Accepted: 27 March 2023 DOI: 10.1111/chso.12731 ORIGINAL ARTICLE Role conflicts of preschool teachers: From children’s perspectives in a changing cultural setting in Taiwan Yuling Su | Ya-hui Chung Department of Early Childhood Education, University of Taipei, Taipei, Taiwan Correspondence Yuling Su, Department of Early Childhood Education, University of Taipei, Bo-Ai Campus: No. 1, Ai-Guo West Road, Taipei 100234, Taiwan. Email: yulingsu1024@gmail.com Funding information National Science and Technology Council, Taiwan, Grant/Award Number: MOST 108-2628-H-845-001 Abstract This study investigated children’s perspectives on preschool teachers’ roles in Taiwan’s changing cultural setting. This study attempted to examine how aspects of culture and time are interwoven in children’s lives. Data were collected through children’s drawings, the photographs they captured and drawing-photo elicitation interviews. The data were analysed to identify the eight types of teachers’ roles and the time points at which the children observed the roles. The children noted that their teachers were ‘regulators’; however, they also discovered teachers’ interactive and democratic roles in less structured settings during the time of learning areas and while outdoors. KEYWORDS Chinese culture, children’s perspectives, culture, preschool, teachers’ roles INTRODUCTION Studies have reported that teachers’ roles reflect society’s expectations of them (Darling-Hammond & Hyler, 2020; Sprott, 2019) and are socially constructed (Keiler, 2018; Richards, 2015). Over time, the implications of teachers’ roles change within the different cultures influencing them (Makovec, 2018; Teo, 2019; Teo & Huang, 2019). Culture is complex to define (Murphie & Potts, 2017). Bruner (1996) stated that culture shapes lifestyles and minds, and people understand themselves and others by understanding the cultural system’s values. Different cultures emphasize different values, which shape how people think—for example, about curriculum content and © 2023 National Children’s Bureau and John Wiley & Sons Ltd. 1484 wileyonlinelibrary.com/journal/chso Children & Society. 2023;37:1484–1502. 1485 teaching methods. According to Hsu and Hwang (2015), the role of teachers in Chinese culture from the past to the present includes transmitting knowledge and instilling cultural values. In Chinese culture, teachers also frequently appear to the public as curriculum producers and to parents as caretakers. These roles suggest that teachers are passive course craftspeople or agents for parents; they lack the teachers’ perspectives and those of the children in the class. However, today, in addition to the role of ‘transmitter’, teachers have adopted the roles of ‘learners’, ‘thinkers’, ‘researchers’ and ‘partners’, representing the changes in teachers’ roles that the West has influenced (Ministry of Education, 2017; Phillipson & Lin, 2013; Yang & Li, 2019a). In Western culture, the roles of preschool teachers are those of co-creators of curriculum and participants with young children in preschool life (Busch et al., 2022; Dahlberg et al., 2007; Pastorek Gripson et al., 2022). From sociocultural perspectives, preschool teachers in the West collaborate with their students in constructing knowledge. Preschoolers are active knowledge builders; they learn continually and construct their own knowledge of the world through interactions with their sociocultural environments and communication with peers and adults (Vygotsky, 1978). Far from being passive learners, preschoolers are capable citizens who possess subjectivity rights (Einarsdottir, 2007, 2014). Preschoolers learn to respect, listen to and identify with each other; moreover, they develop knowledge and work together to solve problems (Dahlberg et al., 2007). In this study, we regarded both traditional and changing Chinese culture as influencing the roles of preschool teachers. The changing cultural setting represents preschool that encompass a fusion of the East and the West. Under the influence of traditional Chinese culture, teachers in Taiwan emphasize the values of empathy, contentment and care (Chen & Chung, 2019; Su & Chung, 2022a, 2022b). Teachers still place importance on traditional Chinese culture but, at the same time, respect their students’ agency, allocate time for students to engage in self-discovery and organize dynamic activities in a timely manner. The Western model of child-centred education shapes teachers’ roles, emphasizing preschoolers’ agency. We have argued that Taiwanese preschoolers in this changing cultural setting hold their own perspectives on teachers’ roles. They do not learn solely from teachers’ instruction, but also from self-discovery and interaction with others in free play. Traditional Chinese culture emphasizes teachers’ roles as transmitting knowledge and instilling cultural values. However, preschool children may perceive teachers’ roles as changing over time. Therefore, we agree with the suggestion from Bragg et al. (2019) that childhood studies must embrace important areas, including relational, historical and cultural factors. Childhood also comprises in the interaction with time and space of school environment (Hohti, 2016). Children’s views of teachers’ roles may not be the same as the teachers’ views of their roles as the changing culture influences them. According to the need for future childhood studies focusing on multiple relationships between children and the changing societies they habituate, we examined teachers’ roles from children’s perspectives and discovered the roles that children observe. This research was both a continuation of previous research into children’s perspectives on teachers’ roles (Einarsdottir, 2014; Pramling Samuelsson & Johansson, 2009) and an expansion of other studies that related to children’s perspectives in a changing Eastern culture in Hong Kong (Wu, 2019). Findings illuminated how Taiwanese preschoolers perceive teachers’ roles. 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SU and CHUNG SU and CHUNG LITERATURE REVIEW The influence of culture values on teacher roles Preschool teachers are members of a culture, and their roles are not only as scaffolders of young children’s learning but also as inheritors of cultural values. According to Cerulo et al. (2021), social customs and cultural norms affect individuals’ actions and cognitive schema. Meanwhile, beliefs borne of sociocultural interactions can also change social customs and cultural norms. Therefore, when cultural change, so do individuals’ values and modes of interaction. In particular, teachers’ roles are adjusted accordingly. An ethnic Chinese society reflects Confucianist influences and emphasizes collectivist beliefs. Individuals are parts of a group that attaches great importance to ethics and interpersonal relations. These beliefs have shaped the goals of education and the roles of teachers. Phillipson and Lin (2013) analysed the four goals of education in an ethnic Chinese society as follows. The first goal is to cultivate students’ ethical knowledge and passion for learning. The second goal is to universalize education. Confucius argued that universalizing education serves to preserve social order; education puts social values into practice and equips individuals with the skills necessary for supporting their families and contributing to society. The third goal is to develop students’ ability in the Six Arts, namely, rites, music, archery, chariotry, calligraphy and mathematics. The intention of the Six Arts is to allow learners to cultivate perseverance, acquire knowledge and various skills, learn deeply and solve problems effectively. According to Hsu and Hwang (2015, p. 242), in an ethnic Chinese society, teachers are not only disseminators of knowledge and implementers of educational goals, but also ‘teachers of people’. The society tasks them with caring for students and guiding students to become virtuous individuals. Although Confucianism serves to drive favourable development and develop social values in individuals, it has exerted negative effects on ethnic Chinese society in several aspects. Because Confucian education and the Confucian examination system have facilitated or affected improvements in the social status of individuals and families, preparation for assessments has become the sole purpose of studying. Today, parents in ethnic Chinese communities focus excessively on their children’s examination scores. This contradicts the initial purpose of Confucian education— to cultivate people’s intrinsic value. Parents’ excessive focus on children’s exam scores poses challenges to teachers holding to the philosophy of being ‘teachers of people’. Traditional and changing Chinese culture influences the goals of education and teachers’ roles in Taiwan (Wu, 2015). From the historical standpoint of Confucian education, teachers disseminate knowledge. This is consistent with the Tan dynasty scholar Han Yu’s argument, in the essay titled ‘On Teaching’, that teachers’ roles include functioning as disseminators of social values, providers of knowledge and skills learning and counsellors for alleviating students’ confusion. Traditional Chinese culture holds teachers in high esteem, and their roles are comparable to those of fathers. Over time, learning objectives and methods have changed to centre on lifelong learning, which encompasses the development of students’ self-understanding, self-management and problem-solving skills. In other words, aside from presenting the necessary learning content, teachers must also be aware of students’ interests, cultivate their potential, demonstrate practical problem-solving methods and engage them in discussion. In short, teachers are instructors as well as learners, thinkers, researchers and partners (Phillipson & Lin, 2013; Yang & Li, 2019a). Following the widespread influence of sociocultural learning theories, Taiwan’s education systems shifted the emphasis to learners’ self-inquiry, peer and teacher scaffolding and parental involvement in education (Hsu & Hwang, 2015; Tsai, 2015). In Taiwan, the Early Childhood Education 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1486 1487 Curriculum Framework addresses the roles of preschool teachers (Ministry of Education, 2017), who must understand their students’ learning experience, establish relationships of mutual trust with their students, provide an