The Strategy Of Conflict by Thomas C. Schelling

Author:Thomas C. Schelling [Schelling, Thomas C.]
Language: eng
Format: epub
Publisher: Pickle Partners Publishing
Published: 2015-01-26T20:00:00+00:00

or

Thus the effective range for π in this example is given by

And there is no threat at all worth making if there is no room between these two limits, if

or

Only a “fractional” threat—a threat with π less than 1—is worth making if:

or

Here is a case, then, in which the fractional threat is superior to the certainty threat, and in which the latter could be not worth making at all while the former were. The argument hinges on the risk of failure, a risk that has been assumed independent of the size of π itself. This is a somewhat special assumption. If we interpret P as the probability that we have misjudged our adversary and exaggerate his preference for avoiding the lower right cell, our assumption implies a bimodal distribution of payoffs in the population. It implies that we have either a man whose payoffs are adequately represented by the numbers in our matrix, or a man whose payoffs are so different that no relevant threat—within the range of values up to π = 1—will dissuade him. If instead we supposed that the ratio of column payoffs in the upper and lower right-hand cells showed a bell-shaped frequency distribution within the population, and that our particular adversary had been drawn at random, the probability that our threat would succeed would vary directly with the value of π itself. The probability that a burglar drawn at random from the universe of burglars will be deterred by some specified probability of apprehension and conviction presumably varies directly with the latter probability; the simple model analyzed above treats burglars as divisible into two classes—those, let us say, who steal for money and are certainly deterred in accordance with the numbers of the matrix, and those who steal for fun and are beyond reach of any threat of the magnitude entered in the lower right-hand cell. On the other hand, if our probability of failure reflected, say, a breakdown of communication with the adversary, there might be better reason for supposing the probability of failure to be independent of the particular threat being communicated.

It is interesting to notice that attaching a probability of fulfillment to our threat is, in the above model, substantially equivalent to scaling down the size of the threat more directly. To see this, interpret X in the lower right-hand cell as a fine that will be levied on both Row and Column, or a number of lashes with the whip or days of imprisonment that both will suffer if the threat is fulfilled. If X is the maximum number of dollars, lashes or days that Row can threaten, let π be interpreted as Row’s specification of what fraction of the maximum permissible penalty is to be exacted; if π is set at 0.5, for example, both Row and Column receive exactly half their maximum punishments. If we interpret the matrix in this way, and ask what value of π provides the optimum threat from Row’s point of view, we go through

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