Euler's Analysis of the Genoese Lottery - Fair Prize Values (continued)

The question Euler examines in Problem 7 is that of determining a fair rate of return for the bank. Euler mentions that the lottery must hold back something in order to cover its expenses, and observes that if the lottery is being used to finance important state needs, then further discounts on the prizes above and beyond what he recommends may be needed. He suggests that on prizes with only one number matched, the amount the bank should hold back be no more than 10% of the revenue, as a greater discount "would be too obvious and disgust the participants". Thus, 90% of \(F_{k,i}\) should be returned to the gambler for each \(k\).

On the other hand, larger profit margins for larger values of \(i\) are both justified by the risk and "will hardly be noticed, given that few people are in a position to calculate the fair value." Accordingly, he suggests returning 80%, 70%, 60% and 50% of \(F_{k,i}\) to the player when \(i=2,3, 4\) and \(5\), respectively. These profit margins of 10%, 20%, and so on, are far more modest than the 31%, 49% and 71% margins spelled out in Roccolini's proposal (see the final column of Table 1).

As with Problem 6, Euler's recommendations in Problem 7 do not have the force of mathematical proof behind them; they amount to little more than the recommendations and educated guesses of a respected scholar.

It is interesting to compare Euler's recommendations to the rules used in a modern lottery, where the pari-mutuel system insulates the bank from any real risk. In such a situation, prize money is concentrated on the highest values of \(i\), rather than the lowest, because large jackpots attract players to the game. Let's return to the case of New York State, where \(t=6\), and \(k=6\) is the only choice that players have. The first difference we notice is that the bank's cut is a uniform 62% across the board, rather than being graduated according to the size of \(i\). Secondly, the weights are dramatically shifted in favor of the highest value of \(i\), in stark contrast to Euler's second and third method of weighting. Here are the values of \(\alpha_{6,i}\) used in the New York State Lotto [8]; the figures don't add up to 100% because 7.25% of the prize money goes to those who win the second prize by matching the "bonus number," as described in section 2.

Having determined the bank's cut and the weighting of prize money, all that remains for Euler is to plug in values of \(n\) and \(t\), and to calculate the size of the prizes under each of the three methods. In Problem 8 he does this in the canonical case of \(n=90\) and \(t=5\). Table 5 contains these recommended prizes.

In Problem 9, he uses the values \(n=100\) and \(t=9\), curiously close to the case \(n=100\) and \(t=10\) which he outlined for Frederick in 1749. Once again, he calculates the prize money under all 3 methods for each value of \(k \leq 5\).

Editor’s note: This article was published in April 2004.