Nowadays how to provide efficient on-demand data access has attracted more and more attentions in wireless networks. Aiming at this problem, based on broadcasting nature of wireless channels and multiuser resource allocation, game theory is considered to be one effective approach to deal with wireless data access. Recently Yeung et. al  proposed a method based on game theory. However, it did not consider the user request collisions, which cannot be neglected in wireless access. In this paper, we employ the similar idea of the game theory and propose an new approach, in which the effect of collisions is taken into account. To quantify the performance gain of our developed approach compared with that in , we present the theoretical analysis of its Nash equilibrium and evaluate its performance by numerical analysis and simulations. It indicates that our developed approach has about 5% performance gain in terms of the total desired data quantity for all clients, average lifetime and the average request number with a little complexity cost.