Boosting trust by facilitating communication: A model of trustee investments in information sharing

Abstract

Trust problems hamper many social and economic exchanges. In such situations, there are often institutions that enable trustors to share information on the performance of trustees. While the benefits of such institutions have been researched extensively, little is known about their emergence. This article presents a game-theoretic model for the understanding of investments by trustees in establishing information sharing between trustors. The model allows for a simultaneous analysis of investments in and effects of institutions for information sharing. It captures two mechanisms by which a trustee’s investment can promote trust. First, a trustee’s investment in establishing information sharing can enable network effects that facilitate trust and trustworthiness. Second, it can promote trust by serving as a signal of intrinsic trustworthiness. The analysis of the model implies predictions for how characteristics of the interaction situation affect whether these mechanisms motivate a trustee to establish information sharing.

Introduction

Exchange often requires that one party—the “trustor”—exposes herself to the risk of abuse by the other party—the “trustee” (Coleman, 1990; Dasgupta, 1988; Kreps, 1990). Buying a used car is a well-known example. The buyer (trustor) does not know the history of the car and the seller (trustee) may benefit from concealing important information. If the buyer places trust and buys, she risks paying too much. Trust problems are also salient in online trade: the buyer typically has to pay upfront without having any real guarantee that the seller will ever ship the product. Such trust problems pose a social dilemma. Both parties would benefit from fair exchange, but the exchange may not take place because the trustor has reason to mistrust the trustee (Akerlof, 1970; Coleman, 1990; Kollock, 1998). Fortunately, institutions that enable trustors to share information about trustees—as, for example, word-of-mouth networks or online reputation systems—allow actors to escape the dilemma (Buskens and Raub, 2002, 2013; Cook et al., 2009; Feinberg et al., 2014; Milinski, 2016; Resnick et al., 2000).

While it is well understood how institutions for information sharing facilitate exchange, little is known about the emergence of such institutions. There is a general theoretical idea that if certain institutions or networks have value for actors to achieve their goals, actors should be likely to set up such institutions or networks (Flap, 2004; Lin, 2002: Chap. 8; Prendergast, 1999). It also suggests itself that many institutions for information sharing have been set up purposively to facilitate exchange. In line with this idea, Guseva and Rona-Tas (2001) suggest that trust problems in credit markets led to the installment of credit bureaus in the United States and that these trust problems led Russian bankers to form extended word-of-mouth networks (see Klein, 1997 for related case studies). Still, there is a lack of theoretical models for the understanding of investments in establishing institutions for information sharing as a means to facilitate trust and trustworthiness. This article addresses this gap in the literature. It investigates theoretically under what circumstances it is likely that a trustee makes a costly investment to enable information sharing between trustors. By treating institutions for information sharing as endogenous, this article furthermore brings to attention effects of such institutions that have previously been overlooked. The article thus contributes to the understanding of the emergence and effects of institutions for information sharing.

The game-theoretic model presented in this article captures two mechanisms by which a trustee’s investment in establishing information sharing can promote exchange. First, there are the well-known network effects. Information sharing facilitates exchange by enabling trustors to learn about trustees from the experiences of other trustors. In addition, it gives trustees incentives to act honestly because developing a bad reputation could inhibit many future exchanges or require a trustee to lower his price. ¹ These network effects are well-established theoretically as well as empirically (Abraham et al., 2016; Bohnet et al., 2005; Bohnet and Huck, 2004; Bolton et al., 2004; Buskens et al., 2010; Buskens and Raub, 2002, 2013; Cook and Hardin, 2001; DiMaggio and Louch, 1998; Feinberg et al., 2014; Frey and Van de Rijt, 2016; Hillmann and Aven, 2011; Huck et al., 2010; Przepiorka, 2013; Snijders and Weesie, 2009; Sosis, 2005). In the current study, I model how these network effects can motivate a trustee to set up an institution for information sharing between trustors.

Second, the model shows that a trustee’s investment in establishing information sharing can promote trust by serving as a signal of intrinsic trustworthiness. Some people are intrinsically trustworthy and behave well also in the absence of contextual factors that deter trust abuse. ² But how could a trustee convince a trustor that he is intrinsically trustworthy given that also those with bad intentions want to be trusted? Signaling theory (e.g. Bliege Bird and Smith, 2005; Gambetta, 2009; Przepiorka and Berger, 2017; Spence, 1973; Zahavi, 1975) maintains that taking some costly, observable action can allow an actor to credibly communicate unobservable characteristics, such as intrinsic trustworthiness (Bacharach and Gambetta, 2001; Barclay, 2004; Bliege Bird and Power, 2015; Gambetta and Przepiorka, 2014; Paik and Woodley, 2012; Przepiorka and Berger, 2017; Przepiorka and Diekmann, 2013; Raub, 2004). This article shows that a trustee’s investment in establishing information sharing can serve as such a signal of intrinsic trustworthiness.

I model these network and signaling effects and how they can motivate a trustee to invest in establishing information sharing using the framework of finitely repeated Trust Games (TGs) with incomplete information (see, e.g. Buskens et al., 2017; James, 2002). The model focuses on a scenario in which several trustors interact with the same trustee. The trustors do not know whether the trustee is of the friendly type or of the opportunistic type. A trustee of the friendly type is intrinsically trustworthy and has no incentive to abuse trust. A trustee of the opportunistic type has an incentive to abuse trust, at least in the short run. At the beginning of the game, the trustee can make a costly investment to set up an institution for information sharing between the trustors. I identify under what conditions there exist (sequential) equilibria in which the trustee pledges this investment to allow the network effects or to signal that he is of the friendly type.

To my knowledge, Frey et al. (2015) and Raub et al. (2013) are the only studies that investigate how network effects can motivate actors to establish information sharing. The paper by Frey et al. (2015) is a companion paper to the current study. It investigates in a very similar game-theoretic model investments in information sharing by trustors rather than trustees. Raub et al. (2013) study a game-theoretic model that allows also investments by trustees, but their model assumes complete information about the incentives of trustees and indefinite repetition. A core result of Frey et al. (2015) and Raub et al. (2013) is that there is an inverse U-shape in the relation between the incentives to invest in establishing information sharing and the size of the trust problem. The incentives to invest are small for trust problems that are very small or very large, while the incentives to invest are large for trust problems of intermediate size. The results of this article regarding investments motivated by the network effects are in line with this prediction.

That a trustee’s investment in establishing information sharing can serve as a signal of intrinsic trustworthiness has not been studied previously. This innovation leads to new predictions on investments in information sharing as well as effects of information sharing. Furthermore, the article contributes to the broader literature on signaling trustworthiness (see Przepiorka and Berger, 2017 for a recent overview). I analyze the model for two types of friendly trustees: friendly trustees who would suffer from an internal sanction if abusing trust and friendly trustees who are trustworthy because they receive an internal reward if honoring trust. These scenarios subsume various motivations for intrinsic trustworthiness, such as inequity aversion, guilt aversion, altruism, or warm-glow. ³ We will see that the possibility to signal intrinsic trustworthiness depends crucially on what motivation leads to intrinsic trustworthiness and that results concerning signaling equilibria generalize beyond the specific context of investments in information sharing.

