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.