  In two-player games on graph, the players construct an infinite path through
the game graph and get a reward computed by a payoff function over infinite
paths. Over weighted graphs, the typical and most studied payoff functions
compute the limit-average or the discounted sum of the rewards along the path.
Beside their simple definition, these two payoff functions enjoy the property
that memoryless optimal strategies always exist.
  In an attempt to construct other simple payoff functions, we define a class
of payoff functions which compute an (infinite) weighted average of the
rewards. This new class contains both the limit-average and discounted sum
functions, and we show that they are the only members of this class which
induce memoryless optimal strategies, showing that there is essentially no
other simple payoff functions.
