Table 2 Interpretation of the model’s key parameters.

\(r^j\)

Reputation \(r^j\) for an agent j indicates how others may perceive j’s overall identity taking into account j’s fusion with the group G. Individual reputations have been widely accepted as providing an explanation for different forms of collective action⁴⁹, based on agent identities being independent (i.e., mutually exclusive). To generalise this we let \(r^j\) incorporate identity fusion², involving an individual’s isolated personal identity overlapping with that of a group through its fusion level \((f_j)\). This was originally conceived for human assessment of identity fusion⁵⁴, and is based on the subject intersecting circles representing their personal identity and the group’s identity. The extent of intersection of these circles has been found to strongly correlate with the subject’s identity fusion \(f_j\)¹⁷, leaving a proportion of \(1-f_j\) of j’s personal identity that is excluded from overlap with group G’s identity on the Venn diagram. We use this representation as a first proxy to represent an integrated identity for j, based on reputations aligned with personal identity \((r_j)\) and group identity \((r_G)\), to derive j’s integrated reputation \(r^j\), where \(r^j = (1-f_j) r_j + f_j r_G\). Conservatively, we assume this occurs only when the donor is itself fused (i.e., \(f_i > 0\)) and therefore can place value in the group’s reputation. Otherwise i perceives j’s reputation \(r^j\) as \(r^j = r_j.\) This ensures that when i is unfused \((f_i =0)\) its behaviour defaults to a base model⁵³ for indirect reciprocity involving only individual isolated reputations. This guards against inadvertently promoting fusion during the agent interaction stage based on implicit assumptions of agent awareness of group G

\(s_i, u_i, d_i\)

These binary variables define the current strategy for agent i’s decision-making in the donation game, and are based on the social comparison of the potential recipient’s reputation with that of the donor, i. Without consideration of identity fusion, the strategy of donating to those with a similar or greater reputation is known to evolve and sustain cooperation (i.e., \(s_i = u_i = 1, d_i = 0\))⁵³. Originating from Festinger^55,56, it is evident that self-referential evaluation frequently influences decision making from a social perspective^57,58,59. Social comparison is also phylogenetically ancient⁶⁰ and embedded in evaluating competitors and assessing whether or not to commit resources in wide ranging contexts^{61,62,63,64,65,66,67,68}. These variables are subject to evolution and coevolve with an agent’s identity fusion \(f_i\).

\(S_i\)

The in-group for an agent i, is defined as those with at least the same level of identity fusion as i, or greater. The in-group is introduced to accommodate possible effects concerning homophily^28,69, allowing an agent to preference interaction with others that have a common social identification (i.e., at least the same level of identity fusion). \(S_i\) controls the probability of i playing the donation game with a randomly selected partner j from the in-group during step (1) of the model, as opposed to j being randomly selected from the whole population with probability \(1-S_i\). Homophily is known to establish itself through evolutionary means^25,28,70,71 and here we control \(S_i\) exogenously, using a range of values including \(S_i = 0\). While we are not aware of research establishing that fusion heightens the probability of in-group interaction, we accommodate this possibility by also considering the hypothetical case that \(S_i = f_i\).

\(ctr_i\)

This represents an additional form of personal reputation for fused agents, noting that under identity fusion, personal identity remains salient for fused agents. This assesses the relative contribution (proportion of actions from a generation’s start) that an agent is making in support of group G, thus reflecting their in-group behaviour. It enables fused agents, who are assumed to have heightened sensitivity to pro-group behaviours, to observe the contribution of other fused agents. This leaves agents open to experiencing inconsistencies, relative to themselves, concerning support for G. Through a ‘visceral sense of oneness’^21,54,72,73 with G, we assume a highly fused agent i may experience vicarious hypocrisy⁷⁴ when a higher fused agent j is observed contributing less support to G than i (i.e., \(ctr_i > ctr_j\)). We assume this invokes cognitive dissonance^75,76,77 for i, which is received through a group connection with j due to fusion with G^78,79. In response, agent i is assumed to invoke a form of ostracism^2,80, because this is the lowest cost mitigation⁸¹ that an agent can make to reduce its dissonance. This aligns with a heightened disposition to maximise in-group advantage⁸² despite the additional overhead to agent i.

p

p controls the probability that an agent checks the extent that another agent’s contribution towards G (i.e., agent i compares \(ctr_i\) against a given agent j’s contribution \(ctr_j\)) is justifiable, again based on self-referential social comparison^55,56. We assume \(p = f_i\), which aligns with an agent having a heightened incentive to consider the pro-group behaviour of other agents when its own fusion level is greater⁸². This assumption results in an increased chance of checking as fusion increases and reflects an increasing pro-group motivation due to an agent’s increased oneness with the group G^21,54,72,73. This also ensures that the personal identity of fused agents remains salient as their fusion increases.

T

Checking the contribution of other agents incurs an overhead for the agent involved and is more relevant to those with greater dependency on fusion towards the group for their own identity. T defines a threshold when agent interest in checking the contribution of others is triggered, and enables the exploration of highly fused sub-groups, which are important to consider^5,6,7,8,9.