Scrip is a generic term for any substitute for real currency; it can be converted into goods or services sold by the issuer. In the classic scrip system model, one agent is helped by another in return for one unit of scrip. In this paper, we present an upgraded model, the one-to-n scrip system, where users need to find n agents to accomplish a single task. We provide a detailed analytical evaluation of this system based on a game-theoretic approach. We establish that a nontrivial Nash equilibrium exists in such systems under certain conditions. We study the effect of n on the equilibrium, on the distribution of scrip in the system and on its performance. Among other results, we show that the system designer should increase the average amount of scrip in the system when n increases in order to optimize its efficiency. We also explain how our new one-to-n scrip system can be applied to foster cooperation in two privacy-enhancing applications.