$$ \newcommand \Bundle {\mathrm{Bundle}} $$

Bundles

On receiving a bundle \( \Bundle(r_k, p_k, s_k, v) \) a player

• Ignores* it if \( Bundle(r_k, p_k, s_k, v) \) is malformed or trivially invalid.

• Ignores it if

  • \( r_k \neq r \) or
  • \( r_k = r \) and \( p_k + 1 < p \).

• Otherwise, observes the votes in \( \Bundle(r_k, p_k, s_k, v) \) in sequence. If there exists a vote that causes the player to observe some bundle \( \Bundle(r_k, p_k, s_k, v’) \) for some \( s_k \), then the player relays \( \Bundle(r_k, p_k, s_k, v’) \), and then executes any consequent action; if there does not, the player ignores it.

Specifically, if the player ignores the bundle without observing its votes, then

$$ N(S, L, \Bundle(r_k, p_k, s_k, v)) = (S, L, \epsilon); $$

while if a player ignores the bundle but observes its votes, then

$$ N(S, L, \Bundle(r_k, p_k, s_k, v)) = (S’ \cup \Bundle(r_k, p_k, s_k, v), L, \epsilon); $$

and if a player, on observing the votes in the bundle, observes a bundle for some value (not necessarily distinct from the bundle’s value), then

$$ N(S, L, \Bundle(r_k, p_k, s_k, v)) = (S’ \cup \Bundle(r_k, p_k, s_k, v), L’, (\Bundle^\ast(r_, p_k, s_k, v’), \ldots)). $$