its relaxation or strengthening of the ratchet effects, privacy on (im)persistency implies an improvement or distortion in the equilibrium revelation of preference. I also characterize the respective implications on the optimal contracts. Such privacy is not welcomed by all types of the agent, for it redistributes the information rent among different types. Privacy on the evolution of preference sustains in equilibrium, as it is not optimal for the principal to perfectly screen the (im)persistency per se.
1 Introduction
A consumer may have evolving preference for goods that are demanded periodically without long-term commitment. Not only is the consumer’s preference his private information, how his preference evolves may also be his private information. For instance, the consumer’s past consumption pattern with other sellers or his previous search records reflect whether he tends to be one with persistent preference or not. In the digital era where rich data on past activities that infer the consumer’s evolution of preference can be detailedly documented, lack of privacy protection may lead to leak of information that hints on the consumer’s (im)persistency.[1] Under some privacy protection protocol that prevents the seller to obtain or share such information, the evolution of preference remains privately known to the consumer himself. I study how this privacy on (im)persistency interacts with the traditional screening problem in a contract theoretical framework with asymmetric information between a seller-principal and a consumer-agent.
Consider a seller contracting with a consumer on the delivery of a product, when they can commit only to short-term contracts in each of the two periods: current and future. The consumer’s preference (valuation of the product) in each period is his private information realized at the beginning of such period. Lack of privacy protection, whether the consumer is persistent or impersistent in preference is common knowledge. This, for instance, can be a result of a technology that the seller invested to track the consumer’s previous shopping and search pattern elsewhere. By truthfully revealing his current preference, the consumer allows the seller to infer his future preference perfectly through her knowledge of the consumer’s evolution of preference. The consumer’s future information advantage is completely extracted unless he lies with a positive probability in the current period. On the contrary, if the consumer is under some privacy protection to easily hide his tracks, the evolution of preference remains privately known. The seller is unable to infer the consumer’s future preference with incomplete information on the (im)persistency, even if the consumer truthfully reveals his current preference with probability one. Information on the evolution of preference may seem redundant in the sense that it does not carry any additional piece of information that preferences in each period jointly do. Privacy on this seemingly redundant information, however, preserves the asymmetric information structure in the future. It is as if an implicit commitment of the seller not to exploit the current information against the consumer in the future.
On the ground of this observation, I characterize the optimal contract with possibly mixed revelation strategies to study how and through what channel privacy on the evolution of preference affects the consumer’s incentive to reveal the payoff-relevant information on preference, and how such privacy affects consumption efficiency and welfare among different types of consumer.
Privacy on the evolution of preference preserves future information asymmetry, which affects to what extent the consumer’s current revelation of preference manipulates the seller’s future belief and the consumer’s future information rent. Abusing the term in the literature of dynamic screening with limited commitment, privacy on (im)persistency manipulates the ratchet effects. How such privacy shapes the optimal contracts to screen the consumer’s preference depends on how the ratchet effects are manipulated. With commonly known evolution of preference, the magnitude of the ratchet effects depends on the distribution of the consumer’s initial preference; with privately known evolution of preference, the magnitude of the ratchet effects depends on the distribution of (im)persistency. Implications of privacy on (im)persistency are then divided into the following three scenarios.
If the distribution of the consumer’s initial preference is not too skewed and his (im)persistency follows a sufficiently even distribution, ratchet effects are mild regardless of privacy on the evolution of preference. The seller optimally induces the consumer to truthfully reveal his current preference with probability one, with or without privacy on the evolution of preference. Given completely truthful revelation, privacy on (im)persistency preserves future information asymmetry, which restores the future rent-efficiency tradeoff that were absent with commonly known evolution of preference. This reduces the potential gain of future information rent from lying and further relaxes the ratchet effects. Without distortion on revelation, privacy on the evolution of preference has no effect on the current consumption; due to the restored rent-efficiency tradeoff, it distorts the future consumption. The preserved future information asymmetry and the relaxed ratchet effects allow the seller to implement a lower intertemporal information rent for the consumer who persistently has a high valuation and a higher intertemporal information rent for the consumer who has impersistent preference. The consumer of different (im)persistency have conflicting opinions about privacy protection.
If the consumer’s initial preference follows a sufficiently skewed distribution and the distribution of (im)persistency is sufficiently even, privacy on (im)persistency improves revelation of preference-related information. With commonly known evolution of preference, the consumer anticipates a significant gain of future information rent by mimicking a similar preference type at a small current loss. Ratchet effects are binding such that it is optimal for the seller to induce the consumer to take a mixed revelation strategy. With privacy on the evolution of preference, the seller cannot distinguish consumers of different (im)persistency who have the same current preference. The consumer anticipates a much lower gain of future information rent by lying in the current period. Privacy on (im)persistency relaxes the ratchet effects, to an extent that completely truthful revelation is optimally induced. Mixed revelation due to binding ratchet effects is implemented with a steeper distortion of current consumption. Improvement of revelation due to privacy on (im)persistency accompanies improvement of current consumption efficiency. In addition, with commonly known evolution of preference, the seller is able to exploit the impersistent consumer who anticipates a positive future information rent, by extracting his potential rent earlier. With privacy on (im)persistency, the seller cannot distinguish the persistent consumer from the impersistent who has the same current preference. This protects the impersistent consumer against exploitation. Along with the improvement of current consumption efficiency, the impersistent consumer is better off when his impersistency is privately known.
If the distribution of the consumer’s initial preference is not too skewed and his (im)persistency follows a sufficiently skewed distribution, privacy on (im)persistency distorts revelation of preference-related information. Ratchet effects are mild with commonly known evolution of preference, as the consumer anticipates a relatively small gain of future information rent by mimicking a very different preference type at a large current loss. The seller induces completely truthful revelation of current preference. With privacy on evolution of preference that is skewedly distributed, the seller’s Bayesian updated belief on the consumer’s future preference is highly sensitive to the consumer’s current revelation, so is the future rent extraction. Value of mis-reporting the current preference for future gain in information rent is high for the consumer whose evolution of preference is in the majority.[2] Privacy on (im)persistency strengthens the ratchet effect for the consumer whose evolution of preference is in the majority, to an extent that the consumer is induced to take a mixed revelation strategy.[3] Mixed revelation due to the binding ratchet effect is implemented with a steeper distortion of current consumption. Meanwhile, the restored future rent-efficiency tradeoff due to preservation of future information asymmetry distorts future consumption. Privacy on (im)persistency results in distortion of consumption efficiency in both periods. Preservation of the future information asymmetry improves the welfare of the impersistent consumer who anticipates a positive future information rent, while the consumption distortion accompanies a lower intertemporal information rent to the consumer who persistently has a high valuation. The consumer of different (im)persistency disagree on privacy protection.
