Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 114, pp. 1-11. Title: p-harmonious functions with drift on graphs via games Author: Alexander P. Sviridov (Univ. of Pittsburgh, PA, USA) Submitted October 26, 2010. Published September 6, 2011. Math Subject Classifications: 35Q91, 35B51, 34A12, 31C20. Key Words: Dirichlet problem; comparison principle; mean-value property; stochastic games; unique continuation.