Game theory in the classroom: low cooperative relationships identify bullying patterns in elementary schools

The study examines how cooperative relationships intersect with bullying dynamics in elementary school classrooms. Prior work highlights the importance of social relationships for students’ emotional well-being and academic development, while also documenting the roles of aggression and victimization in shaping social hierarchies. Beyond victims and perpetrators, the literature identifies "bully-victims," who both experience and engage in bullying, often facing isolation and neglect. The authors focus on how cooperative tendencies relate to bullying involvement, especially the bully-victim profile. They hypothesize: (1) lower peer cooperation is associated with a higher likelihood of being identified as a victim, potentially due to reduced social status and stigma; (2) peer cooperation toward bullies could vary, with dominance possibly attracting cooperation but aggressive behaviors also inviting social sanctions; and (3) bully-victims are likely to receive low peer cooperation, reflecting compounded stigma and social rejection. The study aims to test these hypotheses using a classroom-wide, dyadic social dilemma game to map cooperation networks and relate them to bullying roles measured via validated scales and peer nominations.

Bullying and victimization: Bullying is repeated physical or psychological aggression intended to establish dominance and cause harm, affecting academic and psychological outcomes. Developmental and gender differences exist, with middle school intensifying hierarchical behaviors; males are often more victimized and aggression types differ by gender. Bullies often show low empathy and traits like high extraversion and low agreeableness, linked to disruptive behaviors. Bullying and social implications: Bullies may use both coercive and prosocial strategies to gain popularity and can maintain larger social networks, indicating a complex relationship between aggression and social acceptance. Classroom environments and teacher interventions can moderate these dynamics; positive relational settings are associated with reduced bullying and improved peer relations. Cooperation and experimental game theory: The study leverages experimental game theory to capture cooperative tendencies in a controlled yet ecologically valid way. A dyadic social dilemma game simulates real-life trade-offs between collective welfare and individual gain, allowing construction of classroom cooperation networks. Prior research supports the use of such games to infer social preferences and norms, and to connect networked interaction patterns to educational outcomes.

Design and setting: Cross-sectional observational study conducted during the 2017 school year in a public school district in central metropolitan Chile (San Bernardo), chosen for its School Vulnerability Index (IVE = 0.72) closely matching the national urban mean (0.73). Data were collected from entire classes simultaneously using a digital game and questionnaires. Sample: 47 classrooms across 14 schools; grades 3–5; average age 9.78 years; 57% female, 43% male. In Chile, many students remain classmates since first grade, providing stable peer networks. School records (GPA, attendance) were obtained. Ethics and consent: Approved by the Universidad del Desarrollo IRB (IT15110079; 4 May 2016). Active consent from principals and teachers; passive consent from parents/guardians. Confidentiality emphasized. Procedure and implementation: Classrooms were set up with students seated separately; tablets and headphones used to minimize interference and protect privacy. The field team oversaw standardized administration. Anonymity of survey responses was stressed to reduce bias. Instruments:

• Bullying and victimization: University of Illinois Bully Scale (UIBS) for bullying behaviors and University of Illinois Victimization Scale (UIVS) for victimization in the past 30 days. Responses on a 1–4 scale (Never to Almost always). Reliability in this study: Bully scale α = 0.828; Victimization scale α = 0.802. Discriminant validity supported (UIVS not significantly correlated with UIBS in prior work).
• Peer nominations: Popularity (popular minus not popular nominations), Aggressiveness (start fights + laugh at others), Prosociality (kind + cooperative), Social preference (most liked minus least liked), Friendshipness (best friend nominations). Detailed procedures in Supplemental Materials (S6).
• Academic and administrative data: GPA and attendance from official school records. Bullying-role classification (dependent variables): Self-reported victimization scores (SRVS) and bullying scores (SRBS) were standardized globally by gender (z-scores). Threshold of 0.5 SD above gender mean used for categorization: Bully-Victim (both SRBS and SRVS ≥ 0.5), Bully (SRBS ≥ 0.5, SRVS < 0.5), Victim (SRVS ≥ 0.5, SRBS < 0.5), Non-involved (both < 0.5). Gender standardization addressed gender differences in mean scores (e.g., SRBS males 1.90 vs females 1.55; SRVS males 2.49 vs females 2.28). Game-based cooperation mapping: Each student played dyadic rounds with every classmate (n−1 rounds), onymous (non-anonymous). In each round, both players were Allocator and Receiver simultaneously, each endowed with 10 tokens. Sent tokens were doubled for the receiver. Payoff per round equaled tokens kept plus double the tokens received from the partner. Full mutual cooperation yields 20 tokens per player; unilateral defection yields 30 for the defector and 0 for the cooperator. Token flows from i to j defined a directed, weighted edge. Network measures computed per student and normalized by n−1 to range [0,1]: In-strength (sum of incoming edge weights; received cooperation) and Out-strength (sum of outgoing edge weights; sent cooperation). Analytic strategy: Descriptive analyses of token distributions and dyadic patterns. Multilevel (hierarchical) logistic regressions with random intercepts for classrooms for one-vs-all classifications (e.g., bully-victim vs others), accounting for nested data structure. Independent variables were standardized (grand mean centering). Models controlled for individual-level covariates (gender, GPA, attendance, peer-nominated popularity, prosociality, aggressiveness, friendshipness, social preference) and classroom-level variables (class gender ratio, mean class GPA, mean class attendance, grade level). Robustness checks included adding out-strength as a control and employing causal inference techniques (statistical matching). Sensitivity analyses examined different quantile thresholds for bully-victim classification.

