Modelling trust evolution within small business lending relationships

Tang, Ying; Moro, Andrea; Sozzo, Sandro; Li, Zhiyong Article Modelling trust evolution within small business lending relationships Financial Innovation Provided in Cooperation with: Springer Nature Suggested Citation: Tang, Ying; Moro, Andrea; Sozzo, Sandro; Li, Zhiyong (2018) : Modelling trust evolution within small business lending relationships, Financial Innovation, ISSN 2199-4730, Springer, Heidelberg, Vol. 4, Iss. 1, pp. 1-18, https://doi.org/10.1186/s40854-018-0105-1 This Version is available at: https://hdl.handle.net/10419/237135 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ RESEARCH Open Access Modelling trust evolution within small business lending relationships Ying Tang 1 , Andrea Moro 2 , Sandro Sozzo 3 and Zhiyong Li 1* * Correspondence: [email protected] 1 School of Finance, Southwestern University of Finance and Economics, Chengdu, China Full list of author information is available at the end of the article Abstract Trust is a key dimension in the principal-agent relationship and it has been studied extensively. However, the dynamics, evolution, and intrinsic motivation and mechanisms have received less attention. This paper investigates the intrinsic motivation of trust and it proposes a theoretical model of trust evolution that is based on the notion of ‘trust response’and ‘trust spiral’. We then specifically focus on trust within the lending relationship between banks and small businesses, and we run numerical simulations to further illustrate the evolution of involved mutual trust over time. Our model provides implications for future research in both trust evolution and small business lending relationships. Keywords: Trust, Spiral, Small business lending, Reciprocity, Altruism Introduction Since the 1990s, trust has received considerable attention across many disciplines, ranging from psychology to economics. As a crucial element of any social system, trust is introduced to achieve optimal contracts and improve institutional efficiency. Even when confronted with the potential risk of being betrayed, a trustor is willing to endow trust in a trustee, which cannot be explained by traditional economic theories under the assumption that they are rational economic entities who are only concerned about their own monetary payoff. A large quantity of literature has studied fairness and social welfare in human decision making and has found that theories of reciprocity and altruism provide supportive explanations for these non-selfish behaviours (Fehr and Schmidt, 2006). This paper develops an evolving model of trust based on the notions of ‘trust response’(Pettit, 1995)and‘trust spiral’(Nooteboom, 2002). After analysing the motivation of trust and trust propensity, we further propose a new trust utility function that will be used as the basis of a trust spiral model. The development of trust in this paper is transformed into the evolution of trust propensity along with time t. Our main standpoint lies in the fact that behavioural trust is driven by potential motivations and is caused by trust antecedents on the basis of the subjective perception of trust, as supported by Das and Teng (2004). We focus on trust within a lending relationship between banks and small businesses, as suggested by the empirical research (Moro and Fink, 2013; Howorth and Moro, 2006; Howorth and Moro, 2012; Saparito and Gopalakrishnan, 2009; Harhoff and Körting, 1998). Trust plays an incredibly important role in borrower-lender Financia l Innovation © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Tang et al. Financial Innovation (2018) 4:19 https://doi.org/10.1186/s40854-018-0105-1 relationships. In fact, lenders, or more correctly, loan managers, are asked to take decisions about whether to lend, even when they will be repaid quite far into the future. In other words, they are asked to deal with a very high level of uncertainty about the future behaviour of the borrower. They are typically asked to extrapolate future behaviour based on information about the borrower (Moro et al., 2015). More specifically, they are asked to evaluate whether the borrower has the capability to ‘properly’use the money so that they will be able to repay the loan, as well as the willingness to repay the loan. Interestingly enough, past behaviour can help the lender to develop an idea/opinion about the borrower’s capability to repay the loan (according to their skills as managers) but it can only provide marginal information about the willingness to repay it. Thus, the final decision implies a decision about whether the lender trusts the borrower. The lending decision process (and one can generalise, the bank activity) can be reduced to the following question: Can the borrower be trusted? This process is even more relevant in the case of a small business. In the case of the lending process for medium and large organisations, lenders can rely on large amounts of consolidated data that can help them to develop a proper picture of the borrower, in particular about their ability to repay the loans. In addition, this kind