Information on (im)persistency can be regarded as an early arrival of future preference-related information, which the seller may have incentive to offer a more complicated menu of separating contracts to screen the evolution type. It is, however, not feasible for the seller to propose such menu of separating contracts to induce the consumer to truthfully reveal his (im)persistency with certainty. Privacy on (im)persistency thus remains asymmetrically known in equilibrium. Intuitively, the impersistent consumer is indistinguishable from different preference types of the persistent consumer in each period. Without long-term commitment, to induce the impersistent consumer to truthfully reveal his impersistency, it requires the same information rent to different preference types of the persistent consumer in each period. The same information rent must be implemented by the same consumption plan. Otherwise, there is room for the consumer to mis-report both his current preference and his evolution of preference.
1.1 Literature Review
I study the impact of private information on the evolution of preference in a dynamic screening framework with short-term commitment. In the literature of dynamic screening with limited commitment, seminal papers by Laffont and Tirole (1987, 1988 found that pooling occurs due to ratchet effect, when privately known types are persistent and it is common knowledge that types are persistent.[4] Bester and Strausz (2001) and Doval and Skreta (2019) developed concepts à la revelation principle.[5] Skreta (2006) proposed that sequential price posting is an optimal mechanism in a general environment even when the Bester–Strausz revelation principle does not apply. The focus in these papers are mainly the role of limited commitment on the characteristics and the performance of contracts.
As for factors that shape contracts with limited commitment, Shin and Strausz (2014) studied in a production relationship that future private information on productivity of capital input granted through ownership allocation improves truthful information revelation earlier on the consistent productivity of labor input. Fiocco and Strausz (2015) analyzed how strategic delegation with mandated weights on firm’s profit and consumer’s welfare reduces ratchet effect and improve dynamic efficiency of short-term regulatory contracts. Deb and Said (2015) incorporated a second cohort of agents who only arrive in the future and found that their presence reduces the principal’s ability to extract rents via the dynamic contract with limited commitment. Strategic delay of contract occurs in equilibrium as an endogenous commitment to future terms of contract. Accompanying the previous literature, I observe that information asymmetry on the evolution of preference preserves future information asymmetry even with full information revelation. This prevents the principal from completely extracting future information rent, which manipulates the ratchet effect. Applying the Bester–Strausz revelation principle, the optimal revelation strategy and the optimal contract are characterized.
Loginova and Taylor (2008) shared similar idea to this paper that a consumer would strategically use his private information on future preference. In a price-posting model, their focus was on how a buyer strategically rejects a posted price to conceal or to signal information, with a benchmark comparison to a non-strategic buyer. As a result, the seller posts an equilibrium price that preempts information transmission. In this paper, the focus is on truth-revealing incentives and the intertemporal distortion of contracts in a dynamic screening framework, with a benchmark comparison to commonly known pattern of preference.
With full commitment, Baron and Besanko (1984), Courty and Li (2001), Pavan, Segal, and Toikka (2014), and Krähmer and Strausz (2015) provided classical treatments of dynamic contracts with asymmetric information. Boleslavsky and Said (2013) and Skrzypacz and Toikka (2015) studied sequential screening with full commitment when the stochastic sequence of preference shocks as well as the conditional density of the stochastic sequence are the agent’s the private information. The contract is designed to screen the parameters of the conditional density of the stochastic sequence, as well as the realization of the stochastic sequence periodically based on which the consumer’s preference evolves. Evolution of preference in the current paper differs from the stochastic evolution of preference in the above literature in terms of its role on preserving future information asymmetry.
The results in this paper also have implications on privacy protection, which has been actively debated in the digital era. Lack of a sound definition, the literature on privacy diverged in topics[6] but is heavily studied in the effects of protection and disclosure of personal information on the market and organizational outcomes. In Fudenberg and Tirole (1998), Villas-Boas (2004), Acquisti and Varian (2005), and Conitzer, Taylor, and Wagman (2012), privacy decision was modeled as an investment in technology or an action to identify the consumer, to track or to untrack the consumer’s past actions and choices. Consumer’s anonymity affects the seller’s ability to utilize the information on previous consumption that reveals persistent preference type. With multiple sellers of correlated goods, Taylor (2004) and Calzolari and Pavan (2006) further examined trade of such preference-revealing information among the sellers. The current paper distinguishes itself from the above literature in the sense that the consumer’s chosen option in the earlier contract is identified and remembered in the future at no cost. The key discussion is on whether and how privacy protection on the seemingly redundant information facilitates the revelation of crucial information.
2 Model
A seller-principal randomly matches with a consumer-agent to contract on the delivery of a non-durable product that is demanded for two periods, current and future. They are able to commit only to a short-term contract, so that in each period t = 1, 2, the seller proposes a take-it-or-leave-it contract to the consumer who then accepts or rejects the short-term contract. The seller specifies consumption output q
t
and associated payment p
t
in the contract proposed at period t, denoted as
The consumer may have evolving preference defined as the change of privately known taste. His preference at period t is measured by the marginal value of the product in such period, denoted as v t ∈ {v l , v h } with 0 < v l < v h < ∞. A consumer is said to be persistent (denoted as evolution type P) if he has persistent preference v 1 = v 2, while he is said to be impersistent (denoted as evolution type I) if his preference switches within the two periods v 1 ≠ v 2. (Im)persistency can be viewed as the consumer’s intrinsic personality, which is perfectly known by himself. For ease of expression, I phrase the persistent consumer with high valuation and that with low valuation as the persistent-high type and the persistent-low type respectively. The impersistent consumer who has a higher valuation in the earlier period (v 1 = v h > v 2 = v l ) and that who has a higher valuation in the later period (v 1 = v l < v 2 = v h ) are phrased as the descending type and the ascending type respectively.
The consumer receives a private and perfect signal on his evolution of preference and his first-period preference prior to the first-period contracting.[7] The prior distribution of the consumer’s first-period preference and that of his evolution of preference are common knowledge. With probability σ ∈ (0, 1) the consumer values the product highly in the first period, and with probability ϕ ∈ (0, 1) the consumer is persistent. One can consider that there is a continuum of consumers of mass one, in which ϕ ⋅ σ (respectively ϕ ⋅ (1 − σ)) of them have persistently high (resp. low) valuation over the product and (1 − ϕ) ⋅ σ (resp. (1 − ϕ) ⋅ (1 − σ)) of them perceive a higher valuation over the product at an earlier (resp. a later) date.[8] For ease of description, the persistent (resp. impersistent) consumer is said to be in the minority of evolution type if ϕ < 1 − ϕ (resp. ϕ > 1 − ϕ), and said to be in the majority if otherwise.
