Understanding the rise, spread, and fall of large-scale states in the ancient world has occupied thinkers for millennia. However, no comprehensive mechanistic model of state dynamics based on their insights has emerged, leaving it difficult to evaluate empirically or quantitatively the different explanations offered. Here I present a spatially- and temporally-resolved agent-based model incorporating several hypotheses about the behavior of large-scale (>200 thousand km2) agrarian states and steppe nomadic confederations in Afro-Eurasia between the late Bronze and the end of the Medieval era (1500 BCE to 1500 CE). The model tracks the spread of agrarian states as they expand, conquer the territory of other states or are themselves conquered, and, occasionally, collapse. To accurately retrodict the historical record, several key contingent regional technological advances in state military and agricultural efficiencies are identified. Modifying the location, scale, and timing of these contingent developments allows quantitative investigation of historically-plausible alternative trajectories of state growth, spread, and fragmentation, while demonstrating the operation and limits of the model. Under nominal assumptions, the rapid yet staggered increase of agrarian state sizes across Eurasia after 600 BCE occurs in response to intense military pressure from 'mirror' steppe nomadic confederations. Nevertheless, in spite of various technological advances throughout the period, the modeled creation and spread of new agrarian states is a fundamental consequence of state collapse and internal civil wars triggered by rising 'demographic-structural' pressures that occur when state territorial growth is checked yet (warrior elite) population growth continues. Together the model's underlying mechanisms substantially account for the number of states, their duration, location, spread rate, overall occupied area, and total population size for three thousand years.