The problem of analyzing the effect of privacy concerns on the behavior of selfish utility-maximizing agents has received much attention lately. Privacy concerns are often modeled by altering the utility functions of agents to consider also their privacy loss [4, 14, 20, 28]. Such privacy-aware agents prefer to take a randomized strategy even in very simple games in which non-privacy-aware agents play pure strategies. In some cases, the behavior of privacy-aware agents follows the framework of Randomized Response, a well-known mechanism that preserves differential privacy. Our work is aimed at better understanding the behavior of agents in settings where their privacy concerns are explicitly given. We consider a toy setting where agent A , in an attempt to discover the secret type of agent B , offers B a gift that one type of B agent likes and the other type dislikes. As opposed to previous works, B ’s incentive to keep her type a secret isn’t the result of “hardwiring” B ’s utility function to consider privacy, but rather takes the form of a payment between B and A . We investigate three different types of payment functions and analyze B ’s behavior in each of the resulting games. As we show, under some payments, B ’s behavior is very different than the behavior of agents with hardwired privacy concerns and might even be deterministic. Under a different payment, we show that B ’s BNE strategy does fall into the framework of Randomized Response.