• Cooperative behavior in the game showed bimodality: many students sent either all 10 tokens or none (~25% each). About half of students varied contributions widely across peers (minimum 0 to maximum 10).
• Dyadic outcomes clustered into mutual high cooperation (e.g., 10–10), mutual low cooperation (e.g., 0–0, 1–1), and asymmetric exchanges (e.g., 10–0). Although 0–0 pairs occurred (~4.5%), they were only half as frequent as mutual high-cooperation pairs (~9.18%).
• Bully-victims exhibited the lowest average in-strength (received cooperation) among categories.
• In multilevel one-vs-all models, in-strength strongly predicted bully-victim classification: a one SD increase in in-strength reduced the odds of being a bully-victim by 38% (log-odds −0.47, p<0.001; OR ≈ 0.62). Moving from minimum to maximum in-strength corresponded to an order-of-magnitude (≈ tenfold) decrease in bully-victim probability.
• Victims also showed a significant negative association between in-strength and victim classification; bullies showed no significant association with in-strength.
• For bully-victims, higher aggressiveness and greater friendship nominations increased the likelihood of classification; prosociality and GPA were negatively associated. Effects were more pronounced for bully-victims than for other categories.
• For victims vs non-involved, in-strength was negatively associated with victim status, while friendshipness was positively associated.
• For bullies vs non-involved, GPA and social preference were negatively associated with bully status, while aggressiveness was positively associated; no game-network metric (including in-strength or out-strength) significantly predicted bully status.
• In-strength correlated positively with peer-nominated popularity, prosociality, friendshipness, and social preference; and with GPA and attendance from official records. In-strength correlated negatively with aggressiveness, physical aggression, and relational aggression. All correlations were significant.
• Out-strength correlated moderately with in-strength (r ≈ 0.272) but did not capture the multidimensional student characteristics; controlling for out-strength did not change the main results, ruling out alternative explanations based on sent cooperation.
• Sensitivity analyses using higher quantile thresholds (0.65, 0.75, 0.85) for bully-victim classification showed the in-strength effect persisted and intensified with higher thresholds. Overall, receiving fewer tokens (lower in-strength) is tightly linked to being a bully-victim or a victim, suggesting that received cooperation is a salient indicator of social vulnerability and peer standing in classrooms.

The study demonstrates that the amount of cooperation students receive from classmates in a dyadic social dilemma (in-strength) is a strong marker of their bullying involvement, particularly the bully-victim profile, and to a lesser extent victimization. This addresses the central hypothesis that diminished peer cooperation is associated with increased vulnerability to bullying roles characterized by social isolation and rejection. The absence of a significant relationship between in-strength and being a bully suggests that aggressive dominance does not necessarily reduce received cooperation, potentially due to mixed processes of fear, strategic affiliation, and social sanctioning. The pattern of associations implies that in-strength encapsulates multiple facets of peer recognition and status—popularity, prosociality, friendship ties, social preference—as well as academic engagement (GPA, attendance), while inversely tracking aggression. Thus, received cooperation functions as a multidimensional proxy for students’ social integration and standing. These findings explain why bully-victims, who combine perpetration and victimization and face compounded stigma, receive the least cooperation. Practically, mapping classroom cooperation networks can identify students at risk of victimization or dual-role bullying. Since in-strength is robust to controls and alternative specifications (including out-strength and matching), it offers a reliable indicator for monitoring classroom climate and targeting supports. The results highlight that interventions should enhance cooperative norms and inclusive peer dynamics, potentially disrupting feedback loops of isolation and aggression that characterize bully-victims.

The paper introduces a game-theoretic, classroom-wide measure of cooperation to illuminate bullying dynamics among elementary students. By constructing cooperation networks via a non-anonymous dyadic social dilemma and linking them to validated bullying/victimization measures, the authors show that low received cooperation (in-strength) robustly predicts bully-victim and victim statuses, even after controlling for socio-behavioral, academic, and classroom factors. In-strength aligns positively with peer recognition and academic engagement and negatively with aggression, underscoring its value as a multidimensional marker of social integration. These contributions suggest that integrating cooperation-focused assessments and practices could help predict and reduce bullying. Future research should extend to diverse socioeconomic and cultural settings, incorporate longitudinal designs with repeated measurements, and triangulate self-reports with observational and teacher-reported data. Intervention studies leveraging cooperative learning and network-informed targeting are promising paths to improve classroom harmony and reduce bullying.

• Measurement limitations: Reliance on a single wave of self-reported bullying/victimization and peer nominations within the school year may miss temporal changes. Self-report bias may persist despite privacy safeguards (seating separation, headphones, confidentiality assurances). Future work should triangulate with teacher reports and observational data.
• External validity: The study was conducted in urban, middle-income Chilean schools within a district whose IVE matches the national urban average. Generalization to contexts with substantially different vulnerability indices or to non-urban/wealthier/poorer settings requires caution, as socioeconomic and cultural capital influence aggression and victimization.
• Scope of variables: While models controlled for many individual and classroom factors, unmeasured confounding (e.g., familial factors, broader community influences) may remain. Gender, race/ethnicity, and socioeconomic status warrant deeper analysis in future work.
• Data availability: Data are part of an ongoing project and will be released with related publications (planned 2025), limiting immediate replication by external researchers.