Fix a bounded domain $\Omega \subset {\mathbb R}^d$ , a continuous function $F:\partial \Omega \rightarrow {\mathbb R}$ , and constants $\epsilon >0$ and $1 \lt p,q \lt \infty$ with $p^{-1} + q^{-1} = 1$ . For each $x \in \Omega$ , let $u^\epsilon(x)$ be the value for player I of the following two-player, zero-sum game. The initial game position is $x$ . At each stage, a fair coin is tossed, and the player who wins the toss chooses a vector $v \in \overline{B}(0,\epsilon)$ to add to the game position, after which a random noise vector with mean zero and variance $({q}/{p})|v|^2$ in each orthogonal direction is also added. The game ends when the game position reaches some $y \in\partial \Omega$ , and player I's payoff is $F(y)$ .
¶ We show that (for sufficiently regular $\Omega$ ) as $\epsilon$ tends to zero, the functions $u^\epsilon$ converge uniformly to the unique $p$ -harmonic extension of $F$ . Using a modified game (in which $\epsilon$ gets smaller as the game position approaches $\partial \Omega$ ), we prove similar statements for general bounded domains $\Omega$ and resolutive functions $F$ .
¶ These games and their variants interpolate between the tug-of-war games studied by Peres, Schramm, Sheffield, and Wilson [15], [16] ( $p=\infty$ ) and the motion-by-curvature games introduced by Spencer [17] and studied by Kohn and Serfaty [9] ( $p=1$ ). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about $p$ -capacity and $p$ -harmonic measure