People must integrate disparate sources of information when making decisions, especially in social contexts. But information does not always flow freely. It can be constrained by social networks and distorted by zealots and automated bots. Here we develop a voter game as a model system to study information flow in collective decisions. Players are assigned to competing groups (parties) and placed on an 'influence network' that determines whose voting intentions each player can observe. Players are incentivized to vote according to partisan interest, but also to coordinate their vote with the entire group. Our mathematical analysis uncovers a phenomenon that we call information gerrymandering: the structure of the influence network can sway the vote outcome towards one party, even when both parties have equal sizes and each player has the same influence. A small number of zealots, when strategically placed on the influence network, can also induce information gerrymandering and thereby bias vote outcomes. We confirm the predicted effects of information gerrymandering in social network experiments with n = 2,520 human subjects. Furthermore, we identify extensive information gerrymandering in real-world influence networks, including online political discussions leading up to the US federal elections, and in historical patterns of bill co-sponsorship in the US Congress and European legislatures. Our analysis provides an account of the vulnerabilities of collective decision-making to systematic distortion by restricted information flow. Our analysis also highlights a group-level social dilemma: information gerrymandering can enable one party to sway decisions in its favour, but when multiple parties engage in gerrymandering the group loses its ability to reach consensus and remains trapped in deadlock.