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 $B$.