Broken Telephone: Analysis of a Reinforced Process

We consider the following $L$ player co-operative signaling game. Nature plays from the set ${0,0'}$. Nature's play is observed by Player 1 who then plays from the set ${1,1'}$. Player 1's play is observed by Player 2. Player 2 then plays from the set ${2,2'}$. Player 2's play is observed by player 3. This continues until Player L observes Player L-1's play. Player L then guesses Nature's play. If he guesses correctly, then all players win. We consider an urn scheme for this where each player has two urns, labeled by the symbols they observe. Each urn has balls of two types, represented by the two symbols the player controlling the urn is allowed to play. At each stage each player plays by drawing from the appropriate urn, with replacement. After a win each player reinforces by adding a ball of the type they draw to the urn from which it was drawn. We attempt to show that this type of urn scheme achieves asymptotically optimal coordination. A lemma remains unproved but we have good numerical evidence for it's truth.