  We study the price of anarchy in a class of graph coloring games (a subclass
of polymatrix common-payoff games). In those games, players are vertices of an
undirected, simple graph, and the strategy space of each player is the set of
colors from $1$ to $k$. A tight bound on the price of anarchy of
$\frac{k}{k-1}$ is known (Hoefer 2007, Kun et al. 2013), for the case that each
player's payoff is the number of her neighbors with different color than
herself. The study of more complex payoff functions was left as an open
problem.
  We compute payoff for a player by determining the distance of her color to
the color of each of her neighbors, applying a non-negative, real-valued,
concave function $f$ to each of those distances, and then summing up the
resulting values. This includes the payoff functions suggested by Kun et al.
(2013) for future work as special cases.
  Denote $f^*$ the maximum value that $f$ attains on the possible distances
$0,\dots,k-1$. We prove an upper bound of $2$ on the price of anarchy for
concave functions $f$ that are non-decreasing or which assume $f^*$ at a
distance on or below $\lfloor\frac{k}{2}\rfloor$. Matching lower bounds are
given for the monotone case and for the case that $f^*$ is assumed in
$\frac{k}{2}$ for even $k$. For general concave functions, we prove an upper
bound of $3$. We use a simple but powerful technique: we obtain an upper bound
of $\lambda \geq 1$ on the price of anarchy if we manage to give a splitting
$\lambda_1 + \dots + \lambda_k = \lambda$ such that $\sum_{s=1}^k \lambda_s
\cdot f(|s-p|) \geq f^*$ for all $p \in \{1,\dots,k\}$. The discovery of
working splittings can be supported by computer experiments. We show how, once
we have an idea what kind of splittings work, this technique helps in giving
simple proofs, which mainly work by case distinctions, algebraic manipulations,
and real calculus.
