<p>This paper studies the risk-sensitive average reward criterion for the semi-Markov game with compact state and action spaces. Under some suitable conditions (slightly weaker than the existing ones), we introduce a parametric operator, verify that the corresponding spectral radius is an eigenvalue of it by the nonlinear Krein-Rutman theorem, and further show the continuity of the spectral radius in the parameters. By the continuity and the intermediate value property, we prove that the Shapley equation admits a non-trivial solution, and then establish the existence of the value and a stationary saddle point. Furthermore, we present an iteration algorithm for computing (at least approximating) the value of the game. Finally, we give two examples to illustrate our conditions and algorithm.</p>