At the beginning of each period t, the consumer receives a private and perfect signal on his period-t preference prior to contracting. The seller proposes a take-it-or-leave-it contract
The consumer with preference type v
i
∈ {v
l
, v
h
} in period t earns rent u
ti
= v
i
⋅ q
ti
− p
ti
if he takes the option (q
ti
, p
ti
), t = 1, 2 and i = l, h. The seller earns profit π
ti
= p
ti
− c(q
ti
) in period t when the consumer takes the option (q
ti
, p
ti
), t = 1, 2 and i = l, h. Let there be no discount between the payoffs in two periods to abstract from the situation where the players have exogenous preference over the first or the second period.[10] For future reference, denote the efficient consumption output when contracting with a consumer of valuation v
i
in period t as
It is worthwhile to briefly discuss the intuition on the role of asymmetric information on the evolution of preference and its implication on privacy protection. For example, the consumer’s intrinsic personality, whether he tends to be persistent or not, can be inferred from his past interactions with other firms. With the popularity of online shopping, search engine, and social media platform … etc., rich data on past activities that infer intrinsic personality can be very detailed.[11] If such information can be costlessly acquired without privacy protection, this corresponds to the case in Section 3.1 where the evolution type is common knowledge. The consumer, by truthfully revealing the first-period preference to the seller, lost his information advantage in the second period. For instance, the seller knows that a persistent consumer will be willing to make a high payment in the second period when he is induced to truthfully reveal his high valuation in the first period. Alternatively, if the consumer is under some privacy protection against documentation or share of personal information, or that it is sufficiently costly to acquire information on the consumer’s past activities that infer his intrinsic personality, the consumer’s evolution type is his private information. Even when the first period preference is truthfully revealed, the consumer in the second period still maintains some information advantage due to his private information on the evolution of preference. For instance, the seller only knows that with probability ϕ the consumer who truthfully revealed a high valuation is still willing to make a high payment in the second period. Asymmetric information on the consumer’s evolution of preference is a potential source of the consumer’s information advantage in the future.
3 Contracts
3.1 Commonly Known Evolution of Preference
As a benchmark, consider only in this section that the consumer’s evolution of preference is common knowledge. Applying the revelation principle à la Bester and Strausz (2001), the seller proposes the first-period contract
The seller’s problem can be separated into three stages: (1) a second-period contracting problem given belief ν k (v i ), (2) a first-period contracting problem that is incentive compatible and individually rational to induce an arbitrary revelation strategy n ki , and (3) the optimal revelation strategy to implement that maximizes the seller’s intertemporal payoff.
By backward induction, the conventional revelation principle applies to the seller’s second-period contracting problem with belief ν
k
(v
i
). Following the consumer’s first-period choice of (q
1i
, p
1i
), the seller proposes a standard contract
subject to individual rationality u
2j
≥ 0 for both j = l, h, and incentive compatibility (v
h
− v
l
) ⋅ q
2h
≥ u
2h
− u
2l
≥ (v
h
− v
l
) ⋅ q
2l
. For later reference, denote
Anticipating the second-period contract, it is individually rational[12] for the persistent consumer to accept the first-period contract if
It is incentive compatible for the persistent consumer to reveal his true preference with probability n Pi > 0 in the first period if
with equality if n Pi ≠ 1 is implemented. Meanwhile, it is individually rational for the impersistent consumer to accept the first-period contract if
It is incentive compatible for the impersistent consumer to reveal his true preference with probability n Ii > 0 in the first period if
with equality if n Pi ≠ 1 is implemented.
Denote ρ
k
= σ ⋅ n
kh
+ (1 − σ) ⋅ (1 − n
kl
) as the probability that option (q
1h
, p
1h
) is chosen by the consumer of evolution type k. The seller proposes the first-period contract
subject to
Condition 1
Preference and the distribution of initial preference satisfy
Lemma 1
The first-period contract with a consumer of evolution type k = P, I satisfies binding
Proof
Appendix A.1. □
Implied by the slacking incentive compatibilities in Lemma 1 given Condition 1, the consumer who has a low valuation in the first period reveals his preference truthfully with probability one, regardless of his evolution type. Condition 1 resembles the slacking monotonicity constraint in standard contracting. Satisfaction of the traditional monotonicity[13] is not sufficient for us to neglect the incentive compatibility of the consumer who has a low valuation in the first period. The anticipated future information rent depends on the first-period revelation, so incentive compatibility is more restrictive than the traditional.
Intuitively, there is a ratchet effect for the persistent-high type and the ascending type of consumer to mis-lead the seller’s future belief. Contracting with a persistent consumer, a higher information rent to the persistent-high type of consumer to cope with the binding ratchet effect may introduce the persistent-low type to lie to take the first-period benefit and reject the unfavorable second-period contract. Contracting with an impersistent consumer, the ascending type has incentive to lie about his first-period preference in order to protect his future information advantage. Satisfaction of Condition 1 rules out the binding incentive for the persistent-low type to take the instant benefit and run, and it implies a non-binding ratchet effect for the ascending type.
Evolution of preference being commonly known, to what extent the consumer’s revelation manipulates the seller’s future belief and hence the consumer’s future information rent depends on the difference in the consumer’s taste and the distribution of the initial preference, so is Condition 1. It holds when (i) the consumer’s preference in each period is sufficiently diverse, and/or (ii) the distribution of the consumer’s initial taste is not too skewed towards low valuation.
Lemma 2
Satisfaction of Condition 1, the optimal first-period contracts induce
Proof
Appendix A.1. □
Contracting with a persistent consumer, it is incentive compatible for the persistent-high type of consumer to truthfully reveal his first-period preference with a positive probability only if he receives in advance the potential future information rent had he possessed information advantage in the second period by mis-reporting his preference in the first period. Satisfaction of Condition 1-1 implies that the persistent-low type of consumer does not find such information rent attractive enough to mimic the persistent-high type. The optimal first-period contract thus induces completely truthful revelation, with efficient consumption for the consumer who reveals high valuation and downward distorted consumption for whom revealing low valuation, due to the conventional rent-efficiency tradeoff. This is summarized in Lemma 2.
Violation to Condition 1-1 implies that the persistent-low type has incentive to lie to take the first-period benefit and reject the future contract given completely truthful revelation. There are two channels for the seller to cope with such take-the-money-and-run incentive:[14] consumption distortion and revelation distortion. To induce the persistent-low type to reveal his true preference with a positive probability, the seller distorts the first-period consumption upwards for the consumer who reveals high valuation, and further downwards for whom revealing low valuation. Efficiency loss from such steeper consumption distortion welcomes the implementation of mixed revelation. This is summarized in Lemma 3 below.
Contracting with an impersistent consumer, the standard static information rent is sufficient to induce the descending type of consumer to truthfully reveal his first-period preference with a positive probability, anticipating a zero second-period information rent regardless. Satisfaction of Condition 1–2 or 1–3 implies that the ascending type of consumer does not find the potential future information advantage valuable enough to outweigh the first-period loss from mimicking the descending type. However, this does not guarantee the optimality of completely truthful revelation in the first period. The ascending type of consumer is willing to accept an exploitative first-period contract implementing a negative interim information rent, had he anticipated that rejection leads to a forgone future information rent. This is possible only if the descending type is induced to lie with a positive probability such that the ascending type has a positive future information rent given truthful revelation. This exploitative contract is attractive to the seller in the sense that she can not only exploit the ascending type but also reduce the information rent yielded to the descending type by incentive compatibility. It is at a cost of revelation efficiency, as it takes more than a marginal distortion in the revelation strategy. An exploitative contract is optimal only if
When Conditions 1–2 and 1–3 are violated, the ascending type has incentive to lie to protect his future information advantage as the value of such advantage is sufficiently high to outweigh the first-period loss of lying. There are two channels for the seller to cope with such binding ratchet effect: consumption distortion and revelation distortion. To induce the ascending type to reveal his true preference with a positive probability, the seller distorts the first-period consumption upwards for the consumer who reveals high valuation, and further downwards for whom revealing low valuation. Efficiency loss from such steeper consumption distortion welcomes the implementation of mixed revelation. This is summarized in Lemma 3 below.
