To incentivize users' participations and steer their online activities, online social networks start to provide users with various kinds of rewards for their contributions to the sites. The most frequently distributed rewards include account levels, reputation scores, different kinds of badges, and even material awards like small gifts and cash back, etc. Attracted by these rewards, users will spend more time using the network services. In this paper, we will mainly focus on "badges reward systems" but the proposed models can be applied to other reward systems as well. Badges are small icons attached to users' homepages and profiles denoting their achievements. People like to accumulate badge for various reasons and different badges can have specific values for them. Meanwhile, to get badges, they also need to exert efforts to finish the required tasks, which can lead to certain costs. To understand and model users' motivations in badge achievement activities, we will study an existing badge system launched inside a real-world online social network, Foursquare, in this paper. At the same time, to maximize users' contributions to online social networks, social network system designers need to determine the optimal badge system mechanism carefully. Badge system mechanism describes various detailed aspects of the system and can involve many parameters, e.g., categories of existing badges, number of badges available as well as the minimum contributions required to obtain the badges, which all need to be designed with meticulous investigations. Based on the model of users' badges accumulating activities, in this paper, we will also study how to design the badge system that can incentivize the maximum users' contributions to the social networks.