$$ \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathbb{E}} \newcommand{\Csmooth}{\bar{C}} \newcommand{\Craw}{C} \newcommand{\tasktype}{\tau} \newcommand{\bps}{\mathrm{BPS}} $$

Security Analysis

Sybil Resistance

BPE’s Sybil resistance derives from two mechanisms: a minimum per-sink stake \(S_{\min}\) and a concave capacity cap \(\text{cap}(S) = \sqrt{S/u}\).

An attacker splitting total stake \(S\) into \(n\) identities, each with stake \(S/n\) (plus \(S_{\min}\) overhead per identity), achieves total capacity: $$ \begin{equation} C_{\text{total}}(n) = n \cdot \sqrt{\frac{S/n}{u}} = \sqrt{n} \cdot \sqrt{\frac{S}{u}} \end{equation} $$ at cost \(n \cdot S_{\min}\) in minimum stake overhead. The marginal capacity gain \(\partial C_{\text{total}} / \partial n = \frac{1}{2\sqrt{n}} \cdot \sqrt{S/u}\) is decreasing, while the marginal cost \(S_{\min}\) is constant. The attack becomes unprofitable when: $$ \begin{equation} S_{\min} > \frac{1}{2\sqrt{n}} \cdot \sqrt{S/u} \cdot p \end{equation} $$ where \(p\) is the payment rate per unit capacity.

We validate this analysis in simulation (Evaluation, Experiment E3).

Capacity Truthfulness

Under the BPE mechanism with slashing:

• Over-reporting: Declaring \(\Craw > C_{\text{true}}\) attracts payment \(F > C_{\text{true}}\). The excess accumulates as verifiable underperformance, triggering slashing. Expected payoff: \(p \cdot C_{\text{true}} - s \cdot (\Craw - C_{\text{true}})\) where \(s\) is the slash penalty per unit over-report.

• Under-reporting: Declaring \(\Craw < C_{\text{true}}\) reduces payment proportionally. Lost revenue: \(p \cdot (C_{\text{true}} - \Craw)\). No slashing risk.

For \(s > p\), truthful reporting is a dominant strategy: over-reporting is penalized more than the payment gained, and under-reporting sacrifices revenue.

MEV Resistance

The commit-reveal protocol prevents front-running of capacity updates:

1. Capacity values are hidden during the commit phase (only a hash is visible).

2. The 20-block reveal window limits timing attacks.

3. EWMA smoothing further dampens the impact of any single update, reducing the value of manipulating a single reveal.

An attacker observing a commit transaction learns only that some capacity update is coming, not its magnitude or direction. The EWMA ensures that even perfectly timed front-runs shift pool weights by at most \(\alpha\) (30%) of the capacity change.

Bayesian-Nash Incentive Compatibility

We now prove that truthful capacity reporting is a Bayesian-Nash equilibrium (BNE) under the combined mechanism of EWMA smoothing, dynamic pricing, and slashing.

Strategy space.

Each sink \(k\) with true capacity \(C_k\) chooses a reporting strategy \(\sigma_k \in \mathcal{S} = \{\text{truthful}, \text{over}(\varepsilon), \text{under}(\varepsilon)\}\) where \(\varepsilon > 0\) is the deviation magnitude. The reported capacity is: $$ \begin{equation} \hat{C}_k = \begin{cases} C_k & \text{if } \sigma_k = \text{truthful}, \ C_k + \varepsilon & \text{if } \sigma_k = \text{over}(\varepsilon), \ C_k - \varepsilon & \text{if } \sigma_k = \text{under}(\varepsilon). \end{cases} \end{equation} $$

Payoff function.

Let \(p(\tasktype, k, t)\) be the price from Eq. (Price), \(F_k\) the flow rate routed to sink \(k\) (proportional to smoothed capacity share), \(s\) the slashing penalty rate, and \(T_d\) the expected detection time for overperformance monitoring. The per-epoch expected payoff for sink \(k\) is: $$ \begin{equation} \label{eq:payoff} \pi_k(\sigma_k, \boldsymbol{\sigma}_{-k}) = p(\tasktype, k) \cdot \min(F_k, C_k) - s \cdot \max(F_k - C_k, 0) \cdot \mathbf{1}[t > T_d] \end{equation} $$ where \(F_k \propto \Csmooth(k) / \sum_j \Csmooth(j)\) is the flow share and \(\mathbf{1}[t > T_d]\) indicates that slashing activates after detection.

Proposition (Truthful BNE)

Under the BPE mechanism with slashing rate \(s\), price \(p\), EWMA parameter \(\alpha\), and detection time \(T_d\), truthful reporting \(\sigma_k = \text{truthful}\) is a Bayesian-Nash equilibrium for all sinks when: $$ \begin{equation} \label{eq:bne-condition} s \cdot S_k > p \cdot \varepsilon \cdot T_d \end{equation} $$ where \(S_k\) is sink \(k\)’s stake, for all profitable deviations \(\varepsilon > 0\).

Proof

Proof. We compare each deviation against truthful reporting.

Case 1: Over-reporting (\(\hat{C}_k = C_k + \varepsilon\)). After EWMA smoothing, the effective capacity increase is \(\alpha \varepsilon\) per update. This attracts additional flow \(\Delta F \propto \alpha \varepsilon\). For epochs \(t \leq T_d\), the excess flow \(\Delta F\) exceeds true capacity, accumulating unmet demand. After \(T_d\), the completion tracker detects the shortfall: completion rate \(r_k < 0.5\) triggers slashing of \(s_{\text{slash}} = 10\%\) of stake per 3 consecutive failing epochs. The expected gain from over-reporting over the detection window is at most \(p \cdot \alpha \varepsilon \cdot T_d\). The expected loss from slashing is \(s \cdot S_k\) (where \(s = s_{\text{slash}}\)). By condition Eq. (BNE Condition), the loss exceeds the gain, making over-reporting unprofitable.

Case 2: Under-reporting (\(\hat{C}_k = C_k - \varepsilon\)). No slashing risk, but the sink receives reduced flow \(\Delta F \propto -\alpha \varepsilon\), forfeiting revenue \(p \cdot \alpha \varepsilon\) per epoch indefinitely. Since this is strictly worse than truthful reporting, under-reporting is dominated.

In both cases, no unilateral deviation improves expected payoff, establishing truthful reporting as a BNE. ◻

Parameterization.

With the deployed parameters (\(s_{\text{slash}} = 10\%\) of stake, \(T_d = 3 \times 300\text{s} = 900\text{s}\), minimum stake \(S_{\min}\)), condition Eq. (BNE Condition) holds when \(0.1 \cdot S_{\min} > p \cdot \varepsilon \cdot 900\). For typical parameters (\(S_{\min} = 1000\) tokens, \(p = 10^{-3}\) tokens/unit/s), this bounds profitable over-reporting at \(\varepsilon < 111\) units—far below practical capacity ranges, confirming incentive compatibility for realistic deviations.