Sarah Talboom
email: sarah.talboom@vub.ac.be

Jo Pierson
email: jo.pierson@vub.ac.be

Understanding Trust within Online Discussion Boards: Trust Formation in the Absence of Reputation Systems

Keywords: Online trust, trust, online discussion platforms, information search, advice

Trust plays an important role when it comes to sharing and picking up information on the Internet. Even more so when advice is being exchanged and decisions are being made based on what at first sight seems to be interpersonal communication between people who do not know each other. Research concerned with the role of trust in making an assessment of online user-generated information, often focuses on online reputation systems rather than on the way individuals try to evaluate peer recommendations themselves. In this paper the attention goes out to the construction of trust when people are looking for advice and know-how via online textual discussion boards. The study offers an understanding of online trust by defining the concept of trust, by formulating a trust typology -based on both the object of trust and the stages within a trust relationship -and by identifying trust antecedents. All this while keeping a close eye on the context of online forums and their users.

INTRODUCTION

The study starts from the idea that an online community can play a significant role when it comes to making important individual decisions. When people are looking for information -such as how-to knowledge or pre-buy advice -, they can use User Generated Content (UGC). This way the information search process is becoming a social event, where Internet users can be acknowledged valuable information sources. This is also how Facebook is framing their 'Graph Search' function, namely by socializing search, although in their case making use of friends. 1 UGC is often seen as an important support tool for an assertive, self-reliant and informed Internet user, since it gave people the opportunity to create and share content themselves. Yet because of its open publication principle, online users are being confronted with information that has not been verified by traditional gatekeepers. Anyone with the right skills and hardware can publish content online. Combining this finding with the fact that many online contributions cannot be directly linked to a particular person (in comparison to the Facebook example) raises some questions concerning trust. Before people will share and pick up online information, they must experience some level of trust [START_REF] Ridings | Some antecedents and effects of trust in virtual communities[END_REF][2] [START_REF] Ljung | People, profiles & trust: On interpersonal trust in web-mediated social spaces[END_REF]. However, literature shows that in a traditional situation, trust will emerge slowly during the process where people get to know each other ([4] based on the work of Blomqvist). Identity, reputation and experience are important antecedents that are not always that easy to deduce online. Online communication does not always offer enough knowledge on the skills and capacities of online information providers. This insecurity about the identity of others, combined with the lack of a common interaction history, makes trust development less evident. However, when users succeed in building their own identity and get an understanding of the identity of others, they are much more able to anticipate on the answers of others and thus to build trust (Henderson & Gilding, 2004 and Tanis & Postmes, 2005 as mentioned in [START_REF] Blanchard | A model of online trust. The mediating role of norms and sense of virtual community[END_REF]). Within this paper we use a more sociological approach to trust with a focus on people and how they behave individually as well as collectively within a specific online context: the online bulletin board.

When looking at research concerning online trust, we often find links with e-commerce [START_REF] Gefen | E-commerce: the role of familiarity and trust[END_REF][7] -where a high degree of trust in an online retailer would have a positive effect on sales figures. Although we couldat first sight -make a leap from e-commerce to our own research theme, it seems that many studies focus on electronic recommendation agents rather than on peer consumers as recommenders [START_REF] Smith | Online peer and editorial recommendations: Trust and choice in virtual markets[END_REF]. Also the importance of trust when releasing personal data online is increasingly being studied [START_REF] Dwyer | Trust and privacy concern within social networking sites: A comparison of Facebook and MySpace[END_REF]. The link between online interpersonal communication and trust is less often discussed in scientific research, as well as the role of trust for selecting and evaluating information sources [START_REF] Hertzum | Trust in information sources: Seeking information from people, documents and virtual agents[END_REF]. Exceptions are the importance of trust for the cooperation within virtual teams [START_REF] Jarvenpaa | Toward contextualized theories of trust: The role of trust in global virtual teams[END_REF] and the use of online user-generated health information by patients [START_REF] Sillence | A framework for understanding trust factors in web based health advice[END_REF]. This last domain often emphasizes the actual reliability and accuracy of the disseminated information. Within this study however, the attention goes out to the way trust is build in a (semi) anonymous online environment between people, irrespective of the actual accuracy of the information.

Within both public debate and scientific research, a lot of attention is often given to interpersonal communication via social media like Facebook and Twitter. Yet a person who wishes to gather information about various topics does not necessarily have to turn to people within his own network. The reason why we focus on online textual platforms has to do with the problematic nature of such environments. Within discussion boards, information is exchanged and decisions are made based on what at first sight seems to be interpersonal communication between people who do not know each other (that well). This contrasts highly with social media platforms, where it is generally assumed that you use your own personal details. In online discussion groups, people should not even have real conversations with other members; they can get started by just reading the conversations of others. An online forum, based solely on text, lacks some clues that are present in physical contexts or in online surroundings where you meet up people you know offline. This lack of cues is further emphasized by the fact that conversations do not occur synchronously. Finally, research has also shown that text-based conversations create a greater sense of social distance and that feelings of trust would build up slowly [START_REF] Bos | Effects of four computer-mediated channels on trust development[END_REF].

Within this paper, we focus on the construction of trust when people are looking for both implicit and explicit know-how within a mainly textual online environment: What is the role of this concept when picking up information and sharing knowledge? Which factors ensure or prevent trust to emerge online? The study offers an understanding of trust in online discussion forums by defining the concept of trust, formulating a trust typology and identifying trust antecedents while keeping a close eye on the context in which trust occurs.

THE CONCEPT OF TRUST

Research on the concept of trust has been around for over 60 years. Various disciplines sought to understand trust from different perspectives [START_REF] Corritore | On-line trust: concepts, evolving themes, a model[END_REF]. As a result, there exists a multitude of conceptualizations depending on the context and the domain in which the concept is used. McEvily, Weber, Bicchieri and Ho [START_REF] Mcevily | Can groups be trusted? An experimental study of trust in collective entities[END_REF] state that despite this heterogeneity between different definitions, some common elements can still be identified. These common elements are important conditions for trust to emerge and are therefore key elements in defining the concept.

A first key element is interdependence [START_REF] Mcevily | Can groups be trusted? An experimental study of trust in collective entities[END_REF] [START_REF] Rousseau | Not so different after all: A cross-discipline view of trust[END_REF]. When involved in a trusting relationship, at least one party chooses to depend on the other party [START_REF] Banerjee | An ethical analysis of the trust relationship[END_REF]. If you want to reach a certain goal for which you need to depend on [e.g. the advice of] another person, than the responsibility to actually achieve this goal is partly under control of someone else. The individual who is willing to be dependent -the trustor -bases his reliance on a positive expectation [START_REF] Rousseau | Not so different after all: A cross-discipline view of trust[END_REF] [START_REF] Barber | The logic and limits of trust[END_REF]. You are only willing to rely upon a specific person because you expect that person whom you trust -the trustee -to be competent, reliable and honest, which ensures that the set goals will be met. Yet, you are never 100% sure regarding the outcome of this trust: you take a certain risk when you put your trust in someone. According to leading authors such as Luhmann [START_REF] Luhmann | Familiarity, confidence, trust: problems & alternatives[END_REF], Gambetta [START_REF] Gambetta | Can we trust trust?[END_REF] and Sztompka [START_REF] Sztompka | Trust: A sociological theory[END_REF], risk is unavoidable when it comes to the development of trust. Without risk there is no need for trust. The same goes for insecurity or ignorance [START_REF] Mcevily | Can groups be trusted? An experimental study of trust in collective entities[END_REF][17] [START_REF] Sztompka | Trust: A sociological theory[END_REF] [START_REF] Castaldo | Trust in market relationships[END_REF]. Trust is only necessary when absolute knowledge is missing, which makes it not possible to make an accurate estimation of the result of this trusting behavior. Incomplete knowledge nourishes insecurity and increases feelings of risk and thus the need for trust ( [START_REF] Castaldo | Trust in market relationships[END_REF] based on the work of Simmel & Gambetta). When these feelings of risk are too high, people will not proceed to trusting and risk taking behavior [START_REF] Mayer | An integrative model of organizational trust[END_REF]. This shows that both trust and risk are strongly related to action. In one respect, there is no risk without action, risks are taken and subjected [START_REF] Luhmann | Familiarity, confidence, trust: problems & alternatives[END_REF][20] [START_REF] Castaldo | Trust in market relationships[END_REF]. Yet on the other hand, a lack of trust will ensure that certain behavior will not occur [START_REF] Castaldo | Trust in market relationships[END_REF]. Because of this interdependence and since you are never sure of the desired outcome, you put yourself in a vulnerable position [START_REF] Mcevily | Can groups be trusted? An experimental study of trust in collective entities[END_REF] [START_REF] Rousseau | Not so different after all: A cross-discipline view of trust[END_REF]. It is quite possible that the trustee is not able or even not willing to provide you with adequate advice, which means that your goal will not be reached. Trust is therefore only necessary when the outcome could also turn out bad [START_REF] Luhmann | Trust and power[END_REF].

An often-mentioned definition that combines these key concepts interdependence, expectations, risk, insecurity, action and vulnerability, is the one from Mayer, Davis and Schoorman [START_REF] Mayer | An integrative model of organizational trust[END_REF]. They see 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'. The last part of this definition -irrespective of the ability to monitor or controlrefers to vulnerability, risk and uncertainty. '[People] have a certain degree of freedom to disappoint our expectations' [START_REF] Gambetta | Can we trust trust?[END_REF]; therefore we have no guarantee that our expectations will be met. You cannot check whether the person whose advice you take into consideration, is in the right position to give proper and sincere advice. Accepting the advice of people you have never met before can therefore clearly be seen as a manifestation of trust.

THE CONTEXT OF TRUST

Before we go deeper into the different typologies and antecedents of trust, it is important to emphasize the context in which trust will [or will not] occur.

