Tree-Based Formalization of Multi-Agent Complementarity in Human-AI Interactions

2026-06-03

AI Summary

Existing approaches struggle to model how complementarity in multi-agent human-AI interaction (HAI) depends on collaborative processes. This work introduces, for the first time, a formal framework based on ordered role assignments and planar binary trees with predictive-vector leaf nodes, defining tree-dependent complementarity functionals through recursive local binary composition rules. By integrating barycentric coordinate charts, Tamari lattice-based reparameterization, and loss analysis, the study proves that selector-type HAI systems cannot achieve complementarity. In regression tasks, complementarity is shown to be equivalent to minimizing Euclidean distance, and the optimal linear pooling weights are derived for the case of N=2. Conversely, under endpoint-monotonic losses in classification tasks, complementarity is theoretically unattainable.

Abstract

Complementarity is the case in which a human--AI interaction (HAI) outperforms the best prediction benchmark available among its members. Although this idea is central in HAI research, formal work on complementarity remains limited. Existing frameworks do not model how agents' predictions compose into workflow-sensitive multi-agent protocols. We close this gap by introducing a tree-based formalization of complementarity in multi-agent HAI. An HAI protocol is represented by an ordered agent-role configuration together with a rooted planar binary tree whose leaves are decorated by prediction vectors. A local binary composition rule is evaluated recursively along the tree, yielding a tree-relative complementarity functional relative to a pointwise-min oracle benchmark. We prove four results. First, selector-based HAIs, including self- or AI-reliance, cannot achieve complementarity regardless of task, loss, or prediction quality. Second, in regression under squared loss, complementarity is equivalent to Euclidean distance minimization from the ground-truth vector; for $N=2$, the optimal linear-pooling weight has a closed form and a residual-correction interpretation. Third, under linear local composition, every protocol tree defines a barycentric coordinate chart on the simplex of leaf weights; Tamari-cover reparameterizations of protocol trees preserve complementarity, and for $N=4$, they satisfy the pentagon identity. Fourth, in binary classification, no internal local composition can achieve complementarity under endpoint-monotone losses, including standard Bregman and many finite Bernoulli $f$-divergence losses; an analogous obstruction holds for multiclass aggregation under cross-entropy. In summary, our framework shows that complementarity is attainable in multi-agent regression, but obstructed in classification under natural conditions on local aggregation and loss functions.

Problem

• complementarity
• human-AI interaction
• multi-agent systems
• prediction composition
• workflow-sensitive protocols

Innovation

• tree-based formalization
• multi-agent complementarity
• human-AI interaction
• local composition rule
• barycentric coordinates