SELECT battle.match_id
     , winner.screen_name AS winner_name -- <- really player.X
     , loser.screen_name  AS loser_name  -- <- really player.X
FROM   battle
     , player AS winner   -- <---------------- see? same table
     , player AS loser    -- <---------------- used twice
WHERE  battle.winner = winner.player_id
  AND  battle.loser  = loser.player_id;