First of all the focus is on the use of a text-only discussion forum for the exchange of ideas and support, and the gathering of information before a given action will be taken. This exchange of information situates itself at both the level of explicit and tacit [START_REF] Polanyi | Personal Knowledge Towards a Post-critical Philosophy[END_REF] or organic knowledge -knowledge based on everyday personal experiences [START_REF] Rosenberg | The Our House DIY Club: Amateurs, leisure knowledge and lifestyle media[END_REF]. On the other hand, attention is paid to the idea of trust in the opinions of people you do not know [that well]. Traditionally an information seeker will try to make an assessment of the knowledge, skills and veracity of the information provider, in order to decide whether or not to grant trust. However, if you are looking for advice or experience of others, it is not unrealistic that your search will be accompanied by feelings of insecurity and risk. The amount of online information is enormous, different users contain different levels of expertise [START_REF] Mondal | Privacy, Security and Trust in P2P environments: A Perspective[END_REF] and not all the information that you encounter is usable, reliable or correct. An online world is traditionally more muted than offline social spaces. According to Fisher [START_REF] Fisher | Studying social information spaces[END_REF], people do have some cues, but these are more difficult to find. Additionally, contributions on discussion platforms are all displayed in a similar way. This may create some kind of leveling effect where all authors are considered equally credible (Burbulus, 1998 as mentioned in [START_REF] Metzger | Making sense of credibility on the web: Models for evaluating online information and recommendations for future research[END_REF]). The fact that you do not see [a visual] difference between various online contributions emphasizes again the risk for the information seeker in deciding which experts to consult. Many Web 2.0 and 3.0 applications try to solve this by 'calculating' trust based on information from previous interactions. By ascribing more weight to the contributions of trustworthy people, platforms try to use trust as a kind of filter. This way people -who do not have a clear link with each other -can still make an estimation of the quality of the information and thus decide to proceed to trusting behavior [START_REF] Golbeck | Introduction to computing with social trust[END_REF]. It is important to emphasize that we are not looking for a solution for this difficult relationship between trust and 'text-only platforms' -it is clear that this already exists in the form of the abovementioned trust and reputation systems. We merely want to understand how and based on which aspects trust is being shaped within this type of environment. How do you assess whether your source has the necessary knowledge and skills? Overall, two main issues are often raised when it comes to trust within textual online communities. First of all you have the fact that the exchange of conversations takes place between people who will probably never meet physically. Grasping the identity of your discussion partner becomes a difficult assignment, especially when looking at the anonymous nature of some exchanges [START_REF] Golbeck | Introduction to computing with social trust[END_REF]. Furthermore, people looking for advice based on online discussion forums often lack experience with the trustee. This while past experience, reputation and the duration of a relationship, along with [inter alia] insight into skills and competences are important antecedents of trust [21][30]. [START_REF] Erickson | Social translucence: Using minimalist visualisations of social activity to support collective interaction[END_REF]. The visual hints shared within an online discussion forum are different than those shared in a traditional conversation. Besides the absence of some important physical, verbal and behavioral cues, people also miss out on identity indications such as age, education, lifestyle and status. Online cues would not be as rich as their face-to-face counterparts [START_REF] Riegelsberger | The researcher's dilemma: evaluating trust in computer-mediated communication[END_REF]. However, people tend to trust others when they are similar to themselves [START_REF] Sztompka | Trust: A sociological theory[END_REF][29] [33][34], so cues do play an important role. Question is whether this similarity between people really takes shape on the basis of traditional cues and individual characteristics of members. Furthermore some new cues emerge online. Elements such as writing style, a nickname, the intensity of contributions or sharing personal information can shed light on the identity of forum members. The ease, with which these new signals can be picked up, seems strongly related to the type of user. According to Blanchard, Welbourne and Boughton [START_REF] Blanchard | A model of online trust. The mediating role of norms and sense of virtual community[END_REF]: 'It simply takes a good deal more time and communication effort for an appropriate amount of cues to be accumulated in computer-mediated-communication'. In addition, Fisher [START_REF] Fisher | Studying social information spaces[END_REF] states that there are few signs aimed at new readers. This is especially problematic when a user just quickly wants to pick up credible information and than leaves again. People who repeatedly return to the discussion forum -to interact with others but also just to read the conversations -get to know more about the group itself [START_REF] Fisher | Studying social information spaces[END_REF] and about individual members. Also Ridings, Gefen and Arinze [START_REF] Ridings | Some antecedents and effects of trust in virtual communities[END_REF] mention that repeated interactions over timetogether with the fact that contributions are publicly available -may allow trust to emerge.

'In the digital world we are socially blind […]'

Blomqvist states that in literature the dominant view prevails that trust is built up slowly. 'Trust needs close and repeated interaction to build up. Parties learn about each other's competences and intentions (Nooteboom, 2002) in repeated interactions […]' [START_REF] Bijlsma-Frankema | Trust under pressure: trust and trust building in uncertain circumstances[END_REF]. Past experience, evidence and reputation play an important role when it comes to assigning trust. Trust is therefore highly dependent on the assumed knowledge of an individual, his reputation and his behavior in similar situations [START_REF] Busacca | Trust as market-based resource: Economic value, antecedents and consequences[END_REF]. Keeping our context in mind, we can link this lack of experience to different user types. On the one hand you have people who regularly participate by adding a contribution or by frequently visiting the discussion forum. This group of people builds up a certain level of experience, both with specific group members as with the community itself. On the other hand there are people who less frequently visit the community in order to just pick up information. The latter category cannot fall back on a large amount of previous encounters with forum members and thus will have to find other ways to collect cues regarding reliability and competence. As stated before, we always interact on the basis of insecurity; we never fully know how trustworthy someone truly is [START_REF] Kramer | Trust as situated condition: an ecological perspective on trust decisions[END_REF]. This uncertainty is apparently stronger highlighted with new or less frequent forum users.

TYPOLOGIES OF TRUST

A typology of trust can be constructed based on the object of trust -in whom or what we trust -or on the stages within a trust relationship.

Typology Based on Objects of Trust

When focusing on the object of trust, a distinction can be made between general trust in people, trust in a specific person and trust with respect to an institution -or so called system trust. According to McKnight and Chervany [START_REF] Mcknight | Reflections on an initial trust-building model[END_REF] these three trust types are inextricably linked: the general level -tendency to trustexerts influence on the contextual construction -system trust -, which in turn has an impact on interpersonal trust. First of all, trust is strongly related to the personality of an individual [START_REF] Sztompka | Trust: A sociological theory[END_REF] and can therefore be seen as a social-psychological construct. This general propensity / disposition to trust is often referred to in literature as a moral impulse, a capacity for trust or basic trust ( listed by [START_REF] Sztompka | Trust: A sociological theory[END_REF] based on the work of Wilson, Hardin and Giddens) and 'generalized' trust [START_REF] Hardin | Trust and trustworthiness[END_REF]. It is a kind of basic level of trust, existing independently from specific contexts or people. Yet, due to the more sociological and communication studies perspective in this paper, we focus entirely on interpersonal and collective trust: members of and lurkers on discussion forums can either trust people or they can see the community itself as an object of trust.

Trust upon an interpersonal level refers to situational trust between two ore more individuals and is thus both context-and actor-specific: one trusts person A to do B within situation C. When one is looking for advice or information on discussion forums, one often interacts with specific forum members by participating in the conversation or by reading through the conversation threads. As a lurker you do not have real contact -i.e. a conversation -with forum members but you might experience virtual personal trust; or an apparent intimacy with distant others [START_REF] Sztompka | Trust: A sociological theory[END_REF]. Based on these conversations a reader or participant can decide to follow up the advice of a specific member in order to accomplish a certain goal. When dealing with interpersonal trust, the focus will be on individual characteristics of the trustee.

Literature states that people -due to the emergence of more temporary, airy and impersonal relationshipsincreasingly rely upon institutional trust; characterized by rules and structures in order to support the interpersonal aspect [START_REF] Van De Ven | Relying on trust in cooperative inter-organizational relationships[END_REF]. The development of online reputation systems is an example where one is trying to embed rules and structural guaranties within online communities. System trust however transcends the individual level. It is no longer about the personal properties of people but more about a belief in abstract principles and procedures [START_REF] Sydow | How can systems trust systems? A structuration perspective on trust-building in inter-organizational relations[END_REF]. Möllering [START_REF] Möllering | Rational, Institutional and Active Trust: Just Do It!?[END_REF] states that Giddens and Luhmann see institutional trust as trust based on visual control and performances. He stresses the importance of the entry point; where an actor makes an assessment of the complete system through the interaction with other actors. A positive first contact with a forum member will be extended to the entire community. Also McEvily, Weber, Bicchieri and Ho describe an evolution where trust in individual members would lead to the perception that the community can be trusted. Trust in a discussion forum as a collective could than be re-transferred to other individual members and could be used as a strategy when detailed information about specific members is missing or limited [START_REF] Mcevily | Can groups be trusted? An experimental study of trust in collective entities[END_REF]. Group members are considered reliable because of their membership. Every individual contribution adds up to the reliability and trustworthiness of the community and visa versa; individual contributions are perceived to be credible because of association with the community [START_REF] Brown | Word of mouth communication within online communities: Conceptualizing the online social network[END_REF]. Thus, a shared group identity will be an important trust antecedent for community members and readers of community content. An interesting concept that is rather close to the previous idea is the notion of 'institutional isomorphism' from DiMaggio and Powell (1983, as mentioned in [START_REF] Möllering | Rational, Institutional and Active Trust: Just Do It!?[END_REF]). Here, one unit is forced to conform to other units when they are operating in the same environment. In other words trust breeds trust or you trust because others trust as well. Within online discussion forums, this can be related to the fact that exchanges -threads -are often aimed at the entire group and therefore publicly available for anyone to read. The idea of institutional isomorphism relies greatly on what Möllering [START_REF] Möllering | Rational, Institutional and Active Trust: Just Do It!?[END_REF] mentioned earlier in his article; actors place [institutional] trust because they take it for granted.

