Evolutionary conflict and arms races are important drivers of evolution in nature. During arms races, new abilities in one party select for counterabilities in the second party. This process can repeat and lead to successive fixations of novel mutations, without a long-term increase in fitness. Models of co-evolution rarely address successive fixations, and one of the main models that use successive fixations-Fisher\'s geometric model-does not address co-evolution. We address this gap by expanding Fisher\'s geometric model to the evolution of joint phenotypes that are affected by two parties, such as probability of infection of a host by a pathogen. The model confirms important intuitions and offers some new insights. Conflict can lead to long-term Sisyphean arms races, where parties continue to climb toward their fitness peaks, but are dragged back down by their opponents. This results in far more adaptive evolution compared to the standard geometric model. It also results in fixation of mutations of larger effect, with the important implication that the common modeling assumption of small mutations will apply less often under conflict. Even in comparison with random abiotic change of the same magnitude, evolution under conflict results in greater distances from the optimum, lower fitness, and more fixations, but surprisingly, not larger fixed mutations. We also show how asymmetries in selection strength, mutation size, and mutation input allow one party to win over another. However, winning abilities come with diminishing returns, helping to keep weaker parties in the game.

Multiple organisms can sometimes affect a common phenotype. For example, the portion of a leaf eaten by an insect is a joint phenotype of the plant and insect and the amount of food obtained by an offspring can be a joint trait with its mother. Here, I describe the evolution of joint phenotypes in quantitative genetic terms. A joint phenotype for multiple species evolves as the sum of additive genetic variances in each species, weighted by the selection on each species. Selective conflict between the interactants occurs when selection takes opposite signs on the joint phenotype. The mean fitness of a population changes not just through its own genetic variance but also through the genetic variance for its fitness that resides in other species, an update of Fisher\'s fundamental theorem of natural selection. Some similar results, using inclusive fitness, apply to within-species interactions. The models provide a framework for understanding evolutionary conflicts at all levels.