The game

This section describes the game $\mathrm{\Gamma }$ and the difference between the two versions that I consider—the version in which friendly trustees are guilt-avoiding trustees $({\mathrm{\Gamma }}^{\mathrm{ga}})$ and the version in which friendly trustees are reward-seeking trustees $({\mathrm{\Gamma }}^{\mathrm{rs}})$ . Throughout, I use the superscripts ga and rs to let notation refer specifically to either of these versions of $\mathrm{\Gamma }$ . I omit these superscripts if I refer to both versions simultaneously.

Actors, moves, and information

$\mathrm{\Gamma }$ has two types of actors: $m\ge 2$ trustors and one trustee. It proceeds as illustrated in Figure 1. First, the trustee’s type is determined and the trustee learns his type. Then, the trustee decides whether to invest in establishing information sharing. Finally, the trustors and the trustee interact in TGs.

Figure 1.

Timeline of $\Gamma$ .

More specifically, in period 0.1, a random move of a pseudo-actor Nature determines the type of the trustee. With probability $\pi$ the trustee is of the friendly type (the intrinsically trustworthy type) and with probability $1-\pi$ he is of the opportunistic type, where $0<\pi <1$ . Friendly and opportunistic trustees differ in their payoffs in the TG as described below. The probability $\pi$ is common knowledge (each actor knows $\pi$ , knows that the others know that (s)he knows $\pi$ , and so on). The trustee also knows his actual type, but the trustors are not informed about the trustee’s type.

In period 0.2, the trustee decides whether to invest the cost c to establish an institution for information sharing between the m trustors.

Finally, one binary TG is played in each of the periods 1 to mN. In every TG, the trustor at play, first, decides whether to place trust. If the trustor does not place trust, the TG ends. If the trustor places trust, the trustee chooses whether to honor or abuse trust and the TG ends.

The order in which the trustors play their TGs is random but each of the m trustors plays precisely N TGs with the trustee. The analysis does not require that the order of interaction is known in advance. It suffices to assume perfect recall (i.e. that players never forget information they once acquired), so that actors remember information on outcomes in specific periods and who played in which period.

If the trustee established information sharing in period 0.2, all trustors are always informed about the choices that were made in all past TGs. Information is shared truthfully among the m trustors directly after every TG. By contrast, if the trustee did not establish information sharing, each trustor knows only the outcomes of her own past TGs. I will refer to the continuation of $\mathrm{\Gamma }$ after the trustee did not establish information sharing in period 0.2 as (continuation game) ${\mathrm{\Gamma }}^{-}$ and to the continuation of the game after the trustee established information sharing as (continuation game) ${\mathrm{\Gamma }}^{+}$ . Which continuation game is played, that is, whether the trustee has established information sharing, is common knowledge.

Payoffs

The payoffs in any TG are as follows. If the trustor i who is at play does not place trust, trustor i and the trustee earn the payoffs $P_1$ and $P_2$ , respectively. If i places trust, i gets $R_1>P_1$ if the trustee honors trust and $S_1<P_1$ if the trustee abuses trust. The trustee gets $R_2>P_2$ if he honors trust and $T_2>R_2$ if he abuses trust. These are the material payoffs and I assume that trustors and trustees of the opportunistic type are exclusively concerned with own material payoffs. An opportunistic trustee would thus maximize his utility in the focal TG by abusing trust. Friendly trustees, however, have non-material motivations such that if i places trust, a friendly trustee maximizes his utility by honoring trust. I consider two scenarios with respect to the non-monetary motives of friendly trustees.

• In ${\mathrm{\Gamma }}^{\mathrm{ga}}$ , a trustee of the friendly type is a guilt-avoiding trustee. A guilt-avoiding trustee has no incentive to abuse i's trust because if he abuses trust, he suffers from an internal sanction $\theta$ that offsets the material incentive to abuse trust $(T_2-\theta <R_2)$ .

• In ${\mathrm{\Gamma }}^{\mathrm{rs}}$ , a trustee of the friendly type is a reward-seeking trustee. A reward-seeking trustee has no incentive to abuse i's trust because he receives an internal reward $\upsilon$ if he honors trust such that $T_2<R_2+\upsilon$ .

An actor’s total payoff for $\mathrm{\Gamma }$ (i.e. ${\mathrm{\Gamma }}^{\mathrm{ga}}$ as well as ${\mathrm{\Gamma }}^{\mathrm{rs}}$ ) is the sum of the undiscounted payoffs that the actor earns in the TGs in the periods 1 to mN minus—in case of the trustee—the cost c of an investment in establishing information sharing if such an investment has been pledged.

Analysis of the game

The aim of the analysis is to identify under what conditions there exist equilibria in which the trustee invests in establishing information sharing. A rational trustee will weigh the cost of investment c against the expected “return on investment,” namely, his expected payoff in ${\mathrm{\Gamma }}^{+}$ minus his expected payoff in ${\mathrm{\Gamma }}^{-}$ . Section “Behavior and payoffs in the TGs” specifies the equilibria and expected payoffs of ${\mathrm{\Gamma }}^{+}$ and ${\mathrm{\Gamma }}^{-}$ . This section shows that network effects can lead to more trust and higher earnings in ${\mathrm{\Gamma }}^{+}$ than in ${\mathrm{\Gamma }}^{-}$ . Furthermore, we shall see that the trustee’s expected payoff also tends to be higher if, at the beginning of the TGs, the trustors are more convinced that the trustee is of the friendly type. Section “Investments in establishing information sharing” investigates under what conditions a trustee invests in establishing information sharing to benefit from the network effects or to convince the trustors that he is of the friendly type.

Throughout, I focus on sequential equilibria—a refinement of Nash equilibrium for games with incomplete information (Kreps and Wilson, 1982b; see Rasmusen, 1994: Chap. 6, for a textbook). For the analysis of investments in information sharing, I additionally employ the passive conjectures refinement (Rasmusen, 1994; Rubinstein, 1985), which will be explained below. I will often refer to sequential equilibria and sequential equilibria involving the passive conjecture for brevity as “equilibria.”