Lemma 3
Violation to Condition 1, the optimal first-period contracts induce mixed revelation of first-period preference with steeper consumption distortion
Proof
Appendix A.1. □
With commonly known evolution of preference, information on the second-period preference is indirectly revealed through completely truthful revelation in the first period. If completely truthful revelation is optimally induced, the seller has degenerate belief on the consumer’s second-period taste, so the optimal second-period contract has a single option with efficient consumption and zero information rent. Violation to Condition 1, mixed revelation is implemented in the first period. Asymmetric information on the second-period preference remains. The seller has non-degenerate belief on the consumer’s second-period taste, so the optimal second-period contract is a menu of options which exhibits the static rent-efficiency tradeoff. The consumer with a high valuation in the first period enjoys a positive information rent. The ascending type of consumer has his future information rent completely extracted in the first period either through the induced truthful revelation or the exploitation with mixed revelation.
3.2 Privately Known Evolution of Preference
With private information on the evolution of preference, we focus on the contracts that implement mixed revelation of preference à la Bester and Strausz (2001). The seller proposes the first-period contract
The seller’s problem can be separated into three stages: (1) a second-period contracting problem given belief μ(v i ), (2) a first-period contracting problem that is incentive compatible and individually rational to induce an arbitrary revelation strategy m ki , and (3) the optimal revelation strategy to implement that maximizes the seller’s intertemporal payoff.
By backward induction, conventional revelation principle applies to the seller’s second-period contracting problem with belief μ(v
i
). Following the consumer’s first-period choice of (q
1i
, p
1i
), the seller proposes a standard contract
subject to individual rationality u
2j
≥ 0 for both j = l, h, and incentive compatibility (v
h
− v
l
) ⋅ q
2h
≥ u
2h
− u
2l
≥ (v
h
− v
l
) ⋅ q
2l
. For later reference, denote
Anticipating the second-period contract, it is individually rational for the (im)persistent consumer to accept the first-period contract if
It is incentive compatible for the (im)persistent consumer to reveal his true preference with probability m ki > 0 in the first period if
with equality if m ki ≠ 1 is implemented.
Denote ρ = σ ⋅ϕ⋅m
Ph
+ σ ⋅ (1 − ϕ) ⋅m
Ih
+ (1 − σ) ⋅ϕ⋅ (1 − m
Pl
) + (1 − σ) ⋅ (1 − ϕ) ⋅ (1 − m
Il
) as the probability that option (q
1h
, p
1h
) is chosen by the consumer of any type. The seller proposes the first-period contract
subject to (IR
ki
) and (IC
ki
) for both k = P, I and i = l, h. Let π
1(m) be the seller’s first-period expected payoff as a solution to the above contracting problem, given the consumer’s revelation strategy. Anticipating the contracts
Condition 2
The distribution of evolution type and that of initial preference satisfy
Lemma 4
The first-period contract satisfies binding (IR Pl ). The relevant incentive compatibility constraints are those of the majority of evolution type. The contract satisfies binding (IC Ph ) when ϕ > 1 − ϕ, or binding (IC Ih ) when 1 − ϕ > ϕ. Constraint (IC Pl ) or (IC Il ) is binding when ϕ > 1 − ϕ or when 1 − ϕ > ϕ respectively only if Condition 2 is violated.
Proof
Appendix A2. □
Implied by the slacking incentive compatibilities in Lemma 4, the consumer who is in the minority of evolution type reveals his first-period preference truthfully with probability one. Intuitively, the seller is unable to distinguish the consumer’s evolution of preference and anticipates a lower probability to trade with the minority. If the majority is persistent, the seller believes that the consumer who has revealed v l in the first period remains as a low valuation type in the second period with a higher probability than is the consumer who has revealed v h . The second-period information rent would be higher following a first-period revelation of v l . This implies a more restrictive incentive compatibility for the persistent consumer. If the majority is impersistent, the seller believes that the consumer who has revealed v l in the first period becomes a high valuation type in the second period with a higher probability than is the consumer who has revealed v h . The second-period information rent would be higher following a first-period revelation of v h . This implies a more restrictive incentive compatibility for the impersistent consumer. Asymmetric information on the evolution of preference allows the minority of evolution type to stay under the radar and enjoy a spillover information rent above the minimum requirement for incentive compatibility. Privacy on the evolution of preference weakens the ratchet effect of the minority type to an extent that it is non-binding. The minority type thus fully reveals his preference in the first period if he has private information on the evolution of preference.
Among the majority of evolution type, slacking incentive compatibilities in Lemma 4 given Condition 2 implies that the consumer who has a low valuation in the first period reveals his preference truthfully with probability one. Condition 2 resembles the slacking monotonicity constraint in standard contracting. Satisfaction of the traditional monotonicity is not sufficient for us to neglect the incentive compatibility of the consumer who has a low valuation in the first period. The anticipated future information rent depends on the first-period revelation, so incentive compatibility is more restrictive than the traditional.
If the majority is persistent, there is a binding ratchet effect for the persistent-high type of consumer to mis-lead the seller’s future belief. A higher information rent to the persistent-high type to cope with the ratchet effect may introduce the persistent-low type to lie to take the first-period benefit, anticipating a zero second-period information rent regardless of revelation. Satisfaction of Conditions 2-1 and 2-2 rules out the binding incentive for the persistent-low type to take the instant benefit. If the majority is impersistent, the ascending type of consumer has incentive to manipulate the seller’s future belief through the first-period revelation in his own favor. This ratchet effect for the ascending type is non-binding if Condition 2-3 is satisfied.
Evolution of preference being privately known, to what extent the consumer’s revelation manipulates the seller’s future belief and hence the consumer’s future information rent depends on the distribution of evolution type and the distribution of the initial preference, so is Condition 2. It holds when (i) the distribution of evolution type is sufficiently even, and/or (ii) the distribution of the consumer’s initial taste is not too skewed towards low valuation.
