Complementarity Potential (CP)

• Complementarity Potential (CP) is a measure of performance synergy achieved by combining agents whose diverse strengths reduce individual errors.
• In neutrino physics, CP quantifies enhanced sensitivity to oscillation parameters by integrating distinct experimental data from setups like DUNE, T2HK, and HK.
• In human-AI collaboration, CP guides the design of systems by leveraging information and capability asymmetries to optimize decision-making outcomes.

Complementarity Potential (CP) refers to the theoretical gain in performance achievable by combining two or more agents (experiments, decision-makers, or models) such that their joint performance on a given set of tasks surpasses that attainable by any agent individually. The concept has explicit mathematical formalizations in both human-AI collaborative decision-making and in the joint analysis of neutrino oscillation experiments, where it quantifies the additional sensitivity to physical parameters or predictive loss reduction made possible by leveraging differences in information, capabilities, or systematic constraints across the contributing agents or experiments.

1. Formal Definitions and Quantification

In decision science, as formalized by Hemmer et al. (Hemmer et al., 2024), Complementarity Potential (CP) is defined on a concrete set of tasks $T=\{(x^{(i)},y^{(i)})\}_{i=1}^N$, with two agents—human ($H$) and AI ($AI$)—providing predictions $\hat y_H^{(i)}$ and $\hat y_{AI}^{(i)}$, and a joint mechanism $I$ yielding team decisions $\hat y_I^{(i)}$. Given a loss $l(\hat y, y) \ge 0$, the mean loss $L_D$ for each $D\in\{H, AI, I\}$ is

$L_D = \frac{1}{N} \sum_{i=1}^N l_D^{(i)},$

where $l_D^{(i)} = l(\hat y_D^{(i)}, y^{(i)})$. Let $T^*$ be the better individual (i.e., with smaller $L$):

$L_{T^*} = \min(L_H, L_{AI}).$

The complementarity potential is

$CP = L_{T^*} = \min(L_H, L_{AI}),$

This expresses the gap between zero loss and the best single-agent performance—a ceiling that only synergistic collaboration could, in principle, surpass.

In long-baseline neutrino physics, CP denotes the extra sensitivity to oscillation parameters (mass hierarchy, $\theta_{23}$ octant, CP-phase) acquired by the combined fit of datasets from DUNE, T2HK, and HK atmospheric neutrinos, beyond the direct sum of individual sensitivities. Quantitatively, the synergy factor $S$ for observable $O$ is

$$S \equiv \Delta\chi^2_{\text{combined}}(O) - \left[\Delta\chi^2_{\mathrm{T2HK}}(O)+\Delta\chi^2_{\mathrm{HK(atm)}}(O)+\Delta\chi^2_{\mathrm{DUNE}}(O)\right],$$

and the ratio

$$R \equiv \frac{\Delta\chi^2_{\text{combined}}(O)}{\sqrt{[\Delta\chi^2_{\mathrm{T2HK}}(O)]^2+[\Delta\chi^2_{\mathrm{HK(atm)}}(O)]^2+[\Delta\chi^2_{\mathrm{DUNE}}(O)]^2}},$$

where $R>1$ indicates strong complementarity (Fukasawa et al., 2016).

2. Components and Sources of Complementarity Potential

Inherent and collaborative sources delineate the structure of CP in decision-making (Hemmer et al., 2024):

• Inherent Complementarity ($CP^{inh}$): Reflects gains from one agent correcting the other's errors on particular instances. For $AI$ generally stronger ($L_{AI} \le L_H$):

$$CP^{inh} = \frac{1}{N} \sum_{i=1}^N \max\left(0, l_{AI}^{(i)} - l_H^{(i)}\right)$$

This term captures the loss reduction the weaker judge can provide on instances where it outperforms the stronger.

• Collaborative Complementarity ($CP^{coll}$): Captures the burden remaining where both make errors individually; potential gains here require mechanisms that improve over both individually:

$$CP^{coll} = \frac{1}{N} \sum_{i=1}^N \min\left(l_H^{(i)}, l_{AI}^{(i)}\right)$$

Together,

$CP = CP^{inh} + CP^{coll}$

Two primary drivers of $CP^{inh}$ are identified:

• Information Asymmetry: Provided when either agent has access to exclusive information or features (e.g., contextual knowledge for humans unavailable to AI or vice versa).
• Capability Asymmetry: Stemming from different inductive biases, learning algorithms, or model architectures, causing non-overlapping error regions.

In the neutrino context, sources of complementarity include differences in baselines (DUNE at 1300 km, T2HK at 295 km), energy spectra, matter effects, and systematic uncertainties across detectors, which affect the experimental degeneracies (hierarchy–CP, octant–CP) in distinct, partially orthogonal ways (Agarwalla et al., 2022, Fukasawa et al., 2016).

