Price of Anarchy for Graph Coloring Games with Concave Payoff

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.