A history of lotteries and betting

From lot-drawing as an allocation device to number games, public-finance lotteries, the balanced book and the pari-mutuel pool.

Drawing lots before drawing prizes

Drawing lots is far older than betting on the draw. For long stretches of recorded history the drawing of marked tokens was an administrative device: a way of allocating land, dividing inheritances, assigning duties or filling offices without an argument about who chose. What made the practice useful was precisely that nobody could steer it. The transition that matters for this subject is the point at which people began to stake money on which token would emerge, converting an allocation procedure into a priced game.

Number games in the city-states

That transition is visible in the Italian city-states of the fifteenth and sixteenth centuries, where public offices were filled by drawing names from a pool of eligible candidates. Wagering on which names would be drawn became widespread enough that the draw was eventually abstracted from the appointments altogether: the names were replaced by numbers, the pool was fixed at ninety, and five were drawn. The resulting structure - a fixed pool, a fixed count of numbers drawn, and prizes for matching some of them - is the direct ancestor of every draw game since, and its mathematics is the counting problem worked in the chapter on prize funds.

Lotteries as public finance

From the sixteenth century onward European authorities ran lotteries as a way of raising money for specific projects: harbour works, bridges, waterworks, fortifications, and later the endowment of schools and hospitals. The appeal to a treasury was that a lottery collected revenue from people who volunteered to pay, and the structural feature that made it work is the one described earlier: because the prize fund is a fixed share of stakes, the surplus is known in advance and is independent of who wins.

Two recurring problems followed the form throughout this period. The first was side-betting: unofficial wagers on which numbers the official draw would produce, which required no licence, paid no share to the treasury, and grew large enough in several jurisdictions to threaten the official game. The second was the administration of the draw itself, where the incentive to interfere was obvious and the means of detecting interference were weak. Both problems are the origin of most of what regulation later became.

Suppression and revival

Through the nineteenth century a series of scandals involving both private lotteries and their official counterparts produced waves of prohibition across Europe and North America. In many jurisdictions lotteries were abolished outright and remained so for the better part of a century. Their return, from the middle of the twentieth century onward, was driven by the same logic that created them: a state facing revenue pressure and a large unofficial market already operating found it easier to license and tax the activity than to suppress it. The modern state lottery is the nineteenth-century form re-established with administrative controls attached.

Fixed odds and the balanced book

Betting on events developed along a separate line. Its characteristic figure is the layer, who quotes a price for each outcome and accepts stakes against it, carrying the risk of the result. The essential technique is the balanced book: a set of prices whose implied probabilities sum to more than one, with stakes distributed across outcomes so that the layer retains the excess whatever happens. Prices moved with the weight of stakes rather than with new information about the event, which is a practice that has never really changed.

Two outcomes priced at 1.91 each

  1/1.91 + 1/1.91  =  1.047120       overround 4.71 %  per market

If stakes arrive in proportion to those implied
probabilities, the layer retains

  0.047120 / 1.047120  =  4.50 %                       per unit staked

The pool as an alternative

In the 1860s a different arrangement appeared in France for betting on races, and it spread widely because it solved the layer's central problem. Instead of quoting prices in advance, all stakes are combined into a pool, a fixed share is deducted, and the remainder is divided among winning tickets. The operator's return is fixed by the deduction and is independent of the result, so the operator carries no risk on the outcome at all. Mechanical totalisators from the late nineteenth century onward made the continuous recomputation practical, and pool betting became the standard arrangement for racing in many jurisdictions.

The twentieth century onward

Three developments carried the subject into its present form. Casino gambling was licensed rather than prohibited in a growing number of jurisdictions from the 1930s onward, which created a regulated industry where an unregulated one had existed. Electromechanical and then fully electronic gaming machines replaced physical mechanisms with specified mappings and computed returns, which made the arithmetic of the previous chapters explicit and auditable for the first time. And from the mid-1990s remote betting and gaming detached the activity from a physical venue entirely, which raised jurisdictional questions that the regulatory history had never had to answer.

What did not change across any of it is the structure. A pool with a deduction, a book with an overround, and a machine with a specified return are three implementations of a single arrangement: a fixed proportion of amounts staked retained by the operator, fixed before play and independent of the result.