Right-sided bias in fallow deer terminating parallel walks: evidence for lateralization during a lateral display

► Explores the relationship between lateralization and parallel walking in ungulates. ► Investigates lateralization within game theoretic models of competition. ► Establishes that the decision to terminate parallel walks is highly lateralized. ► Shows that lateralization is related to body size in ungulates. ► Smaller less lateralized individuals are more likely to escalate to fighting.