Combining these perceptions on interpersonal and system trust, we can expect that similarity on an interpersonal level will -especially at start of a new relationship -be largely based on individual sociodemographic characteristics. Homogeneity on a collective level is strongly linked to a shared group identity based on shared beliefs and interests [START_REF] Wellman | Net surfers don't ride alone: Virtual communities as communities[END_REF]. 'People on the net have a greater tendency to develop feelings of closeness on the basis of shared interest rather than on the basis of shared social characteristics.' These feelings of closeness, described by Wellman and Gulia [START_REF] Wellman | Net surfers don't ride alone: Virtual communities as communities[END_REF], will only develop through regular participation. Certain expectations considering the behavior of group members are being created due to membership within a certain group, which makes it easier to build trust [START_REF] Gargiulo | The dark side of trust[END_REF]. Finally, feelings of togetherness and the sense of virtual community are not confined to active group members but can also be experienced by lurkers [START_REF] Nonnecke | Silent participants: Getting to know lurkers better[END_REF].

Typology based on stages of trust

A division can also be made based on the stage of a relationship. Here initial and ongoing trust are considered important concepts.

In the work of McKnight and Chervany [START_REF] Mcknight | Reflections on an initial trust-building model[END_REF], initial trust is described as a phase in which people do not know each other: '[People] have little solid, verifiable information about each other, and what they do know is not from first-hand, personal experience.' Here, people can use reputation, in-group categorization and stereotyping as antecedents or strategies to build initial trust. Initial trust is also an important element when it comes to online discussion forums, since the achieved results will determine how a new user will deal with the forum and with the specific trustee in the future. Unlike long-term relationships of trust, which are rather resistant against small trust fractures, initial trust is rather fragile: it lacks feelings of affection so that once broken it will be difficult to build trust again [START_REF] Mcknight | Reflections on an initial trust-building model[END_REF]. Hardin [START_REF] Hardin | Trust and trustworthiness[END_REF] [45] refers to initial trust as 'as-if trust behavior'. As a new user or member trust is necessary when you are looking for information. Yet, you do not have anything to base trust upon, therefore you can choose to act as if you trust someone. If all goes well -i.e. when the trustee meets the expectations of the trustor -, this temporary situation eventually leads to real or ongoing trust. Looking at ongoing trust, both partners depart from a common interaction history. Thick [START_REF] Kramer | Trust as situated condition: an ecological perspective on trust decisions[END_REF] [START_REF] Williams | Formal structures and social reality[END_REF] or ongoing trust is based on knowledge and is build step by step by means of interactions over time. Here the key factor is having experience with the trustee [START_REF] Mcknight | Reflections on an initial trust-building model[END_REF]. Two other interesting concepts, which are highly consistent with the idea of stages within a trust relationship, are swift and slow trust. The concept of swift trust was introduced by Meyerson, Weick and Kramer [START_REF] Meyerson | Swift trust and temporary groups[END_REF] in response to an increase in project work within companies. According to the authors, this type of trust will be activated in a context where temporary groups with little or no prior history have to work together within a short period of time in order to achieve certain projects. Here, team members behave as if trust is already present (i.e. as-if trust) because there is insufficient time to go through the traditional stages of trust. Swift trust is therefore not based on experience with team members but rather on characteristics that allow people to be associated with certain categories or groups (category driven information [START_REF] Meyerson | Swift trust and temporary groups[END_REF]). Information about categories makes an identification based on stereotypical impressions possible, which in return leaves room for trust: 'Seeing oneself as similar to other people (or as part of a common group or category) provides a basis for assuming that these individuals have similar values and will therefore behave in similar and predictable ways.' [START_REF] Adams | Swift trust in distributed ad hoc teams[END_REF].

The concept of swift trust leaves some space to reflect upon the formation of trust within online discussion forums. New and less-frequent members come to the forum without a common background on which they can base their trust. Moreover, trust signals are usually not directed at newcomers and group value is something difficult to grasp based on irregular, occasional and brief visits. Swift trust clearly corresponds to the idea of initial trust. Temporary groups -such as [new] forum users seeking urgent advice -can rely on a huge amount of information and expertise but they do not immediately know or do not have enough time to find out who knows what [START_REF] Meyerson | Swift trust and temporary groups[END_REF]. Of course initial trust and swift trust are not entirely the same. Swift trust originates from an organizational environment where people are mutually dependent, where they are obliged to work together, keeping a deadline in mind. Within discussion boards, other rules apply. Here people do not even have to have real interactions with each other. On the other hand, trust is also granted ex ante. 2 When people -unfamiliar with the discussion platform and its members -are looking for information, they will need to develop trust in a quick manner, based on swift trust cues.

In line with initial trust, the emphasis is on in-group categorization and stereotyping. People will try to categorize the contributions of forum members based on small role indications such as a profile picture or even writing style (e.g. the use of jargon as an indication of expertise). When you are looking for some tips on parenting, you will pay more attention to the advice of a mother of three, than to the opinion of a man without children. When you want information about a painting job, you will tend to follow the advice of someone who just painted his house or from someone whose profession as a painter is reflected in his nickname. Seeing people as part of the same group as yourself can foster trust as well: 'this person is -just like myself -a member of a home improvement forum, so he must know something about renovations or at least he is in the same situation as I am'. The idea of in-group categorization is therefore twofold: people can be seen as members of a group of experts ['mums' or 'painters'] or as members of the same group ['they are renovating their house, just like me']. Also the idea that trust builds slowly -i.e. slow trust -fits perfectly with the concept of ongoing trust. Information about individual skills of forum members is collected through repeated social exchanges and regular visits to the discussion forum. Linking back to our objects of trust, we can assume that through these interactions people do not only develop interpersonal trust, they also learn how the online group deals with questions and comments, whether the group is helpful, how to pose questions and whether you can expect a clear and reliable answer. Both interpersonal and collective trust are built through use of and experience with the online community.

ANTECEDENTS OF TRUST Fig.1 Antecedents of trust

Within thriving online discussion boards there exists a huge amount of different users, each containing different levels of expertise. Therefore you cannot assign the same weight to every peer opinion. The degree to which a user will trust someone depends on a number of aspects. As a user you will try to estimate to what extent someone is credible, honest, accountable and benevolent in order to reduce feelings of risk. In order to grasp these dimensions of trust, people can use indications of expertise, reputation and identity. Important to notice is that these antecedents of trust do not take away the actual risk, it offers -as Luhmann [START_REF] Luhmann | Trust and power[END_REF] puts it -'a spring board for the leap into uncertainty'. Identity can be seen as an overarching concept, including [indications of] reputation and expertise. Reputation -as a second layer -contains indications of expertise.

Identity

'One reason for uncertainty in mediated environments is that, without visual and auditory social cues, people are not sure whether or not they can trust other people to be who they claim to be.' [START_REF] Baym | Personal connections in the digital age[END_REF] An online identity refers to the role forum members adopt when they make contributions online and to how users try to grasp the identity of others. Before people will trust someone, they will try to neutralize insecurity by making an assessment of the person. By exchanging [amongst other things] sociodemographic data, people can better understand each other's attitudes and beliefs, and predict behavior, causing reduced feelings of insecurity (see uncertainty reduction theory [START_REF] Berger | Some explorations in initial interaction and beyond: Toward a developmental theory of communication[END_REF]). Sztompka [START_REF] Sztompka | Trust: A sociological theory[END_REF] states that observed similar characteristics are important predictors of trust. Also: 'Having a known identity can be critical when assessing important information' [START_REF] Jessen | Aggregated trustworthiness: Redefining online credibility through social validation[END_REF]. In traditional face-to-face conversations people form impressions from each other's personality based on a wide range of both nonverbal and verbal cues. But what when people do not meet physically and what when you cannot fall back on a long-term relationship? Baym [START_REF] Baym | Personal connections in the digital age[END_REF] argues that when people cannot link a physical appearance to a particular behavior, the authenticity of this behavior becomes less clear. People tend to trust others when they consider them as being equal. However, these similarities between trustor and trustee can -besides being physical or linked to socio-demographics -also be based on shared preferences, taste or lifestyle [START_REF] Smith | Online peer and editorial recommendations: Trust and choice in virtual markets[END_REF][29] [START_REF] Brown | Word of mouth communication within online communities: Conceptualizing the online social network[END_REF]. Walther [START_REF] Walther | Selective self-presentation in computer-mediated communication: Hyperpersonal dimensions of technology, language, and cognition[END_REF] states that within a computer mediated communication environment, people idealize their interlocutors based on elements in messages that may indicate a minimal resemblance. According to Baym [START_REF] Baym | Personal connections in the digital age[END_REF] small pieces of information are very important in an online environment that contains little cues. Even when these chunks of information where not distributed with the intention to release information about your personality. Besides unwittingly releasing cues about their identity; people of course also release certain aspects of their person in a highly controlled and socially desirable way [START_REF] Walther | Selective self-presentation in computer-mediated communication: Hyperpersonal dimensions of technology, language, and cognition[END_REF].

