Suppose that players I and II want to jointly employ
two secretaries successively one-by-one from a set of
$n$ applicants. Best ability of management (foreign language)
is wanted by I (II). We assume that these two kinds of abilities
are mutually independent for every applicant. Applicants
present themselves one-by-one sequentially. Facing
each applicant, each player chooses either to Accept
or to Reject. The game ends either when the second time
of choice-pair A--A happens getting the payoffs predetermined
by the game rule, or when $n-2$ applicants except the last
two are rejected. If choice-pair is A--R or R--A, then
arbitration comes in and forces players to take the
same choice as I's (II's) with probability $p\ (\bar{p}),
\frac12\leq p\leq 1$. Each player aims to maximize
the expected payoff he can get. Explicit solutions
are derived to this $n$-stage game, for the cases where
abilities of each applicant are observed as bivariate
random variables with full-information and with
no-information. Some numerical results are presented.