We present a general construction of a rational secret-sharing protocol
that
converts any rational secret-sharing protocol to a protocol with an expected constant-round reconstruction.
Our construction can be applied to protocols for synchronous channels,
and preserves a strict Nash equilibrium of the original protocol.
Combining with an existing protocol,
we obtain the first expected constant-round protocol
that achieves a strict Nash equilibrium with the optimal coalition resilience
$\ceil{\frac{n}{2}}-1$, where $n$ is the number of players.
Our construction can be extended to a construction that preserves the \emph{immunity} to unexpectedly behaving players.
Then, for any constant $m \geq 1$, we obtain an expected constant-round protocol that achieves a Nash equilibrium
with the optimal coalition resilience $\ceil{\frac{n}{2}}-m-1$
in the presence of $m$ unexpectedly behaving players.
The same protocol also achieves a strict Nash equilibrium with coalition resilience $1$.
We show that our protocol with immunity achieves the optimal coalition resilience
among constant-round protocols with immunity with respect to both Nash and strict Nash equilibria.
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