Pietraszewski argues that the representation of a group is defined by four ways a third person can be drawn into a dyadic conflict. We agree that such triadic interactions can lead to group representations, but propose that representations of groups are defined in terms of an abstract, recursive utility calculus (Kleiman-Weiner, Saxe, & Tenenbaum, Reference Kleiman-Weiner, Saxe and Tenenbaum2017; Powell, Reference Powell2021). By recursive utility we mean: People represent individuals as valuing (i.e., adopting, or weighting) the utilities of other individuals.

Representing recursive utilities licenses the same inferences as the triadic behavioural primitives, in the conflicts Pietraszewski considered. If A and C put high weight on each other's utilities, but relatively lower or even negative weight on B's utilities, then observers are licensed to predict: C will attack B, defend A, not be attacked by A, and possibly be attacked by B. Observing one of these patterns allows an observer to infer the relative weights these individuals place on each other's utilities, and therefore to predict their future behaviour.

However, using recursive utilities allows the theory to extend beyond Pietraszewski's account in three ways.

First, the recursive utility representation accounts for how people represent and learn about groups in non-conflict situations. For example, given the pattern of utilities described above observers can predict that C is more likely to respond to A's needs than to B's needs, outside of conflict. Observing C helping or caring for A would provide some evidence of how much C values A's utilities (Powell, Reference Powell2021). Then, at the first sign of a conflict between A and B, an observer would predict C's role, even though no previous conflict-related behaviour had ever been observed.

Second, the recursive utility representation accounts for how people represent and learn about asymmetric relationships. The inferential links between Pietraszewski's primitives are symmetric: When C is drawn into the conflict between A and B, and attacks B, all of three remaining primitives are predicted to the same degree. However, C attacking B provides more evidence of how much C values A relative to B, than about how much A values C. The weights individuals put on each others' utilities do not need to be symmetric (Powell, Reference Powell2021). Recursive utilities allow observers to represent cases where C would always ally with A against B, but A would not reciprocate.

Third, representing conflicts in terms of abstract utilities could account for how people make predictions based on what the conflict is about. Two individuals may become allies when parts of their utilities overlap, and they act to maximize joint utilities (Kleiman-Weiner, Ho, Austerweil, Littman, & Tenenbaum, Reference Kleiman-Weiner, Ho, Austerweil, Littman and Tenenbaum2016). Unlike fully adopting another individual's utilities, alliances built on joint utilities might be limited to specific conflicts. For example, hunters and environmentalists may find common cause in defending public access to wildlands but be opposed on animal rights. To predict future behaviour, it is critical to identify not only who is fighting, but also what is at stake.

A computational theory of groups in terms of abstract, recursive utilities is thus appealing for its expressive range and inferential flexibility. Note that implementing an actual computational model with these properties is very challenging. No existing computational model distinguishes and/or combines both recursive and joint utilities and their interactions.

Yet this proposal shares key limitations with Pietraszewski's.

Both Pietraszewski's behavioural primitives, and the recursive utility representation, represent triads of individuals who have equal power and responsibility. To predict human behaviour in conflict situations, though, asymmetries of power and responsibility are indispensable. For example, if a parent defends their small child from attack, we do not predict that the child is equally likely to defend their parent (because of differences in responsibility) (e.g., Fiske, Reference Fiske1992); when a person is insulted by his boss, rather than his co-worker, we can predict that fewer observers will come to his defense (because of differential power). These asymmetries cannot be reduced to (or derived from) triads of alliance or recursive value, and so require distinct cognitive machinery.

The other major challenge is how people generalize group-constitutive roles from specific observed individuals to collections of unobserved individuals. Pietraszewski calls this “substitution”: commonly, the parties to a conflict are not literal individuals, but potentially large sets of individuals who are substitutable for one another in the conflict roles. But how do observers identify these sets? In the recursive utility framework, the same problem arises: How do observers infer the recursive utilities of new individuals? The intuitive answer here is: Observers use observable (“ancillary”) cues and culturally specific intuitive theories (O'Connor, Reference O'Connor2019) to guess whose utilities are substitutable. These inferences may be well-founded (e.g., in an international war, most individuals will have higher reciprocal weights on the utilities of people who share their nationality, than people from the opposing nationality), but may be wrong (e.g., in every war, some people sympathize with the individual or collective good of people from the “other” nationality); and because they are always based on limited evidence, these inferences are often stereotypes (e.g., people who share a racial or gender or religious identity may be viewed as “substitutable” in absurd and offensive ways) (e.g., Bruneau, Kteily, & Falk, Reference Bruneau, Kteily and Falk2018). Neither account explains the role of shared historical knowledge or cultural learning in these inferences.

To determine who is substitutable in a conflict role, observers could generalize across people based on theories and observations of social organization. Yet this solution seems to require all of the machinery of group membership that Pietraszewski claimed to avoid. To infer which individuals have relevantly similar utilities, or are substitutable in conflict roles, observers need to generalize between people based on dyadic shared properties, rather than triadic conflict interactions.

In summary, a representation of abstract, recursive utilities could make the same predictions as Pietraszewski, and additionally account for how people represent and learn about groups in non-conflict interactions, represent asymmetric relationships within groups, and make predictions based on what the conflict is about. However, two key limitations of Pietraszewski's approach are shared by our proposal, and thus remain to be addressed.