environment conducive to learning and prepare teaching materials according to their specific sociocultural context. Teachers are managers of classroom culture and learning contexts, guides in learning, partners to preschoolers in living and learning and partners of parents. However, differences between parents’ expectations of teachers and teachers’ self-perceived roles have emerged. Parents and teachers might have disagreed on educational beliefs and teaching methods, and teachers have had to manage and respond to parents’ criticisms. Teachers have encountered great challenges to their roles in ethnic Chinese society (Hsu & Hwang, 2015). The influence of both tradition and cultural changes has diversified these roles, and the goals of education now involve focusing equally on group and individual development. The literature above provides a lens for international readers to understand diverse roles in ethnic Chinese societies over time. Reviewing the literature also provides us with a reference for interpreting and discussing the study’s findings. Moreover, we formed our argument from the literature review. Finding teachers’ roles at various grade levels and time points at which young children observed them has contemporary significance. This finding results from children’s points of view on teachers’ roles and could enable teachers and parents to understand teachers’ roles. Listening to children’s points of view on teachers’ roles Concerning children’s rights, according to the United Nations Convention on the Rights of the Child (United Nations, 1989), children possess considerable status and subjectivity in any society. Children have the right to communicate their views and participate in decision-making that affects their lives (United Nations, 2005). Listening to children and researching with them has been important because children have the right to communicate and lead adults to understand them better (Bolshaw & Josephidou, 2018). In addition, advocates of sociocultural constructivism argue that children are active individuals and critical members of a society who share, learn and construct culture and knowledge with adults. According to Einarsdottir’s study in Iceland (2014), the evidence on children’s views of teachers’ roles is scant, warranting further investigation. Through such research, teachers and members of the public can understand children’s perspectives. In turn, this provides children with opportunities to participate in society and influence others. It also helps teachers grow professionally and change their modes of teaching and interaction. Harris and Barnes (2009) found that teachers’ perceptions of their roles are different from children’s views in Australia. For instance, one of the roles in which teachers viewed themselves was as a ‘planner’, but the children were not aware of this role in their study. The children identified their teachers as players, who care for children’s well-being, who teach and organize teaching environments, who do everything, serving as administrators and workers. They suggested the need for further research on listening to children’s perspectives on teachers’ roles, to support teachers’ planning of teaching environments and content. Einarsdottir (2005) examined children’s descriptions of teachers’ roles and their beliefs regarding what constitutes a good preschool in Iceland. The participants stated that a good preschool must both provide them with opportunities to choose the types of games they play and allow them to decide the game content. Moreover, they asserted that teachers must serve as partners, caregivers, guides and helpers. Studies have also reported that children consider teachers to be supporters in their games. Sandberg (2002) noted that Swedish 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SU and CHUNG SU and CHUNG children believed that their teachers must facilitate conflict resolution as well as play games with them, particularly when a child does not have a playmate. Pramling Samuelsson and Johansson (2009) explored why preschoolers invite their teachers to play games with them and revealed that Swedish children regard teachers in games as participants, helpers, explainers, encouragers and conflict resolvers. Teachers also offer explanations for unclear or uncertain concepts in classrooms. Children ask their teachers to join their games because they desire the teacher to affirm their abilities and make the game more challenging. In short, teachers both participate in their students’ games and implement instructional scaffolding, thereby enabling the children to understand their learning circumstances and develop their skills. Kernan (2011) stated that children seek more support from their peers in outdoor activities and from their teachers in indoor activities (e.g. games and tasks), respectively, because outdoor games involve fewer rules than classroom activities. The children in that study believed that teachers serve as critical guides in schools and stated that they could decide certain types of games themselves. Not only teachers’ participation in games but also the personality traits that children like or dislike in their teachers in England (Hutchings et al., 2008), Denmark and the United States (Kragh-Müller & Isbell, 2011) reveal children’s perspectives on teachers’ roles. Children become aware of their teachers’ personality traits through their interactions with others and, in turn, form a perspective on teachers’ roles. Interviews have revealed that children typically consider good teachers to be instructors who express concern, help and encourage others, provide children with options and are good-tempered. The literature on teachers’ roles above indicated that teachers have many roles from children’s perspectives, including as players, conflict resolvers, explainers, encouragers, caregivers, guides, helpers, organizers and administrators. The literature presents the perspectives of young children in Western countries with cultures different from Taiwan’s. Research is lacking regarding Taiwanese children’s perspectives on teacher roles in a classroom infused with East and West. This motivated our further exploration of Taiwanese preschoolers’ perspectives on teachers’ roles and finding the time points at which children observed them. The contribution of our research will be to help other researchers understand young children’s perspectives from the East. METHOD Research setting and participants We invited children from a public preschool representing the change in ethnic Chinese culture to participate in this study. The invitation criterion was based on whether a preschool illustrates the cultural change encompassing a fusion of West and East. The preschool in our study changed its classroom environment and curriculum in the past 15 years based on Western sociocultural learning theory. It transitioned from a teacher-centred to a learner-centred and play-based learning approach. It had no subject-based teaching, created various classroom learning areas and the proportion of collective teaching time in the day was tiny. In the preschool, the children spend about 50 min doing outdoor activities, 40 min in snack and teeth-brushing time, 20–30 min in group-circle time, 100 min playing in learning areas and 20 min sharing learning experiences in the morning. Children eat lunch and brush their teeth during the noon hour and then nap for 70–90 min. After nap time, children have free play inside or outside and an afternoon snack before going home. Chinese lunches and snacks are rich and differ from light Western meals, offering biscuits, fruits, or sandwiches. For example, the snack after naptime for Taiwanese children might be mushroom chicken porridge or meat with vegetables and noodle soup. It still disseminates social values, such as contentment and care influenced by Confucianism, during daily routines. 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1488 1489 The research participants were 25 preschoolers aged 5.7 years on average. We obtained informed consent from the teachers, the children’s parents and the children. We explained our research to the teachers first and obtained their consent. Then, we introduced our study to the parents and obtained their consent. All the teachers and the parents agreed to participate. To build relationships with the children, we were volunteers reading stories at the beginning of the research period and active observers in the children’s classroom for 2 weeks. We sought ongoing consent from the children as well (Flewitt, 2005). Thus, we built relationships that considered power relations (Merewether & Fleet, 2014), to respect and meet our ethical responsibilities when performing research with children. We introduced the children to our study, including the reasons for our interest and the methods we planned to use, thanked them for their help, and answered their questions from the beginning. We also explained to them that they could stop participating in this study at any time. During data collection, we explained the study to the children again and verbally invited them to say whether they were willing to participate on days when our activities included data collection. All the children agreed to participate. Data collection In our study, children shared perspectives through our interviews built on their drawings and photography. According to MacDonald (2009) and Puroila et al. (2012), collecting narrative data on children’s perspectives can occur by examining children’s drawings, widely employed for understanding preschoolers’ perspectives. They are a familiar means through which children in this age group can express themselves (Einarsdottir et al., 2009; Kragh-Müller & Isbell, 2011). The tools with which children are familiar and which they use regularly in their environment can be supportive and encouraging (Clark, 2017). Therefore, because the children regularly drew every morning in class, we used drawings first and photographs later. Drawing was the tool that these children often used, and they were competent and comfortable representing their ideas in drawings. We asked the children to express their views on teachers’ roles through drawings and photographs. Kragh-Müller and Isbell (2011) collected children’s drawings and interviewed them with such questions as, ‘What is best about being in your childcare center?’ and ‘Is there anything that you do not like here?’ Einarsdottir (2014) gave children cameras and invited them to take pictures of school life. The photographs presented what they were doing during the day, what they liked, what they did not like and what the adults in the preschool were doing. These methods of data collection inspired us to invite the children to share aspects of teachers’ roles in their favourite and least favourite activities in school life. Thus, we tried to create a more child-led sharing environment during this study. Children easily share with others their likes and dislikes at school, and doing so makes them feel at ease, rather than having to answer adult-led questions about their teacher’s role in class. In addition, we provided children with different ways to present their views, reflecting the diversity in their lives (Bolshaw & Josephidou, 2018). The topics of the four-weekly drawing sessions were: 1. Sessions 1–2: The children’s favourite class activities in which to engage and what their teachers were doing during those activities; 2. Session 3–4: The children’s least favourite class activities and what their teachers were doing during those activities. In each session held in the morning, we supervised children during the session, and after they completed the drawings, we asked about the characters, activities and things they depicted. We then simply wrote the descriptions in their drawings. In the afternoon interviews, we invited and 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SU and CHUNG SU and CHUNG asked the children to tell us more about what they drew, for example, what happened and what teachers were doing. There were four interviews, each lasting about 5–10 min. In this way, we attempted to make children experts in their own lives (Clark, 2010). We asked for clarification by saying, ‘Could you help me understand this?’ and expressed our appreciation. After completing the drawing data collection, we invited the children to use action cameras for 2 weeks to photograph the activities in which they engaged with their teachers. We did not invite the children to photograph their least favourite activities, as the children indicated from their drawings that those least favourite activities occurred during snack time, lunchtime and nap time, times when children using cameras would be neither appropriate nor convenient. Photography has been extensively used to understand preschoolers’ perspectives because it makes research fun for children (Punch, 2002), promotes their generating rich discussion with others (Baird & Grace, 2021; Phelan & Kinsella, 2011) and presents a belief in young children’s competence (Pyle, 2013). We first demonstrated for the children how to use an action camera. Specifically, we invited them to ask questions, and they demonstrated the camera functions. The researchers also engaged in discussion with them on showing politeness when taking photographs of others. We conducted interviews with each child, building on their photographs. For two afternoons a week over 2 weeks, the researchers interviewed the children, asking them to select their favourite photographs first, suggesting but not limited to 10. We let children feel in control of their participation (Koch, 2017). The children explained the locations where they took the photographs, as well as the people, activities and things captured and what teachers were doing. The photograph-elicitation interviews created a positive atmosphere and allowed children to feel respected (Phelan & Kinsella, 2011). The photographs were then downloaded onto a computer, archived and coded. Recorder pens were used to record the interviews for transcription purposes. There are 50 interview transcripts, of which 25 contain four drawing-elicitation interviews and 25 contain four photography-elicitation interviews. Data analysis Analysing the data took three analytical stages. We employed qualitative content analysis to find the types of teachers’ roles and the time points at which the children perceived them. Content analysis is a method that may be used with qualitative data in an inductive or deductive way (Elo & Kyngäs, 2008; Hsieh & Shannon, 2005), both descriptively and interpretively, to reveal research participants’ thoughts (Lindgren et al., 2020). We first used an inductive approach to code children’s perspectives on teachers’ roles in stage 1. Two researchers coded data separately and discussed disparities. When disagreement occurred, further discussion ensued until they reached consensus (Ferreira et al., 2018; Hsieh & Shannon, 2005; Søndergaard & Reventlow, 2019). In stage 3, we used quantitative means to find the number of occurrences and changes in teachers’ roles at different time points. We explained the stages as follows. During stage 1: (1) First, two researchers familiarized themselves with the data and separately coded each child’s drawings and transcripts of drawing-elicitation interviews inductively. Drawings and interview transcripts were regarded as a single data unit. We used several codes, including teacher’s behaviours, time points and children’s activities. These codes appeared to address the research questions. At the end of this step, two researchers discussed the disparities, such as adding or missing codes. (2) Then, we compared codes across the cases to find categories of teachers’ roles. For example, in the code for teachers’ behaviours, a child depicted and expressed that ‘my teachers told me to try certain food’; another said in interviews, ‘my teachers asked me to close eyes during the nap time’. We sorted similar codes to form a role category (‘regulator’) and developed the definitions 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1490 TA B L E 1 Definitions of the roles identified in this study. Roles Definitions Companions Teachers are being with children or playing with them during daily routines Regulators Teachers ask children to obey rules or complete a routine task, which reflects cultural values Spectators Teachers watch children from the side without participating in the children’s activities Guides Teachers orally explain to children the procedures of games, or orally initiate questions to discuss with children about solutions for problems in games or during play Indirect Interveners Teachers provide tools to encourage children to engage in problem solving or learn new things without verbally explaining or providing direct physical help to the children Helpers Teachers provide direct physical help, for example, unscrambling Legos or to dressing a wound when a child is injured Listeners Teachers listen to children share their thoughts about their work or play experiences Outsiders Children who think that the teachers do not know about their secrets or ideas of this role (Table 1). In discussions, two researchers validated the categories we identified. We added components that the researchers did not consider—for instance ‘teacher role as outsider’. Through the discussion and examination of coding and grouping categories, the researchers reduced the possibility of data distortion. (3) After grouping and defining the roles, we re-examined each child’s data to identify teacher roles and time points at which the children observed each role. During stage 2: We refined our categories at this stage. The eight types of roles created in stage 1 became a frame to analyse the photographs and photo-elicitation interview data. Each child had four interviews to share their photos. We conducted the analysis within cases and examined whether any new roles emerged separately, then discussed the disparities. In stage 2, we found that no new roles emerged but noticed that no children mentioned the role of ‘outsiders’. We also re-examined each child’s data to make notes on teacher roles and derived from photo data the time points at which children observed each role. During stage 3: In stages 1 and 2, from children’s drawings, photographs and interviews, we discovered teachers’ roles and the time points at which each role was observed. Based on the analysis of previous stages, at stage 3, we counted and calculated the number of occurrences of each role from the drawings and photographs of each child, then summed up all the numbers from all children for each role. We used a bar graph to show the total numbers of each teacher role. Subsequently, we summed up the number of time points at which each teacher role was observed, to create a line graph. The line graph suggested not only categories of the roles at each time point but also the numbers of occurrences and changes in teachers’ roles at different time points. Using the ‘regulators’ role as an example, we examined each time point at which the role was observed in learning areas, outdoors, and at lunch, nap and transition times, summing the occurrences of ‘regulators’ at each time point. FINDINGS Types of roles The preschool teachers were perceived to play the roles of companions, regulators, spectators, guides, indirect interveners, helpers and listeners. Furthermore, the role of ‘outsiders’ was noted. 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1491 SU and CHUNG SU and CHUNG In this study, the children did not mention the roles of authority figures or skill instructors. Overall, eight teachers’ roles were identified and explanations of the roles were presented as follows. Several examples of the children’s drawings and photographs indicating teachers’ roles appear in Figure 1. Companions The participants indicated that they enjoyed having their teachers’ company, including as ‘playmates’ during activities. The children from the preschool enjoyed picking vegetables in the school garden, reading stories and playing with blocks with their teachers (Figure 1a). Regulators The children expressed their dislike of their teachers’ regulation of daily routine or playtime. Specifically, they did not like being told to try certain food, such as cauliflowers, tomatoes and chicken skin. They also disliked being instructed to exercise outdoors for a sunbath. Some of the children disliked going to the zoo as a field trip and being told to close their eyes during naptime when they are not sleepy (Figure 1b). However, the children said the teachers want them to be healthy, cherish food and become caring, reflecting cultural values. Spectators The children from the preschools expressed that they enjoyed having their teachers watch them during free playtime, including in learning areas and outdoors. During that time, teachers did not actively participate in those activities (Figure 1c). Guides The children noted that they enjoyed having their teachers guide them in their activities. Guidance involved explaining and demonstrating the activities, as well as asking questions to scaffold children’s learning. When the children played Chinese checkers, their teachers would challenge them to think about strategies for winning. Teachers invited and asked the children how to build a Lego house in a play area (Figure 1d). Indirect interveners The teachers from the preschool did not help their students directly during games when they encountered difficulty. Most of the children stated that they did not immediately ask for their teachers’ help when their block towers collapsed; instead, they tried to fix them or use methods their teachers suggested in the past. The supportive environment showed that preschool teachers did not direct lectures on how to do artwork with step-by-step demonstrations or on reading and writing skills. In the art and weaving area, the teachers prepared relevant books and cards for weaving demonstrations (Figure 1e). 