People can seize upon two aspects to present themselves online or to grasp the identity of others. (1) Platform-dependent tools are instruments embedded in the online environment itself. The most fundamental tool, embedded in discussion boards, is a user profile containing aspects such as a nickname, an avatar, demographic information, a link to a personal website… According to Ma and Agarwal [START_REF] Ma | Through a glass darkly: Information technology design, identity verification and knowledge contribution in online communities[END_REF] persistent labeling -or the use of one label to present yourself -is important. By constantly joining a forum with the same user ID, other members can form an impression of the person. It allows community members to recognize and indicate central figures within the platform. A profile often also contains information about the frequency of use and your reputation as a contributor. This is what Ma and Agarwal [START_REF] Ma | Through a glass darkly: Information technology design, identity verification and knowledge contribution in online communities[END_REF] call deep profiling -or the digital organization of social information. Other items that could yield identity information are membership lists, rankings, 'who did what'-functions and a possibility to browse the archive [START_REF] Ma | Through a glass darkly: Information technology design, identity verification and knowledge contribution in online communities[END_REF]. (2) Platform-independent tools also allow understanding of a person's identity but this insight is based on what can be abstracted from conversations. Language itself can ensure that you will be noticed online [START_REF] Baym | Personal connections in the digital age[END_REF]. Besides the fact that it is much more pleasant to read witty contributions, language also allows you to tell something about yourself by means of self-disclosure. Self-disclosure or self-description is cited by both Baym [START_REF] Baym | Personal connections in the digital age[END_REF] and Leary (1996, as mentioned in [START_REF] Ma | Through a glass darkly: Information technology design, identity verification and knowledge contribution in online communities[END_REF]) as an important way to construct identity online. By telling others something about yourself, you lose the label of newbie or stranger. It allows relationships to establish and to be maintained [START_REF] Baym | Personal connections in the digital age[END_REF]. Not only does an act of self-disclosure provides personal information which allows others to construct your identity; also trust in people who disclose personal information is generally higher [START_REF] Ridings | Some antecedents and effects of trust in virtual communities[END_REF].

Both tools mentioned can be seen as indications of a personal identity but they mainly give an initial impression of your discussion partner. The main way people actually gain insight in the identity of others is of course based on experience with members themselves.

Reputation

Several authors cite reputation as an important reference that allows to figure out the degree of integrity and the capacities of people and thus to build trust [20] [54] [55] [56] [57]. A reputation is 'an expectation about an agent's behavior based on information about or observations of its past behavior.' [START_REF] Alfarez | A distributed trust model[END_REF]. It is a record of past actions [START_REF] Sztompka | Trust: A sociological theory[END_REF]. Through these past actions, you can try to make a prediction of future behavior. A reputation is often build upon direct observations and therefore rarely ahistorical (Good 1988:33 as mentioned in [START_REF] Sztompka | Trust: A sociological theory[END_REF]). The longer you know someone, the more opportunities you have to assess this person's reliability and the greater the chance that a bond of trust will form [START_REF] Sztompka | Trust: A sociological theory[END_REF] [START_REF] Dasgupta | Trust as a commodity[END_REF]. Yet, direct observations and personal experience are often problematic, since we regularly come in contract with people we do not know [that well]. Here people will switch to certain references like second-hand testimonies or achievements that demonstrate trust [START_REF] Sztompka | Trust: A sociological theory[END_REF]. Also Scott and Johnson [START_REF] Scott | Bowling alone but online together: Social capital in e-communities[END_REF] talk about assigning a reputation based on personal experience, recommendations and association. Yet, the authors associate these elements with a typical community of place. Understanding reputation in online communities, can be done by looking at aggregated reputation scores [START_REF] Scott | Bowling alone but online together: Social capital in e-communities[END_REF]. Yet we do believe that -for example -sharing a picture online of a homemade cake, may lead others to think that you know a lot about baking. A similar example is when you share your profession as an electrician in your online user ID on a home improvement discussion forum. Although both individuals will not receive an overall reputation score, showing their achievements can nevertheless contribute to their reputation.

On online discussion platforms, a reputation can be formed on the basis of direct observations. The opinion of a specific member becomes highly important when previous discussions with this person gave proof of expertise. Somewhere in the past, you have asked a question on the forum and you followed the advice of a specific member with success as a result. However, the idea of direct observation is somewhat more complex in a textual community since conversations remain publicly available. When someone publicly demonstrates trust -e.g. by saying he is going to try out the advice -and this trust becomes subsequently confirmed -e.g. by thanking the forum member for the good advice -, other members can see this display of trust, which allows the reader of this conversation to decide to proceed with trusting behavior himself. This entire idea fits strongly with the following quote from Sztompka [START_REF] Sztompka | Trust: A sociological theory[END_REF]: 'Being visibly trusted by some may be an argument for others to grand trust too.' Forum users also have the possibility to derive a reputation based on indirect observations. On the one hand, one can use references from other forum members. For example when during a forum discussion a member refers to another member as being an expert in a certain domain. Another type of reference includes what we have explained earlier with the example of the homemade cake. Showing a picture of something you have made or talking about your experience with certain tasks indirectly shows your expertise/experience within a certain domain and can help to shape your online reputation. New forum users -those who cannot fall back on their own experiences with other members -can also rely on online reputation mechanisms. Some online communities provide their members with explicit information about other users based on rankings, ratings, collaborative filtering and peer-based systems [START_REF] Jensen | Finding others online: Reputation systems for social online spaces[END_REF]. In this way they attempt to provide members with a less anonymous image.

Expertise

Expertise is also an important reason for people to assign trust [START_REF] Smith | Online peer and editorial recommendations: Trust and choice in virtual markets[END_REF][13] [START_REF] Brown | Word of mouth communication within online communities: Conceptualizing the online social network[END_REF][59] [START_REF] Willemsen | Credibility assessments of online product reviewers: The effects of perceived expertise and perceived trustworthiness[END_REF]. Expertise indicates whether a particular source / user is competent or not: Does this person have the necessary knowledge to judge the situation in a correct way or to give proper advice? It is the extent to which a source is regarded as being capable to formulate valid statements [START_REF] Willemsen | Credibility assessments of online product reviewers: The effects of perceived expertise and perceived trustworthiness[END_REF]. According to Mackiewicz [START_REF] Mackiewicz | Assertions of expertise in online product reviews[END_REF] expertise gives people a reason to trust. She sees it as a warranty people can use to make an assessment. Sharing your own experience with using a consumer good or with the performance of a particular task, placing a picture on a discussion forum as proof, using technical or site-specific language or disseminating difficult to find information… These are all examples that can lead to people being granted with the status of an expert. They give insight in the fact whether a person has the necessary skills and knowledge to know what he is talking about. According to Mackiewicz [START_REF] Mackiewicz | Assertions of expertise in online product reviews[END_REF] forum members can express expertise by using specialized terminology or by making an explicit claim of expertise. This explicit claim of expertise can be made by telling about your experience with a certain product, a service or a task through a forum post. Yet we believe that less explicit portrayals of expertise can be important as well; for example showing of your crafts via a picture can be the ultimate proof that you know what you are talking about. A claim of expertise can also be made on the basis of a relevant role: 'reviewers construct […] an identity for themselves that included a role particularly relevant to the product.' [START_REF] Mackiewicz | Assertions of expertise in online product reviews[END_REF]. According to Mackiewicz [START_REF] Mackiewicz | Assertions of expertise in online product reviews[END_REF], roles can be based upon people's experience from a hobby or employment in a profession. They can also be based upon the fact that the member had a formal training or has done a lot of research -which the author calls an instant expert -and upon second hand experiences -the forum member does not talk about his own experiences but about his connection with another person who has relevant experience.

CONCLUSION

Throughout this paper, an attempt was made to gain insight in the way people build trust within a semianonymous online environment such as online discussion boards. These surroundings are typically characterized by the exchange of mainly written messages between people who often do not know each other physically/offline. Important to mention is the absence of any reputation system, which makes computing with or for trust impossible. The fact that we are dealing with a cue-poor environment where a large part of the population cannot fall back on previous, intensive experiences with forum members, makes the construction of trust less easy to understand. Grasping the identity of the person who's message you are reading seems to become a rather difficult task.

In order to form a better understanding of trust, we identified six central concepts, essential when formulating a definition: interdependence, expectations, risk, insecurity, action and vulnerability. When you decide to follow up the advice of a forum member, you take up a dependent position with respect to this member and this on the assumption that following up his advice would lead to a desired result. Yet by acting upon someone's advice, you take a certain risk. You never fully know whether the person is adequate to formulate advise, which makes that you find yourself in a rather vulnerable position. Furthermore we discussed a typology of trust based on the object of trust -e.g. interpersonal trust aimed at a specific forum member versus system trust directed at the discussion platform as a whole -and based on the stages within a relationship -e.g. initial or swift trust experienced by new members and people who less frequently visit the forum in order to gather information versus ongoing or slow trust experienced by loyal members.

Looking back on this paper it appears that user types runs like a thread throughout the entire work. Interpersonal trust seems especially important within the initial trust stage where 'new' members try to make an evaluation of the knowledge, skills and veracity of other members. These users cannot count on experiences within the community and will therefore have to make an assessment of trustworthiness based on the individual characteristics people render in their messages. In order to gain an insight into the identity, the reputation or the level of expertise of specific members, one can depend on in-group categorization and stereotyping. One can pick up small role indications from user IDs or elements of self-disclose in the messages of members. System trust seems to be more reserved for loyal users, people who have developed an ongoing [trust] relationship with the community an its members. Here we are not longer dealing with personal features but it is more a general idea that you can count on the community to answer your questions with appropriate advice. Trust is being created through membership. Based on exchanges over time, users form an idea about the extent to which members share the same beliefs and interests.

Still some work has to be done to make both user types and trust antecedents more explicit. An empirical study, based on interviews with forum members and an analysis of forum conversations, would yield a more detailed user typology instead of a mere contradiction between new and old members or information seekers and social contact seekers. It would also help to get a deeper understanding of possible trust antecedents. It is not enough to say that people could use cues in user IDs or messages. In stead the focus should be on what they actually use and whether they do this rationally. An analysis of forum conversations would also allow us to see how people come to consensus.

https://www.facebook.com/about/graphsearch

Interdependence, interaction, task-oriented and a clear deadline are characteristics of the temporary organizational setting where trust will be granted ex ante.[START_REF] Meyerson | Swift trust and temporary groups[END_REF] expertise reputation identity


Tim Muller
email: tim.muller@uni.lu

Patrick Schweitzer
email: patrick.schweitzer@uni.lu

On Beta Models with Trust Chains

In a type of interactions over the Internet, a user (the subject) is dependent on another user (the target), but not vice versa. The subject should therefore form an opinion about the target, before possibly initiating an interaction. The scenario wherein a subject only relies on information obtained from past interactions with the target, is wellstudied and understood. In this paper, we formally analyze the implication of allowing recommendations (statements of a third party) as source of information. We identify the family of valid models that admit recommendations. This allows us to verify particular existing models that admit recommendations.

