$$ \newcommand \Vote {\mathrm{Vote}} \newcommand \Late {\mathit{late}} \newcommand \Down {\mathit{down}} \newcommand \Next {\mathit{next}} $$
Votes
On receiving a vote \( \Vote_k(r_k, p_k, s_k, v) \) a player
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Ignores* it if \( \Vote_k \) is malformed or trivially invalid.
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Ignores it if \( s = 0 \) and \( \Vote_k \in V \).
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Ignores it if \( s = 0 \) and \( \Vote_k \) is an equivocation.
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Ignores it if \( s > 0 \) and \( \Vote_k \) is a second equivocation.
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Ignores it if
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\( r_k \notin [r,r+1] \) or
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\( r_k = r + 1 \) and either
- \( p_k > 0 \) or
- \( s_k \in (\Next_0, \Late) \) or
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\( r_k = r \) and one of
- \( p_k \notin [p-1,p+1] \) or
- \( p_k = p + 1 \) and \( s_k \in (\Next_0, \Late) \) or
- \( p_k = p \) and \( s_k \in (\Next_0, \Late) \) and \( s_k \notin [s-1,s+1] \) or
- \( p_k = p - 1 \) and \( s_k \in (\Next_0, \Late) \) and \( s_k \notin [\bar{s}-1,\bar{s}+1] \).
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Otherwise, relays \( \Vote_k \), observes it, and then produces any consequent output.
Specifically, if a player ignores the vote, then
$$ N(S, L, \Vote_k(r_k, p_k, s_k, v)) = (S, L, \epsilon) $$
while if a player relays the vote, then
$$ N(S, L, \Vote_k(r_k, p_k, s_k, v)) = (Sā \cup \Vote(I, r_k, p_k, s_k, v), Lā, (\Vote_k^\ast(r_k, p_k, s_k, v),\ldots)). $$