Three models of a three-nation arms race are presented: an all-against-all model, a model in which two nations are allied against the third, and a model in which one of the nations has a secret alliance with the other two. In the all-against-all situation, the model predicts an infinitely explosive arms race as each country fears an alliance of the other two. In the two-sided alliance, the model predicts that there will be an mitial increase until the weaker side approaches parity, after which there will be a mutual deescalation. If one nation signs secret treaties with each of the others, then it immediately deescalates. The other two nations then enter a two-sided arms race in which there will be an initial increase until the weaker nation reaches parity followed by mutual deescalation. These models suggest that two-way arms races would not infinitely escalate if it were not for the fear of third party alliances.