Lemma 5
Satisfaction of Condition 2, the optimal first-period contract
Proof
Appendix A.2. □
When the majority is persistent, it is incentive compatible for the persistent-high type of consumer to truthfully reveal his preference with a positive probability only if he receives in advance the potential future gain of information rent from mis-reporting his first-period preference. Satisfaction of Condition 2-1 or 2-2 implies that the persistent-low type of consumer does not find such additional benefit attractive enough to bear the cost of lying in the first period. Inducing the persistent-high type to take a mixed revelation strategy manipulates the seller’s future belief and reduces such potential gain of information rent. If Condition 2-1 holds, the probability of the consumer having a low valuation in the second period is sufficiently high. The marginal reduction in the potential gain of information rent is sufficiently low that it is optimal for the seller to induce completely truthful revelation. If Condition 2-2 holds instead, it is optimal for the seller to induce the persistent-high type of consumer to take a mixed revelation strategy to reduce the future gain of information rent from lying, which reduces the first-period information rent. Truthful or mixed revelation induced, the optimal first-period contract consists of efficient consumption for the consumer who reveals high valuation and downward distorted consumption for whom revealing low valuation, due to a rent-efficiency tradeoff in terms of consumption and of mixed revelation under Condition 2-2. This is summarized in Lemma 5.
When Conditions 2-1 and 2-2 are violated, the persistent-low type of consumer has incentive to lie to take the additional benefit in the first period, anticipating a zero second-period information rent regardless. The seller relies on consumption distortion and revelation distortion to cope with such incentive. To induce the persistent-low type to reveal his true preference with a positive probability, the seller distorts the first-period consumption upwards for the consumer who reveals high valuation, and further downwards for whom revealing low valuation. Efficiency loss from such steeper consumption distortion welcomes the implementation of mixed revelation for the persistent consumer, with the impersistent consumer revealing his first-period preference truthfully. This is summarized in Lemma 6 below.
When the majority is impersistent, the standard static information rent is sufficient to induce the descending type of consumer to truthfully reveal his first-period preference with a positive probability, anticipating a zero second-period information rent regardless. Satisfaction of Condition 2-3 implies that the ascending type of consumer does not find the potential future information advantage valuable enough to bear the cost of lying in the first period. The optimal first-period contract induces completely truthful revelation, with efficient consumption for the consumer who reveals high valuation and downward distorted consumption for whom revealing low valuation, due to the conventional rent-efficiency tradeoff. This is summarized in Lemma 5.
When Condition 2-3 is violated, the ascending type of consumer finds the potential future information advantage sufficiently valuable to outweigh the first-period loss of mis-reporting. The seller relies on consumption distortion and revelation distortion to cope with such binding ratchet effect. To induce the ascending type to reveal his true preference with a positive probability, the seller distorts the first-period consumption upwards for the consumer who reveals high valuation, and further downwards for whom revealing low valuation. Efficiency loss from such steeper consumption distortion welcomes the implementation of mixed revelation for the impersistent consumer, with the persistent consumer revealing his first-period preference truthfully. This is summarized in Lemma 6 below.
Lemma 6
Violation to Condition 2, the optimal first-period contract
Proof
Appendix A.2. □
With privacy on the evolution of preference, the second-period preference is asymmetrically known even with completely truthful revelation of first-period preference. The seller has non-degenerate belief on the consumer’s second-period taste, so the optimal second-period contract is a menu of options which exhibits the static rent-efficiency tradeoff, Condition 2 satisfied or not. The persistent-high type and the ascending type of consumer enjoys his second-period information rent even when he is induced to reveal his first-period preference truthfully. If the majority is persistent (impersistent), the descending type (persistent-high type) of consumer enjoys a first-period information rent partly due to his private information on the high valuation and partly attributed to being in the minority. In addition, the impersistent consumer is protected from first-period rent exploitation regardless of the implemented revelation strategy. The seller in the first period alone is unable to distinguish the ascending type from the persistent-low type and the descending type from the persistent-high type of consumer. The contract to induce participation of the persistent-low type implies zero rent exploited from the ascending type, which by incentive compatibility, implies no exploitation from the descending type either.
4 Implications of Privacy on (Im)persistency
Difference in the contracts with commonly known and privately known evolution of preference marks the maximum effects of privacy protection of the consumer’s traits. If the distribution of initial preference is not too skewed and the distribution of (im)persistency is sufficiently even, privacy on the evolution of preference only affects the contract to induce completely truthful revelation of preference, but not the optimality of completely truthful revelation.
Proposition 1
Contracts with Truthful Revelation. When Conditions 1-1, 1-2, 2-1 and 2-3 are satisfied, completely truthful revelation of preference is induced with or without privacy on the evolution of preference. Privacy on (im)persistency has no effect on the first-period consumption but distorts the second-period consumption efficiency. The persistent consumer is weakly worse off; the impersistent consumer is weakly better off, strictly better off if the majority of consumer is persistent.
When the distribution of initial preference is not too skewed and the distribution of (im)persistency is sufficiently even, truthful revelation has a marginally small effect on the potential future information rent, with or without privacy on the evolution of preference. The ratchet effects are sufficiently mild that neither first-period revelation nor consumption is distorted beyond the conventional rent-efficiency tradeoff. Truthful revelation in the first period induces a degenerate belief of the seller with commonly known evolution of preference, whereas with privacy on (im)persistency, second-period preference remains privately known by the consumer following truthful revelation in the first period. The seller in the latter scenario finds it optimal to distort the second-period consumption to reduce the second-period information rent à la static rent-efficiency tradeoff. Privacy on (im)persistency results in future consumption distortion.
Privacy on (im)persistency preserves the consumer’s future information advantage, especially when he has a high valuation in the second period. This directly improves the welfare of the ascending type of consumer, who is free from complete rent extraction in the second period following truthful revelation in the first. If the impersistent consumer is in the minority, the descending type also benefits from the seller’s inability to distinguish him from the persistent counterpart, earning a spillover information rent beyond the minimum to induce truthful revelation in the first period. However, such information asymmetry restores the second-period rent-efficiency tradeoff and relaxes the ratchet effect of the persistent-high type,[19] who is worse off from a lower first-period information rent in advance to induce truthful revelation, as well as a distorted second-period information rent. Privacy on the evolution of preference results in redistribution of intertemporal information rent to the consumer of different evolution types. This echos the conflicting opinions for privacy protection even among the consumers.[20]
If the distribution of initial preference is sufficiently skewed and the distribution of the evolution type is sufficiently even, privacy on the evolution of preference improves the first-period revelation of preference.
Proposition 2
Improved Revelation of Preference. With violation to Conditions 1-1 and 1-2 but satisfaction of Conditions 2-1 and 2-3, mixed revelation is induced with commonly known evolution of preference, while completely truthful revelation is induced with privately known evolution of preference.
When the distribution of initial preference is sufficiently skewed, the ratchet effect with commonly known persistency is so strong that it takes a large in-advance information rent to induce the persistent-high type of consumer to truthfully reveal his preference in the first period, which motivates the persistent-low type to take the instant benefit by lying with probability one. The strong ratchet effect is also present with commonly known impersistency. The ascending type of consumer has a larger future information rent forgone than his loss from mimicking the descending type in the first period. To cope with such binding ratchet effects, the seller induces mixed revelation strategies. With privately known evolution of preference distributed sufficiently evenly, the seller cannot identify the consumer’s evolution type, and her Bayesian-updated belief has a sufficiently even distribution over the consumer’s preference. The value of lying for future gain in information rent is thus lower for both the persistent-high type and the ascending type of consumer. The ratchet effects are relaxed, so is the incentive for the persistent-low type to take the instant benefit. The seller optimally induces completely truthful revelation. Privacy on the evolution of preference relaxes the ratchet effects, which improves equilibrium revelation of preference-related information.
