Asymmetric Mediator Scenario

• The asymmetric mediator scenario is a framework where a mediator induces non-symmetric, directional interactions between primary sectors in fields such as physics, economics, and causal inference.
• In particle physics and dark matter studies, it drives radiative neutrino mass generation and asymmetry transfer through extended scotogenic models with precise parameter constraints.
• In strategic and online decision settings, it underpins mechanism design and feedback systems by leveraging two-sided incentive compatibility and abstention-informed algorithms.

The asymmetric mediator scenario refers to a class of models and analytical frameworks where an intermediate agent, field, or process (the "mediator") establishes an essential, typically non-symmetric, relationship between primary sectors or agents. Depending on context, this may involve physical (particle physics, cosmology), strategic (mechanism and game theory), or statistical/causal (mediation analysis) formalisms. Across disciplines, the asymmetric mediator scenario introduces a directionality or sequence to mediation, produces novel constraints, and enables rich stratification of qualitative and quantitative effects.

1. Asymmetric Mediator in Beyond Standard Model Physics and Cosmology

In particle physics and cosmology, the asymmetric mediator scenario most prominently arises in extended scotogenic models and asymmetric dark matter theories. The scotogenic model (Asai et al., 16 Dec 2025, Asai et al., 2022) extends the Standard Model (SM) by introducing new $Z_2$-odd Majorana fermions $N_i$, an $SU(2)_L$ doublet scalar $\eta$ (the mediator), and a singlet scalar $\sigma$ (the dark matter candidate). The asymmetric mediator $\eta$ enables transfer of an asymmetry generated in the visible sector (via leptogenesis and Majorana fermion CP-violating decays) to the dark sector in the form of a dark-matter relic density.

Key structural features:

• Radiative neutrino masses are induced at one-loop order with Yukawa couplings $h_{\alpha i}$ and $\lambda_8$ coupling in the potential.
• Asymmetry transfer: Lepton-number asymmetry generated in $N_i$ decays produces equal but opposite asymmetries in SM leptons and the $\eta$ doublet; the $\eta$ decay then relays the asymmetry to the dark sector.
• Separation of timescales and processes: The requirement that $\eta$ decays after most symmetric-annihilation and before BBN imposes cosmological bounds and leads to a nontrivial allowed region for parameters such as $m_\eta$ and $\mu$.
• Joint origin of $\Omega_{\mathrm{DM}}/\Omega_{B}$: Both baryon and DM densities are sourced from the same primary asymmetry.

Allowed parameter space: $$700\,\text{GeV}\lesssim m_\eta\lesssim10\,\text{TeV},\quad 10^{-11}\,\text{GeV}\lesssim\mu\lesssim10^{-8}\,\text{GeV},\quad m_\sigma\simeq1.8\,\text{GeV}$$ with collider constraints $m_{\eta^\pm}\gtrsim660$ GeV and $\lambda_8\sim10^{-8}$ and $h_{\alpha i}\sim10^{-3\text{–}4}$ (Asai et al., 16 Dec 2025).

The dynamics are governed by coupled Boltzmann equations that track the evolution and transfer of the asymmetry, with initial conditions set by the primordial lepton-sector asymmetry.

2. Asymmetric Mediators in Asymmetric Dark Matter and Freeze-In

Asymmetric mediator constructions are central in many models of asymmetric dark matter (ADM). In minimal renormalizable completions (Wise et al., 2014), DM is a Dirac fermion $\chi$, while the mediator is a scalar $\phi$. The asymmetric mediator bridges annihilation and decay channels that are essential to the efficient transfer and depletion of both visible and dark sector asymmetries.

Key aspects:

• Static Yukawa potential: $\phi$ exchange gives rise to bound-state formation in DM sectors, with the stability regime determined by $\alpha_\chi m_\chi/(2m_\phi) > 0.84$.
• Early Universe formation: The out-of-equilibrium decay and scattering processes involving $\phi$ are critical for both ADM relic density and cogenesis (joint generation of baryon and DM asymmetries).
• Direct detection phenomenology: The coherence enhancement of direct detection cross-sections due to mediator-induced bound states.

In asymmetric freeze-in scenarios (Unwin, 2014), the mediator's inability to remain in equilibrium is necessary to prevent cancellation of the generated asymmetry. The condition $\Gamma_\phi(T)\lesssim H(T)$ at $T\sim m_\phi$ ensures freeze-in of $\phi$ and enables cogenesis: $Y_{\Delta\chi}\sim 10^{-10}$ for portal couplings $\sim10^{-8}$ and $m_\phi\sim 10$ GeV.

