Audit games are an important variant of the Stackelberg security game, a widely studied game-theoretic model over the past years. It has been acknowledged that a pre-audit phase can notably enhance the audit's efficiency by informing and directing the following audit procedures. In this paper, we model the above process with a two-stage audit game. The game encompasses two stages: an investigation stage where the auditor gathers information about potential policy breaches, and an audit stage where the auditor allocates the audit resources based on the investigation results. We formulate the problem as a set of mathematical programs. Due to the non-convexity of the programs, we consider a restricted strategy space and show that the optimal strategy in the restricted space can be determined by solving a polynomial number of convex optimization problems. Finally, we conduct extensive experiments to evaluate the effect of introducing the initial investigation stage and our algorithm. Our experiments show that even a small budget for the initial investigations can significantly enhance the defender's utility.

Consider a market with a seller and many buyers. The seller has a kind of item for sale to the buyers. The items have a quality and each buyer has a private type. The quality is only known to the seller, and the buyers only have a prior belief of the quality. A third party (e.g., intermediaries or product reviewers) is able to reveal information about the actual quality by using a so-called signaling scheme. After receiving the information, buyers can update their beliefs accordingly and decide whether to buy the items. We consider the third party's problem of maximizing the purchasing probability by sending signals. However, the optimal signaling scheme has implementation issues, as the number of signals in the optimal scheme is the same as the number of buyer types, which can be exceedingly large or even infinite. We therefore investigate whether a finite and limited set of signals could still approximate the performance of the optimal signaling scheme. Unfortunately, our results show that with a finite number of signals, no signaling scheme can achieve a certain fraction of the performance of the optimal signaling scheme. This limitation persists even with the regularity or the monotone hazard rate assumption. Nevertheless, we identify a mild technical condition under which the third party can approximate the optimal performance within a constant factor by employing only two signals. We also conduct extensive experiments to substantiate our theoretic results. These experiments compare the performance of using a small signal set across different value distributions. Despite the negative results, our experiment results show that using only a small number of signals is able to achieve a fairly reasonable performance in average cases.