Behavior and payoffs in the TGs

We can lend from earlier work for the analysis of ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ . Camerer and Weigelt (1988) have analyzed reputation building in sequential equilibria in finitely repeated TGs with incomplete information (see also Anderhub et al., 2002; Bower et al., 1997). In a nutshell, if one trustor and one trustee play a finitely repeated TG together, there is typically a sequential equilibrium such that trust is placed and honored in early periods and, then, there is an “endgame” in which trust may break down. In early periods also a trustee of the opportunistic type typically honors trust, balancing the short-term incentive to abuse trust and the long-term benefit of having a reputation for being of the friendly type. Toward the end of the game, however, the long-term benefit of having a good reputation decreases, the opportunistic trustee begins to abuse trust with positive probability, and trust may break down.

If there is no information sharing $({\mathrm{\Gamma }}^{-})$ , this equilibrium applies to the N TGs of each trustor separately, as if there were no other trustors who interact with the trustee (cf. Buskens, 2003; Frey et al., 2015). By contrast, in ${\mathrm{\Gamma }}^{+}$ , where the trustors can learn from each other’s experiences with the trustee, this sequential equilibrium applies to the total number of mN TGs as if it was one single trustor who plays all these TGs (Buskens, 2003; Camerer and Weigelt, 1988; Frey et al., 2015). ⁴ Lemma 1 in Appendix 1 specifies the sequential equilibria of the continuation games ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ formally. In the following, I describe the sequential equilibrium in more detail, focusing on the interactions between one trustor and the trustee in ${\mathrm{\Gamma }}^{-}$ . I also describe how the equilibrium applies to ${\mathrm{\Gamma }}^{+}$ and specify the expected payoffs of the trustee for ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ . Note that the sequential equilibria of ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ are the same in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}}$ . The reason for which a friendly trustee always honors trust is irrelevant for the equilibrium of the TGs. ⁵

We will need some further notation. Let ${\pi }^{-}$ and ${\pi }^{+}$ denote the beliefs of the trustors about the trustee’s type that enter the continuation games ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ , respectively. In addition, let $\mathrm{RISK}:=(P_1-S_1)/(R_1-S_1)$ measure the risk a trustor incurs when placing trust and let $\mathrm{TEMP}:=(T_2-R_2)/(T_2-P_2)$ measure an opportunistic trustee’s temptation to abuse trust (see Snijders, 1996).

Behavior and payoffs in ${\mathrm{\Gamma }}^{-}$

If there is no information sharing, what happens in the TGs between one trustor and the trustee is independent of what happens in the other trustors’ TGs. In ${\mathrm{\Gamma }}^{-}$ , the sequential equilibrium in each trustor i's TGs is, therefore, the same as if there was only one trustor playing N TGs with the trustee. Anderhub et al. (2002) and Camerer and Weigelt (1988) provide detailed and accessible descriptions of this equilibrium. The equilibrium evolves typically over three phases. First, trustor i places trust and the trustee honors trust until trustor i and the trustee have ${\tau }^{-}$ TGs left to play together, where ${\tau }^{-}$ is the smallest integer for which it holds that ${\pi }^{-}\ge \mathrm{RIS}{K}^{{\tau }^{-}}$ , that is

$${\tau }^{-}:=\lceil \frac{\log {\pi }^{-}}{\log \mathrm{RISK}}\rceil \mathrm{for}0<{\pi }^{-}<1$$

(1)

Then, in the next TG of trustor i, the trustee—if he is of the opportunistic type—starts to randomize. Still one TG later, trustor i begins to randomize too, placing trust with probability TEMP. In this second phase, the “randomization phase,” trustor i and the trustee (if he is of the opportunistic type) randomize until i does not place trust or until the trustee abuses $i’\mathrm{s}$ trust. ⁶ After the first instance that i did not place trust or that the trustee abused $i’\mathrm{s}$ trust, the third and final phase starts: i does not place trust anymore. If i still places trust in her last TG, an opportunistic trustee abuses trust.

The equilibrium does not always evolve over all these phases. If the trust problem is very severe, there may be no trust at all. As equation (1) shows, ${\tau }^{-}$ tends to be larger (the randomization tends to start earlier) if ${\pi }^{-}$ is smaller or RISK is larger. If ${\pi }^{-}$ is so small or RISK so large that ${\tau }^{-}=N$ , the opportunistic trustee randomizes already in the first TG with trustor i. If ${\pi }^{-}$ and RISK are even such that ${\tau }^{-}>N$ , i never places trust (because, given the parameters, there are not enough interactions with trustor i for the trustee to start building a reputation).

The equilibrium also does not exhibit a randomization phase if ${\pi }^{-}=0$ or ${\pi }^{-}=1$ . In these cases, it is as if trustor i had complete information about the trustee’s incentives, and ${\tau }^{-}$ is not specified by equation (1). If i is totally convinced that the trustee is of the opportunistic type $({\pi }^{-}=0)$ , the logic of backward induction implies that i should never place trust, irrespectively of how long the game is. Therefore, we say that ${\tau }^{-}=\infty$ if ${\pi }^{-}=0$ . In contrast, if ${\pi }^{-}=1$ , i should place trust even in a one-shot TG. In the repeated game, i will then in equilibrium place trust in every TG. If the trustee is indeed of the friendly type, he always honors trust. If—contrary to i's belief—the trustee is of the opportunistic type, he waits with abusing trust until i's last TG. This is the same equilibrium course of play as if $\mathrm{RISK}<{\pi }^{-}<1$ , and thus, by equation (1), ${\tau }^{-}=1$ . We, therefore, say that ${\tau }^{-}=1$ if ${\pi }^{-}=1$ .

We can now specify the expected payoffs of friendly and opportunistic trustees in ${\mathrm{\Gamma }}^{\mathrm{ga}-}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}-}$ .

Proposition 1

The expected payoffs of a friendly trustee in ${\mathrm{\Gamma }}^{\mathrm{ga}-}$ $(U_F^{{\mathrm{\Gamma }}^{\mathrm{ga}-}})$ and ${\mathrm{\Gamma }}^{\mathrm{rs}-}$ $(U_F^{{\mathrm{\Gamma }}^{\mathrm{rs}-}})$ are

$$U_F^{{\Gamma }^{ga-}}=\{\begin{array}{cc}m((N-{\tau }^{-})R_2+{\tau }^{-}{P}_2+(T_2-P_2)(1-TEM{P}^{{\tau }^{-}}))& if{\tau }^{-}\le N\hfill \\ mN{P}_2\hfill & if{\tau }^{-}>N\end{array}$$

$$U_F^{{\Gamma }^{rs-}}=\{\begin{array}{cc}m\left(\left(N-{\tau }^{-}\right)(R_2+\nu )+{\tau }^{-}{P}_2+(T_2-P_2)\frac{R_2+\nu -P_2}{R_2-P_2}(1-TEM{P}^{{\tau }^{-}})\right)& if{\tau }^{-}\le N\\ mN{P}_2\hfill & if{\tau }^{-}>N\end{array}$$

