Elsevier

Games and Economic Behavior

Volume 100, November 2016, Pages 240-256
Games and Economic Behavior

Reminder game: Indirectness in persuasion

https://doi.org/10.1016/j.geb.2016.09.011Get rights and content

Highlights

  • The sender tries to persuade the receiver with limited memory by sending a costless message.

  • Given a private signal, the receiver chooses whether to remember an alternative to form her consideration set.

  • The sender then decides whether to send a costless message that can remind the receiver of the alternative.

  • In spite of the strong conflict of interest, the sender's costless message can be informative in equilibrium.

  • Being more “indirect” conveys more positive information in any informative equilibrium.

Abstract

A seller wants a buyer to choose a good whose value is the seller's private information. The buyer's memory is limited, and she decides whether to remember the good conditional on a signal about the value. The seller then decides whether to send a costless message that can remind the buyer of the good. Since the reminder could convey the seller's private information in equilibrium, whether to send a reminder is a non-trivial question. It is shown that costless messages can be informative in equilibrium in spite of the strong conflict of interest between the players. In any informative equilibrium, silence conveys positive information about the value, whereas the reminder conveys negative information.

Introduction

A decision maker with limited memory often makes a choice from the set of alternatives she remembers. In fact, there is ample evidence that consumers often make a choice from a subset of feasible choices called a “consideration set.” When a decision maker uses a consideration set, one of the important roles of persuasion is making the decision maker aware of a choice. For example, salespersons approach potential clients to make them aware of their products; job applicants contact potential employers to be considered as candidates. However, effective persuasion methods might be subtler than this in practice. Introspection suggests that aggressive persuasion often makes a decision maker cautious or even skeptical. The purpose of this paper is to analyze how and when “indirectness” could be an effective persuasion strategy.

To illustrate the idea of this paper, suppose that a manager looks for an alternative programmer since the current one, programmer A, did not meet his expectation. One day, the manager went to a business exhibition and encountered programmer B. She explained her project that could be useful for the manager's firm, and the manager casually mentioned that the firm might need a new programmer. Since the manager is busy, he remembers programmer B only if he was really impressed. After one week, programmer B has not heard anything from the manager. Is it always a good idea for her to send a reminder to the manager if she wants the job?

If the manager could forget programmer B as a candidate, sending a reminder seems the first step of persuasion. However, the effective persuasion could be subtler. This paper shows that there can be an equilibrium in which a “high quality type” does not send any reminder since it can be interpreted as “lack of confidence.” To see the idea, first, note that if the programmer really has a high ability, the manager is likely to be impressed and thus remembers her. Since the programmer believes that the manager would contact her soon, she comfortably remains silent. On the other hand, if she does not have the high quality, she knows that the manager is likely to be unimpressed and forget her as a candidate. She then sends a reminder since, at least, it can remind the manager of her availability. Note that even though sending a reminder reveals her type, the reminder is still beneficial since the firm could need a decent programmer urgently.

An important observation in the above example is that the subtlety of persuasion arises from the manager's use of the “consideration set.” That is, if he has unlimited memory and remembers programmer B independently of the impression he had, her reminder is just cheap talk under a strong conflict of interest, which cannot convey any information about her quality.

This paper studies a communication game in which a decision maker with limited memory forms a consideration set. There is a seller (he) and a buyer (she). The buyer wishes to replace her status quo choice with an alternative. Since the buyer's memory is limited, it is costly to remember the alternative she encounters. She then decides whether to remember the alternative to maximize her payoff conditional on an exogenous signal about the alternative's value. Without knowing whether the buyer chose to remember the alternative, the seller decides whether to send a costless reminder or remain silent. The buyer then makes a choice from her consideration set.

A novel feature of the model is that the consideration set is determined as an equilibrium memory; the buyer takes into account the two benefits of remembering an alternative when deciding whether to remember the alternative with a memory cost given the seller's communication strategy. The first benefit of remembering an alternative is the option value: if the seller does not send a reminder, remembering the alternative ensures the availability at the decision stage. Since the value of the status quo depends on a taste/technology shock, the buyer has an incentive to keep the availability even if it is less attractive than the status quo at the consideration stage. The second benefit of remembering an alternative is the information value. At first glance, there is no benefit from remembering the alternative if the seller sends a reminder since it makes the alternative always available. However, since the buyer who chooses to remember the alternative also remembers the value of the exogenous signal, the benefit of remembering is thus the informed choice at the decision stage.