of organisation is run by a management team with the support of middle management staff. Therefore, a firm’s capacity to repay the loan does not depend on the behaviour of just one person or very few people and, therefore, the risk in terms of the lack of willingness to repay the loan is also reduced. Consequently, the information asymmetry and the adverse selection risk are reduced (Jensen and Meckling, 1976; Williamson, 1988). In the case of small firms and micro firms, the action of a single person (the entrepreneur) is essential for the success/failure of the venture and the ability to repay the loan. Similarly, the lack of willingness to repay a loan made to just one person (the entrepreneur) can adversely affect the possibility of the lender being able to recover the loan. The problems are amplified by the fact that information available to the lender is reduced, less reliable and is not necessarily updated. Interestingly, the increase in the level of information asymmetry and the increased risk of adverse selection increases the role of lender trust in the borrower. However, irrespective of the relevance of the topic and empirical evidence, research has not properly developed a theoretical model to describe the evolution of this process. Our research contributes to trust and small business lending literature by looking at a quantitative and endogenous perspective of trust evolution. We also provide a suggestive explanation for its evolution from the perspective of motivation of trusting behaviour and trust propensity. The rest of this paper is organized as follows. Section “Literature review”reviews the literature. Section “Model”explains trust propensity and constructs the trust utility function. Section “Application in small business lending”discusses the application of the trust spiral model in small business lending. The results of the model simulation are presented in Section “Simulation results”. Finally, Section “Conclusions”concludes this paper. Literature review Relationships based on trust are rooted in imperfect information: if the trustor has access to all the information and can systematically monitor the behaviour of the trustee, then there is no point to take a leap of faith or, to put it in other words, there is no Tang et al. Financial Innovation (2018) 4:19 Page 2 of 18 need to trust others. Mayer et al. (1995, pp712) define trust as “the willingness of a party to be vulnerable to the actions of another party based on the expectation that the other will perform a particular action important to the trustor, irrespective of the ability to monitor or control that other party.”Nooteboom’s(2002) definition of trust stresses the expectation that a partner will not engage in opportunistic behaviour, even in the face of countervailing short-term opportunities and incentives. Research suggests that information and trust have a positive and non-monotonic relationship. Under the condition of complete information, trust is redundant. Meanwhile, under the situation of incomplete information with some risk and uncertainty, trust is likely to be introduced (Fisman and Khanna, 1999; Uzzi and Lancaster, 2003). A set of institutional or environmental factors that influence the floor level of information is needed to enter a trusting relationship, including the ceiling level of information above which trust becomes redundant (Rooks et al., 2000). The hypothesis of self-interest is considered to be a basic assumption in traditional economic theories (Williamson, 1988). However, this hypothesis has increasingly been challenged by a large number of experimental findings during the recent two decades. These findings show that people are not only concerned with their own monetary payoff but also are concerned with the payoff of others. Based on this experimental evidence, complicated social preferences have been proposed to be rooted in human behaviour and this can help to provide convincing explanations for those non-selfish phenomena (e.g. Andreozzi, 2013). Among these social preferences, reciprocity, altruism, and inequality aversion all supply distinct perspectives, which can be classified into two strands. First, reciprocity stresses mutual intentions. During interactions, a first mover’s behaviour showing trust is likely to induce the counterparts’fulfilment of his or her anticipation, and vice versa. Thus, trusting behaviours tend to be reciprocated with positive responses that validate trust. This is consistent with Pettit’s(1995) notion of ‘trust-responsiveness’. The studies by Dufwenberg and Kirchsteiger (2004), and Falk and Fischbacher (2006) have improved the theory of psychological games proposed by Rabin (1993), mainly in the respect that each player’s kindness function depends on his or her available equitable payoff. These studies have also made a significant contribution to the development of reciprocity theory. The second strand is based on a positive attitude towards human nature. This strand stresses the fact that people consider altruism in their behaviour and tend to be inequality adverse. In other words, people are not only concerned