1 In the literature, the subject and the target are also referred to as trustee and trustor.

This terminology may however lead to the incorrect conclusion that the trustee is being trusted and the trustor is trusting.

Introduction

With the advent of the Internet, new types of interactions between different people arose. It is now possible, if not even common, to provide sensitive personal information to parties about which virtually nothing is known. For example, anyone can purchase goods from complete strangers on eBay. Contrary to purchasing goods in ordinary shops, buyers cannot inspect the commodities they acquire from an e-commerce website. Instead the shoppers have to wait and hope that everything will be delivered as ordered. In this paper, we focus on these kind of interactions, i.e. interactions where one party alone determines whether the outcome is beneficial or harmful to the other party. We call such interactions, interactions between a passive and an active party. The passive party attempts to avoid interactions with an active party that is likely to harm it. As a consequence, before potentially initiating an interaction, the passive party would like to estimate the likelihood with which the interaction outcomes are beneficial. We refer to such an estimate as a trust opinion. If a potentially passive party establishes a trust opinion about a potentially active party, the former is the subject, and the latter is the target 1 .

In interactions over the Internet, the information which a subject has about (alleged) past behavior of a target is limited. Hence it might be beneficial to ask for the help of third parties. Third party statements about the target are called recommendations, hence we call these third parties recommenders. Trust opinions constructed with the help of recommendations are called chained trust opinions. In this paper, we formally study the implications of such recommendations.

In the past, numerous formal models that derive trust opinions based on information about past behavior of active parties have been proposed. There exist simple models that allow a subject only to use his own past interactions with the target for information (see [START_REF] Staab | Towards Trust-Based Acquisition of Unverifiable Information[END_REF] for an effective method of gathering and filtering such information). For these approaches, a formal model, called the Beta model (or beta reputation system), has been derived [START_REF] Jøsang | The beta reputation system[END_REF][START_REF] Mui | A computational model of trust and reputation[END_REF].

To illustrate the Beta model, we introduce a simple running example.

Running Example. An economy teacher wants to teach her students about ecommerce with the help of a turn-based game. To set up the game, the teacher secretly distributes a random value p i ∈ [0, 1] to each student c i for 1 ≤ i ≤ 30.

The value p i represents the integrity of each student, and, similar to the integrity of users on an e-commerce website, it is unknown to the other players. On an ecommerce system this parameter models how likely the outcome of an interaction is to be successful. Each turn of the game follows the following pattern. First, in the turn of student c i , the teacher assigns another student c j to c i . Then, c i has the choice between trusting or not trusting c j . In case c i chooses to trust c j , c i gains two points with probability p j , i.e. with the probability corresponding to the other student's integrity parameter. With the remaining probability of 1p j , c i loses one point. If c i chooses not to trust c j , then he neither gains nor loses points. On an e-commerce platform winning points corresponds to a successful interaction (a success), losing points to a failed interaction (a failure). After every turn, the teacher updates the students' points, only revealing the outcome to c i .

Like in e-commerce, trusting someone with high integrity has a high probability to result in a successful interaction; trusting someone with a low integrity has a high probability to result in an unsuccessful interaction. The classroom game can easily be analyzed within the Beta model. Assume that c i previously had s+f interactions with c j . Of these s+f interactions, s were successes and f were failures. With the help of the Beta model [START_REF] Jøsang | Artificial reasoning with subjective logic[END_REF] we estimate the probability of a success when trusting c j to be s+1 s+f +2 , and the expected value of trusting c j to be 2 s+1 s+f +2 -1 f +1 s+f +2 points. When not trusting c j , the points remain constant.

Suppose the next day, the teacher changes the rules of the game and allows c i to query a classmate about his experience with c j before having to choose whether or not to trust c j . That expansion of the classroom game can no longer be expressed in the Beta model (as it does not admit recommendations), it requires an extension.

To overcome this challenge, many modern trust models use the Beta model as a foundation, and increase the model's expressivity and its (practical) applicability by including recommendations. We say that a model which uses the Beta model as a foundation is in the Beta family. If a model is in the Beta family and also supports trust chains, we say it is in the Beta family with trust chains. Many models in the Beta family with trust chains are ad-hoc. By ad-hoc models, we understand models in which the inventors define chained trust opinions according to their intuition. The existence of ad-hoc models is supported by the fact that the research community has not yet settled on one trust model [START_REF] Krukow | Trust models in ubiquitous computing[END_REF], not even under the assumption that the trust model is in the Beta family [START_REF] Jøsang | Exploring different types of trust propagation[END_REF].

Rather then proposing a new model in the Beta family, we rigorously prove properties of trust chains valid in all models in the Beta family. We show the following properties. Chained trust opinions are modular (Proposition 4 and Theorem 3), meaning that complex trust opinions can be constructed from simpler ones. Every trust model makes implicit or explicit assumptions about how a recommender lies or about the number of interactions between users (Corollary 3). Chained trust opinions resulting have a different shape from the trust opinions in the Beta model (Theorem 4). Furthermore, Subjective Logic, an expressive ad-hoc extension of the Beta model, is not in the Beta family with trust chains (Corollary 5). The same conclusion can be derived for models similar to Subjective Logic, such as TRAVOS [START_REF] Teacy | TRAVOS: Trust and Reputation in the Context of Inaccurate Information Sources[END_REF] and CertainTrust [START_REF] Ries | Certaintrust: a trust model for users and agents[END_REF] (Corollary 4).

In Section 3, we formalize the notion of recommendations and add it to the Beta model, effectively formalizing all models in the Beta family with trust chains. Then, in Section 4, we study the most basic trust chains in the Beta family with trust chains. In Section 5, we prove that all models in the Beta family with trust chains have the property that trust opinions can be constructed modularly from the most basic trust chains. Finally, in section 6, we characterize trust models in the Beta family with trust chains, and show that existing models based on the Beta model are not in the Beta family with trust chains.

The Beta model

In this section, we introduce the Beta model. The formulation of the Beta model relies on well-known techniques from probability theory (see e.g. [START_REF] Billingsley | Probability and measure[END_REF][START_REF] Gut | Probability: A Graduate Course[END_REF]). There are two concepts in particular that are important for our analysis. The first is conditional independence: Definition 1 (Conditional independence of variables [START_REF] Bouckaert | Bayesian belief networks and conditional independencies[END_REF]). Let (Ω, F, P ) be a probability space and let X, Y , Z be random variables (from Ω) with values in the measurable spaces (E i , E i ), i ∈ {X, Y, Z}. Two random variables X and Y are conditionally independent given the variable Z if

P (X ∈ A, Y ∈ B|Z ∈ C) = P (X ∈ A|Z ∈ C)P (Y ∈ B|Z ∈ C). for each A ∈ E X , B ∈ E Y and C ∈ E Z .
As shorthand we write (X ⊥ ⊥ Y )|Z or even X ⊥ ⊥ Y |Z. Note that the definition is equivalent to P (X|Y, Z) = P (X|Z).

And the second is the concept of beta distributions:

Definition 2 (Beta distribution). A beta distribution is a family of continuous probability distributions in the interval [0, 1], parameterized by two positive parameters, α, β ≥ 1. The probability density function of a beta distribution with parameters α and β is

β(x; α, β) = x α-1 (1 -x) β-1 1 0 y α-1 (1 -y) β-1 dy
.

The expression under the fractions is known as the beta function on α and β, and for positive integers α and β, the beta function fulfills B(α, β) = (α-1)!(β-1)! (α+β-1)! . We define the Beta model in a similar fashion to [START_REF] Muller | A Formal Derivation of Composite Trust[END_REF]. We first define a series of random variables. Let A denote a set of agents. For A, C ∈ A and a set of events Ω, we then define:

• E C : Ω → {s, f} is a discrete random variable modeling the outcome of the corresponding interaction with target C. • R C : Ω → [0, 1] is a continuous random variable modeling the (hidden) integrity parameter of target C which defines the probability of success.

• O A C : Ω → N × N
is a discrete random variable modeling the interaction history of subject A about target C, representing the past interactions (number of successes and failures) between A as passive party and C as active party.

Running Example. In the classroom game, E C models the outcome of an interaction with student C. The variable R C describes the secret parameter initially assigned by the teacher to C and O A C expresses how many times student A interacted successfully with student C, and how many times it was a failure.

A trust opinion is a distribution over the integrity parameter of a target, based on the interaction history about the involved active parties. Hence, if a subject A establishes a trust opinion about a target C, the probability density function is of the form f R C (x|O A C , ϕ), where ϕ may express additional conditions. Next, we provide the assumptions of the Beta model, in the shape of dependencies and independencies of random variables, as formulated in [START_REF] Muller | A Formal Derivation of Composite Trust[END_REF]. For a more concise formulation of the (in)dependencies, we introduce sets of random variables.

E := {E C : C ∈ A}, R := {R C : C ∈ A}, O := {O A C : A, C ∈ A}, W := E ∪ R ∪ O.
The size of the interaction histories is unknown. We therefore model it with a distribution λ, called the entanglement. Let c ∈ [0, 1], x s , x f ∈ N and λ : N → [0, 1] be a probability distribution. For all agents A, C ∈ A we set up the following dependency and independency relations as our assumptions. D1 R C is the uniform distribution on [0, 1].

If we know nothing about the integrity of C, we assert all values equally likely. For specific applications, statistical data about behaviors of agents may be used to construct an alternative distribution. A suitable distribution has a probability density function that is non-zero on (0, 1).

D2 P (E C =s|R C =c) = c.
We assume that the probability of good behavior of A is determined by an integrity parameter a.

D3 P (O A C =(x s , x f )|R C =c) = xs+x f xs c xs (1 -c) x f λ(x s + x f ).
Assumes that the probability that A and C had an interaction history with size x s + x f is λ(x s + x f ), and that each past interaction had success probability c.

