Arbitration Vector: Interdisciplinary Applications

• Arbitration vector is a mathematical and algorithmic construct that encodes allocation and blending of decisions across diverse technical domains.
• It enables fast, parallel arbitration in NoC systems, adaptive model ensembling, shared autonomy blending, and secure dispute resolution via cryptographic proofs.
• Its design leverages signal processing, probabilistic inference, and geometric arbitrage theory to enhance performance, security, and interpretability in distributed systems.

An arbitration vector is a mathematical or algorithmic construct central to diverse fields including electronic arbitration in computing systems, ensemble modeling, financial mathematics, shared autonomy, and blockchain-based dispute resolution. It systematically encodes the allocation, selection, or blending of choices among competing agents, models, or claims, facilitating robust, efficient, or fair decision-making across distributed or uncertain environments.

1. Arbitration Vectors in Wave-Interference Bus Arbitration

The wave-interference-based arbitration scheme for Network-on-Chip (NoC) environments introduces an $N$-bit arbitration vector $r=[r_1, r_2, \ldots, r_N]$ to achieve fast, parallel bus arbitration (Li, 2017). In this system, a "home" node injects $N$ orthogonal frequency carriers ("tokens") onto a shared transmission line. Each contending node can destructively cancel its designated token frequency by injecting a $180^\circ$ out-of-phase signal. The resulting signal on the bus, after all cancellations, directly encodes which tokens have been "stolen" (i.e., which nodes are competing).

At each node—including the home node—real-time demodulation and thresholding of each carrier yields the arbitration vector $r$, where $r_j = 0$ if token $j$ was canceled, $r_j = 1$ otherwise. This vector is identical at all nodes and allows all participants to acquire global, up-to-date arbitration information after a single round-trip at near-light speed.

This approach sharply contrasts with sequential or multi-round protocols by leveraging signal orthogonality, frequency selectivity, and speed-of-light propagation, enabling sub-nanosecond, fully parallel arbitration for up to $\sim$100 concurrent nodes within the available bus bandwidth and SNR limits (Li, 2017).

2. Arbitration Vectors in Adaptive Model Aggregation

In the context of adaptive ensembling, notably Synapse for time series foundational models, the arbitration vector is the weight vector $\mathbf{w}_t = (w_{1,t}, \ldots, w_{N,t})^\top$, dynamically assigning credence to each model $M_i$ at forecast step $t$ (Das et al., 7 Nov 2025). Each component $w_{i,t} \geq 0$, with $\sum_i w_{i,t} = 1$, reflects the recent performance (e.g., average CRPS) of $M_i$ over a rolling window. This arbitration vector is recalculated adaptively, using inverse error or a softmax fallback in degenerate cases, with normalization ensuring probabilistic interpretation.

The outputs of the TSFMs are then combined via weighted predictive sampling: each $M_i$ is allocated $n_i$ samples proportional to $w_{i,t}$, creating a pooled predictive distribution from which arbitrary quantiles can be extracted. This approach preserves distributional properties and enables rigorous probabilistic forecasts, outperforming both static ensembles and best-single-model approaches, especially for long forecast horizons (Das et al., 7 Nov 2025).

Key Steps in Synapse’s Arbitration Vector Construction:

Step	Description	Mathematical Formulation
1	Compute rolling errors $s_{i,t}$	$$s_{i,t} = \frac{1}{W}\sum_{u=t-W}^{t-1} \text{CRPS}_{i,u}$$
2	Transform errors to raw weights $\tilde{w}_{i,t}$	$1/s_{i,t}$ or $\exp(-s_{i,t}/\tau)$ (softmax fallback)
3	Normalize to obtain arbitration vector $\mathbf{w}_t$	$w_{i,t} = \tilde{w}_{i,t} / \sum_j \tilde{w}_{j,t}$

3. Arbitration Vectors in Shared Autonomy and Human-Robot Interaction

In shared autonomy for teleoperation, the arbitration vector is commonly reduced to a scalar interpolation parameter $\alpha_t \in [0,1]$ representing the real-time blend between user and autonomous agent commands. More nuanced arbitration schemes aggregate multi-source uncertainties (e.g., intent inference, autonomy confidence) into a composite arbitration weight (Li et al., 2020). The arbitration weight is computed as a product of closed-form confidence functions derived from Bayesian or heuristic fusion of real-time sensory variances, yielding a smooth, interpretable transition between user and robot control.

An alternative approach based on machine-learned arbitration uses recurrent neural networks (LSTM) to output $\alpha_t$, conditioned on robot state, intent prediction, and user command, trained via data aggregation with hindsight labels to optimize seamless control allocation (Oh et al., 2019).

Blending Equation Example:

Shared-control action $a_s = \alpha_t a_r + (1-\alpha_t) a_u$ or via rotation-based interpolation. Here, $\alpha_t$ is learned or computed analytically to achieve efficient and user-friendly collaboration.