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1492 (a) Companions (b) Regulators In the school garden, I like harvesting At lunchtime, I dislike being told to try vegetables with my teacher. certain foods, such as cooked chicken meat and skin. Chicken Skin (c) Spectators (d) Guides When we play outdoors, I enjoy having my I was invited and asked by my teacher how teacher watch over us. to build a house using Lego blocks in a manipulative area. To show her my construction, I created a wall. My Teacher (e) Indirect Interveners (f) Helpers My class has an art and weaving area. The My teacher helped me knot my beading teacher provided instruction cards and necklace. materials so we could learn how to weave. (g) Listeners (h) Outsiders In dramatic play, I presented my homemade I am not sure how to convey my thoughts to paper-cup pudding to my teacher. In the my teacher, such as my dislike of strong classroom, I placed it on the display table. sunlight outdoors. FIGURE 1 Children’s perspectives on the preschool teacher’s roles. (a) Companions. (b) Regulators. (c) Spectators. (d) Guides. (e) Indirect Interveners. (f) Helpers. (g) Listeners. (h) Outsiders. [Colour figure can be viewed at wileyonlinelibrary.com] 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1493 SU and CHUNG SU and CHUNG Helpers The children indicated that their teachers helped them manage small incidents in everyday life—for example, when one child had an injured finger, or the children’s teachers helped them disassemble their block towers. The children also mentioned that teachers assisted them with beading, knitting and weaving (Figure 1f). Listeners The children expressed that their teachers listened to them—for example, when they introduced their toys or shared their work with the class. The children mentioned that they showed their Lego boat and shared handmade paper-cup puddings in dramatic play with their teachers (Figure 1g). Outsiders The children indicated that their teachers sometimes did not understand them. They also revealed that they did not share certain information with their teachers, such as their secrets or quarrels with friends. At times, they did not know how to share their favourite toys with their teachers or tell them certain information, such as their least favourite food or dislike of sunlight (Figure 1h). The climate in Taiwan is very warm and humid, with an annual average of 27°C (79°F); relative humidity averages 82% annually. There are few truly tropical and sultry months, but in the summertime, the climate often hits 35°C (95°F) during the day. Time points for different teachers’ roles We further analysed the total numbers of each teacher role and the time points that the children mentioned for each. The roles were identified 245 times in total (Figure 2). The children most frequently mentioned the roles of ‘companion’ (N = 55) and ‘regulators’ (N = 51). Regarding the time points (Figure 3), the children observed the most diverse teachers’ roles in free play time in learning areas, possibly because that was the children’s favourite time in school. Notably, the changing culture influenced the teachers from the preschool, who thoroughly arranged learning-area activities based on children’s interests and changed their roles depending on children’s needs. The second greatest number of time points for the children’s observations of teachers’ roles were outdoors and during transition times. During outdoor time, the children required their teachers’ guidance and company but were also aware that their teachers were regulating their behaviour. The teachers reminded the children of the rules during transition time and kept the children company. Notably, the children are aware of diverse teachers’ roles in less structured settings but not in the collective teaching activity. However, at lunchtime, the children were aware that they were under stricter regulation than at any other time, and they observed the teacher’s role of ‘regulators’ most frequently then. The roles of ‘indirect interveners’ and ‘listeners’ were not observed at this time point. 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1494 60 55 51 50 Companions 43 Regulators 40 Spectators 30 30 28 Guides Indirect Interveners Helpers 21 20 10 9 8 Listeners Outsiders 0 FIGURE 2 Teachers’ roles identified by the children. [Colour figure can be viewed at wileyonlinelibrary.com] Companions Regulators Spectators Guides Indirect Interveners Helpers Listeners Outsiders 35 30 25 20 15 10 5 0 LEARNING AREAS TIME OUTDOOR TIME LUNCHTIME NAP TIME TRANSITION TIME F I G U R E 3 Time points at which teachers’ roles were observed at the preschool. [Colour figure can be viewed at wileyonlinelibrary.com] DISCUSSION Based on the above-mentioned roles, we further discussed the remaining and changing roles. We illuminated diverse roles by comparing them to the traditional and changing Chinese culture. The children’s views on diverse teachers’ roles, such as guides and indirect interveners, also reflect social constructivism, which argues that children learn through interaction and are active members of society. 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1495 SU and CHUNG SU and CHUNG A longstanding role in culture: Regulators The children from the preschool noted that their teachers enforced certain rules in class. In particular, the teachers served as regulators during lunchtime, outdoor activities and transition activities. The children noted that they were required to follow rules at lunchtime, such as trying various types of food. Regarding outdoor time, they were instructed to sunbathe or take field trips to the zoo together. During transition time, the teachers reminded the children to follow certain rules. For example, the children were told to keep their eyes closed during naptime. The role of ‘regulators’ revealed that a society with a changing culture still retained the influences of traditional Chinese cultural values on teachers’ roles. In traditional Chinese culture, teachers were responsible for fostering virtues in their students, such as contentment, perseverance and care (Chen & Chung, 2019: Su & Chung, 2022b), typically accomplished through the implementation of school rules during daily routines (Su & Chung, 2022a). Therefore, teachers urged their students to avoid food waste, instructing them to try various foods instead of discarding them. In traditional Chinese culture, the saying goes, ‘it is difficult to draw a perfect circle and square without a pair of compasses and a set square’. In other words, rules must be established to foster individuals’ good character and the skills they require to realize their goals. In the present context, preschool routines reflected values, and the teachers were identified as regulators. This finding revealed that teachers in Chinese ethic society might have carefully considered the cultural and contextual situations before they put approaches into practice (Yang & Li, 2019a). Children in Einarsdottir’s study mentioned the roles of preschool teachers as ‘controllers’ and ‘rulers’, and that children think teachers must make sure things go smoothly among them (Einarsdottir, 2014). In our study, we found a ‘regulator’ role close to her finding. However, the role of ‘regulators’ revealed that the perspectives of preschoolers in traditional Chinese culture differ from those of preschoolers in other cultures. Teachers ask children to obey rules or complete a routine task reflecting cultural values. The children in our study were aware of teachers’ values reflected in the rules but still disliked the rules. In addition, this study expanded on previous studies by finding time points at which the children perceived the role of ‘regulators’. Children particularly noted the role at lunchtime, transition time and outdoor time. According to Kernan (2011), indoor activities are more strictly controlled than outdoor activities. However, the present study found that the children considered teachers ‘regulators’, even at lunchtime and outdoors. A changing role in culture: Guides and indirect interveners Chinese changing culture has influenced teachers’ roles in Taiwan (Wu, 2015). As times change, preschoolers have begun to see teachers as ‘guides’ and ‘indirect interveners’ which is conflict to ‘regulators’. The teachers did not directly participate in the children’s favourite activities but guided them through oral explanations, by raising scaffolding questions or providing supportive learning environments during play. Sandberg (2002) indicated that Swedish preschoolers regard teachers as supporters in their play. Both the children from relevant studies conducted in other parts of the world (Einarsdottir, 2005, 2014) and the children from the preschool in our study considered their teachers guides in play; the explanation is cultural changes occurring over time and the influences of social constructivism. Confucian culture has strongly influenced society’s views of teachers’ respected and authoritative roles (Llasera, 1987). The teachers now are not 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1496 viewed as direct instructors who maintain authoritative and expert