I1' For W ∈ W \{O A C }, it holds that O A C ⊥ ⊥ W |R C .
The interaction history is completely determined by its size, and the probability of a success in a single interaction (by Dependency D3).

I2' For W ∈ W \{R C }, it holds that R C ⊥ ⊥ W |E C ∩ D∈A {O D C }. The only indicators of the integrity parameter of C, are interactions with it. I3' For W ∈ W \{E C }, it holds that E C ⊥ ⊥ W |R C .
The behavior of C is completely determined by its integrity parameter (by Dependency D2).

A trust opinion of A about C can now be seen as the probability density function given by f R C (c|ϕ), where ϕ represents all knowledge of A about C, modulo the relations of the random variables. Typically, ϕ is equal to O A C , provided there are no recommendations. In this case, we call f R C (c|ϕ) a simple trust opinion, to be able to distinguish it from trust opinions involving recommendations.

Theorem 1 (Axiomatization of the Beta model [START_REF] Muller | A Formal Derivation of Composite Trust[END_REF]). The Beta model adheres to Dependencies D1-D3 and Independencies I1'-I3'. The simple trust opinion obtained from an interaction history with x s successes and x f failures is the beta distribution β(c; x s + 1, x f + 1).

Suppose there are two concurrently held trust opinions based on two different interactions with a single agent. It is desirable to combine these two trust opinions into a single trust opinion based on both interactions. We introduce a trust aggregation operator to accomplish that:

Definition 3 (Aggregation of trust opinions). The aggregation of trust opinion

T = f (c) and T = g(c) is T ⊕ T = f (c)×g(c) 1 0 f (c)×g(c) dc ∝ f (c) × g(c).
The trust aggregation operator correctly combines simple trust opinions: Lemma 1. Given trust opinions T and T based on (x s , x f ) and (y s , y f ), respectively, the aggregate trust opinion T ⊕ T is based on (x s + y s , x f + y f ).

Proof. T ⊕T ∝β(c; x s +1, x f +1)×β(c; y s +1, y f +1)∝β(c; x s +y s +1, x f +y f +1) Our assumptions regarding simple trust opinions are in line with the Beta model. They are in fact sufficient to derive it (Theorem 1). Hence, those assumptions can be seen as valid for the numerous models that use the Beta model as a foundation [START_REF] Jøsang | Artificial reasoning with subjective logic[END_REF][START_REF] Teacy | TRAVOS: Trust and Reputation in the Context of Inaccurate Information Sources[END_REF][START_REF] Ries | Certaintrust: a trust model for users and agents[END_REF].

Beta family with trust chains

According to the Beta model, a subject A constructs his trust opinion using only his own information, when planning to interact with a target C. Depending on the constructed trust opinion, A chooses to interact or not. Suppose that A wants to make a more informed decision. Then, the subject A may ask a third party, a recommender B, for advice. A recommender could provide an honest recommendation, or lie. Chained trust opinions are based on the notion that a trust opinion on the recommender B is a valid measure for the likelihood that B provides an honest recommendation about C. More formally: Definition 4 (Chained trust opinions). Every recommender (like every target) has an integrity parameter that determines the probability of a successful interaction. In case of a successful interaction, their recommendation is their trust opinion about the target. Chained trust opinions are trust opinions based on recommendations from recommenders.

We add recommendations to the classroom game:

Running Example. After a number of turns, the students realize that the Beta model can be applied to construct a trust opinion about other students. This allows all students to make optimal choices. To keep the game interesting, as well as make it a more realistic emulation of e-commerce, the teacher adds recommendations in the following way: In the beginning of every turn, the teacher not only assigns a subject c i ∈ {c 1 , . . . , c 30 } =: S and a target c j ∈ S, but also a set of recommenders R ⊆ S \ {c i , c j } if c i has never interacted with c j . Every recommender c k ∈ R has to honestly provide their past interactions with c j with probability p k , or construct and provide a fake past history with c j with probability 1 -p k . Again, students with a high integrity p k are more likely to provide the past interactions rather than fake interactions. For a subject to construct the most accurate trust opinion, he needs to incorporate his opinion of c k and the recommendation by c k , for all c k ∈ R.

To formally model recommendations in the Beta model, we introduce another random variable.

• S B

C : Ω → N × N is a discrete random variable modeling recommendations of the recommender B about the target C, representing the alleged past interactions between B as passive party and C as active party.

We also introduce additional sets of random variables:

S := {S B C : B, C ∈ A}, W := W ∪ S. Let a, b, x ∈ [0, 1], n, k ∈ N and λ : N → [0, 1] as well as χ B : [0, 1] × N × N → (N × N → [0, 1]
) , where B ∈ A be probability distributions. For all agents A, B, C ∈ A we set up the following additional dependency and independency relations as our assumptions. In fact, Independencies I1'-I3' from the initial Beta model only need to be generalized to encompass recommendations.

D4 P (S B C =(w s , w f )|E B =s, O B C =(w s , w n )) = 1
Assumes that good behavior of B implies that the recommendation of B corresponds to his interaction history with C.

D5 P (S B C =(y s , y f )|E B =f, R B =b, O B C =(w s , w f )) = χ B (b, w s , w f )(y s , y f ) Defines the lying strategy χ B of agent B.
The lying strategy is a function, from a parameter and an interaction history (k , nk ) to a distribution of recommendations. A recommender (probabilistically) selects its fake recommendations. There are models that are inspired by the Beta model, and that include an operator dealing with recommendations, but that are not models in the Beta family with trust chains. We argue that such models either are not Beta models or that is not a trust chaining operator. If a model violates any of the Dependencies D1-D3 or Independencies I1'-I3', it is not a Beta model. We distinguish the possible violations of an assumption for each remaining assumption separately. If a model violates D4, then the model does not support trust chaining. D5, then another assumption must also be violated. This is due to the fact that under Dependencies D1-D4 and Independencies I1-I4 there exists a χ B such that χ B (b, w s , w f )(y s , y I2, then the model either violates Independency I2', or it assumes that some R C are dependent with S D E given all observations of C. This is not in the spirit of the Beta model as the collection of all interactions with C should be an optimal estimator for R C . I3, then the model either violates Independency I3', or it assumes that some E C are dependent with S D E (for D = C) under all observations of C, which is not in the spirit of the Beta model as the probability of success of an interaction (given the integrity) should not be influenced by recommendations of others. I4, then in this model recommenders differentiate their strategy either on information they cannot know (e.g. interactions that the recommender did not participate in) or on information that is irrelevant for the recommendation (e.g. his opinion on yet another agent).

I1 For W ∈ W\{O A C }, it holds that O A C ⊥ ⊥ W |R C . Similar to Independence I1', except recommendations are also independent. I2 For W ∈ W\{R C }, it holds that R C ⊥ ⊥ W |E C ∩ D∈A {O D C }. Similar to Independence I2', except recommendations are also independent. I3 For W ∈ W\({E B } ∪ D∈A {S B D }) , it holds that E B ⊥ ⊥ W |R B . Similar to Independence I3', except recommendations not from B are also independent. I4 For W ∈ W\{S B C }, it holds that S B C ⊥ ⊥ W |E B =f ∩ R B ∩ O B C .
Not every model in the Beta family with trust chains is formalized our way. A model is already in the Beta family with trust chains when the assumptions can be reformulated to fit the assumptions up to isomorphisms.

Basic Trust Chains

The most basic scenario that involves trust chains, involves exactly one recommendation. This recommendation is given about a target with which the subject has no prior interactions. In other words, the recommendation is the only source of information that a subject has. This scenario is called basic trust chaining.

It is studied in this section. In Section 5, we then prove that more complicated scenarios can be reduced to scenarios with basic trust chains.

Definition 6 (Basic trust chain, basic chained trust opinion). A basic trust chain consists of three agents: the subject A, the recommender B, and the target C. The subject has an interaction history x = (x s , x f ) with the recommender. The recommender provides a recommendation y = (y s , y f ) about the target and, in reality, has an interaction history w = (w s , w f ) with the target. The trust opinion of subject A about target C with recommendations by recommender B is the basic chained trust opinion. It is depicted in Figure 1.

Running Example. In the classroom game, basic trust chains appear when the teacher assigns only one recommender. Then, the subject is c i ∈ S, the target is c j ∈ S \ {c i } and the set of recommenders is {c k } ⊂ S \ {c i , c j }.

We may now formulate the basic chained trust opinion of A about C with recommendations given by B as f R C (c|O A B =(x s , x f ), S B C =(y s , y f )). In other words, to formulate a trust opinion about the target, the subject uses its interaction history about the recommender as well as the (possibly fake) recommendation given be the recommender. If A has never directly interacted with B, the pair (x s , x f ) equals (0, 0). Theorem 2 (Basic chained trust opinion). Dependencies D1-D5 and Independencies I1-I4 are sufficient to derive the basic chained trust opinion of A about C with recommendations by B as:

f R C (c|O A B =(x s , x f ), S B C =(y s , y f )) = eq 1 (y s , y f ) × eq 2 + w∈O B C (eq 1 (w s , w f ) × eq 3 ×(1 -eq 2 )), (1)
where,

eq 1 (ϕ s , ϕ f ) = β(c; ϕ s + 1, ϕ f + 1), eq 2 = eq 4 ×(x s + 1) eq 4 ×(x s + 1) + w ∈O B C eq 5 (w ) × (x f + 1)
, Proof. The equations eq 1 -eq 5 represent the following probabilities:

eq 3 = eq 5 (w s , w f ) w ∈O B C eq 5 (w s , w f ) , eq 4 = λ(y s + y f ) × y s + y f y s × y s !y f ! (y s + y f + 1)! eq 5 (ϕ s , ϕ f ) = 1 0 χ B (b, ϕ s , ϕ f )(y s , y f ) × β(b; x s + 1, x f + 2) db × λ(ϕ s + ϕ f ) × ϕ s + ϕ f ϕ s × ϕ s !ϕ f ! (ϕ s + ϕ f + 1)! A C B O A B = (x s , x f ) S B C = (y s , y f ) C B O B C = (w s , w f )

Basic chained trust opinion

eq 1 (ϕ) = P (R C =c|O A B =x, S B C =y, E B =u, O B C =w), eq 2 = P (E B =s|O A B =x, S B C =y), eq 3 = P (O B C =w|O A B =x, S B C =y, E B =f), eq 4 = P (S B C =y|O A B =x, E B =s), eq 5 (ϕ) = P (S B C =y, O B C =ϕ|O A B =x, E B =f).
A proof of correctness of eq 1 -eq 5 can be found in [START_REF] Muller | On beta models with trust chaining[END_REF]. The correctness from Formula (1) follows from the correctness of eq 1 -eq 5 , given that, for all W ∈ W:

S B C ⊥ ⊥ W |E B =s ∩ O B C
follows from Dependency D4. Although Formula 1 may seem complicated, it can abstractly be viewed as a (infinite) weighted sum of beta distributions: Proposition 1. For every entanglement and lying strategy, a basic chained trust opinion is a weighted sum of beta distributions.

