Retweeting is a featured mechanism of some social media platforms such as Twitter, Facebook, and Weibo. Users share articles with friends or followers by reposting a tweet. However, the ways in which retweeting affects the dominant behaviors of users is still unclear. Therefore, we investigate the influence of retweeting on the behaviors of social media users from a networked, game theoretic perspective; in other words, we attempt to clarify the ways in which the presence of a retweeting mechanism in social media promotes or diminishes the willingness of users toward posting articles and commenting. We propose a retweet reward game model that has been derived by adding a retweeting mechanism to a reward game, which is a simple social networking service model. Subsequently, we conduct some simulation-based experiments to understand the effects of retweeting on the behaviors of users. We observe that users are motivated to post new articles if there is a retweeting mechanism. Furthermore, agents in dense networks are motivated to comment on the articles posted by others because articles spread widely among users, and thus, users can be incentivized to post articles.