We provide a general construction of a rational secret-sharing
protocol in which the secret can be reconstructed in expected three rounds.
Our construction converts any rational secret-sharing protocol
to a protocol with an expected three-round reconstruction in a black-box manner.
Our construction works in synchronous but non-simultaneous channels,
and preserves a strict Nash equilibrium of the original protocol.
Combining with an existing protocol,
we obtain a rational secret-sharing protocol
that achieves a strict Nash equilibrium with the optimal coalition resilience
of $\ceil{\frac{n}{2}}-1$ for expected constant-round protocols,
where $n$ is the number of players.
Although the coalition resilience of $\ceil{\frac{n}{2}}-1$ is shown to be optimal
as long as we consider constant-round protocols,
we circumvent this limitation by considering players
who do not prefer to reconstruct \emph{fake} secrets.
By assuming such players,
we construct an expected constant-round protocol that achieves a strict Nash equilibrium
with coalition resilience of $n-1$.
We also extend our construction to a protocol that preserves \emph{immunity}
to unexpectedly behaving (or malicious) players.
Then we obtain a protocol that achieves a Nash equilibrium
with coalition resilience of $\ceil{\frac{n}{2}}-t-1$
in the presence of $t$ unexpectedly behaving players for any constant $t \geq 1$.
The same protocol also achieves a strict Nash equilibrium in the absence of malicious players.
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