3. Asymmetric Mediator in Strategic and Mechanism Design Settings

In economic and algorithmic settings, an asymmetric mediator denotes an agent/intermediary with strategic preferences or an asymmetry in information or incentivization. In the strategic facility location problem with agents and mediators (Babaioff et al., 2015), agents' preferences are aggregated through mediators who act strategically on behalf of their subgroups. Asymmetry arises both from the communication structure (agents communicate only through their mediator) and in the interplay of incentive constraints.

Key results:

• No dominant-strategy mechanism exists when both agents and mediators are strategic; thus, mechanisms achieve only two-sided incentive compatibility (IC): an agent’s dominant strategy is truthfulness given a truthful mediator, and vice versa.
• Deterministic and randomized mechanisms achieve minimal loss in approximation: deterministic (3-competitive) and randomized (2-competitive) mechanisms under two-sided IC.
• Hierarchical mediation: As the mediator structure deepens (multiple levels), the approximation ratio deteriorates exponentially, and stronger IC concepts are lost.

The table below summarizes core mechanism types:

Mechanism	Approximation Ratio	Incentive Compatibility
Weighted Median (WMM)	3	Two-sided IC
Randomized (TRM)	2	Two-sided IC
Hierarchical (IWMM)	$2^s-1$ (depth $s$)	Naive IC (much weaker)

4. Asymmetric Mediator in Statistical and Causal Mediation Analysis

In causal inference, the asymmetric mediator scenario refers to settings with multiple, possibly sequential or interdependent mediators where interventions, mediation pathways, and decompositions reflect ordering or inter-mediator dependencies.

• Ordered causal mediation: In two-sequential mediator systems $A \to M_1 \to M_2 \to Y$, the total effect (TE) can be decomposed into direct, indirect, and interaction pathways. Gao et al. (Gao et al., 2020) introduce the natural counterfactual interaction effect (NCIE), capturing interaction between mediators:

$$TE = PNDE + NIE_1 + NIE_2 + NCIE_{12} + \text{(reference interactions)}$$

Identification relies on extended g-formula expressions, generalizing standard mediation analyses.

• Clinical and high-dimensional settings: For $k$ mediators with arbitrary DAGs (Sun et al., 2020), practical interventions can only be performed on one mediator at a time ("asymmetric" manipulation). The controlled direct and indirect effects (CDE, CIE) are defined so that the total effect decomposes into

$TE = CDE_j(0) + sCIE_j$

where $sCIE_j$ encodes prevalence-weighted indirect effects through $M_j$, naturally aligning with interventions feasible in real-world or clinical trials.

5. Asymmetric Mediator in Online Decision Mediation

In machine learning, the asymmetric mediator scenario describes online decision frameworks in which a sequential mediator policy intermediates between human and oracle action, electing dynamically among acceptance, intervention, or abstention (querying the oracle) (Jarrett et al., 2023). Here, the asymmetry is operational: only the abstention action reveals the oracle action, making learning fundamentally asymmetric.

Key features:

• UMPIRE algorithm: Minimizes cumulative system regret by uncertainty-modulating the threshold for oracle queries, with dynamic cost adjustment based on expected learning value measured by mutual information gain.
• Abstentive feedback: Only abstention provides training data, leading to theoretical and empirical distinctions from classical contextual bandits, with proven consistency and lower regret compared to baselines.
• Metrics: System-level regret, model error, and granular mediator intervention statistics.

In practical terms, this sets the paradigm for real-world AI decision support, particularly in clinical or high-stakes applications where mediation must balance autonomy, oversight, and active learning opportunities.

6. Phenomenological and Methodological Implications

The asymmetric mediator scenario, across all described domains, results in:

• Nontrivial parameter or effect bounds arising from the interplay between mediation timing, strength, and environmental constraints (e.g., BBN, collider signatures in cosmology; statistical identifiability in causality).
• Novel stratification of effect pathways in ordered mediation analyses, enabling more granular partitioning of direct, indirect, and interaction effects.
• Algorithmic challenges and innovation in mechanism design and online learning, requiring new solution concepts (e.g., two-sided IC, abstentive feedback modeling).
• Enhanced interpretability and programmatic interventions in mediation analysis and clinical applications, particularly where complex dependencies and practical actionability limit feasible interventions to single mediators or specific timepoints.

The asymmetric mediator scenario thus represents a unifying conceptual thread linking mediator-induced asymmetry in physical models, strategic intermediaries in economics and computer science, and methodological innovations in causal inference and statistical learning (Asai et al., 16 Dec 2025, Asai et al., 2022, Wise et al., 2014, Unwin, 2014, Babaioff et al., 2015, Gao et al., 2020, Sun et al., 2020, Jarrett et al., 2023).