The expected payoffs of an opportunistic trustee in ${\mathrm{\Gamma }}^{\mathrm{ga}-}$ $(U_O^{{\mathrm{\Gamma }}^{\mathrm{ga}-}})$ and ${\mathrm{\Gamma }}^{\mathrm{rs}-}$ $(U_O^{{\mathrm{\Gamma }}^{\mathrm{rs}-}})$ are

$$U_O^{{\mathrm{\Gamma }}^{\mathrm{ga}-}}=U_O^{{\mathrm{\Gamma }}^{\mathrm{rs}-}}=\{\begin{array}{cc}m((N-{\tau }^{-})R_2+T_2+({\tau }^{-}-1)P_2)\hfill & \mathrm{if}{\tau }^{-}\le N\hfill \\ \multicolumn{1}{c}{\mathrm{mN}{P}_2}& \mathrm{if}{\tau }^{-}>N\hfill \end{array}$$

The proofs of all propositions are in Appendix 1. For the scenario that ${\tau }^{-}\le N$ , one can think of the trustee’s expected payoff for the TGs with each of the m trustors as consisting of two components. The first component pertains to the phase during which trust is placed and honored with certainty. In ${\mathrm{\Gamma }}^{\mathrm{ga}}$ , this is $(N-{\tau }^{-})R_2$ for both trustees; in ${\mathrm{\Gamma }}^{\mathrm{rs}}$ , this is likewise $(N-{\tau }^{-})R_2$ for the opportunistic trustee and it is $(N-{\tau }^{-})(R_2+\upsilon )$ for the friendly trustee. The second component, the reminder of the formulas in Proposition 1, pertains to the ${\tau }^{-}$ last TGs that the trustee plays with each trustor i, the second and third phase of the equilibrium (see the proof of Proposition 1 for details).

Behavior and payoffs in ${\mathrm{\Gamma }}^{+}$

In ${\mathrm{\Gamma }}^{+}$ , network effects facilitate trust and trustworthiness. The trustors can learn from each other’s experiences. Therefore, the trustee has to take into account that his choice in one TG will affect the future choices of all trustors. Given the assumption that information is always shared truthfully, the trustors should condition their behavior on third-party information in the same manner as on personal experience (cf. Bolton et al., 2004). An opportunistic trustee’s incentive to honor trustor i's trust in a given period n is, therefore, the same as if he played all remaining $\mathrm{mN}-n$ TGs with that trustor (even if he plays some or all of these TGs with different trustors). Hence, the equilibrium that applies to the interactions between one trustor i and the trustee in ${\mathrm{\Gamma }}^{-}$ applies to the interactions between all trustors and the trustee in ${\mathrm{\Gamma }}^{+}$ (Buskens, 2003; Camerer and Weigelt, 1988; Frey et al., 2015).

Specifically, in ${\mathrm{\Gamma }}^{+}$ , trust is placed and honored until there are in total ${\tau }^{+}$ TGs left, where ${\tau }^{+}$ is calculated as ${\tau }^{-}$ using equation (1), but with ${\pi }^{+}$ instead of ${\pi }^{-}$ (hence, if ${\pi }^{+}={\pi }^{-}$ , then ${\tau }^{+}={\tau }^{-}$ ). Next, the randomization begins. And after the first instance that the trustee abuses the trust of one trustor or that one trustor does not place trust, no trustor ever places trust again. Analogous to the situation in ${\mathrm{\Gamma }}^{-}$ , the trustors never place trust in ${\mathrm{\Gamma }}^{+}$ if ${\tau }^{+}>\mathrm{mN}$ , and we also assume that ${\tau }^{+}=\infty$ if ${\pi }^{+}=0$ and ${\tau }^{+}=1$ if ${\pi }^{+}=1$ . Proposition 2 specifies the expected payoffs of a trustee in ${\mathrm{\Gamma }}^{\mathrm{ga}+}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}+}$ .

Proposition 2

The expected payoffs of a friendly trustee in ${\mathrm{\Gamma }}^{\mathrm{ga}+}$ $(U_F^{{\mathrm{\Gamma }}^{\mathrm{ga}+}})$ and ${\mathrm{\Gamma }}^{\mathrm{rs}+}$ $(U_F^{{\mathrm{\Gamma }}^{\mathrm{rs}+}})$ are

$$U_F^{{\Gamma }^{ga+}}=\{\begin{array}{cc}(mN-{\tau }^{+})R_2+{\tau }^{+}{P}_2+(T_2-P_2)(1-TEM{P}^{{\tau }^{+}})& if{\tau }^{+}\le mN\\ mN{P}_2\hfill & if{\tau }^{+}<mN\end{array}$$

$$U_F^{{\mathrm{\Gamma }}^{\mathrm{rs}+}}=\{\begin{array}{cc}(\mathrm{mN}-{\tau }^{+})(R_2+\upsilon )+{\tau }^{+}{P}_2\hfill & \mathrm{if}{\tau }^{+}\le \mathrm{mN}\hfill \\ \multicolumn{1}{c}{+(T_2-P_2)\frac{R_2+\upsilon -P_2}{R_2-P_2}(1-\mathrm{TEM}{P}^{{\tau }^{+}})}\\ \multicolumn{1}{c}{\mathrm{mN}{P}_2}& \mathrm{if}{\tau }^{+}>\mathrm{mN}\hfill \end{array}$$

The expected payoffs of an opportunistic trustee in ${\mathrm{\Gamma }}^{\mathrm{ga}+}$ $(U_O^{{\mathrm{\Gamma }}^{\mathrm{ga}+}})$ and ${\mathrm{\Gamma }}^{\mathrm{rs}+}$ $(U_O^{{\mathrm{\Gamma }}^{\mathrm{rs}+}})$ are

$$U_O^{{\mathrm{\Gamma }}^{\mathrm{ga}+}}=U_O^{{\mathrm{\Gamma }}^{\mathrm{rs}+}}=\{\begin{array}{cc}(\mathrm{mN}-{\tau }^{+})R_2+T_2+({\tau }^{+}-1)P_2\hfill & \mathrm{if}{\tau }^{+}\le \mathrm{mN}\hfill \\ \multicolumn{1}{c}{\mathrm{mN}{P}_2}& \mathrm{if}{\tau }^{+}>\mathrm{mN}\hfill \end{array}$$

The comparison of equilibrium behavior and expected payoffs in ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ reveals the network effects. All else equal (including the trustors’ belief about the trustee’s type at the beginning of the TGs), trust tends to be placed and honored during more TGs and the trustee earns more in ${\mathrm{\Gamma }}^{+}$ than in ${\mathrm{\Gamma }}^{-}$ . If trust is also possible in ${\mathrm{\Gamma }}^{-}$ , the trustee can avoid having the randomization phase with each trustor separately in ${\mathrm{\Gamma }}^{+}$ . If trust is not possible in ${\mathrm{\Gamma }}^{-}$ , the network effects can make a phase with honored trust possible in ${\mathrm{\Gamma }}^{+}$ . Another important observation is that the trustee will also be trusted more and earn a higher payoff if the trustors are at the beginning of the TGs more convinced that they are dealing with a friendly trustee (because a higher belief ${\pi }^{-}$ and ${\pi }^{+}$ implies a smaller ${\tau }^{-}$ and ${\tau }^{+}$ , respectively).

