We consider the problem of using logged data to make predictions about what would happen if we changed the `rules of the game' in a multi-agent system. This task is difficult because in many cases we observe actions individuals take but not their private information or their full reward functions. In addition, agents are strategic, so when the rules change, they will also change their actions. Existing methods (e.g. structural estimation, inverse reinforcement learning) assume that agents' behavior comes from optimizing some utility or that the system is in equilibrium. They make counterfactual predictions by using observed actions to learn the underlying utility function (a.k.a. type) and then solving for the equilibrium of the counterfactual environment. This approach imposes heavy assumptions such as the rationality of the agents being observed and a correct model of the environment and agents' utility functions. We propose a method for analyzing the sensitivity of counterfactual conclusions to violations of these assumptions, which we call robust multi-agent counterfactual prediction (RMAC). We provide a first-order method for computing RMAC bounds. We apply RMAC to classic environments in market design: auctions, school choice, and social choice.