By introducing a specified definition of the 
equilibrium values of two-person two-choice games, 
a non-zero-sum multistage arbitration game
is formulated and solved.  At each random offer 
$X_i, i=1, 2, \cdots, n$, comes up, two players must 
decide either to accept it terminating the game, 
or to reject it expecting that a larger random value 
may come up in the near future.  Arbitration comes 
in when they choose different choices.  Each player
aims to maximize the expected reward he can get.
It is shown that if $X_i$ is unifomly distributed 
in $[0, 1]$, then even when arbitration stands 
100 percent in favor of the accepting side, 
the advantage for the players is only one percent.
It is also shown that players are more advantageous 
when arbitration favors the rejecting side than when 
it favors the accepting side.