Proof. If we collect factors that do not contain the variable c in the scalars k and k ws,w f , Formula (1) simplifies to

k • c ys (1 -c) y f + ws,w f ∈N×N k ws,w f c ws (1 -c) w f . (2)
Furthermore, for some specific models in the Beta family with trust chains, the formula significantly simplifies. Particularly, for a lying strategy that consists of constructing truthful recommendations (see dash-dotted graph in Figure 2), the trust opinion is a beta distribution: Proposition 2. If χ B (b, w s , w f )(y s , y f ) = 1 iff (w s , w f ) = (y s , y f ), then the trust opinion from Formula (1) simplifies to β(c; y s + 1, y f + 1).

Taking an arbitrary entanglement λ and a lying strategy that consists of constructing completely informationless recommendations (see dashed graph in Figure 2), the trust opinion is a weighted sum of a beta distribution and the uniform distribution:

Proposition 3. If χ B (b, w s , w f )(y s , y f ) = 1 ys+y f +1 iff w s + w f = y s + y f , then the trust opinion from Formula (1) simplifies to xs+1 xs+x f +2 β(c; y s + 1, y f + 1) + x f +1 xs+x f +2 .
An immediate consequence of Theorem 2 and Proposition 1 is that a model that supports basic chained trust opinions, makes assumptions about the entanglement and lying strategies. Proof. Proposition 2 and 3 are not equal, hence the choice of χ B matters.

Running Example. In terms of the classroom game, the corollary states that it is relevant how many turns have been played and how students lie. If a recommendation states "8 successes and 2 failures", but each stunted has played 9 turns, the recommendation is clearly fake, whereas the same recommendation may be likely true when each student has had 100 turns. Suppose, a student c k provides a recommendation to c i that is likely to be fake. If c k and c i are good friends outside of the game, c k might have a lying strategy of creating fake recommendations that strongly resemble the truth. Otherwise, c k provides recommendations unrelated to the truth. Then, it is wise for c i to rely on the recommendation of his friend, but not on recommendations of other arbitrary classmates.

Corollary 1 implies that without assumptions on λ and χ B , no model can provide trust opinions. Therefore, any trust model in the Beta family with trust chains either implicitly or explicitly makes assumptions about numbers of interactions and about the lying strategy of recommenders. We believe that making implicit assumptions about lying strategies is critical, as it obfuscates the analysis of a model or hides undesirable consequences of a model. Hence, we suggest that new proposals for models in the Beta family with trust chains explicitly (and formally) provide the lying strategy of the recommenders.

Corollary 2. For every entanglement λ and lying strategy χ B , the subject can calculate the basic chained trust opinion.

Proof. Apply Formula (1), with the relevant instantiations of λ and χ B . Thus, when the number of turns in the classroom game is known, and it is known what kind of lying strategy each student has, the subject can correctly compute the trust opinion, whenever the teacher assigns only one recommender.

A positive consequence of Corollary 2 is that defining the entanglement and the lying strategy is sufficient to explicitly define a model in the Beta family with trust chains. Not only is it mathematically possible, but we have developed a tool named Canephora2 that can compute basic chained trust opinions, when χ B and λ are provided. The tool is a proof of concept, that creating a model in the Beta family with trust chains is merely a matter of defining an entanglement and lying strategies. It is a prototype that allows the numerical comparison between different models (i.e. different choices of entanglements and lying strategies).

In Section 5, we see that defining the entanglements and the lying strategies is sufficient to explicitly define models in the Beta family with trust chains (not just models restricted to basic trust chains). Determining the entanglement λ is usually simpler than finding the lying strategy. On many e-commerce systems, the number of interactions between users is known to the system. For example, eBay knows if a product is sold, even if it does not know whether the transaction was a success for the subject. Or in the classroom game, the teacher announces the number of turns, explicitly providing λ. Even if the entanglement is unknown, by restricting the choices of χ B , the entanglement λ can be eliminated from Formula (1).

Lemma 2. For some lying strategies, the entanglement has no impact on the basic chained trust opinion.

Proof. Consider the basic chained trust opinion given by Formula (1). For all b ∈ R, and w s , w f , y s , y f ∈ N such that w s +w f = y s +y f , take χ B (b, w s , w f )(y s , y f ) = 0. Then, λ(ϕ s + ϕ f ) cancels out of eq 5 unless ϕ s + ϕ f = y s + y f . In the reduced term, we can substitute λ(ϕ s +ϕ f ) for λ(y s +y f ). Then λ(y s +y f ) is a scalar that appears in every summand in the numerators and denominators of eq 2 and eq 3 . Thus λ cancels out of Formula (1).

Running Example. If a recommender makes a recommendation of which the size was impossible (or very unlikely), a student can identify the recommendation as a fake (or likely a fake). If all students take care never to fall into the pitfall of sizing fake recommendations according to a different distribution than the real interactions, sizing becomes irrelevant. Hence, the entanglement cancels out.

Modular Construction of Trust Opinions

In Section 3, the assumptions of the Beta model were formally extended to include trust chaining. We have formally derived a parameterized trust opinion in the case of basic trust chains. However, it is possible that a subject receives more than one recommendation, or that the subject also has a simple trust opinion of the target. Recall trust aggregation from Definition 3. We first prove that a basic chained trust opinion can be aggregated with a simple trust opinion.

Later, we prove that more complicated trust opinions can also be aggregated with basic trust opinions. The notion that aggregation of these trust opinions is possible, is called modularity.

Running Example. Imagine that the subject c i constructs a trust opinion about the target c j based on his past interactions (z s , z f ) with c j . However, the teacher also provides a recommender c k , with which the subject has an interaction history of (x s , x f ). The student c k himself, give the recommendation (y s , y f ) about c j . From the Beta model, the subject can construct his (simple) trust opinion based on (z s , z f ). From Section 4, the subject can construct his (basic chained) trust opinion based on (x s , x f ) and (y s , y f ). The subject wants to construct a trust opinion based on (x s , x f ), (y s , y f ) and (z s , z f ). We prove the subject merely needs to aggregate both trust opinions.

Many trust models in the Beta family with trust chains (such as Subjective Logic) assert modularity. A priori, it is not obvious that the assertion of modularity is justified.

Running Example. Consider a situation in the classroom game where a student first constructs a trust opinion T d directly from all his information. Then he tries an alternative approach and constructs simple trust opinions based on only parts of his information. These simple trust opinions he then aggregates into a trust opinion T i . Assume that the subject c i has a strongly positive opinion T about the target c j , and a mildly positive opinion T about the only recommender c k . Assume further that the lying strategy of c k is probabilistic and unrelated to the actual interactions of c k with c j and that λ(n) is irrelevant (Lemma 2). Moreover assume, the recommender c k gives a mildly negative opinion R about the target c j .

Constructing his trust opinion T d directly, the subject c i concludes that, even though he expected the recommender to give honest recommendations more often than fake ones, this particular recommendation is nearly certainly fake. The subject expects the recommendation to be fake because he is quite certain that c j has a high integrity (due to his trust opinion T ). In other words c i does not think it likely that c k has more failed than successful interactions with c j (which honesty of R would entail). Therefore, in the resulting trust opinion T d , the recommendation R does not have a large impact.

If the subject constructs his trust opinion T i modularly, then he aggregates T with a basic chained trust opinion T c based on T and R, without applying his own experience with c j . If the subject does that, he will accept (in T c , thus in T i ) that it is likely that the recommender provided an honest opinion about the target.

In conclusion, we may expect that T i is more influenced by R than T d .

The naive intuition that a modularly constructed opinion (T i ) differs from a directly constructed opinion (T d ), is proven incorrect in Proposition 4 and Theorem 3. First, we prove modularity between a simple trust opinion and a basic chained trust opinion: Proposition 4. For all models in the Beta family with trust chains, the chained trust opinion

f R C (c|O A B =(x s , x f ), S B C =(y s , y f ), O A C =(z s , z f ))
is the aggregate of the simple trust opinion f R C (c|O A C =(z s , z f )) and the basic chained trust opinion f R C (c|O A B =(x s , x f ), S B C =(y s , y f )). Proof. We require Independence I1 and Dependence D1.

f R C (c|O A B =(x s , x f ), S B C =(y s , y f ), O A C =(z s , z f )) = P (O A B =(x s , x f ), S B C =(y s , y f ), O A C =(z s , z f )|R C =c) × f R C (c) P (O A B =(x s , x f ), S B C =(y s , y f ), O A C =(z s , z f )) I1 = P (O A B =(x s , x f ), S B C =(y s y f )|R C =c) × P (O A C =(z s , z f )|R C =c) × f R C (c) P (O A B =(x s , x f ), S B C =(y s , y f ), O A C =(z s , z f )) D1 ∝ P (O A B =(x s , x f ), S B C =(y s , y f )|R C =c)×f R C (c)×P (O A C =(z s , z f )|R C =c)×f R C (c) P (O A B =(x s , x f ), S B C =(y s , y f )) × P (O A C =(z s , z f )) =f R C (c|O A C =(z s , z f )) × f R C (c|S B C =(y s , y f ), O A B =(x s , x f
)) Similar to Proposition 4, we can even prove that modularity holds for all trust opinions. Let ϕ be a collection of basic trust chains and potentially the interaction history between the target and the subject. In other words, for some n, let ϕ be given by: C =(w s , w f ), R B =b) = P (ϕ|R C =c), which follows from Independencies I2 and I3. A more detailed proof can be found in [START_REF] Muller | On beta models with trust chaining[END_REF].

