Wager Mage
Wager Mage
Photo: Natalie Birdy
The Reverse Martingale Strategy, like the Martingale Strategy, is a very simplistic strategy involving doubling bets. However, where the Martingale Strategy doubled our bet when we lost, the Reverse Martingale Strategy doubles our bet when we win.
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Read More »Before we can truly breakdown what these probabilities mean, we need to note an incredibly significant caveat. Previously, we used the fact that a probability of bankruptcy greater than 50% meant that it was guaranteed to be a losing strategy. This was due to the fact that we knew we would have to stop when our bankroll hit exactly 2 x k. If that were the case, then we’d know for certain that a probability of bankruptcy greater than 50% would mean we are more likely to lose money than gain money, since we would lose k over 50% of the time and gain k less than 50% of the time. In this instance, this is not true however, as we stop when our bankroll is at least 2 x k. As a result, we cannot use our 50% value as a test. Instead, we can bound our values in a slightly different way. Let’s consider the maximum bankroll we could ever finish with. This would involve maximising our bankroll and final bet size such that both is as close to 2 x k as possible. First, let’s maximise the bankroll. If our bankroll is equal to 2 x k, then we have reached our exit condition and will place no further bets. So, the maximum bankroll we could theoretically have is (2 x k) – 1. Now consider the maximum bet size we could place. If our final bet was greater than (2 x k) – 1 then we would not be able to place it. Thus, our final bet can be at most (2 x k) – 1. Since our finishing bankroll is equal to our final bankroll plus our final bet size, we can conclude our maximum finishing bankroll is equal to (4 x k) – 2. Since we only consider real values of k which are greater than or equal to 1, we can say that (4 x k) – 2 < (4 x k). This might not seem important, however, we have now successfully created a boundary condition. We know that our finishing bankroll will never exceed 4 x k, so if we can find the probability at which it is impossible for it to be a winning system when our finishing bankroll is exactly equal to 4 x k, then we know it will never be a winning system for the Reverse Martingale Strategy. Pretty intuitively, we should be able to see that the probability at which it is impossible to be a winning system is equal to 75%, anything above this value will be a guaranteed losing system. Consequently, we can immediately discredit any starting bankroll that is greater than or equal to 4 as the probabilities are greater than 75% at this point. Note, this does not mean a starting bankroll of 1, 2 or 3 is profitable, it just means we can instantaneously rule out any starting bankroll greater than or equal to 4 without any further calculation. To determine if starting bankrolls of 1, 2 or 3 are profitable, we will have to calculate their expected return.
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