about their own payoff but are also concerned about the payoff of others. Among those that support this view, Cox (2004) referred to these kinds of preferences over others’monetary payoff in addition to their own payoff as the ‘other-regarding preference’; in the same way that altruism, inequality aversion and vulnerability-responsiveness are the most important drivers for trustworthiness. Barney (1990) argues that an alternative set of assumptions about human nature including altruism is valid and can be integrated into traditional organizational economics. Finally, Mansbridge (1999) suggests that one crucial form of trust is altruistic trust, pointing out that an altruistic nature is involved in each trust relationship (Ferrary, 2003). We should not neglect other motivations of trust like altruism, as pointed out by Dufwenberg and Kirchsteiger (2004). Trust antecedents can fall into two categories, trust propensity and trustworthiness (Mayer et al., 1995). Trust propensity refers to the trustor’s tendency to trust others Tang et al. Financial Innovation (2018) 4:19 Page 3 of 18 and is more likely to be treated as a stable personal characteristic (Mayer et al., 1995; Das and Teng, 2004). With respect to trustworthiness, ability and intentions are accepted as the dominant source of trust, corresponding to ability trust and goodwill trust (eg. Nooteboom, 1996). Mayer et al. (1995) further classify the perceived trustworthiness about a trustee into three different aspects: ability, benevolence and integrity. From the perspective of risk, Das and Teng (2004) differentiate subjective trust and trust antecedents from behavioural trust and they construct a risk-based conceptual framework. Nevertheless, the combination of motivations and antecedents of trust has not been fully studied. Research on trust emphasizes that a high level of trust can not only encourage trustworthy behaviour (Nooteboom, 2002; Pettit,1995) but can also decrease agency problems and transaction costs, including expenses of monitoring and control (Macaulay, 1963; Zand, 1972; Ring and Van De Ven, 1992). A trust relationship is not static but evolves in a dynamic process along with the interaction between trustor and trustee and the previous experience in terms of trust and reciprocation. Thus, the process is a spiral that is a continuously harmful increase or decrease in trust, which gradually gets faster and faster. The spiral of trust can be either upward or downward. The relational trust behaviour, where both parties display trust, tends to spiral upward, creating stronger relationships; distrust reproduces itself in a downward spiral (Friend et al., 2010). Zand (1972) developed a conceptual model that clarifies the logic relationship between participants’behaviour and their counterparts’trust and suggests the importance of support for the spiral-reinforcement model: mutual trust or mistrust, among members of a group, is likely to be reinforced, unless there is marked or prolonged disconfirming behaviour. Pettit (1995) proposes the notion of ‘trust-responsiveness’, based on which Nooteboom (2002) explains how the upward spiral and downward spiral of trust can happen. Lewicki and Wiethoff (2000), Lewicki and Bunker (1996) and Kramer and Lewicki (2010) stress that trust evolves from activity-based trust to relationship-based trust to identification-based trust. However, they stress that the development from one stage to the following, more solid one, is based on testing and re-testing trust via continuous iterations. This can be seen as a spiral where the trusting relationship is retested many times. However, all of these studies are devoted to the conceptual construction of the development of trust, while a detailed modelling of trust evolution and how it spirals up and down has not yet been fully explored. Several studies extend this line of research on the evolution of trust into other areas. For instance, Ybarra and Turk (2009) discovered that the transaction cost perspective of trust may actually serve to make partners suspicious of each other and force companies to invest in specific assets that may actually create a spiral of distrust among the participants. Similarly, Wang and Murnighan (2017) studied the dynamics of punishment and interpersonal trust and found that people trust punishers more than non-punishers, but only when punishers are not motivated by revenge. However, little attention has been paid to the evolution of trust within a bank lending relationship. Research on trust has also looked at the specific context where it arises in the buyer-supplier relationship-performance spiral (e.g. Autry and Golicic, 2010; Celuch et al., 2011; Liu, 2012) and in project business (e.g. Smyth et al., 2010). As far as a lending relationship is concerned, empirical research is quite limited and it is mainly based on the fact that trust can promote cooperation (Gulati, 1995; Das and Teng, 1998) and Tang et al. Financial Innovation (2018) 4:19 Page 4 of 18 facilitates decision making in a situation with scarce information (Luhmann, 2000). Especially in small business lending, a high degree of