roles, only providing knowledge and skills. As contemporary teachers, they had adopted other roles, including as students’ partners and researchers (Fu & Li, 2013). Following cultural changes over time, preschool teachers today are not only instructors; they seek to understand their students’ life experience and ways of learning through observation (Stetsenko, 1999). This finding also corresponded to that of Yang and Li (2019b, p. 15) that the ‘progressive idea of child-centredness and children’s nature and rights have been highlighted to make teaching and learning more child focused rather than teacher-directed’ in Chinese ethnic society. Diverse changing roles reflect democracy and interactive relationship The diverse teachers’ roles that the children identified also reflected social constructivism and cultural changes resulting from individuals’ points of view, generated through their interactions with social environments (Cerulo et al., 2021). This preschool featured an open classroom design, with numerous learning areas and available materials for preschoolers to explore actively. This environment design was based on sociocultural learning theory (Figure 4). Specifically, the teachers facilitated the children’s participation in learning activities; through interactive learning processes, the teachers observed and listened to their students’ opinions and learning experiences. This is consistent with the teachers’ role of ‘helpers’ that Pramling Samuelsson and Johansson (2009) identified. The teaching environment reflected cultural changes, in that the students learned proactively through independent exploration, as well as through discussions and collaborations with their peers in learning areas. The children also observed that their teachers did not intervene directly when they encountered problems. The goals and methods of education in the preschools centred on child-initiated learning and problem-solving skills. This reflected the specification in the Early Childhood Education Curriculum Framework (Ministry of Education, 2017). The teachers created and provided a democratic 2 1 5 4 3 Learning Areas FIGURE 4 Preschool classroom environment. [Colour figure can be viewed at wileyonlinelibrary.com] 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1497 SU and CHUNG SU and CHUNG F I G U R E 5 Photograph of a student’s completed worksheet on counting money. [Colour figure can be viewed at wileyonlinelibrary.com] learning environment, and the children perceived that they learned through interaction and discussion in class. During learning-area time, the children observed the most diverse teachers’ roles. This result reflected that teachers play important roles and interact with children in various ways through children’s play. From children’s perspectives on teachers’ roles, we learned that the changing culture, resulting from Western and Chinese cultures intertwining, affects perceptions of teachers’ roles. The learning environment in the preschool was open, not rigidly structured and conducive to exploration, enabling student-teacher interactions and children’s observations of diverse teacher roles. Some Taiwanese preschools have tables and chairs but no learning areas; the teachers allow students to play with toys only if they have finished their math or writing assignment in class. Assignments include worksheets on recognizing and counting money (Figure 5) and writing Zhuyin symbols and English letters. These worksheets reflect the emphasis on learning outcomes, the primary concern of parents in ethnic Chinese communities. Accordingly, children may view teachers in such roles as ‘authority figures’, ‘knowledge instructors’ and ‘homework examiners’. Chinese parents believe that a good academic start for young children from preschool age is important, and preschool is a place to prepare academic skills. This belief suggests that Chinese parents may evaluate the quality of preschools based on children’s learning outcomes and emphasize the role of teachers as knowledge and skill instructors (Yamamoto & Li, 2012). On the contrary, the children in this study regarded their teachers in diverse roles, not only as instructors of knowledge and skills. Amid sociocultural changes in Taiwan, the interweaving of Western and Chinese cultures has influenced the goals and methods of education, both of which increase the diversity of teachers’ roles (Hsu & Hwang, 2015). Diverse changing roles observed outdoors With the role of ‘companions’ the most apparent during outdoor time, the children also observed the most diverse teachers’ roles then, likely because outdoor activities correspond to an open atmosphere and a less rigidly structured learning context. The space also allowed children’s 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1498 1499 playful engagement with their surroundings. Moreover, outdoor settings were conducive to interactions between the students (Kernan, 2011) and the teachers in our study. The teachers were not only authoritative figures to the children but also companions. Thus, childhood comprised an interactive environment that the children and their teachers co-created. This finding corresponded to Hohti’s (2016) argument of children as an entangled practice, based on Barad’s (2007, p. 182) statement that ‘humans, time, space and matter exist only in relationship with each other’, suggesting the definition of ‘entanglement’. This study revealed that cultural changes in Taiwan have shaped preschool education as well as childhood. The preschoolers’ perspectives on teachers’ roles differed with traditional and changing cultures. Moreover, with less rigid classroom structure, including learning areas and outdoors, the children observed most diverse teacher roles. The findings enable researchers globally to understand preschoolers’ views on teachers’ roles in the entanglements of time and cultural changes. CONCLUSION Our study expanded previous research related to children’s perspectives on teachers’ roles in Western countries, to find teachers’ roles in a changing cultural setting in the East. The changing cultural setting represents a preschool encompassing a fusion of Eastern and Western culture. Based on the children’s perspectives in the study, a total of eight types of teachers’ roles were identified, with the time points at which those roles were observed. Particularly, role conflicts were found that revealed the infusion of Eastern culture by Western culture. The children observed teachers’ roles as regulators, conflicting with their roles as guides and indirect interveners. Teachers’ roles as regulators in lunchtime, outdoor time and nap time reflect the influence of Confucianism with its emphasis on cultivating the values of contentment, perseverance and caring in young children. Meanwhile, the children observed great diversity in such teachers’ roles as companions, guides and listeners, in less structured settings and highly exploratory environments, during playtime spent in learning areas and outdoors. This reflects children’s agency in learning from self-discovery and interaction with others. The children’s perceptions of teachers’ roles at various time points also revealed the influence of the West, focusing on sociocultural learning theory and child-centred education. Altogether, this study highlights the importance of changing cultures and time for childhood. The findings may have implications for future research and teacher education. Understanding teachers’ perceptions of role conflicts and how these affect their pedagogy is warranted. The cultivation of teachers in ethnic Chinese preschools includes not only the Western educational models and teaching methods but also the practice of values of Chinese culture in children’s lives. ACKNOWLEDGEMENTS The authors acknowledge all the research participants who participated in this study. FUNDING INFORMATION This work was supported by National Science and Technology Council, Taiwan under Grant (MOST 108-2628-H-845-001). CONFLICT OF I NTEREST STATEMENT The authors have no conflicts of interest to disclose. 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License SU and CHUNG SU and CHUNG DATA AVAILABILITY STATEMENT Research data are not shared. ETHICS APPROVAL STATEMENT This study was approved by an IRB of University of Taipei. PERMISSION TO REPRODUCE MATERIAL FROM OTHER SOURCES Not applicable. ORCID Yuling Su https://orcid.org/0000-0003-3513-639X REFERENCES Baird, K., & Grace, R. 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(2019b). Changing culture, changing curriculum: A case study of early childhood curriculum innovations in two Chinese kindergartens. The Curriculum Journal, 30(3), 279–297. AUTHOR BIOGRAPHIES Yuling Su, Ph.D., is associate professor in Early Childhood Education at University of Taipei. Her research interests focus on children’s perspectives, professional development of early childhood teachers, experiential teaching and learning, and children’s social-emotional learning. Ya-hui Chung, Ph.D., is assistant professor in Early Childhood Education at University of Taipei. She was a co-project investigator of Taiwanese Cultural Curriculum for Kindergarten commissioned by the Ministry of Education in Taiwan. Her research interests focus on children’s peer culture and interaction, and teacher education. How to cite this article: Su, Y., & Chung, Y.