Price of Anarchy for Graph Coloring Games with Concave Payoff
Lasse Kliemann, Elmira Shirazi Sheykhdarabadi, Anand Srivastav

TL;DR
This paper analyzes the price of anarchy in graph coloring games with complex concave payoff functions, providing tight bounds and new techniques for understanding how equilibrium efficiency varies with different payoff structures.
Contribution
It extends the analysis of the price of anarchy to a broader class of concave payoff functions in graph coloring games, offering new bounds and a novel technique for bounding inefficiency.
Findings
Upper bound of 2 on the price of anarchy for certain concave functions
Upper bound of 3 on the price of anarchy for general concave functions
Matching lower bounds established for specific cases
Abstract
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 to . A tight bound on the price of anarchy of 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 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 the…
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