Relaxation of the ratchet effects from privacy on (im)persistency also improves the first-period consumption efficiency. Binding incentive compatibilities with commonly known evolution of preference is accompanied with steeper consumption distortion in the first period. Privacy on (im)persistency relaxes the ratchet effects such that the upward incentive compatibilities are non-binding. This restores consumption efficiency for the consumer who has a high valuation in the first period and reduces consumption distortion for the consumer who has a low valuation in the first period. The effect on the second-period consumption, however, is ambiguous, depending on the equilibrium revelation strategy with commonly known evolution of preference.
Improved first-period consumption efficiency accompanies an increase in the information rent to the descending type of consumer. On top of that, private information on the evolution of preference protects the impersistent consumer against an exploitative contract. The ascending type is free from rent exploitation when the seller is unable to distinguish him from the persistent-low type, so is the descending type by incentive compatibility. The impersistent consumer is better off from privacy on (im)persistency that improves the revelation of preference as well. With an ambiguous effect on the second-period consumption, the welfare effect on the persistent-high type of consumer is ambiguous.
If the distribution of initial preference is not too skewed and the distribution of the evolution type is sufficiently skewed, privacy on the evolution of preference distorts the first-period revelation of preference.
Proposition 3
Distorted Revelation of Preference. With satisfaction of Conditions 1-1 and 1-2 but violation to Conditions 2-1 and 2-3, completely truthful revelation is induced with commonly known evolution of preference, while mixed revelation is induced for the majority of evolution type with privately known evolution of preference.
When the distribution of initial preference is not too skewed, the ratchet effect with commonly known evolution of preference is sufficiently mild that truthful revelation is optimally induced. With privately known evolution of preference, the consumer whose evolution type is in the minority truthfully reveals his preference in the first period as the seller is unable to distinguish him from the majority. If Condition 2-2 is satisfied, the ratchet effect is insufficiently strong that the upward incentive compatibility is slacking for the consumer with low valuation in the first period. However, it is insufficiently mild either that the seller finds it optimal to induce the persistent-high type of consumer to take a mixed revelation strategy, as the marginal gain of future information rent from lying is sufficiently high when the latter truthfully reveals his first-period preference with a higher probability.[21] Privacy on the evolution of preference distorts the equilibrium revelation of preference-related information for the consumer who persistently has a high valuation.
With sufficiently skewed distribution of the evolution type, Condition 2 is violated. If the majority is persistent, following the first-period revelation of low valuation, the seller believes that with a very high probability the consumer persistently has a low valuation in the second period. Difference in future rent extraction following different first-period revelation is sufficiently large. The value of lying for future gain in information rent is so high for the persistent-high type of consumer that it takes a large in-advance information rent to induce him to truthfully reveal his first-period preference. This instant benefit is attractive for the persistent-low type of consumer to take by lying with probability one. If the majority is impersistent, following the first-period revelation of high valuation, the seller believes that with a very high probability the consumer impersistently has a low valuation in the second period. Difference in future rent extraction following different first-period revelation is sufficiently large. The value of lying for future gain in information rent is sufficiently high for the ascending type of consumer to outweigh the current loss of lying. Privacy on the evolution of preference strengthens the ratchet effects. The seller optimally induces the majority of evolution type to take a mixed revelation strategy. Equilibrium revelation of preference-related information is distorted with privacy on (im)persistency.
Tighter ratchet effects from privacy on the evolution of preference also distorts consumption efficiency. With commonly known (im)persistency, truthful revelation is induced with static rent-efficiency tradeoff in the first period. Following truthful revelation, the seller has a degenerate belief in the second period and offers efficient consumption. Privacy on (im)persistency strengthens the ratchet effects such that in the first period, the upward incentive compatibility of the majority type of consumer is binding. The first-period contract has steeper consumption distortion. Following truthful or mixed revelation, the seller has a non-degenerate belief in the second period and proposes a contract that exhibits the static rent-efficiency tradeoff. Privacy on (im)persistency weakly distorts the second-period consumption.
Privacy on (im)persistency preserves the consumer’s future information advantage, which directly improves the welfare of the ascending type of consumer. The preservation of future information advantage, however, restores the second-period rent-efficiency tradeoff and results in a binding ratchet effect in the first period when Condition 2 is violated. These two effects jointly make the persistent-high type of consumer worse off. The descending type of consumer suffers from the binding ratchet effect when the majority is impersistent; his welfare effect is ambiguous when the majority is persistent, depending on whether the binding ratchet effect or the spillover information rent is stronger in magnitude. With mixed revelation optimally induced, it remains that privacy on the evolution of preference results in redistribution of intertemporal information rent to the consumer of different evolution types.
4.1 Numerical Examples
To see how the above propositions depend on the distribution of preference type and that of evolution type, consider some numerical examples with v
h
= 7 and v
l
= 4. Conditions 1-1 and 1-2 become
5 Screening (Im)persistency
The previous analysis is based on the implicit assumption that the consumer’s evolution of preference per se is not screened by a menu of separating contracts. Inducing the consumer to reveal his preference and his (im)persistency in the first-period is equivalent to inducing the consumer to reveal his second-period preference early in the first period. The following proposition indicates that in equilibrium, privacy on the evolution of preference remains asymmetrically known between the seller and the consumer.
Proposition 4
It is sub-optimal to offer a menu of separating contracts to screen the evolution of preference per se such that the evolution type is completely revealed in equilibrium.[22]
Proof
Appendix A.3. □
Suppose that the first-period menu of separating contracts is offered to induce truthful revelation of both the evolution type and the preference type, each with a positive probability. If it is optimal to induce one type of the consumer to truthfully reveal his (im)persistency with probability one, the opposite type, in terms of different (im)persistency and first-period preference, must be induced to take a mixed revelation of preference. Otherwise, at least one type of the consumer has incentive to mimic the type that the seller believes to be highly unlikely. To illustrate the i.e. consider for example a contract to induce the ascending type to truthfully reveal his impersistency with certainty. If the persistent-high type of consumer is induced to truthfully reveal his high valuation in the first period, following the revelation of “persistent-low type,” the seller believes with probability one that the consumer has a low valuation in the second period, because this message must not be revealed by an ascending type (who truthfully reveals his impersistency) nor by a persistent-high type (who truthfully reveals his first-period taste). If this message were in fact revealed by the consumer who has a high valuation in the second period, he would have earned the maximum second-period rent. The ascending type of consumer thus has incentive to mimic the persistent-low type with probability one. The same idea applies to other types of consumer, which is shown in Lemma A in Appendix A.3.