[O A C =(z s , z f ), ]O A B1 =(x 1 s , x 1 f ), S B1 C =(y 1 s , y 1 f ), . . . , O A Bn =(x n s , x n f ), S Bn C =(y n s , y n f ).
From Theorem 3, we can conclude that the subjects can compute a trust opinion based on their own history with the target, as well as on recommendations of an arbitrary number of other users, provided that the subject can compute basic chained trust opinions for all recommendations. More generally, Theorem 3 allows us to generate the following structures S(λ, θ) = (P, O, g : P → O, c λ,θ : P ×P → O, a : O×O → O), where P is the set of interaction histories, O is the set of opinions, g is the function that maps interaction histories to simple trust opinions, c λ,θ is the function that generates basic chained trust opinions (for entanglement λ and assignment of lying strategies to users θ) , and a represents aggregation of trust opinions. Depending on the choice of the entanglement and the assignment of lying strategies, the structures S(λ, θ) (generally) differ.

Analysis of the Models

The results from the last sections allow us to study the conditions that all trust opinions for in all models in the Beta family with trust chains must adhere to. If an existing trust model violates these conditions, it is therefore not in the Beta family with trust chains. Which, in turn, means that these trust models either break an assumption of the Beta model (on which they are based), or its operator dealing with recommendations does not actually model trust chains according to Definition 5.

First, we point out that the work in Sections 4 and 5 captures all models in the Beta family with trust chains up to isomorphism: Corollary 3. Every model in the Beta family with trust chains is isomorphic to a structure S(λ, θ) for an entanglement λ and an assignment of lying strategies θ.

Proof. The corollary is a direct consequence of Corollary 2 and Theorem 3.

A consequence of the corollary is that if a model is in the Beta family with trust chains, there is a formulation of the model where the entanglement and the assignment of lying strategies are explicitly provided. This entails that if a formulation of a model does not explicitly mention the assignment of lying strategies, it is not an appropriate formulation as it obfuscates the lying strategies.

Furthermore, we prove a restriction on the shape of chained trust opinions: Theorem 4 (Chained trust opinions are not beta distributions). A basic chained trust opinion in any model in the Beta family with trust chains is in general not a beta distribution. An exceptional case arises when the lying strategy and the entanglement are set in such a way that the resulting opinion of a basic trust chain based on a recommendation (y s , y f ) is always (y s , y f ).

Proof. Expression (2) from Proposition 1 can only represent a beta distribution β(c; S + 1, F + 1), if it can be simplified to h • c S (1 -c) F for some S, F ∈ N and h ∈ R + . Rearranging the coefficients specifying how the constants depend on eq 2 (henceforth let p = eq 2 ), we therefore have to prove the following:

p a c + (1 -p) m b m c m = c S (1 -c) F
Comparing coefficients (n < S or n > S + F ), we obtain that pa n + (1 -p)b n has to equal 0. This linear equation can only hold for a specific value of p unless a S = b S = 0. As we prove the general case, we may not restrict p to a specific value, seeing that eq 2 depends on x and y. Hence the sum reduces to

p S+F =S a c + (1 -p) S+F m=S b m c m = c S (1 -c) F
The sum S+F =S a c is proportional to a beta distribution, namely eq 1 (y). The only beta distribution with maximal exponent S + F , and minimal exponent S is β(c; S + 1, F + 1). That means that the recommendation, β(c; S + 1, F + 1), equals the resulting opinion, meaning that the equation can only hold if we are in the exceptional case.

Therefore, any model that represents all its chained trust opinions as beta distributions, is not in the Beta family with trust chains. Corollary 4. CertainTrust [START_REF] Ries | Certaintrust: a trust model for users and agents[END_REF] and TRAVOS [START_REF] Teacy | TRAVOS: Trust and Reputation in the Context of Inaccurate Information Sources[END_REF] are not in the Beta family with trust chains.

TRAVOS is an interesting case, as the authors set out to do essentially the same as is done in this paper. Similar to this paper, they treat the Beta model formally (using random variables for the integrity, for the outcomes and the recommendations) and study the relation between honest recommendations and fake recommendations. However, TRAVOS asserts that the result of a trust chain (in their case called reputation) is a beta distribution. A similar argument hold for Subjective Logic: Corollary 5. Subjective Logic [START_REF] Jøsang | Artificial reasoning with subjective logic[END_REF] is not in the Beta family with trust chains. Proof. Subjective Logic is isomorphic to a model where all trust opinions are beta distributions.

Hence, Subjective Logic breaks an assumption of the Beta model (on which it is based), or its operator dealing with recommendations (called trust transitivity or trust propagation) does not actually model trust chaining. Both can be argued, since in Subjective Logic the trust transitivity operator is based on fuzzy logic, rather than distributions over integrity parameters, yet trust opinions and trust aggregation (called fusion) are based on the Beta model (i.e. based on distributions). The latter, would entail that the fraction of Subjective Logic dealing with trust chaining is not useful; the former entails that usefulness of trust chaining does not follow from the theory surrounding the Beta model.

It is possible to alter Subjective Logic to incorporate a trust chaining operator such that it is isomorphic to a structure S(θ, χ). However, the property of Subjective Logic that a trust opinion equates to a belief triple will no longer hold. Rather, a trust opinion will equate a weighted sum of belief triples, e.g. i k i (b i , d i , u i ). The fusion (trust aggregation) of two trust opinions i k i (b i , d i , u i ) and j k j (b j , d j , u j ) will then be i,j k i × k j ((b i , d i , u i ) ⊕ (b j , d j , u j )), where ⊕ denotes unaltered fusion of belief triples from Subjective Logic. There are several valid variations for transitive trust operators (trust chains), and Proposition 3 shows that the operator need not be complicated.

Conclusion

We study a family of models based on the Beta distributions: the Beta family with trust chains. The models in that family are very similar to the Beta model, but more expressive. In particular, they can express trust chaining.

An important property, proven for all models in the Beta family with trust chains, is that trust chaining operations are modular (Proposition 4 and Theorem 3). So complicated trust opinions can be constructed by aggregating simpler trust opinions. Many existing trust models have asserted this property, which we now proved.

Another commonly asserted property in models inspired by the Beta model, is that all trust opinions can be represented as beta distributions. This property is disproven for models in the Beta family with trust chains (Theorem 4). This result implies in particular that Subjective Logic, TRAVOS and CertainTrust are not in the Beta family with trust chains (Corollaries 5 and 4).

We have proven that, up to isomorphism, every trust model in the Beta family with trust chains implicitly or explicitly makes assumptions about lying strategies and (except in special cases) about the entanglement (Corollary 3). Conversely, we have shown that, up to isomorphism, all trust models in the Beta family with trust chains can be constructed by selecting lying strategies and an entanglement (Corollary 3). Moreover, we have created a tool (Canephora) that calculates chained trust opinions, when instantiations of an entanglement and lying strategies are provided.

In the future we want to study the effectiveness of lying strategies using game theory. That would enable us to calculate the optimal lying strategies of recommenders, providing powerful models. Furthermore, we want to formally extend the Beta family with trust chains with additional operators, such conjunction [START_REF] Muller | A Formal Derivation of Composite Trust[END_REF]; in particular it is interesting to discover whether a modularity result still holds.

  The choice of B for making fake recommendations about C is completely determined by χ B (b, n, m) in Dependence D5.Models in the Beta family with trust chains should adhere to Dependencies D1-D5 and Independencies I1-I4.Definition 5 (Beta family with trust chains).A model is said to be in the Beta family with trust chains, when it satisfies Dependencies D1-D5 and Independencies I1-I4.

  f ) = P (S B C =(y s , y f )|O B C =(w s , w y ), R B = b, E B = f). I1, then the model either violates Independency I1', or it assumes that some S C D are dependent with O A C given R C . This is not in the spirit of the Beta model as the outcomes of the interactions between A and C should depend only on C.

Fig. 1 .

 1 Fig. 1. Left: The view of subject A about target C, including the recommendation S B C

Corollary 1 .

 1 It is not possible to compute basic chained trust opinions without knowledge of the entanglement λ and the lying strategy χ B .

Fig. 2 .

 2 Fig.2. The same trust chain, x = (6, 5) and y = (8, 4), with different lying strategies. Solid: lies opposite of his true opinion. Dashed: lies independent of the his true opinion. Dash-dotted: lies similar to his true opinion. Dotted: lies with a positive bias.

Theorem 3 (

 3 Modularity of trust opinions). For all models in the Beta family with trust chains, the trust opinionf R C (c|O A B =(x s , x f ), S B C =(y s , y f ), ϕ) is the aggregate of the trust opinion f R C (c|ϕ) and the basic chained trust opinion f R C (c|O A B =(x s , x f ), S B C =(y s , y f )).Proof. The only step of the proof in Proposition 4 that cannot be replicated (with ϕ substituted for O A C =(z s , z f )) is the application of Independence I1. Thus:P (O A B =(x s , x f ), S B C =(y s , y f ), ϕ|R C =c) ? =P (O A B =(x s , x f ), S B C =(y s , y f )|R C =c) × P (ϕ|R C =c)The proof obligation can be reduced (with Independencies I1 and I4) to P (ϕ|R C =c, E C =u, O B

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