Occasional reflections on Life, the World, and Mathematics


I’ve commented on the peculiar dissipation in recent times of the moral stench of gambling, particularly as practiced by the quant elite, who seem at times to revel in their role as gamblers. But I discover now that I was preceded by more than 3 centuries by Daniel Defoe, in his brilliant Essay on Projects:

Wagering, as now practised by politics and contracts, is become a branch of assurances; it was before more properly a part of gaming, and as it deserved, had but a very low esteem; but shifting sides, and the war providing proper subjects, as the contingencies of sieges, battles, treaties, and campaigns, it increased to an extraordinary reputation, and offices were erected on purpose which managed it to a strange degree and with great advantage, especially to the office-keepers; so that, as has been computed, there was not less gaged on one side and other, upon the second siege of Limerick, than two hundred thousand pounds.

This last extraordinary remark, that people were wagering vast sums on the outcomes of sieges. (£200 thousand in the 1690s is probably like £20 million today, or $30 million.) And he goes on to use gambling on the outcomes of siege warfare to present a fascinating example of a sort of arbitrage called a “dutch book”: Combining different wagers with different parties so as to obtain a cumulative certain profit. (De Finetti’s “Dutch Book Theorem”, stating that you need to calculate with something indistinguishable from the standard rules of probability if you don’t want to fall victim to someone else’s dutch book, is the basis of certain approaches to the foundations of probability.)

For example: A town in Flanders, or elsewhere, during the war is besieged; perhaps at the beginning of the siege the defence is vigorous, and relief probable, and it is the opinion of most people the town will hold out so long, or perhaps not be taken at all: the wagerer has two or three more of his sort in conjunction, of which always the office-keeper is one; and they run down all discourse of the taking the town, and offer great odds it shall not be taken by such a day. Perhaps this goes on a week, and then the scale turns; and though they seem to hold the same opinion still, yet underhand the office-keeper has orders to take all the odds which by their example was before given against the taking the town; and so all their first-given odds are easily secured, and yet the people brought into a vein of betting against the siege of the town too. Then they order all the odds to be taken as long as they will run, while they themselves openly give odds, and sign polities, and oftentimes take their own money, till they have received perhaps double what they at first laid. Then they turn the scale at once, and cry down the town, and lay that it shall be taken, till the length of the first odds is fully run; and by this manage, if the town be taken they win perhaps two or three thousand pounds, and if it be not taken, they are no losers neither.

It is visible by experience, not one town in ten is besieged but it is taken. The art of war is so improved, and our generals are so wary, that an army seldom attempts a siege, but when they are almost sure to go on with it; and no town can hold out if a relief cannot
be had from abroad.

Now, if I can by first laying 500 pounds to 200 pounds with A, that the town shall not be taken, wheedle in B to lay me 5,000 pounds to 2,000 pounds of the same; and after that, by bringing down the vogue of the siege, reduce the wagers to even-hand, and lay 2,000 pounds with C that the town shall not be taken; by this method, it is plain –

If the town be not taken, I win 2,200 pounds and lose 2,000 pounds. If the town be taken, I win 5,000 pounds and lose 2,500 pounds.

This is gaming by rule, and in such a knot it is impossible to lose; for if it is in any man’s or company of men’s power, by any artifice to alter the odds, it is in their power to command the money out of every man’s pocket, who has no more wit than to venture.

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