If the seller is to offer a menu of separating contracts that induces all types of consumer to truthfully reveal his evolution of preference with probability one, by the argument above, this is incentive compatible only with each type of the consumer taking a completely mixed revelation of first-period preference. The consumer is indifferent between revelation of preference types, so the menu of contracts induces the consumer to truthfully reveal his evolution type with a positive probability only if it implements pooled per-period information rents. Otherwise, the consumer has incentive to mimic the opposite type, to lie on both the evolution type and the preference type with probability one. Intuitively, the impersistent consumer is indistinguishable from different preference types of the persistent consumer in each period. In order to deter the impersistent consumer from lying with probability one, it requires an intertemporal information rent equal to the summation of the per-period information rents to different preference types of the persistent counterpart. For instance, to deter the ascending type from mimicking the persistent-low type with certainty, the intertemporal information rent to the ascending type is implemented as the sum of the first-period rent to the persistent-low type and the second-period rent to the persistent-high type. Without commitment to the future in the first period, equality of intertemporal information rent is implemented by equality of per-period information rent. If the same information rent is implemented by a menu of separating contracts with different levels of consumption, the consumer with high valuation in the first period has incentive to mimic the one with low valuation by choosing the contract that has a higher consumption intended for the consumer with low valuation. The incentive compatible contracts thus have the same level of consumption for the persistent and the impersistent consumer. The optimal menu of contracts is pooled with respect to the consumer’s evolution type.
6 Conclusions
Private information on the evolution of preference preserves asymmetry on future preference-related information. With short-term commitment, this secures the consumer’s future information advantage and manipulates the ratchet effect. How privacy on the evolution of preference shapes the optimal contracts to screen the consumer’s preference depends on how the ratchet effects are manipulated. Different factors play a crucial role in such manipulation under different information structures. With commonly known evolution of preference, the ratchet effects are relatively mild when the distribution of the consumer’s initial preference is not too skewed; with privately known evolution of preference, the ratchet effects are relatively mild when the distribution of (im)persistency is sufficiently even.
When both conditions hold, completely truthful revelation on preference is optimally induced with or without privacy on (im)persistency. Such privacy has no first-period effects on consumption and revelation, but it does redistribute the consumer’s information rent due to its effect on the second-period information structure and hence the relaxation of first-period incentive compatibility. When the former condition holds while the latter is violated, privacy on the evolution of preference strengthens the ratchet effects to an extent that the upward incentive compatibility constraint becomes binding for the consumer whose evolution type is in the majority. Revelation is distorted along with steeper consumption distortion. Privacy on (im)persistency distorts truthful revelation of preference-related information if the distribution of the consumer’s initial preference is not too skewed and the distribution of (im)persistency is sufficiently skewed. When the former condition is violated while the latter holds, privacy on the evolution of preference relaxes the ratchet effects to an extent that the upward incentive compatibility constraints become non-binding. Revelation and consumption efficiency in the first period are improved. Privacy on (im)persistency improves truthful revelation of preference-related information if the consumers have sufficiently skewed distribution of initial preference and the distribution of (im)persistency is sufficiently even.
To demonstrate in a simple model how the privacy on seemingly redundant information improves or distorts the incentive to reveal payoff-relevant information, some potentially interesting research questions were left out. The seller was assumed to be the principal in this paper. Allowing the consumer to have some bargaining power, we will be able to study how the privacy on the evolution of preference affects the informed player’s incentive to actively reveal his preference-related information. Generalization of the model beyond a binary type space is also of interest. The consumer’s private knowledge of the evolution of preference can be generalized to his private information on the function of future preference. For instance, suppose that the consumer’s current preference is v 1 which will possibly differ in the future indicated by the function v 2 = f(v 1). The consumer has private information not only on the realization of v 1 but also on how his preference will evolve represented by the function f(⋅). In practice, privacy on the consumer’s evolution of preference is related to the use of technology to keep track of the consumer’s past activities, as well as the legal restrictions on the use of tracking technology. It was assumed exogenous in this paper. A natural extension is to study the seller’s decision to invest in such tracking technology, and her endogenous choice of privacy protection in response to the legal restrictions.
Funding source: Ministry of Science and Technology, Taiwan
Award Identifier / Grant number: MoST 106-2410-H-305-006-MY2
Acknowledgment
I appreciate comments and suggestions raised by the seminar participants at FUR 2018, East Asian Contract Theory Conference 2018, CEANA 2019, Academia Sinica, National Central University, National Tsing-Hua University, National Taiwan University, and National Dong Hwa University. Financial support from the Ministry of Science and Technology, Taiwan (MoST 106-2410-H-305-006-MY2) is gratefully acknowledged.
-
Research funding: This work was support by the Ministry of Science and Technology, Taiwan (MoST 106-2410-H-305-006-MY2).
Appendix A: Proof
A.1 Proof of Lemma 1 to Lemma 3 in Section 3.1
The consumer with first-period preference v i , i = l, h, and evolution of preference k = P, I reveals truthfully his first-period taste with probability n ki . The seller’s Bayesian updated belief of a high valuation type in the second period following the persistent consumer’s choice of first-period option (q 1h , p 1h ) and (q 1l , p 1l ) are
and
respectively. Her Bayesian updated belief of a high valuation type in the second period following the impersistent consumer’s choice of first-period option (q 1h , p 1h ) and (q 1l , p 1l ) are
and
respectively. Following the consumer’s first-period choice of (q
1i
, p
1i
), the seller proposes the contract
subject to u 2j ≥ 0 for both j = l, h, and (v h − v l ) ⋅ q 2h ≥ u 2h − u 2l ≥ (v h − v l ) ⋅ q 2l .
By backward induction and the standard methods in static contract theory, the optimal contract
Anticipating the second-period contract
subject to
with a persistent consumer, and
subject to
with an impersistent consumer. ρ k denotes the probability that the consumer of evolution type k = P, I reveals a high valuation in the first period.
A.1.1 Contract with a Persistent Consumer
Rearranged from
The optimal contract has at least
If
The contracts and revelation strategy strictly satisfies
If
At
When Condition 1-1 is violated and
subject to
The principal implements
At
A.1.2 Contract with an Impersistent Consumer
The optimal contract has at least
The optimal contract must not have both
If
The contracts and revelation strategy strictly satisfies
Given the above conditions, the principal implements
At
When Condition 1 is violated and
subject to
The principal implements
seller’s payoff between a truth revealing and a lying descending type is higher than or equal to the sum of weighted decrement in the exploited rent and weighted increment in the gain of second-period information rent from lying when inducing the descending type to reveal his true preference with a higher probability. The principal implements
At
A.2 Proof of Lemma 4 to Lemma 6 in Section 3.2
The consumer with first-period preference v i , i = l, h, and evolution of preference k = P, I reveals truthfully his first-period taste with probability m ki . The seller’s Bayesian updated belief of a high valuation type in the second period following the consumer’s first-period choice of option (q 1h , p 1h ) and (q 1l , p 1l ) are
and
respectively. Following the consumer’s first-period choice of (q
1i
, p
1i
), the seller proposes the contract
subject to u 2j ≥ 0 for both j = l, h, and (v h − v l ) ⋅ q 2h ≥ u 2h − u 2l ≥ (v h − v l ) ⋅ q 2l .
