In this paper, we describe a new restricted randomization method called run-reversal equilibrium (RRE), which is a Nash equilibrium of a game where (1) the clinical trial statistician chooses a sequence of medical treatments, and (2) clinical investigators make treatment predictions. RRE randomization counteracts how each investigator could observe treatment histories in order to forecast upcoming treatments. Computation of a run-reversal equilibrium reflects how the treatment history at a particular site is imperfectly correlated with the treatment imbalance for the overall trial. An attractive feature of RRE randomization is that treatment imbalance follows a random walk at each site, while treatment balance is tightly constrained and regularly restored for the overall trial. Less predictable and therefore more scientifically valid experiments can be facilitated by run-reversal equilibrium for multi-site clinical trials.