risk and uncertainty is involved, with more severe information asymmetry. McCabe et al. (2003) suggested that the lending relationship between a bank and an enterprise is a reciprocal trust relationship. Some studies provide empirical support for the role of trust in small business enterprise lending (e.g. Moro and Fink, 2013;HoworthandMoro,2012). Palazuelos et al. (2017), using Spanish data, found that accounting information quality is relevant in trust formation and affects the willingness of a bank to grant credit. Massaro et al. (2017)exploredthejointeffect of trust and control mechanisms on knowledge transfer in the networks of SMEs and they found that trust substitutes for the implementation of boundary, diagnostic and belief tools. Enimu et al. (2017) examined the determinants of loan repayment among agricultural microcredit finance group members in Delta state, Nigeria. No detailed modelling has been proposed in terms of trust evolution. The only model that describes trust in the lending process has been proposed by Howorth and Moro (2006). However, while their model focuses on describing the trust process, it does not explore the dynamics of trust evolution. Therefore, in this paper, we aim to correct this gap in the research. Model Following Mayer et al. (1995), we assume that trust antecedents fall into two aspects, one is the trustor’s perceived trustworthiness of trustee and the other is his or her trust propensity. In the respect of trustworthiness, a trustor could make a subjective evaluation about the trustee’s ability, benevolence and integrity through collecting related hard and soft information. Trust propensity is more likely to be an individual trait. As proposed by Castelfranchi (2008), it is trust attitude that prompts the trustor to make the trust decision and gauge the behaviour of trusting. We propose that it is the trust propensity that determines a trustor’s willingness of take trusting behaviour, which comes from three components, namely: self-interest, reciprocity and altruism. Based on the framework of the psychological game by Rabin (1993), our model is constructed on the basis of trust utility functions. Although we assume that reciprocity and altruism are also involved in the trust behaviour, the goal of utility maximization still holds in our model. Several studies have made important extensions of utility theory with the support of experimental data, including: (1) fairness equilibria (Rabin, 1993); (2) a theory of altruism (Levine, 1998); (3) self-centred inequality aversion (Fehr and Schmidt, 1999); (4) equity, reciprocity and competition (Bolton, Ockenfels, 2000); (5) a social preference model (Segal and Sobel, 2007); and (6) a model of altruism and inequality aversion (Kohler, 2011). Nonetheless, nearly all of the parameters, including trust propensity, are assumed in these models to be constant in these linear utility functions and their evolution has been largely neglected. Self-interest term It is generally accepted that economic rationality exists in all circumstances. This means that the preference of self-interest is rooted in all human beings and it becomes the basic term of all economic behaviours, including trusting behaviour. Specifically, the monetary payoff is the dominating motivation for a trustor when confronted with a Tang et al. Financial Innovation (2018) 4:19 Page 5 of 18 strategy choice. Therefore, the monetary payoff constitutes the first part of a trust utility function, without which the trusting behaviour is unlikely to happen. In addition, we propose a self-interest propensity to capture the significance of self-interest utility or the monetary payoff that is considered in the total trust utility. Therefore, the self-interest term of a trustor’s trusting utility is related to his available monetary payoff from trusting behaviour, which is stated as follows: U0 i¼αiπiaihðÞ;bij hðÞ  ;i≠j;i;j∈E;BðÞ ð1Þ where α i is the self-interest propensity of player i,π i (a i (h), b ij (h)) is his monetary payoff received at history hwhen his strategy is a i (h) and his belief about player j’s strategy is b ij (h). Following the related literature (Rabin, 1993; Levine, 1998; Fehr and Schmidt, 1999; Segal and Sobel, 2007; Kohler, 2011), we propose α i =1. Reciprocity term Tanis and Postmes (2005) present a social identity approach to trust, indicating that interpersonal trustworthiness enhances expectations of reciprocity, which in turn increases trusting behaviour. Their study also provides evidence for the reciprocal motivation of trust. According to the theory of reciprocity (Dufwenberg and Kirchsteiger, 2004; Falk and Fischbacher, 2006), a trustor’s behaviour is perceived to be kind or unkind by a trustee, and it then receives a behavioural reaction. Thus, the reciprocity utility is formalized to consist of a kind (or unkind) treatment, which is represented by the kindness term, and a reaction to that treatment, which is represented by the reciprocation term. In Zand’s(1972) model, the trustee’s perception of a trustor’s intention and choice determines their conclusion about that trustor, as well as his final