Rational Expectation Theory and Ideological Factor of National Security

Deriving the Maximin Principle

The maximin principle ranks states according to the utility of the worst-off actor. The national security function is therefore . The ranking is unchanged when the same monotone increasing transformation is applied to each actor utility. That is, it is invariant under a transformation for which whenever . This means that the maximin principle requires level comparability, because the monotonicity of implies that if and only if .

To derive the maximin principle, one additional property is needed: separability, which states that actors to whom all states look the same play no role in the balance of power. More precisely, let be a subset of actors such that for any tuple of utility functions, is the same for every state . Then and give the same ranking if for all actor not in , for all states . The claim is that given level comparability and the above axioms, the national security function must be the maximin function. Curiously, however, these premises imply only that welfare function is maximin or maximax (21). The latter maximizes the utility of the best-off actor; that is, . To infer a minimax principle, one must rule out the maximax principle on some other ground.

Again the idea of the argument can be conveyed in the two-actors case (22), where separability does not play a role. Let as shown in Figure 2 be an arbitrary utility vector, and let . Divide the plane into regions about the diagonal line as shown. Then it suffices to demonstrate that one of two situations must obtain: all the points in regions and their reflections (shaded area in Figure 3 ) are preferable (or indifferent) to , and all other points are worse than , or all the points in regions and their...

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...tarian calculation meaningful. However, unit comparability remains if the ranking is invariant only under a proper subset of translated rescaling, while the proof assumes invariance under any translated rescaling. In other words, the proof assumes that utilities have unit comparability and no more than unit comparability.

This strong assumption is already very close to utilitarianism. A Rawlsian, for example, would immediately object to it because it makes the comparison of worst-off actors meaningless from the start. If the utility vectors are in state and in state , the Rawlsian prefers because of the higher utility of the worse-off actor.

However, a translated rescaling maps these vectors to and , respectively, in which the Rawlsian preference is reversed. A similar point applies to the derivation of a maximin welfare function from level comparability.