-h. (2023). Role conflicts of preschool teachers: From children’s perspectives in a changing cultural setting in Taiwan. Children & Society, 37, 1484–1502. https://doi.org/10.1111/chso.12731 10990860, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/chso.12731 by Liberty University, Wiley Online Library on [17/09/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 1502 European Journal of Operational Research 305 (2023) 806–819 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Decision Support Solving the inverse graph model for conflict resolution using a hybrid metaheuristic algorithm Yuming Huang a, Bingfeng Ge a,∗, Keith W. Hipel b,c, Liping Fang d,∗, Bin Zhao a, Kewei Yang a a College of Systems Engineering, National University of Defense Technology, Changsha, Hunan 410073, China Department of Systems Design Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada c Centre for International Governance Innovation, Waterloo, ON N2L 6C2, Canada d Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada b a r t i c l e i n f o Article history: Received 9 September 2021 Accepted 24 June 2022 Available online 30 June 2022 Keywords: Group decisions and negotiations Graph model for conflict resolution (GMCR) Hybrid metaheuristic algorithm Inverse analysis a b s t r a c t The paper is concerned with one of the most important questions in conflict analysis: How decision makers can strategically interact in a conflict to reach a specified equilibrium? More specifically, an inverse preference optimization model is formulated to solve the problem of adjusting the preferences of decision makers so as to make a specified state an equilibrium. To this end we define an algorithm which incorporates the hybridization of particle swarm optimization and genetic algorithm. The proposed algorithm adopts a random neighbor strategy for population initialization, the elite reservation and mixed selection operation, and a diversity strategy for population update to improve its efficiency and effectiveness when searching for the required preferences. This approach can help decision makers or third parties to focus their resources on guiding them toward preferences that lead to a specified resolution. Finally, a real-world dispute over exporting bulk water from Eastern Canada is used to demonstrate the applicability and effectiveness of the approach. © 2022 Elsevier B.V. All rights reserved. 1. Introduction Strategic conflict is a ubiquitous and costly interaction between two or more stakeholders with distinct value systems in a wide range of areas, including political negotiation, economics, resource allocation, and military activity. As human interactions expand and intensify, the demand for decision technologies to balance the interests of stakeholders in a peaceful manner will increase sharply (Hipel, Fang, & Kilgour, 2020). As a comprehensive and flexible methodology extended from metagame analysis (Howard, 1971) and conflict analysis (Fraser & Hipel, 1984), the graph model for conflict resolution (GMCR) can be used to systematically investigate challenging disputes (Fang, Hipel, & Kilgour, 1993; Xu, Hipel, Kilgour, & Fang, 2018a). With minimal information requirements on decision makers (DMs), feasible states, and relative preferences over states, GMCR can be used to determine possible resolutions of a conflict, based on a specific stability definition. This procedure is called the forward perspective of GMCR, where the preferences of DMs are known and used to predict the possible equilibria through the stability analysis (Kinsara, Kilgour, & Hipel, 2015a). ∗ Corresponding authors. E-mail addresses: bingfengge@nudt.edu.cn (B. Ge), lfang@ryerson.ca (L. Fang). https://doi.org/10.1016/j.ejor.2022.06.052 0377-2217/© 2022 Elsevier B.V. All rights reserved. On the other hand, if focal DMs, third parties, or analysts want to make a specific state the outcome of a conflict, the inverse perspective of GMCR can determine the required preferences (Hipel & Fang, 2021). Where DMs, states, and original preferences are known, the inverse GMCR can explore the required preferences with minimal preference adjustments. Preferences of the DMs are often entrenched, especially for a dispute that has been going on for a long time (Hipel, Sakamoto, & Hagihara, 2016). In such cases, a preference ranking that differs largely from the original may be hard to realize. The fewer the compromises that DMs have to make on their preferences, the likelier the negotiation is to succeed (Wu, Xu, & Ke, 2019). The goal is therefore to find the best solution with the fewest compromises. However, this–called the optimal preference manipulation problem–involves intensive computations, which is proven to be NP-Hard and hence necessitates an effective optimization method (Rêgo, Silva, & Rodrigues, 2021). Huang et al. (2020) and Tao, Su, & Javed (2021) have made a pioneering attempt to apply the Genetic Algorithm (GA) to solve the inverse graph model problem. However, their works cannot guarantee optimal and robust solutions due to the slow convergence and poor local exploitation capability of the GA algorithm (Juang, 2004). Accordingly, the main objective of this paper is to develop a hybrid metaheuristic algorithm to solve the inverse GMCR problem of reaching a mutually beneficial outcome by identifying the Y. Huang, B. Ge, K.W. Hipel et al. European Journal of Operational Research 305 (2023) 806–819 preferences having the fewest adjustments. To address the weakness of single GA, the hybridization of different algorithms will be an alternative technique worth to explore (Agarwal & Srivastava, 2018). Particle Swarm Optimization (PSO) is a nature-inspired metaheuristic algorithm that has solved diverse optimization problems due to its fast convergence and ready implementation (Dang, Guibadj, & Moukrim, 2013; Jakubik, Binding, & Feuerriegel, 2021; Tang, Zhao, & Liu, 2014). Inspired by the reported hybridization strategy of algorithms for Operations Research, it is reasonable to expect that the combination of PSO and GA (named PSO-GA) can improve the convergence speed, global exploration, and robustness of solving the inverse graph model problem when tackling conflicts of diverse scales. The main contributions of this paper are summarized as follows: R+ (s j ) i C C (s, s1 ) sE   IP P (sE , Sta ) = P 1 , P 2 , . . . , P K Sta ∈ {Nash, GMR, SMR, SEQ } RSta ⊆ {Nash, GMR, SMR, SEQ } AdPk Pori   V k = vk1,s1 , . . . , vk1,sm | . . . |vkn,s1 , . . . , vkn,sm ω c1 , c2 r1 , r2 (1) On the basis of the forward GMCR procedure, a framework of inverse GMCR is proposed. A preference optimization model for the inverse GMCR for the cases of two and n (n > 2) DMs, where the amount of preference adjustments is adopted as the objective function, is established. DMs or third parties are thus able to focus minimal resources on guiding parties to a desired outcome. (2) An improved metaheuristic algorithm is presented for solving the problem. Based on the hybrid PSO-GA, a random neighborhood strategy is designed to initialize the population, which can guarantee randomness, diversity, and quality. The elite reservation and mixed selection operation improve the quality of solutions and enhance the global exploration capability of the algorithm. The diversity strategy for population update aims to diversify the solutions and ensure their robustness. (3) Compared with the benchmark method GA as well as PSO and classical PSO-GA, the effectiveness and the efficiency of the proposed algorithm are verified through a case study investigating a real-world dispute over exporting bulk water from Eastern Canada. 2. Related research Within the GMCR framework, three perspectives have been presented to tackle a conflict: forward GMCR, inverse GMCR, and behavior GMCR (Hipel, Kilgour, Xu, & Xiao, 2021). Here, we focus on the first two, as displayed in Fig. 1. 2.1. Forward graph model problem In the forward perspective, one needs to determine the input information required, such as the DMs, states, and preferences, then carry out a stability analysis to identify the equilibrium results, as displayed at the top of Fig. 1. Forward GMCR has been widely used to handle conflicts within areas such as greenhouse gas emission (He, Kilgour, & Hipel, 2017), environmental management (Taravatrooy, Nikoo, Adamowski, & Khoramshokooh, 2019), and system-of-systems architecting (Ge, Hipel, Fang, Yang, & Chen, 2014). Many new initiatives have been developed based on the original GMCR framework. Preference is the most essential element in modeling, as the research topics reflect: preference structure, such as fuzzy preference (Bashar, Hipel, Kilgour, & Obeidi, 2018), grey preference (Kuang, Bashar, Hipel, & Kilgour, 2015), strength of preference (Hamouda, Kilgour, & Hipel, 2004), and probabilistic preference (Rêgo & dos Santos, 2018); and preference elicitation, such as fuzzy option prioritization (Bashar, Kilgour, & Hipel, 2014), evidence theory (Silva, Hipel, Kilgour, & Costa, 2019), and the technique for preference ranking by similarity to an ideal solution (Zhao & Xu, 2019). Some developments have also been introduced into conflict analysis, such as mixed stabilities (Zhao, Xu, Hipel, & Fang, 2019), mixed coalitional stabilities (Zhao, Xu, Hipel, & Fang, 2021), attitude analysis (Xu, Xu, & Ke, 2018b), status quo analysis (Xu, Kilgour, Hipel, & Kemkes, 2010b), and behavioral analysis (Wang, Hipel, Fang, Xu, & Kilgour, 2019). Over three decades, the forward GMCR has shown great power in enabling analysts or DMs to identify the equilibria of historical or ongoing disputes by performing stability analysis after ascertaining the input data. This procedure can be utilized in two ways: 1) to predict possible evolution paths and resulting outcomes of ongoing conflicts; and 2) to review the underlying mechanisms of historical conflicts. The remainder of this paper is structured as follows. The related work on the graph model and hybrid PSO-GA approach is briefly reviewed in Section 2. Section 3 outlines the basic components of forward GMCR. The overall framework and formal modeling of inverse GMCR are introduced in Section 4. The hybrid PSO-GA algorithm for inverse preference determination is set out in Section 5, followed by a case study in Section 6 to demonstrate how the inverse analysis based on the proposed PSO-GA algorithm can be employed to investigate real-world conflicts. Finally, some conclusions and future work are presented in Section 7. For the convenience of the reader, a list of the notation used in this paper is given below, according to the order of appearance: Notation pbest i gbest G N = {1, 2, . . . , n} S = {s1 , s2 , . . . , sm } Ai Pi = ( pi (s1 ), pi (s2 ), . . . , pi (sm )) P = {P1 , P2 , . . . , Pn } Ri ( s j ) DM i’s unilateral improvement (UI) list A coalition The group of all last DMs in legal sequences from s to s1 The desired equilibrium Set of the optimized preference profiles The specified stability definition Set of stability definitions The adjustment amount of optimized preference profile The original preference profile The velocity of particle k Inertia weight The acceleration constants The random numbers The best position ever found by particle i The global best position ever found by the population The graph model Set of decision makers (DMs) Set of states Set of DM i’s oriented arcs DM i’s preference The preference profile for all DMs DM i’s reachable list 2.2. Inverse graph model