Dubins and Savage’s result is disappointing, as it indicates that you need to play as fast as possible. If what amuses you is gambling, the conclusion obtained is the worst of any that could be discovered.

This is in fact linked to the law of constant loss: When we play a game with a probability p less than 1/2 of winning (for the player), then, whatever the strategy used, we lose “on average »In proportion to what we bet.

All this is sadly simple: the average losses depend only on the sums on the table and are determined by the coefficient p of the game.

The way to play is irrelevant, geometric martingale or bold game or optimal strategy, everything is equivalent.

A consequence of the law of constant loss is also that no martingale ever turns the odds of winning in your favor. In other words, there is no martingale. In particular :

- no advantage can be obtained by playing only on certain draws;
- no advantage can be obtained by the one who decides the end of the game (the right to make Charlemagne is useless);
- even if we want to reduce our hope for each franc wagered, we cannot. Ultimately, the more you play, the more you lose, inevitably.

To lose the least, you need to play as little as possible. In order not to lose at all, you must not play at all.

It’s hard to be completely convinced that taking the past into account is unnecessary, and that just because red has come out 10 times in a row doesn’t mean that black has a better chance on the next move.

However, we have to resolve it: casino games are stupid, and they are indeed commercial enterprises whose profitability is guaranteed in the long term by proven and testable mathematical laws.

All in all, my friend who claimed to be able to win at roulette is wrong: the more you play the more you lose, nothing helps. In his original reasoning, there was a mistake: if you are given a total of sums to bet, there are neither good ways to play it, nor bad ways.

Reduced to the total money you put on the table, you can neither play well nor play badly.

However, as Paul Deheuvels points out, the rules of the game of roulette allow other bets than single bets, for example full numbers, which lead to winning a multiple of his bet or losing it (with a probability of lower payout which maintains a positive payout expectation for the casino). It is not impossible that by fully using the authorized bets, other methods can for example improve a little the 0.90426… found to go from 10 to 11 in French roulette.

Alas, there is not much to nibble, because to go from 10 to 11, with an unfavorable play, the product of the probability of success r by your desired gain (1 franc) will always be lower than your probability of lose your 10 euros, multiplied by this loss. So: 1r <(1 – r) 10, which implies that r <10/11 = 0.90909… (this 10/11 that we get by doing anything when p is equal to 1/2 is a barrier inaccessible when p is less than 1/2).

**Win Differently?**

We have so far assumed that the game is fair, in the sense that the roulette wheel is well balanced: each number has a one in 37 chance of coming out (one in 38 in American roulette). This may not be the case because the roulette wheel is rigged or worn out (the service life of a roulette wheel used every night is about ten years).

Then comes the idea of betting on what falls most often, so as to take advantage of the inequality of chances between numbers. The general idea of exploiting the imperfection of casters is not absurd, and it was indeed used by William Jaggers at the end of the 19th century.

He thus won 1,500,000 euros in Monte-Carlo, following a detailed analysis of the output frequencies of the numbers.

Since then, casinos have understood that their interest is that the wheels are well balanced and they take care of it: the interest of a roulette casino is that there is no bias!