behaviour choice. Thus, the reciprocity utility term consists of two dependent components: kindness function and reciprocity propensity. Kindness function 1 Based on Rabin’s(1993) idea, we define player i ' skindness to player jas zero if he believes j ' smonetary payoff will be the average of the lowest and the highest available toj, which is compatible with ichoosing an efficient move. The same equilibrium payoff function is defined in this paper as Dufwenberg and Kirchsteiger (2004), and Falk and Fischbacher (2006), as follows: πei jbij hðÞ  ¼1 2maxπjaihðÞ;bij hðÞ  þminπjaihðÞ;bij hðÞ  ð2Þ In (2), πei jðbijðhÞÞ is so-called equilibrium payoff for trustoriwhen his belief about trusteej ' sbehaviour is expressed by b ij (h), and could be deemed as a reference point for measuring how kind iis to j. Therefore, the kindness of trustor ito trustee jat history hcould be described by the difference between the actual payoff that trustee jreceives at node hand the equilibrium payoff that he can get, denoted byf ij ; that is, fij aihðÞ;bij hðÞ  ¼πjaihðÞ;bij hðÞ  −πe jbij hðÞ  ð3Þ Under trustor i ' sstrategy a i (h)and his first-order belief b ij (h) j≠i ,iff ij (a i (h), b ij (h)) > 0, it seems that iis kind to j, and vice versa. Tang et al. Financial Innovation (2018) 4:19 Page 6 of 18 After defining kindness, we now turn to reciprocity. Reciprocity means that if iis kind to j, then jwants to be kind to iin return, and likewise, if iis unkind to j,jis likely to take revenge. However,j ' skindness cannot be observed directly because it depends onj ' sbelief, which is unknown to i, but which can be inferred fromj ' smoves. Let f iji describe the perception of player iabout the kindness of trustee jto him or her, defined as follows: fiji bij hðÞ;ciji hðÞ  ¼πibij hðÞ;ciji hðÞ  −πe jciji hðÞ  ð4Þ Under trustor i ' sfirst-order belief b ij (h) j≠i and second-order belief c iji (h) j≠i ,iff iji (b ij (h), c iji (h)) > 0, it seems that ibelievesjis kind to themselves, and vice versa. As stated above,- b ij (h)=a j (h) and c iji (h)=b ji (h)=a i (h)both hold; then, the reciprocity function f iji (b ij (h), c iji (h)) equals to f ji (a j (h), b ji (h)) mathematically. Reciprocity propensity After defining the kindness function, we further introduce the reciprocity propensity, which measures how sensitive a trustor is to reciprocity regarding a trustee. As an indicator of reciprocal preference, the higher the trustor’s reciprocity propensity, the more important a role reciprocity utility plays in his or her total trust utility. Specifically, if it is zero—that is, the trustor has no motivation to engage in any reciprocity behaviour towards the trustee—, then the reciprocity term of the trustor is also zero. Let β i be the reciprocity propensity of player i; then, the reciprocity term of his trust utility is specified as: U00 i¼βifij aihðÞ;bij hðÞ  fiji bij hðÞ;ciji hðÞ  ð5Þ In related literature, the reciprocity propensity is always deemed to be an exogenously positive constant and it is assumed to be stable and common knowledge (Dufwenberg and Kirchsteiger, 2004; Falk and Fischbacher, 2006). In contrast to this viewpoint, we attempt to propose an endogenous explanation of reciprocity propensity in the context of a bank-small business lending relationship. More specifically, inspired by Kalai and Lehrer’s(1995) idea of the subjective game, as well as Hanaki et al.’s(2009) game learning, we assume that the trustor makes his or her decision based on the attractions for each strategy in each period and it is through the evolution of attractions that players learn more about the (repeated) games in the long run. Specifically, this attraction could be reasonably understood as the reciprocity propensity β i , which is supposed to be mainly determined by the average payoff of each player in the past periods, denoted by πiðhÞ, and his or her sensitivity to this average payoff, denoted by e i . Thus, the proposed endogenous reciprocity propensity β i evolves as: βi¼te iπihðÞþcos tðÞ½;πihðÞ¼ 1 t−1X t−1 τ¼1 πit−τðÞ ð6Þ Altruism term Andreoni and Miller (2002) suggest that people do not focus on their own payoff or utility alone but that they also pay attention to the fairness of outcomes and there exists a positive correlation between their payoff and others’, which is so-called altruism. Tang et al. Financial Innovation (2018) 4:19 Page 7 of 18 Similar to reciprocity propensity, we define altruism propensity as the personal concern of a trustor about the trustee’s payoff or utility: it is the weight that is assigned by the trustor to the payoff of the trustee. The higher the trustor’s altruism propensity, the more he is willing to improve the trustee’s payoff. Since Levine (1998) proposes that players’payoffs are in line with their own and their opponents’monetary payoff, linear altruism models have been adopted extensively in related literature (e.g. Ioannou et al., 2013). Kohler (2011) proposes that the altruism propensity is positive and the value interval is [0,1]. A positive altruism propensity indicates that the altruism utility of a trustor increases with the monetary payoff of a