A model of gradual information disclosure
Introduction
As Hayek (1945) claimed, the central issue in a variety of economic and social interactions is how to utilize information efficiently.2 Theoretical and practical developments have contributed many effective and commonly used tools to help achieve this efficiency, including patent protection, contractual enforcement, and property rights allocation. However, these tools are often unavailable, resulting in a hold-up problem that discourages the utilization of information. For example, if an initially uninformed party has learned valuable information from another party, his incentive to pay for the information weakens, as now himself is informed (Arrow, 1962). We provide an equilibrium analysis of the utilization of information in this paper and address how a process of gradual information disclosure helps to alleviate the hold-up problem.
To gain better understanding of this study, consider the scenario that a financial expert, who knows about some investment opportunities, seeks cooperation from an investor to maximize the value of her information. Being aware that the investor may be more motivated to seize her information rather than establish a cooperative relationship, the financial expert may strategically slow down the release of information to reduce the risk of being exploited. As the uncertainty about the investor's motives is gradually resolved, the financial expert eventually becomes confident enough to release all information remaining.
For another example, consider the scenario that an international auto company aims to enter the market of a developing country by cooperating with a local company and transferring its technology. However, because this country lacks an established law system, the auto company's technology is under the risk of being leaked. In response, the auto company may choose to transfer some preliminary technology first, which provides an opportunity to learn about its partner. Contingent on the local company's reactions to these preliminary transfers, the auto company can decide to transfer more technology or exit the market.
These scenarios share some similarities. First, information is divisible and can be transmitted or disclosed in parts, which allow the values of those parts to be realized separately. For instance, preliminary technology can also generate revenues for the auto company. Second, contractual enforcement on information disclosure may be unreliable or even absent, which causes the potential hold-up problem. As a result, the parties' interactions must be self-enforcing, as in the case of the financial expert and the investor. Finally, the cost of time can be an important factor that affects the utilization of information. Investment opportunities lose their values rapidly over time in a volatile market, whereas the auto company may lose market shares to its competitors if it delays the transfer of its technology.
Taking the first scenario as the prominent example in this paper, we develop a dynamic game that examines the optimal utilization of information. A financial expert is endowed with an amount of private information that is valuable in the financial market, but she can only utilize it inefficiently on her own. An investor has the potential to maximize the value of the expert's information, but he lacks the relevant information. As a result, efficient utilization of information requires information disclosure between the parties. Interactions go as follows. In each period, the expert may choose to self-use some information, in which case the investor is inactive. Alternatively, the expert may disclose some information to the investor, who can then either cooperate, which is mutually beneficial, or betray, which benefits only himself. The investor is either aligned or biased. An aligned investor always cooperates, whereas a biased investor may strategically betray. The financial expert is initially uncertain about the investor's type, so she must learn about it over time. Given the time discounting between periods, the expert's goal is to optimize the payoff from her information when external contracts are infeasible.
The financial expert faces two main trade-offs in determining the utilization of information. The first trade-off is whether information should be kept for self-use or disclosed to the investor. Although self-use of information yields an efficiency loss, the rents captured by the biased investor must be considered when choosing to disclose information. If the expert chooses to disclose information, then the second trade-off is the timing of information disclosure. A longer process of information disclosure is more costly in time, but it safeguards more information from the biased investor. A shorter process of information disclosure saves on the cost of time, but it provides a better opportunity for the biased investor to betray.
We characterize explicitly the joint dynamics of the expert's utilization of information and the investor's type revelation. In equilibrium, if the expert's prior belief that the investor is aligned is at least moderately large, the expert discloses her information gradually that induces the biased investor to reveal his type. The process of information disclosure reaches an end either if the investor is revealed to be biased, after which the expert self-uses the information remaining, or if the expert's belief becomes large enough, after which the expert discloses all information remaining to the investor at once. Particularly, the larger is the expert's prior belief, the shorter is the process of information disclosure. On the other side, if the expert's prior belief is relatively small, she self-uses all her information. This characterization gives a clear insight about when and how the expert can alleviate the hold-up problem exerted by the biased investor. In addition, we show that the equilibrium of this game is “essentially” unique.
We consider an extension in which the expert can commit to a deadline for the process of information disclosure. When the deadline period is reached, gradual information disclosure is no longer viable to the expert. Such a restriction lowers the expert's ex post payoff. However, expecting that the expert will disclose all information remaining in the deadline period even her posterior belief is not sufficiently high, the biased investor can reduce the probabilities of betrayal in the periods before the deadline period. This reduction in the probabilities of betrayal, in turn, increases the expert's ex ante payoff. We show that, with moderate prior beliefs, the expert's equilibrium payoff is strictly increased if she commits to a proper deadline.
We also examine the effects on the value of information if the expert can fully commit to a process of information disclosure. In equilibrium, the optimal process with commitment has a property that the biased investor is induced to cooperate in all periods except the last one. In other words, the amount of information disclosed in the final period serves as a reward to the biased investor for exchanging his cooperation up to that period. The expert's problem in determining the optimal process is to trade off between the scale of the reward and the cost of time to deliver it. Consequently, by fully committing to a proper process of information disclosure, the expert's payoff can also be increased.
