papers/lux-wave-protocol/sections/09-comparison-with-slush-and-flare.tex

\section{Comparison with Slush and Flare}

\label{sec:comparison}

\subsection{Slush: No Memory}

Slush has no counters---it simply adopts the sampled majority in each round. While Slush
converges in $O(\log n)$ rounds, it provides no finality guarantee. An adversary can
continually perturb a Slush node by strategically timing equivocal responses.

\subsection{Flare: Consecutive Counters}

Flare tracks consecutive rounds of supermajority agreement. A counter $cnt$ increments
when the supermajority matches the current preference and resets to 0 on a preference
switch. Flare finalizes when $cnt \geq \beta_2$. While more robust than Slush, Flare
is vulnerable to \emph{counter reset attacks}: an adversary who can occasionally force
a preference switch resets the counter, potentially delaying finalization indefinitely.

\subsection{Wave: Cumulative Advantage}

Wave's confidence counters are cumulative---they never reset. Even if the adversary
temporarily shifts some honest nodes, the historical confidence lead is preserved.
This makes Wave strictly more robust than Flare against adaptive adversaries:

\begin{proposition}[Wave Dominates Flare]
\label{prop:dominance}
For any adversary strategy $\sigma$ and any initial configuration, the expected
finalization time under Wave is at most the expected finalization time under Flare.
\end{proposition}

\begin{proof}
Flare finalizes when $\beta_2$ \emph{consecutive} supermajority rounds occur. Wave
finalizes when $\beta_1$ \emph{total} supermajority rounds occur (for the leading color).
Since $\beta_1$ total rounds of supermajority for a fixed color always occur before
$\beta_2$ consecutive rounds (as consecutive is a stricter condition), and Wave needs
only cumulative rather than consecutive counts, Wave's finalization time is stochastically
dominated by Flare's for any $\beta_1 \leq \beta_2$.
\end{proof}

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