3. Complementarity Potential in Physics: Neutrino Oscillation Experiments

The role of CP is critical in the simultaneous analysis of data from DUNE, T2HK, and atmospheric neutrinos at HK. The combined analysis leads to significantly greater resolving power for key unknowns in the PMNS framework:

• Hierarchy sensitivity: Increased from $\sim 1\sigma$ (T2HK) or $8\sigma$ (DUNE) to $15\sigma$ for the joint fit, across all true $\delta_{CP}$ values.
• Octant resolution: The combined setup resolves the $\theta_{23}$ octant for all but a narrow window $(43.5^\circ < \theta_{23} < 48^\circ)$ at $5\sigma$ confidence.
• CP violation (CPV) coverage: The coverage at $5\sigma$ rises from $30\%$ (T2HK) or $20\%$ (DUNE) to $68\%$ with all three experiments.
• Precision: Joint analysis achieves $0.3\%$ on $\Delta m^2_{\mathrm{eff}}$, $2\%$ on $\sin^2\theta_{23}$, and $20^\circ$ on $\delta_{CP}$ at $1\sigma$ (Fukasawa et al., 2016).

The complementarity arises as the degeneracies present in individual experiments (e.g., sign degeneracy in T2HK, octant–CP coupling in DUNE) are resolved only in combination, leading to a superadditive increase in $\Delta\chi^2$ for mass ordering and CPV discovery.

Observable	T2HK	T2HK+HK(atm)	DUNE	T2HK+HK+DUNE
Hierarchy ($\sqrt{\Delta\chi^2}$)	$\sim1\sigma$	$5\sigma$	$8\sigma$	$15\sigma$
Octant (5$\sigma$ window)	excludes $[43^\circ,49^\circ]$	same	same	excludes $[43.5^\circ,48^\circ]$
CPV fraction @ 5$\sigma$	$30\%$	$60\%$	$20\%$	$68\%$

4. Complementarity Potential in Human-AI Collaboration

In human-AI teaming, CP enables teams to reach a level of performance, termed complementary team performance (CTP), that neither human nor AI can reach alone. Hemmer et al. (Hemmer et al., 2024) empirically demonstrated:

• Real-estate appraisal: With unique human contextual information, $CP^{inh}$ increased by $44\%$ and the team realized $45\%$ of that inherent potential, with statistically significant gains only present when the human had unique features unavailable to the AI.
• Noisy image classification: AI models deliberately trained to err on instances where humans are strong (increasing capability asymmetry) yielded $CP^{inh}$ gains of $287\%$, with the team capturing $89\%$ of the inherent complementarity effect—a substantial improvement over the baseline.

The decomposition into realized complementarity effect ($CE$) versus theoretical potential enables diagnosis of where collaboration mechanisms succeed or fail.

5. Practical Measurement and Optimization Strategies

CP and its realization $CE$ offer a principled lens for fine-grained evaluation and design of multi-agent systems:

• Measuring CP/CE: Analysis of individual instance-level errors (and their overlap) is required to precisely partition $CP^{inh}$ and $CP^{coll}$.
• Increasing $CP^{inh}$: Provide information asymmetry (exclusive features) or align models to have disjoint error structures (e.g., adversarial training for complementarity).
• Maximizing $CE$: Design task interfaces and workflows (uncertainty quantification, explicit comparison) enabling agents to leverage their unique strengths on appropriate instances.
• Collaborative Reasoning: $CP^{coll}$ and $CE^{coll}$ quantify the (typically rare) cases where true synergy—beyond either agent's or model's native output—is possible and achieved.

In neutrino physics, similar principles apply: the experimental program is structured to exploit asymmetries in $L/E$, detector principle, and systematic regime to maximize the non-overlapping strengths of each facility. Analysis choices, exposure allocation, and systematic error reduction are optimized jointly to push the envelope of possible sensitivity.

6. Implications and Application in Scientific Practice

Complementarity Potential is central in system design, diagnosis of team effectiveness, and in experimental planning where maximizing collective sensitivity is paramount. In decision-making, $CP$ directly informs the potential value-add of human-AI or AI-AI teaming above straightforward automation or ensembling. In neutrino oscillation physics, CP formally guides the construction and optimization of experimental consortia, exposure allocations, and joint analysis methodologies, as evidenced by explicit $\Delta\chi^2$ computations of synergy and CPV coverage (Fukasawa et al., 2016, Agarwalla et al., 2022).

The theoretical and practical value of CP lies in its explicit quantification of the maximum threshold attainable only through collaboration, the identification of design levers to enhance this threshold, and the diagnostic separation of realized effect from mere potential—establishing a rigorous framework for engineering and analyzing complex multi-agent scientific, technical, and decision-making systems.