Investments in establishing information sharing

In this section, I establish under what conditions there exist equilibria in which the trustee establishes information sharing between the trustors. Let ${\rho }_F$ and ${\rho }_O$ denote the probability with which friendly and opportunistic trustees establish information sharing, respectively. There are only two combinations of ${\rho }_F$ and ${\rho }_O$ that can be part of an equilibrium that involves an investment of the trustee. These are (1) both trustees invest with probability 1 (i.e. ${\rho }_F={\rho }_O=1$ ) and (2) the friendly trustee invests while the opportunistic trustee does not invest ( ${\rho }_F=1$ and ${\rho }_O=0$ ).

For the analysis of investments in information sharing, it is important to realize that a trustee’s expected payoffs for ${\mathrm{\Gamma }}^{-}$ and ${\mathrm{\Gamma }}^{+}$ depend on the trustors’ beliefs entering these continuation games and that these beliefs depend on the investment probabilities ${\rho }_F$ and ${\rho }_O$ . Bayes’ rule implies that ${\pi }^{-}=((1-{\rho }_F)\pi )/((1-{\rho }_F)\pi +(1-{\rho }_O)(1-\pi ))$ and ${\pi }^{+}={\rho }_F\pi /({\rho }_F\pi +{\rho }_O(1-\pi ))$ . Hence, to assess whether some combination of ${\rho }_F$ and ${\rho }_O$ can be part of a sequential equilibrium, one needs to check whether the trustees maximize their expected payoffs by investing with the assumed probabilities ${\rho }_F$ and ${\rho }_O$ , given the beliefs ${\pi }^{-}$ and ${\pi }^{+}$ that are implied by these probabilities. ⁷

In the following, I establish under what conditions it is part of an equilibrium that both trustees establish information sharing ( ${\rho }_F={\rho }_O=1$ ; scenario (1)) or that only the friendly trustee establishes information sharing ( ${\rho }_F=1$ and ${\rho }_O=0$ ; scenario (2)). We shall see that the trustee’s investment promotes trust for different reasons in these two scenarios. In brief, in “pooling equilibria” in which ${\rho }_F={\rho }_O=1$ , the investment enables network effects but does not affect the beliefs of the trustors. By contrast, in “separating equilibria” in which only the friendly trustee invests, the investment leads to higher payoffs for the trustee in ${\mathrm{\Gamma }}^{+}$ exclusively because it signals to the trustors’ that they are dealing with a friendly trustee.

Investments in information sharing to allow network effects

In equilibria in which the trustee invests in establishing information sharing regardless of his type $({\rho }_F={\rho }_O=1)$ , the investment enables the network effects that promote trust and trustworthiness, but it does not signal intrinsic trustworthiness. If ${\rho }_F={\rho }_O=1$ , Bayes’ rule implies that ${\pi }^{+}$ equals the prior probability $\pi$ . If, unexpectedly, the trustee deviates from a conjectured equilibrium in which ${\rho }_F={\rho }_O=1$ and does not invest, the trustors do likewise not learn anything because neither trustee should ever do this. ${\pi }^{-}$ is not determined by Bayes’ rule and, in principle, the trustors could hold any “out-of-equilibrium” belief $0\le {\pi }^{-}\le 1$ . However, I employ the passive conjecture refinement to avoid constructing equilibria that require beliefs ${\pi }^{-}$ that are difficult to justify. ⁸ That is, I assume that the trustors do not change their beliefs upon observing a deviation from a conjectured equilibrium in period 0.2. ⁹ Thus, if ${\rho }_F={\rho }_O=1$ , ${\pi }^{-}={\pi }^{+}=\pi$ and, hence, ${\tau }^{-}={\tau }^{+}$ . The investment does not make the trustors more or less optimistic about the trustee’s type and it can benefit the trustee only due to the network effects. To simplify the discussion of the conditions for the existence of equilibria in which ${\rho }_F={\rho }_O=1$ , I use $\tau$ without a $\mathrm{superscript}-$ or $+\mathrm{to}$ denote simultaneously ${\tau }^{-}$ and ${\tau }^{+}$ .

$\mathrm{\Gamma }$ has an equilibrium in which ${\rho }_F={\rho }_O=1$ if c does not exceed the benefit that the network effects yield for the trustee who benefits least from these effects. That is, there exists an equilibrium in which ${\rho }_F={\rho }_O=1$ if $c\le \overline{c}:=min(U_F^{{\mathrm{\Gamma }}^{+}(\tau )}-U_F^{{\mathrm{\Gamma }}^{-}(\tau )},U_O^{{\mathrm{\Gamma }}^{+}(\tau )}-U_O^{{\mathrm{\Gamma }}^{-}(\tau )})$ , where, for example, $U_F^{{\mathrm{\Gamma }}^{+}(\tau )}$ denotes the expected payoff of a friendly trustee for ${\mathrm{\Gamma }}^{+}$ given the $\tau$ (i.e. ${\tau }^{+}$ ) that results from ${\pi }^{+}=\pi$ .

Proposition 3

${\mathrm{\Gamma }}^{\mathrm{ga}}$ has a sequential equilibrium in which ${\rho }_F={\rho }_O=1$ if and only if $c\le {\overline{c}}^{\mathrm{ga}}$ , where

$${\overline{c}}^{\mathrm{ga}}=\{\begin{array}{cc}(m-1)(\tau (R_2-P_2)-(T_2-P_2))\hfill & \mathrm{if}\tau \le N\hfill \\ \multicolumn{1}{c}{(\mathrm{mN}-\tau )(R_2-P_2)+(T_2-P_2)(1-\mathrm{TEM}{P}^{\tau })}& \mathrm{if}N<\tau \le \mathrm{mN}\hfill \end{array}$$

${\mathrm{\Gamma }}^{\mathrm{rs}}$ has a sequential equilibrium in which ${\rho }_F={\rho }_o=1$ if and only if $c\le {\overline{c}}^{\mathrm{rs}}$ , where

$${\overline{c}}^{\mathrm{rs}}=\{\begin{array}{cc}(m-1)(\tau (R_2-P_2)-(T_2-P_2))\hfill & \mathrm{if}\tau \le N\hfill \\ \multicolumn{1}{c}{(\mathrm{mN}-\tau )(R_2-P_2)+(T_2-P_2)}& \mathrm{if}N<\tau \le \mathrm{mN}\hfill \end{array}$$