problem In the inverse perspective, in order to achieve a desirable or specified equilibrium, one needs to ascertain the necessary inputs, theoretically including all the components of inputs in the forward GMCR, such as the DMs, states, and preferences (Hipel et al., 2021). To the best of the authors’ knowledge, the inverse GMCR analysis currently focuses on preference. There are two lines of research within inverse GMCR: 1) identifying all the possible required preferences with partially or completely unknown preferences, and 2) (continued on next column) 807 Y. Huang, B. Ge, K.W. Hipel et al. European Journal of Operational Research 305 (2023) 806–819 Fig. 1. Two perspectives of the graph model (adapted from Hipel et al. (2021)). exploring the optimal preferences with the fewest preference adjustments with known original preferences, such that a particular state is an equilibrium (Han, Xu, Fang, & Hipel, 2022). Based on robustness analysis, Sakakibara, Okada, & Nakase (2002) first suggested exploring the inverse GMCR problem in the first direction. Similarly, Kinsara et al. (2015a) used an enumeration method to obtain required preferences, but the computational complexity increases exponentially with the increase of conflict size. Meanwhile, Kinsara, Petersons, Hipel, & Kilgour (2015b) introduced inverse analysis into the new decision support system GMCR+, providing a tool for DMs and analysts to study inverse problems. Given the foregoing issues, Wang, Hipel, Fang, & Dang (2018) proposed the matrix representation of inverse GMCR, establishing matrix inequalities for inverse analysis to reduce the computational complexity, but did not provide computational algorithms to find unknown preferences. Then, Han et al. (2022) and Han, Xu, Hipel, & Fang (2021) developed an integer programming approach for solving the inverse GMCR formulation presented by Wang et al. (2018) for the cases of two DMs and multiple DMs, respectively. In the second direction, Wu et al. (2019) made a first attempt to apply a 0–1 nonlinear programming method for computing minimal priority adjustments of preference statements for a two-DM model under the Nash stability, where only the order of the preference statements can be adjusted. Rêgo et al. (2021) further proposed several cost aggregation criteria for the optimal preference manipulation problem, but did not provide computational implementation. Huang et al. (2020) first developed a GA algorithm to solve the inverse GMCR by modeling it as an optimization problem. Each individual is referred to as a preference ranking of all feasible states, where each gene within an individual is encoded with a real value. The value assigned to each gene indicates the ordinal payoff of the associated state. The total changes of ordinal payoffs of all states within an individual are modeled as the fitness to influence the evolution of the population. Then, Tao et al. (2021) designed a solution method using GA algorithm to solve the inverse preference optimization problem under matrix expression, where each individual is encoded as a preference matrix. However, neither study can guarantee the optimality and robustness of the results due to the slow convergence and lack of local search capability, especially for large-scale problems. Since the PSO has shown greater convergence property than other metaheuristic algorithms (Kennedy & Eberhart, 1995; Weiner, Ernst, Li, Sun, & Deb, 2021), this paper aims to use a hybridization of the PSO and GA to balance the superiorities of the two kinds of algorithms to solve the minimum adjustment-based preference optimization problem. The minimum adjustment-based preference optimization analysis can be utilized as an intervention tool to bring about a desired resolution for a particular dispute. Quite often the DMs are entrenched or locked-in on fixed value systems resulting in an unsatisfactory resolution of a conflict (Hipel et al., 2016). Appropriate assistance from a third party who is not an actual stakeholder may be mutually accepted by all involved DMs (Hipel & Fang, 2021). In a given negotiation, a third-party mediator or a DM may explore how a serious conflict can be guided to a win/win resolution by encouraging one or more DMs to change their preferences; or a reluctant DM may wish to understand how to adjust his/her own preference to achieve a more desirable outcome. The algorithm presented in this paper can be utilized by third parties or focal DMs to develop credible and robust strategies for resolving a conflict with minimal preference adjustments. 2.3. Hybrid PSO-GA approach Inspired by the cooperative and competitive behaviors of fish schooling and bird flocks, the PSO algorithm was originally presented by Kennedy & Eberhart (1995) for identifying the optimal value within the search space. In PSO, a swarm is composed of a certain number of particles that are characterized by velocity and position. The position of particle i within an n–dimensional hyperspace means a possible solution for the specific optimization problem, which can be formulated as xi = (xi1 , xi2 , . . . , xin ). Changes to the position of each particle are based on its velocity, as represented by vi = (vi1 , vi2 , . . . , vin ). During the searching procedure, each particle i will store its historically best position as pbest i , and the best pbest in a swarm will be memorized as the globally best position (gbest). Both the positional information of personal best ( pbest i ) and global best (gbest) will be used to update particle i’s velocity and position information. PSO has been popularly applied to solve discrete optimization problems (Aminbakhsh & Sonmez, 808 Y. Huang, B. Ge, K.W. Hipel et al. European Journal of Operational Research 305 (2023) 806–819 2) S = {s1 , s2 , . . . , sm } represents the group of all feasible states for the conflict (2 ≤ m < ∞); 3) Ai ⊆ S × S means the oriented arc set containing the state transitions in one step unilaterally controlled by DM i; and 4) Pi denotes DM i’s relative preference over S. A DM involved in a conflict can be an individual, a company, or a country, that usually controls diverse options and has the full authority to decide which options to choose or not, which constitutes a specific strategy of the DM. The complete combination of the strategies of all DMs constitutes all mathematically possible states, some of which may be infeasible and removed by considering various constraints. A conflict can be modeled as a directed graph where the vertices represent the feasible states and the arcs between vertices are movements that can be one-way or two-way. Thus, a unilateral move (UM) is referred to as a state transition in one step under the control of a specific DM from a given state. In a graph model, DM i ∈ N’s preference on S is usually denoted by a binary relation {i , ∼i }. Taking s1 , s2 ∈ S for example, s1 i s2 indicates that state s1 is definitely preferred to s2 , and s1 ∼i s2 signifies that state s1 is equally preferred to s2 for DM i. It is worth mentioning that both transitive and intransitive preferences are allowed in the graph model (Fang et al., 1993). If the preference is transitive, DM i can order the states from most to least preferred by applying a real-valued function, pi . Note that in this paper, the preference in inverse GMCR is assumed as transitive. Definition 2. Formally, the preference of DM i ∈ N on S can be denoted as Pi = ( pi (s1 ), pi (s2 ), . . . , pi (sm )), pi (s j ) ∈ {1, 2, . . . , m}, where pi (s j ) is called the ranking function, the values of which are assigned in ascending order starting at 1. Fig. 2. Basic procedure for forward GMCR (adapted from Fang et al. (1993) and Xu et al. (2018a)). In other words, the value indicates the position of an associated state in the preference ranking, where the most preferred state is given by the value of 1, meaning in the first position, the least preferred element is in the last position, and the equally preferred states are in the same position (in a sense) with the same value. Therefore, s1 i s2 if and only if (iff) pi (s1 ) < pi (s2 ), and s1 ∼i s2 iff pi (s1 ) = pi (s2 ). It is worth mentioning that the value assigned to each feasible state only represents the relative degree of preference between any pair of states, rather than the absolute strength of preference for a specified state. For example, Pi = (3, 1, 5, 1, 4 ) means that states s2 and s4 are most preferred and ranked in the first position by DM i with Pi (s2 ) = Pi (s4 ) = 1, yet state s3 is least preferred and ordered in the last position with Pi (s3 ) = 5. This example can be equivalent to the preference ranking, s2 ∼ s4  s1  s5  s3 . Note that Pi (s3 ) is given by 3 instead of 2, although states s2 and s4 are in some sense considered to be in the same position. Each preference ranking is paired with one and only one preference generated by Definition 2. The preference rankings of DMs for the feasible states are weak orderings, that is, some states would be divided into an equivalence class (delineated by ∼). Accordingly, the total number of the possible rankings of m feasible states can be determined by the ordered Bell number (Good, 1975): 2016; Gong, Hu, Chen, & Li, 2021; Unler & Murat, 2010; Wang, Zhang, & Zhou, 2021). However, the conventional PSO algorithm tends to be trapped in a local optima when solving large-scale problems. Hybridizations with other metaheuristic approaches are adopted to enhance the exploration capability of PSO. Among the existing hybridizations, embedding the GA into PSO is popular (Agarwal & Srivastava, 2018; Garg, 2016; Juang, 2004). Similar to PSO, the GA algorithm is an evolutionary computational model based on population, which was introduced by Holland (1975). Taking inspiration from the mechanics of natural genetics, the basic implementations of the GA algorithm consist of selection, crossover, and mutation operations, following an iterative procedure. As such, the basic idea of PSO-GA is to utilize the genetic operators of GA to promote the global exploration capability of PSO. A hybridization of the PSO and GA is utilized in this paper. 