Noisy signaling in discrete time

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Abstract

This paper characterizes the equilibrium set of a dynamic noisy-signaling model in discrete time. A seller privately knows the quality of her asset. She can exert a costly effort to generate stochastic returns. Buyers stochastically arrive over time and, after observing the history of returns, they make price offers. In our model, the equilibrium behavior of the buyers is discontinuous: they only make acceptable (high) offers if the posterior about the quality is above a given threshold. As a result, the recursive nature of the model replicates the discontinuity, giving the equilibrium continuation payoff a complex self-replicating structure that may take the form of a devil’s staircase.

Introduction

This paper characterizes the set of perfect Bayesian equilibria of a dynamic noisy-signaling model in discrete time. We show how the recursive nature of dynamic signaling shapes the equilibrium objects, such as payoffs and effort choices. Our analysis reveals the challenges that endogenizing the informativeness of stochastic signals poses in discretized dynamic models, where local discontinuities may have global effects on the equilibrium behavior.

The analysis focuses on the following dynamic trade model. A seller wants to sell an asset, which can have a low or a high underlying quality, also referred to as type. Only the seller observes the quality of her asset, and she can exert an unobservable effort that generates observable noisy returns. The cost of signaling is type-dependent, and absent signaling motives, the efficient effort differs across types. Short-lived buyers, who stochastically arrive over time, observe the history of returns and make offers to the seller. If the seller accepts an offer, the asset is sold, and the game ends. Otherwise, the seller continues managing her asset until the arrival of the next buyer.

We assume that there are no gains from trade for the low-quality asset, and as a result, buyers never make acceptable offers intended only for the low-quality seller. Also, due to Diamond’s paradox, buyers use their local monopoly power to extract all surplus from the high-quality seller in all equilibria. As a result, the different types of the seller pool on the acceptance decision and the high-quality seller has strict incentives to manage her asset optimally. In equilibrium, separation comes from different effort choices across the types of the seller.

If the cost of effort is high enough, our model features a unique equilibrium outcome. In most periods, the low-quality seller randomizes between exerting her efficient (low) managerial effort versus masquerading her type by exerting a suboptimal (high) effort. Intuitively, if the low-quality seller is supposed to exert a low effort, the signal becomes very informative. In this case, high returns would convince future buyers that the quality of the asset is high, so the seller has incentives to undertake a cost-inefficient (but revenue-generating) effort in order to increase her expected revenue from selling the asset. The reverse is true if she was supposed to exert a high signaling effort: since the signal would be uninformative, the incentive to exert a high effort would be very low. The effort exerted by the low-quality seller in mimicking the high-quality seller is high for intermediate posteriors about the quality, as it is there where beliefs are updated fast. Alternatively, if buyers believe that, with a high probability, the quality of the asset is high, the low-quality seller exerts a low effort for some periods, hoping to sell the asset at a high price before buyers become pessimistic.

We explicitly construct the “equilibrium continuation payoff set”, which is the graph of the correspondence that maps each prior about the quality of the asset being high to the corresponding equilibrium continuation payoffs for the seller of the low-quality asset. Since buyers make acceptable offers only when the posterior is above a given threshold, the continuation payoff correspondence is not lower-hemicontinuous at this threshold. The effect of this discontinuity replicates itself due to the recursive structure of the continuation values, giving the equilibrium continuation set a step structure. In particular, it may take the form of a devil’s staircase (or Cantor function), that is, a non-constant continuous function that is flat almost everywhere. As a result, if the prior (i.e., composition of the market) is endogenized by introducing an entry fee, it is discontinuous in a dense set with respect to the entry fee. Furthermore, in this case, the expected equilibrium managerial effort features an infinite number of peaks and valleys.

When the cost of effort is low, there are equilibria where, even if no buyer arrives for a long time, the type of the seller is not revealed. In this case, an offer from a buyer is high only if recent realizations of the returns are also high, which incentivizes the low-quality seller to exert a high effort. So, pooling is sustained in equilibrium by rewarding high managerial effort with a high probability of receiving a high offer.

Applications. There are a number of economically relevant situations where an agent has inside information not only about a state of the world (or type), but she also can undertake actions which cannot be observed by outsiders, and which stochastically affect some signals. A prominent example is the sale of an asset or a patented business idea by an entrepreneur. In this case, potential buyers may only observe successful prototypes, prizes or patents. Similarly, employers may learn about the productivity of a potential employee only through the observation of signals like her grades during her high education, her successful scholarship applications or her student prizes. Finally, politicians may signal their skills through successfully passing their proposals in the parliament or by wining local elections. In these examples, some outcomes are observable or incentive-compatible to disclose, while effort or other costs may be difficult to observe or credibly report. Also, even though the accumulation of signals over time generates a rich signal about the private persistent information, each individual signal is heavily discretized.

The organization of the rest of the paper is as follows. In Section  2, we set our base model and the main results of the paper. Section  3 discusses the literature and concludes. Appendix A Proofs, Appendix B A continuous-time (diffusion) limit provides the proofs of all the results and a continuous-time limit of our base model.

Section snippets

Setting

Time is discrete, t=0,1,2. There is a (female) seller who wants to sell an asset. The asset is either of low quality (θ=L) or of high quality (θ=H). The quality of the asset, also referred to as type, is known to the seller. The seller discounts future payoffs at a discount factor δ[0,1).

As long as the asset has not been sold, in every period t, the seller decides on the effort et{0,1} put into managing the asset. We assume that high effort can seem low, but low effort cannot seem high. More

Relationship with the previous literature

Recently, there has been some interest in dynamic noisy-signaling models. Examples are Gryglewicz (2009) and Daley and Green (2012) with exogenous (type-dependent) signals, and Dilmé (2015) and Heinsalu (2016) with endogenous (effort-dependent) signals, who analyze continuous-time models where an informative signal progressively reveals information over time about the type of the sender.10

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Cited by (0)

Early versions of this paper were titled “Slowing Learning Down.” I thank George Mailath, Andrew Postlewaite, Qingmin Liu and Stephan Lauermann for their comments and suggestions, as well as the participants at the seminars at University of Pennsylvania, Columbia University, University of Bonn, University of Toulouse, University of Barcelona, the 2014 SAET Conference in Tokyo, the 2014 Stony Brook Festival and the EEA-ESEM 2014 Conference in Toulouse.

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