Proposition 3 shows that $\overline{c}$ is similar for ${\mathrm{\Gamma }}^{\mathrm{ga}}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}}$ and distinguishes two scenarios: $\tau \le N$ and $N<\tau \le \mathrm{mN}$ . If $\tau \le N$ , the trustors place trust in some TGs also in ${\mathrm{\Gamma }}^{-}$ . However, establishing information sharing can payoff for the trustee because the network effects enable the trustee to have the randomization phase just once rather than with each of the m trustors separately. Therefore, a trustee earns approximately $(m-1)\tau (R_2-P_2)$ more in ${\mathrm{\Gamma }}^{+}$ than in ${\mathrm{\Gamma }}^{-}$ . An opportunistic trustee benefits somewhat less from the network effects than a friendly trustee because he has the opportunity to abuse trust m times in ${\mathrm{\Gamma }}^{-}$ but only once in ${\mathrm{\Gamma }}^{+}$ . Hence, if $\tau \le N$ , $\overline{c}$ equals how much the opportunistic trustee benefits from the network effects, which is the same in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}}$ .

If $N<\tau \le \mathrm{mN}$ , the trustors never place trust if the trustee does not establish information sharing (because, given the parameters, the sanctioning potential of one trustor alone is not sufficient to motivate an opportunistic trustee to start building a reputation). However, the network effects make honored trust possible in ${\mathrm{\Gamma }}^{+}$ . In this case, a trustee’s return on investment equals approximately his benefit from trust being placed and honored with certainty in $\mathrm{mN}-\tau$ TGs in ${\mathrm{\Gamma }}^{+}$ compared to never being trusted in ${\mathrm{\Gamma }}^{-}$ , that is, approximately $(\mathrm{mN}-\tau )(R_2-P_2)$ . For the friendly trustee, the return on investment is smaller in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ than in ${\mathrm{\Gamma }}^{\mathrm{rs}}$ . Therefore, $\overline{c}$ is somewhat smaller in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ than in ${\mathrm{\Gamma }}^{\mathrm{rs}}$ ( $\overline{c}$ is by $(T_2-P_2)\mathrm{TEM}{P}^{\tau }$ smaller in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ (where $\overline{c}$ equals the friendly trustee’s return on investment) than in ${\mathrm{\Gamma }}^{\mathrm{rs}}$ (where $\overline{c}$ equals the opportunistic trustee’s return on investment)).

The next proposition establishes how parameter changes affect $\overline{c}$ . All parameters affect $\overline{c}$ in the same manner in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}}$ , apart from a minor difference in the effect of changes in $R_2$ . I continue using $\overline{c}$ without a superscript to refer to ${\overline{c}}^{\mathrm{ga}}$ and ${\overline{c}}^{\mathrm{rs}}$ simultaneously.

Proposition 4

The maximum cost $\overline{c}$ for which $\mathrm{\Gamma }$ has a sequential equilibrium in which ${\rho }_O={\rho }_F=1$ depends on the parameters of the game as follows:

• (1) •If $\tau \le N$ , $\overline{c}$ increases weakly if RISK increases (i.e. if $P_1$ increases or $S_1$ or $R_1$ decreases) or if $\pi$ decreases.

•  • If $N<\tau \le \mathrm{mN}$ , $\overline{c}$ decreases weakly if RISK increases (i.e. if $P_1$ increases or $S_1$ or $R_1$ decreases) or if $\pi$ decreases.

• (2) •If $\tau \le \mathrm{mN}$ , ${\overline{c}}^{\mathrm{ga}}$ increases in $R_2$ .

•  • If $\tau \le \mathrm{mN}$ , ${\overline{c}}^{\mathrm{rs}}$ increases in $R_2$ .

•  • If $1<\tau \le \mathrm{mN}$ , $\overline{c}$ decreases in $P_2$ .

•  • If $\tau \le N$ , $\overline{c}$ decreases in $T_2$ .

•  • If $N<\tau \le \mathrm{mN}$ , $\overline{c}$ increases in $T_2$ .

• (3) •If $N<\tau \le \mathrm{mN}$ , $\overline{c}$ increases in m.

•  • If $N<\tau \le \mathrm{mN}$ , $\overline{c}$ increases inN.

List (1) of Proposition 4 concerns the effects of changes in $\pi$ and $\mathrm{RISK}=(P_1-S_1)/(R_1-S_1)$ , that is, the probability of a friendly trustee and the risk a trustor incurs when placing trust. It shows that a trustee’s incentive to establish information sharing changes in an inverse U-shape over $\pi$ and RISK. The trustee’s incentives to invest are small for small trust problems (large $\pi$ or small RISK) where there is a lot of trust even without information sharing and, hence, information sharing increases trust only marginally. The incentives to invest are also small if trust problems are very large (small $\pi$ or large RISK) and honored trust is hardly attainable even with information sharing. However, the incentives to invest are large for trust problems of an intermediate size in which the network effects make a considerable difference for the behavior of trustors and trustee.

To understand this inverse U-shape in more detail, recall that if $\pi$ increases or $\mathrm{RISK}$ decreases, the randomization phase tends to start earlier ( $\tau$ tends to decrease; see equation (1)). Now assume that $\pi$ is not too small and RISK is not too large such that trust is also possible without information sharing $(\tau \le N)$ . In that case, establishing information sharing enables the trustee to have the randomization phase just once rather than with each trustor separately. Obviously, this is more valuable if the randomization phase is longer due to a smaller $\pi$ or larger RISK. More formally, if $\tau \le N$ , the network effects enable the trustee to benefit from trust being placed and honored with certainty in $(m-1)\tau$ additional TGs, which is more valuable if $\tau$ is larger. Hence, as long as $\pi$ and RISK are such that $\tau \le N$ , $\overline{c}$ increases if $\pi$ or RISK changes such that $\tau$ increases. The effects are opposite once $\pi$ and $\mathrm{RISK}$ are such that trust is no longer possible without information sharing. If $N<\tau \le \mathrm{mN}$ , $\overline{c}$ becomes smaller if $\pi$ or RISK further changes such that $\tau$ increases. An increase in $\tau$ then leads to less trust in ${\mathrm{\Gamma }}^{+}$ while there was already no trust in ${\mathrm{\Gamma }}^{-}$ before the increase in $\tau$ .

List (2) of Proposition 4 concerns the TG payoffs of the trustee. $\overline{c}$ is larger if the benefit a trustee derives from honored trust compared to no trust $(R_2-P_2)$ is larger. Furthermore, $\overline{c}$ decreases in the payoff an opportunistic trustee earns when abusing trust $(T_2)$ if $\tau \le N$ , but increases in $T_2$ if $N<\tau \le \mathrm{mN}$ . ¹⁰ Note, furthermore, that if $\tau \le N$ , there may be no equilibrium in which ${\rho }_F={\rho }_O=1$ even for an arbitrarily small cost $c>0$ . This occurs if the loss an opportunistic trustee incurs from having only once the opportunity to abuse trust in ${\mathrm{\Gamma }}^{+}$ rather than m times in ${\mathrm{\Gamma }}^{-}$ is large while his benefit from an extended phase of trust and trustworthiness is small (if $\tau \le N$ and $\tau (R_2-P_2)<T_2-P_2$ , $\overline{c}$ is negative; see Proposition 3).