Since the seller's payoff function is independent of his message, uninformative equilibria always exist. On the other hand, the buyer's equilibrium consideration strategy depends on the memory cost, the quality of the exogenous signal, and the seller's communication strategy. In the pooling equilibrium in which the seller always sends a reminder, the benefit of remembering the alternative is the information value of the exogenous signal. Thus, the buyer tends to remember the alternative in the equilibrium when the exogenous signal reflects the quality well. On the other hand, in the pooling equilibrium in which the seller never sends a reminder, the benefit of remembering is the option value. Thus, the buyer tends to remember the alternative in the equilibrium when it is “promising” given the exogenous signal.

The main question is whether an informative equilibrium can exist. First, it is shown that if the memory cost is moderate and the positive signal is rarely observed for the low quality, an informative equilibrium exists in spite of the strong conflict of interest between the players. Moreover, whenever an informative equilibrium exists, the buyer remembers the alternative only if she observes a positive exogenous signal. Any informative equilibrium turns out to be a “persuasive silence equilibrium,” that is, the lower types send a reminder while the higher types remain silent. In this equilibrium, the seller with a low quality sends a message to get the buyer's “attention” while the seller with a high quality remains silent believing that the buyer chose to remember his good. Thus, in the equilibrium, the fact that the seller needs to send a message reveals the seller's “lack of confidence” in his quality. In other words, our negative reaction to an aggressive persuasion, which is seemingly emotional, could be consistent with the equilibrium reasoning.

Since pooling and informative equilibria can coexist, I also analyze the equilibrium refinement. Equilibrium dominance based refinements are not effective in this game because of the costless message. Thus, I employ perfect sequential equilibrium (PSE). It is shown that if the positive exogenous signal reflects the quality sufficiently well and the memory cost is “moderate,” PSE favors informative equilibria over the most natural pooling equilibrium.

This paper also analyzes whether a higher type could be more “direct” in a more general setting. To investigate the question, I define the degree of directness of a message as the probability that the message makes the buyer aware of the seller's good. The basic model is generalized so that the seller can select the degree of directness as his choice of a message. It is shown that in any informative equilibrium, the seller with higher quality is never more direct than with a lower quality, analogous to persuasive silence equilibria in the basic setting.

The rest of this paper is organized as follows. In Section 2, the model is introduced. Section 3 analyzes the set of equilibria and characterizes them. In Section 4, the set of equilibria is refined. Section 5 generalizes the basic model. Section 6 provides some discussion.

Related literature. Since the sender's payoff function is independent of messages, the communication game of this paper can be categorized into cheap talk games. On the other hand, the game of the current paper departs from the standard cheap talk game, i.e., Crawford and Sobel (1982), in two aspects. First, in the current paper, the receiver forms a consideration set, and the sender can influence the receiver’s consideration set by sending a reminder. Since the sender cannot influence the consideration set by remaining silent, the two messages, the reminder and silence, are not interchangeable when constructing an equilibrium. In other words, silence is more than a label and has an equilibrium meaning that is specific to silence. Second, unlike in Crawford and Sobel (1982), the current paper focuses on the situation where there is a strong conflict of interest between the sender and the receiver. Since a costless message is often uninformative under a strong conflict of interest, the analysis of strategic communication under a strong conflict of interest often relies on the assumption of “verifiability,” e.g., Milgrom (1981) and Grossman (1981). The current paper shows that if the decision maker has limited memory and forms a consideration set, a costless non-verifiable message could be still informative under a strong conflict of interest.1

The “persuasive silence” result in the current paper is somewhat similar to the counter-signaling in Feltovich et al. (2002) where high and low types send “less” signal than a middle type, i.e., a non-monotonic signaling strategy. However, the two models are qualitatively different. In their model, the single crossing property of the payoff function plays a crucial role in signaling whereas the sender's payoff function is independent of his message in the current paper. As a result, their game always has a separating equilibrium in addition to the counter-signaling equilibrium, whereas the persuasive silence is the only informative equilibrium strategy in the current paper.

Consideration set is an important concept in marketing. The basic idea is that decision makers follow the following two-step procedure: (i) decision makers form a small set of options that they will consider for their consumption decision; (ii) they evaluate the options in this set and choose the one they prefer the most. Nedungadi (1990), Alba et al. (1991), and Lynch and Srull (1982) explain the role of limited memory in consideration sets. A contribution of the current paper is formalizing the memory-based consideration set as an equilibrium phenomenon.

In economics, there are two types of papers on consideration sets. The first type focuses on the individual choice problem to understand the property of consideration set procedures, e.g., Manzini and Mariotti (2007) and Masatlioglu et al. (2009). The second type analyzes economic interactions when agents use consideration sets, e.g., Eliaz and Spiegler, 2011a, Eliaz and Spiegler, 2011b. While the current paper belongs to the second type, there are two major differences: first, their papers treat consideration sets as a behavioral assumption, whereas the consideration set is determined as an equilibrium memory in the current paper. Second, their papers focus on how the seller can effectively grab the buyer's attention, whereas the current paper analyzes how attention-grabbing could backfire when the quality of the good is the seller's private information.

