In Part I, Midlarsky develops his theoretical-probabilistic argument, wherein he links scarcity to inequality. Using both exponential and geometric distributions, Midlarsky tries to demonstrate mathematically that under conditions of scarcity, there will be an exponentially declining probability of attaining valued resources. The greater the scarcity of these resources the more rapid the exponential decline, resulting in a far larger portion of a population being unable to obtain these valued resources. Thus, Midlarsky argues that if there is scarcity of desired resources then inequality exists, and given his assumption of equality of opportunity, who gets these valued resources and who does not is a random process. Midlarsky uses the example of agricultural land to illustrate his theory of exponential subdivision under conditions of scarcity.
Agricultural land is a finite resource and beyond a certain point no more can be generated.
It is this resultant extreme inequality that Midlarsky argues will lead to state formation under one set of conditions and lead to state dissolution under a different set of conditions. Besides using the theory of exponential subdivision under scarcity to support his claim that inequality is a product of a random process that is neither linked to structural economic conditions nor to human greed, Midlarsky employs the theory of fractal formation upon expansion to argue that those individuals who do acquire more valued resources in a society do so as a result of a random process based on genetic capability. For example, in the case of sequential expansion over a relatively empty space, inequality emerges because those groups or nation-states who arrived first on the scene are better able to claim the best land and other valued resources than those who arrived at a later date. Similarly, when expansion occurred over occupied land, those conquerors who arrived first with better technological operating advantage would secure the prime lands and resources for their own use. A third fractal pattern of inequality emerges when conquerors resettle depopulated land that has been cleared of its original inhabitants either by killing them directly or indirectly through the introduction of diseases or through the removal of the indigenous population to the periphery.