List (3) of Proposition 4 shows that $\overline{c}$ increases in the number m of trustors and the number N of TGs played with each trustor if $N<\tau \le \mathrm{mN}$ . If $N<\tau \le \mathrm{mN}$ , an increase in m implies that the trustee benefits from honored trust in N additional TGs in ${\mathrm{\Gamma }}^{+}$ while he will not be trusted in these N additional TGs in ${\mathrm{\Gamma }}^{-}$ . If $\tau <N$ , $\overline{c}$ may also increase in m because if there is one additional trustor, trust is placed and honored with certainty in N additional TGs in ${\mathrm{\Gamma }}^{+}$ and only in $N-\tau$ additional TGs in ${\mathrm{\Gamma }}^{-}$ . However, if $\tau <N$ and $\overline{c}$ is negative because $\tau (R_2-P_2)<T_2-P_2$ , $\overline{c}$ decreases even further if m increases because the opportunistic trustee benefits more from having the chance to abuse the additional trustor’s trust than from an extended phase of honored trust.

An increase in N also leads to an increase in $\overline{c}$ if $N<\tau \le \mathrm{mN}$ . If $N<\tau \le \mathrm{mN}$ , an increase in N means adding one TG for each trustor in which the trustee is not trusted in ${\mathrm{\Gamma }}^{-}$ but in which he benefits from honored trust in ${\mathrm{\Gamma }}^{+}$ . However, if trust is also possible in ${\mathrm{\Gamma }}^{-}$ (i.e. $\tau <N$ ), $\overline{c}$ does not change in N. An increase in N then means adding one TG for each trustor in which the trustee benefits from honored trust in ${\mathrm{\Gamma }}^{-}$ as well as in ${\mathrm{\Gamma }}^{+}$ .

Summarizing, the condition for the existence of pooling equilibria in which the trustee establishes information sharing to benefit from network effects is almost the same in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}}$ (i.e. if friendly trustees are guilt-avoiding trustees and if they are reward-seeking trustees). Important results are that the maximum cost $\overline{c}$ for which there are such equilibria is high especially if (1) the size of the trust problem is intermediate— $\pi$ and RISK are neither too small nor too large—such that network effects bring about a considerable increase in honored trust and if (2) the trustee benefits a lot from honored trust compared to no trust ( $R_2-P_2$ is large).

Investments in information sharing as signals of intrinsic trustworthiness

In equilibria in which only the friendly trustee invests in establishing information sharing ( ${\rho }_F=1$ and ${\rho }_O=0$ ), the trustee’s investment is a perfectly discriminating signal of intrinsic trustworthiness. After observing the trustee’s investment decision, the trustors know whom they are dealing with. They always place trust if the trustee invested and never place trust if the trustee did not invest. Formally, ${\rho }_F=1$ and ${\rho }_O=0$ together imply ${\pi }^{-}=0$ and ${\pi }^{+}=1$ and, hence, ${\tau }^{-}=\infty$ and ${\tau }^{+}=1$ . The results concerning such separating equilibria are quite different for ${\mathrm{\Gamma }}^{\mathrm{ga}}$ and ${\mathrm{\Gamma }}^{\mathrm{rs}}$ and, therefore, presented separately. For the game with guilt-avoiding trustees, we have the following proposition:

Proposition 5

In ${\mathrm{\Gamma }}^{\mathrm{ga}}$ , there cannot exist a sequential equilibrium in which ${\rho }_F=1$ and ${\rho }_O=0$ .

An equilibrium in which ${\rho }_F=1$ and ${\rho }_O=0$ requires that the benefit from always being trusted compared to never being trusted is larger for a friendly trustee than for an opportunistic trustee. However, this is not the case in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ . Both trustees earn $\mathrm{mN}{P}_2$ if the trustors never place trust. If the trustors always place trust, the opportunistic trustee earns more than the friendly trustee, namely, $(\mathrm{mN}-1)R_2+T_2>\mathrm{mN}{R}_2$ (because he can abuse trust in the last TG without feeling remorse). Hence, ${\mathrm{\Gamma }}^{\mathrm{ga}}$ cannot have an equilibrium in which ${\rho }_F=1$ and ${\rho }_O=0$ .

A remark is in order. ${\mathrm{\Gamma }}^{\mathrm{ga}}$ could have an equilibrium in which ${\rho }_F=1$ and ${\rho }_O=0$ if friendly trustees could establish information sharing at lower costs than opportunistic trustees. ¹¹ Assuming that different types face different costs of sending some signal is not uncommon (e.g. Aksoy and Gambetta, 2016; Gintis et al., 2001; Patel, 2012). For settings with relatively few trustors, one could defend such an assumption based on arguments along the line that a trustee with social preferences (a friendly trustee) is more adept at bringing people together than a trustee who is exclusively interested in material outcomes or that a friendly trustee even derives some pleasure from doing this. However, I here focus on whether a trustee can signal his intrinsic trustworthiness by establishing information sharing when trustee types differ exclusively in their payoffs in the TG. My last proposition establishes that this is possible in ${\mathrm{\Gamma }}^{\mathrm{rs}}$ where a friendly trustee receives an internal reward $\upsilon$ every time he honors trust.

Proposition 6

In ${\mathrm{\Gamma }}^{\mathrm{rs}}$ , there exists a sequential equilibrium in which ${\rho }_F=1$ and ${\rho }_O=0$ (and in which the trustee is always trusted after having established information sharing but never trusted otherwise) if and only if $\mathrm{mN}(R_2-P_2)+T_2-R_2\le c\le \mathrm{mN}(R_2+\upsilon -P_2)$ .