3. Graph model for conflict resolution The main procedure for forward GMCR, as displayed in Fig. 2, involves two stages: conflict modeling and conflict analysis. Bn(m ) = 3.1. Conflict modeling m    j! j=0 m , j (1)   A graph model for a conflict, including four key components, is formally represented as below. m represents the Stirling number of the second kind that j is given by where Definition 1. Let G = (N, S, (Ai )i∈N , (Pi )i∈N ) denote the graph model, where:   1) N = {1, 2, . . . , n} contains all key stakeholders or DMs engaged in a conflict (2 ≤ n < ∞); m j 809 1  j m (−1 ) j−k k . k j! j = k=0 (2) Y. Huang, B. Ge, K.W. Hipel et al. European Journal of Operational Research 305 (2023) 806–819 stability definitions for DM i ∈ N, one needs first to determine possible responses from all other DMs N\{i} after DM i’s initial move from state s to s1 , and afterwards, DM i’s further moves. Generally, let i ∈ N in Definitions 6–9, where 2 ≤ |N | < ∞. The total number of possible preference rankings of feasible states for a conflict is quite large, which increases exponentially as the number of feasible states increases. Using Definition 2, the preference relations between feasible states for each DM can be determined. Specially, for each i ∈ N and s j ∈ S, a UM (s j , st ) ∈ Ai is referred to as a unilateral improvement (UI) for DM i from state s j iff pi (st ) < pi (s j ). Definition 6. A state s ∈ S is Nash stable for DM i iff R+ ( s ) = ∅. i The intuition behind Nash stability is that the focal DM cannot unilaterally move from a Nash stable state to a more preferred state. Definition 3. For each i ∈ N and s j ∈ S, the reachable list for DM i from s j by unilateral movements (UMs) can be formally written as  Definition 7. A state s ∈ S is GMR stable for DM i iff for each s1 ∈ R+ (s ) there exists s2 ∈ RN\{i} (s1 ) satisfying pi (s ) ≤ pi (s2 ). i Intuitively, all UIs of the focal DM from a GMR stable state are sanctioned by subsequent moves legally carried out by other DMs.  Ri (s j ) = st ∈ S (s j , st ) ∈ Ai . Definition 4. For each i ∈ N and s j ∈ S, the unilateral improvement (UI) list for DM i from s j is defined as  Definition 8. A state s ∈ S is SMR stable for DM i iff for each s1 ∈ R+ (s ) there exists s2 ∈ RN\{i} (s1 ) satisfying pi (s ) ≤ pi (s2 ), and i pi (s ) ≤ pi (s3 ) for all s3 ∈ Ri (s2 ). Compared with GMR, the requirements of SMR are stricter. For a state to be SMR stable, all of the focal DM’s UIs should be sanctioned by the opponents, even after response by the focal DM.  R+ (s j ) = st ∈ S (s j , st ) ∈ Ai , pi (st ) < pi (s j ) . i Within an n-DM (n > 2) conflict, a nonempty subset C ⊆ N is called a coalition. For example, a representative coalition is the group of opponents of a given DM i, represented by N\{i}, in which \ indicates “set subtraction”. Permissible movements by a coalition of DMs in GMCR are defined in the literature (e.g., Fang et al., 1993). The UMs of a coalition are referred to as a sequence of UMs under the control of its members, where a DM may move more than once, but not twice consecutively (Fang et al., 1993). Let C (s, s1 ) ⊆ N denote the group of all last DMs in legal sequences from s to s1 . Definition 9. A state s ∈ S is SEQ stable for DM i iff for each s1 ∈ R+ (s ) there exists s2 ∈ R+ (s ) satisfying pi (s ) ≤ pi (s2 ). i N \ {i } 1 The intuition is similar to GMR stability, except that the sanctions from other DMs have to be UIs. 4. Inverse GMCR: Formal problem formulation 4.1. Overall design Definition 5. Let s ∈ S, C ⊆ N, and C = ∅. A UM by C is a member of RC (s ) ⊆ S, defined inductively by: The forward GMCR, along with many extensions to the preference structure, is only capable of predicting the potential resolutions for DMs and analysts by using the preferences of DMs. However, the preferences intrinsic to the DMs may lead to an undesirable equilibrium. In this regard, inverse GMCR is developed here to help DMs or third parties reach a more satisfactory outcome by the minimum adjustments in preferences. Focal DMs can utilize inverse GMCR to generate a desired equilibrium by changing their own preferences or influencing the preference(s) of their opponent(s); third parties can concentrate their efforts on guiding DMs towards the desired preferences. However, it is still challenging to identify the required preferences with the minimum adjustments due to the large solution space of complex, large-scale conflicts. This paper explores the optimal preference rankings by a hybrid metaheuristic algorithm, rather than by enumeration. Among the artificial intelligence algorithms, the PSO algorithm excels in low-computational cost as well as fast convergence speed, and has been applied in various fields (Garg, 2016). Nevertheless, conventional PSO suffers from lower robustness and premature convergence in addressing large-scale optimization problems. Therefore, a PSO variant incorporating the GA process is presented. The forward GMCR has provided a flexible and efficient framework to model, analyze, and solve conflicts by using available preference information as the input. By incorporating a hybrid PSO-GA algorithm, the procedure for inverse GMCR reveals how real-world conflicts can be solved in an inverse perspective. The overall framework for inverse GMCR modified from forward GMCR is displayed in Fig. 3. The inputs required for this procedure to model and analyze conflicts are listed as follows: 1) assuming C (s, s1 ) = ∅ for each s1 ∈ S; 2) if h ∈ C and s1 ∈ Rh (s ), then s1 ∈ RC (s ) and C (s, s1 ) = C (s, s1 ) ∪ {h}; and 3) if s1 ∈ RC (s ), h ∈ C, and s2 ∈ Rh (s1 ), then, provided C (s, s1 ) = {h}, s2 ∈ RC (s ) and C (s, s2 ) = C (s, s2 ) ∪ {h}. Likewise, a legal sequence of UIs for C, RC+ (s ) ⊆ S, can be formulated following Definition 5 by only replacing Rh (s ) with R+ ( s ). h 3.2. Forward analysis The goal of forward analysis is to predict the potential resolutions and use them for informed decision-making, whose key stages are displayed as follows: 1) individual stabilities for each DM; 2) overall equilibrium results for a conflict; 3) evolution path analysis; and 4) sensitivity analysis. Stability analysis, the fundamental analysis of forward GMCR, aims to identify the most possible outcomes by examining the stabilities of feasible states for all DMs. A state is considered to be stable for a given DM engaged in a conflict iff the focal DM does not take the initiative to move away from it regarding specified stability definitions (or solution concepts). Furthermore, a state that is stable for all DMs involved in the conflict is called an equilibrium, since all DMs lack motivation to move away from that state. A stability definition is used to explain whether a DM would be motivated to make moves or countermoves unilaterally from a specific state. Since different DMs may have diverse decision characteristics, a wide variety of stability definitions can apply to a group of DMs who vary in levels of foresight, knowledge of preferences, disimprovement, and risk attitudes (Fang et al., 1993). Four basic stability definitions–Nash stability, general metarationality (GMR), symmetric metarationality (SMR), and sequential stability (SEQ)– are most often employed in graph model. To formally define these 1) key stakeholders or DMs engaged in a conflict and their available options; 2) feasible states for the conflict; 3) allowable state transitions; 4) original preferences of DMs; and 5) desired equilibria for DMs. 810 Y. Huang, B. Ge, K.W. Hipel et al. European Journal of Operational Research 305 (2023) 806–819 where P k denotes the kth preference profile leading to the equilibrium sE . If the difference between the original and the optimized preference profile is small, the focal DMs or third parties are likelier to achieve the desired solution with less cost. For the inverse graph model, the definition of preference adjustment cost varies in the literature. A simple and practical way is to formulate it as the amount of preference adjustment (Tao et al., 2021). It is also adopted in this paper. Thus, the amount of preference adjustment is defined as the total difference of all states’ payoffs between the original preference profile and the optimized preference profile that reaches the desired resolution. Therefore, the inverse preference optimization under a minimum amount of preference adjustment is established as follows: min AdPk = P k − P ori = n  m    pki (s j ) − pori i (s j ) , (3) i=1 j=1 where AdPk is the adjustment amount of optimized preference profile P k in comparison to the original preference profile P ori , and “| |” represents the absolute value operation. 5. The proposed PSO-GA algorithm for inverse GMCR To resolve conflicts from an inverse perspective, we propose a PSO variant integrating the GA for preference identification. The proposed PSO-GA algorithm mitigates the local optimum and premature convergence issues of conventional PSO, and is powerful in searching for the needed preferences. A primary population is first constructed in the PSO process whose particles are updated based on both the individual best experience and the best experience of the whole population. The position of a particle is referred to as a preference profile, represented by Pk = 1 global exploration capability of the algorithm, a secondary population generated by GA enables the primary population to escape from a local optimum. To improve the algorithm, three strategies– a random neighbor strategy for population initialization, the elite reservation and mixed selection operation, and a diversity strategy for population update–are developed in response to the inverse GMCR. Overall, the proposed algorithm aims to find the global optima robustly while ensuring a diversity of solutions, whose lifecycle is shown in Fig. 4. Fig. 3. Overall framework for inverse GMCR. The italics in Fig. 3 indicate the main differences between the forward and inverse GMCR. The primary function of inverse GMCR in this paper is to inversely analyze possible preferences with the minimum adjustments for the specified equilibria. 5.1. Determinating fitness Fitness not only reflects the quality of a solution but also influences the update of a population. Since the goal is to find the best solution with the fewest compromises, the fitness of a preference profile is inversely proportional to its adjustments. Therefore, to determine the fitness function, it is necessary to define the equilibrium results caused by a given preference profile. 4.2. Formulation of the inverse preference optimization model A third party or DM can use the inverse GMCR to achieve an equilibrium…
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