Randomization of threats and promises

An extract from

T. C. Schelling

Economics Division, The RAND Corporation, P-1716

June 5, 1959




[Preventing deduction and anticipation in a zero-sum game]

In the theory of games of pure conflict ("zero-sum" games) randomized strategies play a central role. It may be no exaggeration to say that the potentialities of randomized behavior account for most of the interest in game theory during the past decade and half (*). The essence of randomization in a two-person zero-sum game is to preclude the adversary's gaining intelligence about one's own mode of play -- to prevent his deductive anticipation of how one may make up one's own mind, and to protect oneself from tell-tale regularities of behavior that an adversary
might anticipate. In the game that mix conflict with common interest, however, randomization plays no such principal role, and the role it does play is rather different.

[Giving fractional/sampled information to avoid total war situations in a general game]

Randomization in the theory of these ("non-zero-sum") games is not mainly concerned with preventing one's strategy from being anticipated. In these games one is often more concerned with making the other player anticipate one's mode of play, and making him anticipate it correctly, than with disguising one's strategy. There may of course be zero-sum components embedded in a larger game. In limited war one may be concerned to communicate rather than to disguise the limits that one proposes to observe, but within those limits may sortie his aircraft in a randomized way to minimize the enemy's tactical intelligence. Again information samples may be exchanged, or agreements enforced on a sample basis, where neither party can afford to yield the other full knowledge. Arms-control agreements, for example, might have to be monitored by a sampling technique that yielded each side enough knwoledge about the enemy's forces to reveal compliance or non-compliance without yielding so much that the possibility of successful suprise attack on those forces were enhanced.

[Enabling divisibility and homogeneity of indivisible and incommensurate objects]

But the main role of randomization in the traditional literature on non-zero-sum games is a different one. It has been a device to make indivisible objects divisible, or incommensurate objects homogeneous. Their "expected values" are divisible by lottery when the objects themselves are not.