In ${\mathrm{\Gamma }}^{\mathrm{rs}}$ , a friendly trustee benefits always more from being trusted in some TG than an opportunistic trustee, irrespectively of whether the latter honors or abuses trust (because $R_2+\upsilon >T_2$ ). It can, therefore, be worthwhile for a friendly trustee to invest in information sharing to signal his type and induce the trustors to place trust while it does not payoff for an opportunistic trustee to mimic the friendly trustee by investing, too. To deter mimicry by the opportunistic trustee, the cost c must be “rather high.” The comparison of Propositions 3 and 6 shows that the lower bound on c for the existence of a separating equilibrium is higher than the maximum cost for which ${\mathrm{\Gamma }}^{\mathrm{rs}}$ has a pooling equilibrium in which ${\rho }_F={\rho }_O=1$ and in which the investment affects the trustors’ behavior less strongly. ¹² Proposition 6 shows that the lower bound for c for which there exists an equilibrium in which ${\rho }_F=1$ and ${\rho }_O=0$ increases in m, N, $R_2-P_2$ , and $T_2$ . On the other hand, c must also be small enough such that it pays off for a friendly trustee to signal his type by investing. The maximum cost c for which this is the case increases in m, N, $R_2-P_2$ , and $\upsilon$ , as can be inferred from Proposition 6. The cost range for which ${\mathrm{\Gamma }}^{\mathrm{rs}}$ has a separating equilibrium is larger if m or N is larger (because a friendly trustee can then receive the internal reward $\upsilon$ more often), if $\upsilon$ is larger, and if an opportunistic trustee’s benefit from abusing trust $(T_2-R_2)$ is smaller.

Summarizing, a trustee’s investment in establishing information sharing can signal that he is of the friendly type if friendly trustees are trustworthy because they derive pleasure from good behavior (as in ${\mathrm{\Gamma }}^{\mathrm{rs}}$ ) but not if they are trustworthy because they would suffer an internal sanction if abusing trust (as in ${\mathrm{\Gamma }}^{\mathrm{ga}}$ ). In the former case, signaling equilibria can exist especially if the cost of investment c is high, the trustee interacts many times with many trustors, friendly trustees derive a large internal reward from honoring trust, and opportunistic trustees cannot gain too much from abusing trust.

Conclusion and discussion

In this article, a game-theoretic model has been developed that advances our understanding of the emergence and effects of institutions for information sharing in trust situations. The model allows for a simultaneous investigation of investments in setting up an institution for information sharing and the effects of such an institution. It focuses on investments in information sharing by a trustee, and it captures two mechanisms by which a trustee’s investment can promote trust. First, the trustee’s investment can allow network effects that facilitate trust and trustworthiness. Second, the trustee’s investment can serve as a credible signal of intrinsic trustworthiness.

A core result is the prediction that there is an inverse U-shape in the relation between the size of the trust problem and the incentives for establishing information sharing in order to allow network effects. A trustee should be particularly likely to establish information sharing to enable network effects if the size of the trust problem is intermediate (if there is an intermediate probability $\pi$ of interacting with an intrinsically trustworthy trustee and if the risk that a trustor incurs when placing trust $(\mathrm{RISK})$ is neither very small nor very large). This could suggest, for example, that someone who sells an ordinary consumer good in a brick and mortar market will not invest in setting up an institution for information sharing because the trust problem is too small. However, someone who sells the same good over the Internet may invest in a reputation system because buyers are more afraid of fraud in online trade.

This inverse U-shape prediction is in line with the results of Frey et al. (2015) and Raub et al. (2013). Frey et al. (2015) report qualitatively the same effects of changes in $\pi$ and $\mathrm{RISK}$ for the scenario that the trustors, rather than the trustee, can establish information sharing. In the Raub et al. (2013) model—a game with indefinite repetition and complete information on all actors’ incentives—network effects depend exclusively on the incentives of trustees (and not at all on the trustors’ incentives). Still, this alternative model, which also covers social dilemmas other than the TG, leads to a similar conclusion: investments in information sharing should be more likely if the incentives to take advantage of others are neither very small nor very large. This article thus reinforces the testable prediction of an inverse U-shape in the relation between the size of the trust problem and the incentives to establish information sharing in order to allow network effects.

A second key result of this study is that a trustee’s investment in information sharing can serve as a credible signal of intrinsic trustworthiness. The analysis suggests that a trustee is particularly likely to establish information sharing to signal his intrinsic trustworthiness if he derives a large internal reward from good behavior, if there are many trustors and many interactions with each trustor, and if the investment cost is high. The analysis, furthermore, shows that signaling can only occur if intrinsically trustworthy trustees derive a larger benefit from being trusted than trustees that have a short-term incentive to abuse trust.

The article also offers new predictions on the effects of institutions for information sharing. Earlier work focused exclusively on network effects. The result that a trustee’s investment in information sharing can serve as a signal of intrinsic trustworthiness suggests that information sharing can affect behavior more strongly than previous work predicts. Network effects tend to affect behavior gradually, as in my model in which they lead to honored trust in some more TGs. Signaling can have more swiping effects. In separating equilibria in which only the intrinsically trustworthy trustee invests, the trustors never place trust if the trustee does not invest while they place trust throughout if the trustee does invest. This suggests that information sharing can promote trust more strongly if it was established by the trustee than if it is given exogenously. Another implication is that a trustee’s investment in information sharing may have a particularly strong effect on trust if the costs of investment are high.

The results on the conditions for separating equilibria also generalize beyond the context of investments in information sharing. In a separating equilibrium, the trustee’s investment is a “wasteful signal.” After having observed the investment, the trustors know for sure that they are dealing with an intrinsically trustworthy trustee. There is no more need for the network effects. Therefore, the trustee could signal his type by “burning resources” in any other way than by creating a context that would promote trust if the trustors were uncertain about the trustee’s type. In this sense, the analysis establishes under what conditions a trustee who interacts N times with m trustors sends any costly signal to convey his intrinsic trustworthiness. The results on how parameters affect the restrictiveness of the conditions for the existence of such separating equilibria are in line with results of Przepiorka and Diekmann (2013) for infinitely repeated TGs.

Although a trustee could take various actions to signal his intrinsic trustworthiness, I believe that establishing information sharing is a particularly attractive option for three reasons. First, trustors have an interest in acquiring information about a trustee’s reputation. It is therefore, for example, unlikely that it escapes the attention of potential buyers that an online seller invested in a reputation system. At least, it is probably more likely that they overlook a note that reports a charitable donation (see Fehrler and Przepiorka, 2013, 2016; Milinski et al., 2002 for studies on charitable giving as a signaling device). That is, an investment in establishing information sharing might be a signal with high “broadcast efficiency” (Bliege Bird and Smith, 2005; Gintis et al., 2001). Second, there may be a chance that trustors do not understand a trustee’s action as a signal of intrinsic trustworthiness. If the signal fails to convey its intended message, the trustee can still benefit from network effects if he tried to signal his type by establishing information sharing. Finally, trustors might not be totally convinced of the trustee’s intrinsic trustworthiness also after having observed the signal (for example, due to heterogeneity in the cost of sending the signal that can make it possible for an opportunistic trustee to afford sending the signal). Network effects can then further increase the level of trust after signaling. These considerations highlight the importance of studying how signaling can drive investments in information sharing. However, these considerations also suggest that it may be a limitation that in the presented model an investment in information sharing facilitates exchange either because it enables network effects or because it signals intrinsic trustworthiness. Future work could modify the presented model to investigate how signaling and network effects can complement one another.