trustee. Both reciprocity and altruism preference are related to social welfare, and social distance is likely to exert an important influence on them (Hoffman et al., 1996). Long-term relationships between banks and small businesses help to reduce the information asymmetry. Engle-Warnick and Slonim (2006) further find that past relationship lengths also affect trust and trustworthiness. For banks, they are more concerned about the payoff or utility of those borrowers that maintain longer and closer relationships with them. For borrowing businesses, they are more likely to show a higher altruism propensity to maintain the hard-earned lending relationship with banks. Similar to reciprocity preference, the altruism preference also lies in the inequity aversion, and fairness of the outcomes may also affect the altruism propensity. In this paper, the lending relationship between a bank and small business is thought to be not competitive but cooperative. A bank focuses mainly on the safety of granted loans, while the small business pays more attention to obtaining the loan and repaying it on time. Therefore, we assume that the bank-firm relationship is a key factor in determining altruism propensity, especially for banks who play a dominant role in the small business credit market. Let dbe the relationship distance between a bank and a small business and kbe the sensitivity of both players to d; then, the endogenous γ i evolves as follows: γi¼e−kidðÞ ;d¼1 tð7Þ Thus, the altruism term of trust utility for player ican be specified as: Ui 000 ¼γiπjaihðÞ;bij hðÞ  ð8Þ Trust utility function Based on previous discussions, we summarise the trust utility function as follows: Ui¼Ui 0þUi 00 þUi 000 ¼αiπiaihðÞ;bij hðÞ  þβifij aihðÞ;bij hðÞ  fiji bij hðÞ;ciji hðÞ  þγiπjaihðÞ;bij hðÞ  þεi αi¼1 βi¼te iπihðÞþcos tðÞ½;πihðÞ¼ 1 t−1X t−1 τ¼1 πit−τðÞ γi¼e−kidðÞ ;d¼1 t 8 > > > > < > > > > :ð9Þ Application in small business lending Regarding how a trust spiral may occur, Pettit (1995) proposes the notion of ‘trust-re- sponse’, which is related to the desire for social recognition and to the notion of social Tang et al. Financial Innovation (2018) 4:19 Page 8 of 18 βi¼te iπihðÞþcos tðÞ½;t≤td te iπihðÞ−cos tðÞ½;t>td ;πihðÞ¼ 1 t−1X t−1 τ¼1 πit−τðÞ ð10Þ In terms of altruism propensity, γ i also evolves after time t d and follows the following function: γi¼e−kidðÞ ;t≤td ekidðÞ ;t>td ;d¼1 tð11Þ Assuming t d = 200, a bank trusts a small business, and the small business abides by the loan contract during the time interval of [0,200], while the small business defaults during the time interval of [201,360]. Simulations are run based on this set of parameters, as stated previously. Figure 4reports the results. Figure 4reveals that trust propensities of the bank and the small business evolve differently. For the bank, its reciprocity propensity (β B ) exhibits an upward spiral during the time interval of [0, t d ], when the bank trusts the small business and the small business rewards the endowed trust by honouring the loan contracts. Meanwhile, after t d ,β B starts to decrease along with time and follows a fast-downward spiral. This indicates that once the small business starts to default, the level of trust built through repeated lending interactions drops dramatically from the bank’s side. This results in the decreasing reciprocity propensity of the bank (β B ). At the same time, the bank’s altruism propensity (γ B ) experiences a similar evolution. During the first stage ([0, t d ]), γ B - increases along with the repeated interactions and approximates 1. Meanwhile, it gradually decreases during the second stage ([t d ,t]) due to the change in the small business’s behaviour strategy. For the small business, its reciprocity propensity (β E ) continues increasing along with time instead of enduring a gradual or sudden decrease. This indicates that the small business’s propensity to trust the bank grows with the ongoing lending interactions, driven by the reciprocity motivation. This is caused by the increased monetary payoffs that the small business receives from its default. Meanwhile, its altruism propensity (γ E ) exhibits a similar change as γ B . Fig. 4 Trust evolution in the case of (T, R-T, D). Note: (T, R-T, D) refers to the situation that the strategy portfolio changes from (T, R) during the period of [0, t d ] to (T, D) during the period of [t d ,t]. Namely, bank decides to trust a small business (T), and small business respects the loan contract (R) during the first rounds ([0, t d ]) and then the small business starts to default (D) after t d . We use the standardized data to generate the simulation results in order to be consistent in scale Tang et al. Financial Innovation (2018) 4:19 Page 15 of 18 Conclusions This paper presents a model of trust spiral and it uses the small business lending relationship as an example to examine a credit game. We simulate repeated interactions between a bank and a small business. We suggest that the trust spiral within the lending relationship between a bank and a small business depends on the strategy behaviour taken by both parties. The model that