4. Arbitration Vectors in Blockchain Dispute Resolution Protocols

In blockchain-based interactive verification protocols, such as BoLD, the arbitration vector is a structured tuple recording both protocol state and cryptographic evidence (Alvarez et al., 2024). For BoLD, the arbitration vector at each node in the protocol graph is $(\ell, l_-, \ell_s, b, s)$, where $l_-$ and $\ell_s$ index the current state span, and $b$, $s$ denote Merkle roots of specific state intervals. Disputes are resolved through a series of deterministic bisections, with each bisection halving the disputed interval and creating two new arbitration vectors.

Unlike prior designs (e.g., Arbitrum Classic), where the arbitration vector could have dimension $O(k)$ and permit multiple dissection choices, the BoLD arbitration vector is dimension 2 (plus indices), admits exactly one valid bisection, and thus eliminates adversarial delay strategies. This deterministic, low-dimensional structure drastically reduces both the complexity and attack surface of the dispute protocol, ensuring fast (fixed to two challenge periods) and cost-efficient (linear in adversarial stake) on-chain arbitration (Alvarez et al., 2024).

5. Arbitration Vectors in Mathematical Finance and Portfolio Construction

In geometric arbitrage theory, the arbitrage vector $\boldsymbol{\alpha}(x,t)$, defined as the gradient of market drift and short rate with respect to portfolio weights,

$$\alpha_i(x,t) = \partial_{x^i}[D \log D(x,t) + r(x,t)],$$

measures the local arbitrage potential available in a stochastic principal geometric bundle formulation of financial markets (Farinelli et al., 2019). Nonzero $\boldsymbol{\alpha}$ encodes the instantaneous possible gain by infinitesimal rebalancing, with zero curvature (i.e., $\boldsymbol{\alpha} = 0$ everywhere) corresponding to no-arbitrage equilibrium.

In relative arbitrage theory and stochastic portfolio theory, the term "arbitrage vector" refers to the portfolio map $\pi:\Delta^n \to \Delta^n$ assigning allocations to each market weight simplex, constructed via functionally generated portfolios derived from a concave generating function $G$ (Pal et al., 2014). These portfolios, by virtue of satisfying multiplicative cyclical monotonicity, guarantee outperformance over the market under diversity and volatility conditions without reliance on parametric assumptions.

6. Arbitration Vectors in Distributed Machine Learning and Device Arbitration

Device arbitration, as in smart home multi-microphone scenarios, utilizes an arbitration vector composed of per-device neural embeddings, aggregated through permutation-equivariant functions (e.g., Deep-Sets) and subsequent MLP scoring to determine the "winning" device (Barber et al., 2021). Each device's signal is embedded as $z_j$, with global context $c = \sum_j z_j$, and arbitration logits $\ell_j$ are computed as $g(z_j, c)$.

This approach enables the arbitration vector to encode complex multi-device signal information in a learnable, permutation-invariant manner, providing substantial accuracy gains over baseline energy methods in speaker localization and device arbitration.

7. Comparative Structure and Cross-Domain Characteristics

Despite significant domain-specific implementation differences, the various notions of arbitration vector share several structural traits:

• Dimensionality and Representation: Ranges from bit vectors and weightings (bus arbitration, ensembling) to geometric or functional portfolio maps (finance) and tupled Merkle-root commitments (blockchain).
• Purpose: Universally, the arbitration vector enables decentralized, interpretable, and/or efficient decision-making among concurrent agents, models, or claims.
• Computation and Adaptation: Construction may be based on analytic computations (wave interference, uncertainty models), online learning (model aggregation, shared autonomy), or cryptographic procedures (blockchain).
• Impact: Arbitration vectors directly mediate access, allocation, blending, or resolution, determining resource usage, performance, or protocol security.

Summary Table: Domains and Archetypes

Domain	Arbitration Vector Form	Role/Significance
NoC/Wave Arbitration	$r \in \{0,1\}^N$	Encodes token capture, parallelizes arbitration
Time Series Model Aggregation	$\mathbf{w}_t \in \Delta^N$	Dynamic model weighting in ensembling
Human-Robot Shared Control	$\alpha_t \in [0,1]$ (scalar/vector)	Real-time command blending (control allocation)
Blockchain Dispute Resolution	$(l_-, \ell_s, b, s)$ (tuple)	Witnesses state spans, enables trustless bisection
Geometric Arbitrage/Finance	$\boldsymbol{\alpha}(x,t)$, $\pi(\mu)$	Measures arbitrage, defines optimal portfolio
Distributed Device Arbitration	$(z_1,\ldots,z_N)$ (embeddings)	Aggregated for device selection/decision

These constructs highlight the universality and adaptability of the arbitration vector paradigm in orchestrating resolution, blending, or allocation tasks across complex technical domains.