TITLE: Probability to be the winner in a tournament
QUESTION [14 upvotes]: In a project in Game Theory we (Ayala Arad and Ariel Rubinstein) are stuck with the following "simple" question. We are sure of the conjecture but we failed to find a (hopefully simple) proof: 
Let $A$ and $B$ be two non-empty finite disjoint sets of players.  Any two players in $A$ are "matched" and $\$2$ are transferred from one to the other.  Any player in $A$ is also matched with any player in $B$ and $\$1$ is transferred from one to the other.  The two possible directions of each transfer are equally likely and independent. No transfers are carried out between players in $B$.
    The winner is the player with the highest net transfers. In the case of a tie, the winner is selected randomly from among the highest scoring players. 
(For example if $|A|=1$ and $|B|=2$ the probability of winning for the player in $A$ is $1/4$ and the probability for the player in $B$ is $3/8$.  If $|A|=|B|=2$ the corresponding numbers are $21/64$ and $11/64$).

Claim:  If $|AUB|>3$ then the probability of winning for any player in $A$ is strictly larger than that of any player in $B$.

REPLY [3 votes]: This was intended to be a comment to Ori's post but it is too long, so I'm posting it as an answer. First of all, let us modify the game a bit by initially giving each player a random score between $0$ and $\varepsilon$. That will break the ties just as needed but will allow us to talk about the winner. 
Now the case $|A|=|B|$ is trivial. Let's do all transactions between $A$ and $B$ first and look at the resulting configurations. They split into natural pairs (swapping $A$ and $B$). Now let $a$ be the top score in $A$ and $b$ be the top score in $B$. Arrange the pair so that $a>b$. Then we need to show that for every configuration the probability that the top score in $A$ will become less than $b$ after transactions in $A$ is less than that the probability that the top score in $B$ will become larger than $a$ if we do the transactions in $B$. Identify $A$ with $B$ in some way so that the top scorers are identified. Any way to do the transactions in $A$ that moves the winner to $B$ should bring the score $a$ of the top scorer in $A$ below $b$ at the very least and that may be insufficient in some configurations. On the other hand, if have one such way and do the inverse transactions in $B$ instead, they'll bring the top scorer in $B$ above $a$ and it is not necessary to move the winner to $B$. That's all one needs to say about the equal cardinalities case. 
Now, like Ori, I have to say that I'll try to give a more complete answer later.