By backward induction and the standard methods in static contract theory, the optimal contract
Anticipating the second-period contract
subject to
A.2.1 Relevant Constraints
With
Incentive compatibility of the impersistent consumer is slacking if
If ϕ > 1 − ϕ and
If 1 − ϕ > ϕ and
A.2.2 Contract and Revelation
Scenario 1. ϕ > 1 − ϕ. If ϕ > 1 − ϕ and (IC
Pl
) is slacking, m
Ii
= m
Pl
= 1 so ρ = σ ⋅ ϕ ⋅ m
Ph
+ σ ⋅ (1 − ϕ). With binding (IR
Pl
) and (IC
Ph
), the optimal contract
The contracts and revelation strategy satisfies (IC
Pl
) if
Given the above conditions, the principal implements
At
When Condition 2 is violated and (IC
Pl
), (IC
Ph
), and (IR
Pl
) are all binding, with ρ = σ ⋅ ϕ ⋅ m
Ph
+ (1 − σ) ⋅ ϕ ⋅ (1 − m
Pl
) + σ ⋅ (1 − ϕ), the optimal contract
subject to
The principal implements
At
Scenario 2. 1 − ϕ > ϕ. If 1 − ϕ > ϕ and (IC
Il
) is slacking, m
Pi
= m
Il
= 1 so ρ = σ ⋅ ϕ + σ ⋅ (1 − ϕ) ⋅ m
Ih
. With binding (IR
Pl
) and (IC
Ih
), the optimal contract
The contracts and revelation strategy satisfies (IC
Il
) if
Given the above conditions, the principal implements
At
When Condition 2 is violated and (IC
Il
), (IC
Ih
), and (IR
Pl
) are all binding, with ρ = σ ⋅ ϕ + σ ⋅ (1 − ϕ) ⋅ m
Ih
+ (1 − σ) ⋅ (1 − ϕ) ⋅ (1 − m
Il
), the optimal contract
subject to
The principal implements
At
A.3 Proof of Proposition 4 in Section 5
The consumer truthfully reveals his evolution type with probability r
ki
and truthfully reveals his first-period preference type with probability m
ki
. Denote ω(k, v
i
) as the seller’s Bayesian updated second-period belief of a high valuation type following revelation (k, v
i
) in the first period. By backward induction and the standard methods in static contract theory, the second-period contract
The first line indicates incentive compatibility to truthfully reveal the preference type with a positive probability, conditional on truthful revelation of the evolution type. The second line indicates incentive compatibility to truthfully reveal the evolution type with a positive probability, conditional on truthful revelation of the preference type. The third line indicates incentive compatibility to truthfully reveal both dimensions of types with a positive probability.
Lemma A
If the seller implements truthful revelation of persistency r Pi = 1, the persistent consumer is induced to truthfully reveal his preference and the impersistent consumer is induced to take a mixed revelation of preference, m Pi = 1 and m Ii ≠ 1 for both i = h, l with binding (Ih − 1) and (Il − 1). If the seller implements truthful revelation of impersistency r Ii = 1, the impersistent consumer is induced to truthfully reveal his preference and the persistent consumer is induced to take a mixed revelation of preference, m Ii = 1 and m Pi ≠ 1 for both i = h, l with binding (Ph − 1) and (Pl − 1).
Proof
Suppose that r
Ph
= 1 is implemented. The seller believes in the second period that whoever reveals to be impersistent is certainly not a persistent-high type. If m
Il
= 1, following the revelation of (I, v
h
) in the first period, the seller believes that with probability one the consumer has v
2 = v
l
, so
If r
Pi
= 1 for both preference types is implemented, binding (Ih − 1) and (Il − 1) imply
Combination of (ki − 3) finds
Constraints (Ih − 2) and (Ph − 2) both hold only if
Suppose that r
Ih
= 1 is implemented. The seller believes in the second period that whoever reveals to be persistent is certainly not a descending type. If m
Pl
= 1, following the revelation of (P, v
h
) in the first period, the seller believes that with probability one the consumer has v
2 = v
h
, so
If r
Ii
= 1 for both preference types is implemented, binding (Ph − 1) and (Pl − 1) imply
Combination of (ki − 3) finds
Constraints (Ih − 2) and (Ph − 2) both hold only if
A.3.1 Truthful Revelation of (Im)persistency
If the seller implements revelation strategy r ki = 1 for all (k, i) so that her second-period Bayesian updated belief is completely degenerate on the evolution type. By Lemma A, the consumer takes a completely mixed revelation of preference m ki ≠ 1 with binding (ki − 1) for all (k, i). Summation of the binding (ki − 1) finds
Combination of (ki − 3) finds
which is satisfied only if
Denote
If
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- Frontmatter
- Research Articles
- Screening with Privacy on (Im)persistency
- Quality, Shelf Life, and Demand Uncertainty
- Transfers and Resilience in Economic Networks
- Technology Adoption under Negative External Effects
- Management Centrality in Sequential Bargaining: Implications for Strategic Delegation, Welfare, and Stakeholder Conflict
- Financial and Operational Creditors in Bankruptcy Resolution: A General Equilibrium Approach Under Three Game-Theoretic Division Rules with an Application to India
- Product Differentiation and Trade
- A Theoretical Analysis of Collusion Involving Technology Licensing Under Diseconomies of Scale
- Product Quality and Product Compatibility in Network Industries
- How the Future Shapes Consumption with Time-Inconsistent Preferences
- Notes
- The Strategic Adoption of Environmental Corporate Social Responsibility with Network Externalities
- Strategic Environmental Corporate Social Responsibility (ECSR) Certification and Endogenous Market Structure
- A Note on a Moment Inequality
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Screening with Privacy on (Im)persistency
- Quality, Shelf Life, and Demand Uncertainty
- Transfers and Resilience in Economic Networks
- Technology Adoption under Negative External Effects
- Management Centrality in Sequential Bargaining: Implications for Strategic Delegation, Welfare, and Stakeholder Conflict
- Financial and Operational Creditors in Bankruptcy Resolution: A General Equilibrium Approach Under Three Game-Theoretic Division Rules with an Application to India
- Product Differentiation and Trade
- A Theoretical Analysis of Collusion Involving Technology Licensing Under Diseconomies of Scale
- Product Quality and Product Compatibility in Network Industries
- How the Future Shapes Consumption with Time-Inconsistent Preferences
- Notes
- The Strategic Adoption of Environmental Corporate Social Responsibility with Network Externalities
- Strategic Environmental Corporate Social Responsibility (ECSR) Certification and Endogenous Market Structure
- A Note on a Moment Inequality