The current paper also contributes to the literature on optimal memory. In Dow (1991) and Rubinstein (1998), an agent optimizes what to remember under a memory constraint. There are two major differences between their models and the current model. First, the nature of limited memory in the current paper is different from that in their models. Their models consider imperfect memory about the value of a variable; that is, their buyers remember the existence of an alternative while they forget the price of the alternative. On the other hand, in the current paper, another type of imperfect memory plays a key role; that is, the buyer becomes unaware of a choice she encountered. The difference between the two types of imperfect memory has been known in computer science, e.g., Hans van Ditmarsch et al. (2009) whereas the difference has not been emphasized in economics. The second difference is that the current paper analyzes an optimal memory in a strategic interaction; that is, the memory is determined in equilibrium. The idea of equilibrium memory can also be found in Benabou and Tirole (2002). In their model, a time-inconsistent agent plays an intrapersonal signaling game. Since the future-self needs a high level of self-confidence to overcome her present bias, the current-self could choose to forget a negative event in order to protect self-confidence in equilibrium. In their model, the memory is chosen to deceive the future-self whereas, in the current paper, the memory is determined to economize the limited memory given the seller's communication strategy.

From the perspective of marketing, the current model can be categorized into a model of “soft sell.” There are some marketing papers that explain how “soft sell” could work in different settings. Mayzlin and Shin (2011) show that a seller with a high quality product could use uninformative advertising in a signaling game with verifiable messages. On the other hand, the current paper considers a communication game in which the seller's message is not verifiable, and the informational content of a message is determined only in equilibrium. Another relevant paper in marketing is Miklós-Thal and Zhang (2013). They show that a seller can choose a low level of marketing activity, i.e., “demarketing,” in a two-period signaling model. Demarketing is a pooling equilibrium in their paper whereas it is explained as an informative equilibrium in the current paper; that is, the buyer's skepticism emerges as an equilibrium phenomenon in the current paper.

Section snippets

Model

There is a seller (he) who wishes to sell a good x1 to a buyer (she). The value of x1, denoted by θΘ=[0,1], is the seller's private information. It is assumed that θ is drawn from distribution F(θ), which has a continuous density f with supp(f)=Θ. The buyer is considering to replace her status quo option x0 with an alternative. When the buyer encounters x1, she observes an exogenous signal sS={0,1}, which is correlated with θ. Specifically, the probability that the buyer observes a positive

Preliminary analysis

Before stating the results of this section, it might be useful to clarify the conditions of equilibrium strategy profile (r,σ,y).

Consider stage 3. First, if a history includes z={x0} and a=N, the buyer's consideration set is X={x0}, and she has no decision to make. Second, if a history includes z={x0,x1}, her consideration set is X={x0,x1}, and the buyer needs to infer θ. Then, PBE requires the buyer to update her belief about θ conditional on a and s given σ. That is, the expected value of θ

Refinement

This section provides the equilibrium refinement. Since the seller's payoff function is independent of the choice of a message, refinements based on “equilibrium dominance” are not effective in this game. Thus, I employ perfect sequential equilibrium (PSE) (Grossman and Perry, 1986), which refines the set of perfect Bayesian equilibria (PBE) by restricting off-equilibrium beliefs to be “credible.” Specifically, once a deviation has occurred, the buyer tries to rationalize the deviation by

Generalization

In the basic model, it is shown that if the buyer decides whether to remember an alternative given her memory cost, sending a reminder is a more negative signal than remaining silent in any informative equilibrium. One natural question is whether this result is robust when the seller can choose the level of “directness” of his message; that is, is it possible to be more persuasive by being more “direct”?

One way to define the degree of directness of a message is the probability that the message

Some extensions as a model of advertising

As in the illustrative example in Section 1, the basic idea of this paper is not specific to the seller-buyer context. However, since advertising is one of the applications, it might be useful to discuss two potential extensions as a model of advertising.

First, since advertising is often costly, suppose that the seller has to pay some cost to send a reminder. It can be shown that if the cost is sufficiently small given the payoff from selling the good, the result of this paper is preserved.

Appendix

This section provides the omitted proofs.

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  • Cited by (1)

    I thank two anonymous referees, Jacob Glazer, Hsueh-Ling Huynh, Barton Lipman, Jonathan Newton, John Wooders, Andriy Zapechelnyuk, and seminar participants at IUPUI, Keio University, Monash University, Queen Mary University of London, Syracuse University, University of Technology Sydney for helpful comments.

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