Who wins? Painter's Servant or Mycosynth Lattice?
The most recently played one takes precidence:

From the rulings on painter's servent:


  5/1/2008  While Painter's Servant is on the
  battlefield, an effect that changes an object's colors will overwrite
  Painter's Servant's effect. For example, casting Cerulean Wisps on a
  creature will turn it blue, regardless of the color chosen for
  Painter's Servant. 
  
  5/1/2008   If something affected by
  Painter's Servant is normally colorless, it will simply be the new
  color. It won't be both the new color and colorless.


In general continuous effects are applied in time stamp order (rule 613. Interaction of Continuous Effects). More info here: http://wizards.custhelp.com/app/answers/detail/a_id/1030