Two players observe a stochastic stream of offers. 
They arrive via continuous-time simple Markov process 
in the time interval $[0, 1]$ and their arrival becomes 
less probable as time passes. At each arrival, players 
must decide either accept (A) or reject (R), immediately 
and independently. Players have equal weights, 
so if both players want to accept a same offer, 
a lottery is used to the effect that each player can 
get it with equal probability 1/2. If one player 
accepts an offer and the other doesn't the game goes on 
as one-person game for the latter. A player wins if he 
accepts the latest offer arriving before time 1. 
Each player aims to find his strategy that maximizes 
his probability of winning. The normal form of the game 
is formulated and the structural form of the solution to 
this game is found to calculate the Nash equilibrium. 
Another best-choice games where each player aims to accept 
an offer later than the opponent is also discussed.