we propose here shows that where the bank trusts a small business and the small business rewards the endowed trust, then the mutual trust level of both the bank and the small business increases along with their mutual interactions. Their trust experiences an upward spiral, which is consistent with Pettit’s(1995) notion of trust-response and Nooteboom’s(2002) trust spiral. Where a bank trusts a small business and the small business takes opportunistic behaviour without repaying the loan, the altruism propensity of both the bank and the small business endures a gradual drop from a high starting value, which approximates 1. For the bank, this fall in trust is mainly due to the distrustful behaviour taken by the small business. Meanwhile, for the small business, trust falls because it is aware that its behaviour will provoke distrust from the bank. The reciprocity propensity of the bank (β B ) follows a gradual downward spiral with a slightly growing radius. However, the reciprocity propensity of the small business (β E )followsagradualincrease and presents an upward spiral due to a high level of kindness and trust from the bank. Where the bank trusts the small business and the small business respects the loan contract in the first round but defaults after t d , the bank’s trust spiral moves upward before t d , and it then falls. Meanwhile, the reciprocity propensity of the small business (β E ) continues to increase instead of decrease. This indicates that the small business’s propensity to trust the bank grows with the ongoing lending activities, driven by the reciprocity motivation. Meanwhile, its altruism propensity (γ E ) increases along with the repeated interactions, which approximates 1 during the first stage. It then gradually decreases during the second stage, after it takes the default action. Our model has observed some limitations. First, we focus on the evolution of reciprocity propensity and altruism propensity without considering the overall trust utility function, based on which each player makes his or her behaviour decision. Second, as Engle-Warnick and Slonim (2004) have suggested, concerns for the future of a relationship with repeated interactions are important for trust to persist. Our simulations are repeated games, and we do not consider that the lending relationship with repeated interaction may in the future affect the generation, development and persistenceoftrust.Finally,wherethesmallbusinessdefaults,weonlyconsiderthe case of the business repaying nothing of the loan and we ignore the possibility of a partial repayment. Nevertheless, irrespective of these limitations, our model represents the first attempt to mathematically model the evolution of trust in a lending relationship after considering the reciprocal and altruistic behaviour of the players. Our model also provides a suggestive explanation for trust evolution based on motivation of trusting behaviour and the trust propensity of the trustor. Our theoretical analysis may also provide implications for experimental researchers who wish to examine trust evolution from motivation and trust propensity in experimental designs. Tang et al. Financial Innovation (2018) 4:19 Page 16 of 18 Endnotes 1 As exemplified in the work of theory of sequential reciprocity, a trustor’s reciprocity and altruism, as well as his or her perception of a trustee’s reciprocity and altruism may change after different histories, and the interaction history his introduced to capture this. For example, a i (h) denotes the updated strategy choice of trustor iat the nod h and b ij (h) represents the updated beliefs of trustor iabout the strategy of trustee jat the nod h. The updated belief is an effective way to deal with the changing preferences when unexpected choices occur. Acknowledgements We thank the feedback received by discussants attending the Twelfth International Conference on Management Science and Engineering Management (ICMSEM) where the previous version of the paper has been published as a proceeding. We acknowledge the National Natural Science Foundation of China for the financial support and three universities of co-authors for providing necessary research resources. We also thank the anonymous referees for their valuable comments to improve the quality of this paper. A previous version of this paper was published in the proceedings of the Twelfth International Conference on Management Science and Engineering Management (ICMSEM), and we have received permission from Springer Nature to reuse part of the content in this journal. We acknowledge the National Natural Science Foundation of China (Grant no. 71703127) to support this research. Funding This research is supported by the National Natural Science Foundation of China (Grant no. 71703127). Availability of data and materials The paper employs no real data or materials in the analysis. Authors’contributions The authors have corresponding contributions as the order indicates. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Publisher’sNote Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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