  We consider an N-player multi-armed bandit game where each player chooses one
out of M arms for T turns. Each player has different expected rewards for the
arms, and the instantaneous rewards are independent and identically distributed
or Markovian. When two or more players choose the same arm, they all receive
zero reward. Performance is measured using the expected sum of regrets,
compared with an optimal assignment of arms to players that maximizes the sum
of expected rewards. We assume that each player only knows her actions and the
reward she received each turn. Players cannot observe the actions of other
players, and no communication between players is possible. We present a
distributed algorithm and prove that it achieves an expected sum of regrets of
near-O\left(\log T\right). This is the first algorithm to achieve a near order
optimal regret in this fully distributed scenario. All other works have assumed
that either all players have the same vector of expected rewards or that
communication between players is possible.
