Here we present proofs that are omitted throughout this article.  The proofs may look difficult, but we used only high school level of combinatorics and mathematical induction.
The proof of Lemma 2.
 
These equations are direct from Definition 2.
 
(1)
  
                                                            
 
 
 
 
 
(2)                                                            
 
 
 
(3)                                                                          
 
 
 
 
 
Remark. Lemma 1, 2 and Theorem 1 were proved by the authors and published in Reference [9].
 
 
 
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The proof of Lemma 1.
 
If the game is to end in the y-th round, the players have to continue the game without becoming the loser from the first round to the (y-1) -th round. The probability to
 
survive the first to the (y-1) -th round are  
 
respectively and the probability of one of the players loses in the y -th round is      
 
              .
 
 
Therefore we have (1) of Lemma 1.
 
 
 
 
The proof of Theorem 1.
 
Clearly the number of rounds is n-m+1 at most. The v -th player plays the v -th, v+p -th, v+2p -th ... v+(t-1)p -th round, where t is the biggest natural number that satisfies v+(t-1)p ≤ n-m+1 and hence
 
 
                                                            
 
Therefore by Lemma 1 we have
 
                                                                            
 
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