In addition to the examples aforementioned, this game is also applicable to many other situations. For instance, if the valuable information refers to research ideas, then the game can address the building and termination of relationship between scientists. More broadly, if what the expert possesses are some valuable assets, the game can be interpreted as a contribution game, in which one party contributes inputs and another party contributes productivity.
Our study on the divisibility of the expert's information and its implications for relationship dynamics is most closely related to Baliga and Ely (2016). Baliga and Ely (2016) consider a model in which a principal uses torture to extract information from a suspect who may or may not be informed. The authors find that, because of the principal's lack of full commitments, the informed suspect concedes to release information gradually. The main feature that distinguishes our work from Baliga and Ely (2016) is that, while in their paper gradual information disclosure arises due to the principal's inability to extract all information at once, in our paper the expert can, but in equilibrium she optimally chooses not to, disclose all information in a single period.
Hörner and Skrzypacz (2016) develop a model in which an agent who knows his competence for a project can reveal this information to a firm in exchange for payments. They show that, without any cost of time, revealing information gradually increases the competent agent's payoff and the process of information revelation exhausts all the time periods. Our work reaches the similar result that gradual information disclosure could be beneficial to the information possessor, but the underlying setup is quite different. Specifically, while the discounting cost and the outside option of self-use of information are crucial to our findings, they have limited or no roles in Hörner and Skrzypacz (2016).
Anton and Yao (1994) show that an inventor can appropriate a sizable share of an idea's market value from a buyer by threatening to reveal the idea to a competitor in the event that the buyer defaults. Anton and Yao (2002) find that a seller can use partial disclosure to signal the full value of an idea and benefit from the buyers' competition for this idea. These papers allow enforceable contracts, but the timing structure of information disclosure is pre-determined. In contrast, in our paper there is no explicit contracts, and the timing of information disclosure is endogenized.
Gradualism also appears as the means to alleviate the hold-up effects in the literature on contribution games, including Admati and Perry (1991), Gale (2001), Lockwood and Thomas (2002), Marx and Matthews (2000), and Compte and Jehiel (2004). However, unlike our work, in these papers there is no asymmetric information about the players' types. Watson (1999, 2002) studies a contribution game with two-sided incomplete information and shows that the relationship between partners generally starts small and grows over time. In these papers, the amounts of contributions along the time horizon are pre-determined before the game starts. As a result, the players' actions at any time are binary: cooperate or betray. In our paper, the expert has infinitely many choices in each period, and the amounts of information disclosure are determined during the process of play.
The gradual revelation of the investor's type is analytically related to the literature on reputation games, including Kreps and Wilson (1982), Milgrom and Roberts (1982), and Fudenberg and Levine (1989, 1992), as well as the literature on the war of attrition with incomplete information, including Abreu and Gul (2000), and Damiano et al. (2012). In these papers, the stage game is repeated and the only variables that change over time are the beliefs about the informed players' types. In our paper, both the belief and the stage game vary over time as the amount of information remaining decreases.
Section snippets
Model
We study a dynamic game with two players: a financial expert (she or E) and an investor (he or I). At the beginning of the game, the expert is endowed with an amount of information, which refers to some investment opportunities that can be exploited in the financial market. A key feature regarding this information is that, although the amount is common knowledge between the players, the detailed contents of the information are initially known only to the expert. Thus, the investor has
Equilibrium analysis
The equilibrium we characterize in this section has some qualitative features as follows. (i) The expert self-uses all information at once if her belief that the investor is aligned falls below a relatively small cut-off value. (ii) The expert discloses all information to the investor at once if her belief goes above a relatively large cut-off value. (iii) For her belief being between these two cut-off values, the expert discloses information to the investor gradually.
These features are
Extensions and discussions
We consider some extensions in this section. Our main focus is how the expert can increase the value of her information if she has some or full commitment power in determining the process of information disclosure.
Conclusions
In this paper, we study a dynamic game in which an expert can utilize her private information either by self-use or by information disclosure to an investor. The investor has the potential to realize the value of the expert's information in a more efficient way, but he may have incentive to hold up the expert for his own benefit. Our main finding is that, in the essentially unique equilibrium, the expert can mitigate the hold-up problem by disclosing her information gradually. We also
References (19)
Monotone games with positive spillovers
Games Econ. Behav.
(2001)- et al.
Reputation and imperfect information
J. Econ. Theory
(1982) - et al.
Predation, reputation, and entry deterrence
J. Econ. Theory
(1982) Starting small and renegotiation
J. Econ. Theory
(1999)Starting small and commitment
Games Econ. Behav.
(2002)- et al.
Bargaining and reputation
Econometrica
(2000) - et al.
Joint projects without commitment
Rev. Econ. Stud.
(1991) - et al.
Expropriation and inventions: appropriable rents in the absence of property rights
Am. Econ. Rev.
(1994) - et al.
The sale of ideas: strategic disclosure, property rights, and contracting
Rev. Econ. Stud.
(2002)
Cited by (0)
- 1
I thank the Editor, Sushil Bikhchandani, the Advisory Editor, and an anonymous referee for helpful comments. I'm also grateful to Marcus Berliant, Philip Dybvig, David Levine, Fei Li, Qingmin Liu, John Nachbar, Maher Said, and participants at various seminars and conferences for their valuable suggestions.