The Nuclear Gamble
What Is Enough?
Almost everyone agrees that nuclear weapons cannot be used to advantage because to do so would be suicide. But the policy of nuclear deterrence requires that those weapons always be ready for use. Deterrence is
The Nuclear Gamble
What Is Enough?
Almost everyone agrees that nuclear weapons cannot be used to advantage because to do so would be suicide. But the policy of nuclear deterrence requires that those weapons always be ready for use. Deterrence is therefore a gamble that what we are always ready to do, we will not ever do. The gamble has worked for the last forty years but, in the long run, is it a good bet?
Probability theory is a natural approach for evaluating the nuclear gamble. The early contributions of Blaise Pascal, Pierre de Fermat, and Christiaan Huygens all had as their main consideration the expected winnings in games of chance.1 From there it was a short leap to expected losses, as in insurance or medicine, and finally to diverse applications from communications satellites to quality control. This paper expands the area of application to the most serious issue of all time - preventing a nuclear holocaust. But, as in the early days, the stage is best laid with a surprising result from gambling.
Coin Tossing. Coin tossing is the archetypal game of chance. Two players, A and B, bet a dollar on whether tossing a fair coin will show heads or tails. A tosses and B calls. If B guesses correctly, A must pay him a dollar and vice versa. A very simple, not too interesting game.
The game becomes more interesting - positively intriguing to some, judging by the sums that have been lost - if B not only guesses the outcome, but also gets to decide the size of the wager on successive tosses. This is the situation in casinos that offer roulette. The gambler chooses red or black and also the size of his bet. Returning to the fair coin toss, A reasons that there is no harm in letting B vary the size of his bet. A fair game is a fair game whatever the size of the wager. But B thinks differently.
B bets a dollar on the first toss and calls heads or tails at random - neither is more likely to win than the other. If he wins, he stops and is a dollar ahead. If he loses, he bets two dollars on the second toss. If he wins the second time, he stops and is a dollar ahead, having lost $1 on the first toss and having won $2 on the second. If he loses on the first two tosses, he doubles his bet again, betting $4 on the third toss, etc.This doubling approach, known as the Martingale Strategy, is one of the oldest "sure win" gambling strategies around: B keeps doubling his bet until he wins.2 When he does, he is a dollar ahead. And he is bound to win eventually. He cannot keep guessing wrong forever, even if he wants to! Try it and see. With high probability, no one will have to wait more than thirty tosses before he wins. Only one in 1 billion should be that unlucky.
While this strategy guarantees that B will inevitably win, there is a flaw: The strategy only works if he has unlimited funds at his disposal.3 With any finite amount of money, there is a small chance of losing the whole sum. This small chance of a large loss exactly offsets the large chance of a small ($1) win, keeping the game fair. But that is not the point of this paper. Now we come to the serious part.
Pistol Roulette. Consider a new game in which A repeatedly tosses the coin and B calls heads or tails each time. The game continues until B guesses incorrectly, at which point he is shot. Just as B was sure to win in the doubling strategy, he is sure to die at this game. Try it - without the gun. Before, no one was likely to go beyond thirty tosses to win a dollar; now, no one is likely to go beyond thirty tosses before he will be shot. The chance of surviving thirty tosses is roughly one in 1 billion!
This suicidal game is like loading one chamber of a two-chambered revolver, spinning the cylinder, putting the gun to your head, and pulling the trigger - a game we will call "pistol roulette." If we change the game to the usual version, with a six-chambered revolver, the probability of being killed with each pull of the trigger is one-sixth instead of one-half. The lower probability changes the time scale until you expect to be killed, but does not change the inevitability of that result. Because the probability per trial is one-third what it was before, you expect to live three times as long. But, if you play this game day in and day out, your death is merely delayed.
In the same way, playing with one bullet in a 600-chambered revolver prolongs the process - you expect to live one hundred times as long as with a six-chambered revolver. But that does not change the inevitability of your death. If you play once each day, you might be lucky enough to live several years. Or you might be unlucky enough to go in the first month - there is roughly a 5 percent chance of that.
Nuclear Roulette. What does pistol roulette have to do with nuclear war? During the Cuban missile crisis, President Kennedy estimated the odds of nuclear war as being "somewhere between one out of three and even." So the Cuban missile crisis was equivalent to nuclear roulette - a version of pistol roulette in which the entire world is at stake - with a two- or three-chambered revolver.
"During the Cuban missile crisis, President Kennedy estimated the odds of nuclear war as being'somewhere between one out of three and even.'"
The events support Kennedy's view: Early in the crisis, most advisors recommended military action to remove the missiles, a so-called "surgical strike." Later assessments by these same advisors concluded that, far from being "surgery," such action almost certainly would have meant a catastrophic war with the Soviet Union.4, 5
George Ball, one of Kennedy's senior advisors, wrote that when he met with the other advisors many years after the crisis, "much to our own surprise, we reached the unanimous conclusion that, had we determined our course of ac