  Recently, Shi et al. (Phys. Rev. A, 2015) proposed Quantum Oblivious Set
Member Decision Protocol (QOSMDP) where two legitimate parties, namely Alice
and Bob, play a game. Alice has a secret $k$ and Bob has a set
$\{k_1,k_2,\cdots k_n\}$. The game is designed towards testing if the secret
$k$ is a member of the set possessed by Bob without revealing the identity of
$k$. The output of the game will be either "Yes" (bit $1$) or "No" (bit $0$)
and is generated at Bob's place. Bob does not know the identity of $k$ and
Alice does not know any element of the set. In a subsequent work (Quant. Inf.
Process., 2016), the authors proposed a quantum scheme for Private Set
Intersection (PSI) where the client (Alice) gets the intersected elements with
the help of a server (Bob) and the server knows nothing. In the present draft,
we extended the game to compute the intersection of two computationally
indistinguishable sets $X$ and $Y$ possessed by Alice and Bob respectively. We
consider Alice and Bob as rational players, i.e., they are neither "good" nor
"bad". They participate in the game towards maximizing their utilities. We
prove that in this rational setting, the strategy profile $((cooperate, abort),
(cooperate, abort)$) is a strict Nash equilibrium. If $((cooperate, abort),
(cooperate, abort)$) is strict Nash, then fairness as well as correctness of
the protocol are guaranteed.
