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.