This paper aims at modeling and inferring the influence among individuals from voting data (or more generally from actions that are selected by choosing one of m different options). The voting data are modeled as outcomes of a discrete random process, that we refer to as the discuss-then-vote model, whose evolution is governed by the DeGroot opinion dynamics with stubborn nodes. Based on the proposed model, we formulate the maximum-a-posterior estimator for the opinions and influence matrix (or the transition matrix) and derive a tractable approximation that results in a convex optimization problem. In the paper, the identifiability of the network dynamics' parameters and the vote prediction procedure based on the influence matrix, are discussed in depth. Our methodology is tested through numerical simulations as well as through its application to a set of the United States Senate roll call data. Interestingly, in spite of the relatively small data record available, the influence matrix inferred from the real data is consistent with the common intuition about the influence structure in the US Senate.