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$A$ and B$B$ be two non-empty finite disjoint sets of players. Any two players in A$A$ are "matched" and $\\\$2$ are transferred from one to the other. Any player in A$A$ is also matched with any player in B$B$ 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$B$. 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$|A|=1$ and |B|=2$|B|=2$ the probability of winning for the player in A$A$ is 1/4$1/4$ and the probability for the player in B$B$ is 3/8$3/8$. If |A|=|B|=2$|A|=|B|=2$ the corresponding numbers are 21/64$21/64$ and 11/64$11/64$). Claim: If |AUB|>3 then the probability of winning for any player in A is strictly larger than that of any player in B.
Claim: If $|AUB|>3$ then the probability of winning for any player in $A$ is strictly larger than that of any player in $B$.
