Poker Digest Vol. 3, No. 3, January 28 - February 10, 2000
We now return to a comment mentioned earlier in this series on expectation. Namely, let's examine the claims made by some people that they have a betting system which can beat a negative expectation game. By the end of this article we shall prove such a claim is false thereby demonstrating people who make such claims are either ignorant, misguided, or outright charlatans.
Let us first clarify what we mean by a betting system so we all know what is under discussion here. For technical reasons, it would be better to call these betting systems ideterministic betting systems. This means that the amount to be bet at any point is determined precisely by the history of the bets and outcomes up to that point. No randomness or choice is involved.
The simplest example is a doubling scheme for a bet which pays 1-to-1. You place a bet of one unit. If you win, you start over again, but if you lose you now place a bet of two units. If you win you are now one unit ahead and you start over again. However, if you lose this bet, you now bet four units. You continue (theoretically) in this way until you win at which point you will be one unit ahead. Then you start at the beginning again.
Another characteristic of a deterministic betting scheme is the fact that at some point the bettor reaches an outcome where he goes back to the starting point. This is necessary if for no other reason than the sheer number of possible sequences of outcomes becoming too immense to handle. Humans can remember only so much.
Let me emphasize the fact no randomness or choice is allowed. People who whimsically vary their bet sizes for a variety of reasons such as, ``I can't possibly lose again'' or ``Red has come up six times in a row,'' are not following a deterministic betting strategy. This analysis does not cover such behavior.
The best way to analyze these betting strategies is to develop a so-called tree diagram. We are going to develop our analysis without the diagrams, but for those of you who know what they are, they are lurking in the background. Let's develop the main idea via an example.
Suppose you're betting heads or tails and getting paid 1-to-1 for your bets. We learned in an earlier article the expectation for this game is zero. Let's assume you have 63 units available and all you want to do is win one unit. You decide to use the above doubling betting strategy. Here are the possible sequences of outcomes for your strategy: W, LW, LLW, LLLW, LLLLW, LLLLLW, and LLLLLL, where L represents a loss and W represents a win. Notice that in the first six sequences you win your one unit and quit, while for the last sequence you lose your stake of 63 units.
Now let's determine the contribution of each of these sequences to the overall expectation (which is zero). The sequence W occurs with probability 1/2 and gives us a win of one unit. Thus, it contributes 1/2 to the expectation. Similarly, the sequence LW occurs with probability 1/4 and gives a win of 1 unit, thereby contributing 1/4 to the expectation. Likewise, LLW contributes 1/8; LLLW contributes 1/16; LLLLW contributes 1/32; and LLLLLW contributes 1/64. Finally, the sequence LLLLLL occurs with probability 1/64 and gives a loss of 63 units, thereby contributing -63/64 to the expectation.
Adding all the contributions to the expectation yields zero as it should. If you now contemplate the simple example we just finished, you will observe several key points which essentially prove there is no deterministic betting strategy which can beat a negative expectation game.
First, notice the contributions to the expectation at each possible conclusion sum to the expectation. Second, since the expectation of this game is zero, if there are conclusions which have a positive contribution to the expectation, there must be conclusions which have a negative contribution to the expectation.
The latter observation is profound. It means there must be a positive probability of concluding with a loss. Therefore, if your strategy yields a large probability of finishing with a profit, you must have a small probability of a large loss. There is no way around this because the expectation is zero. Of course, the situation is worse if the game has negative expectation.
Now let's examine a real game, in particular, roulette. If you watch a typical roulette player, you will observe him placing all kinds of bets. Perhaps he will place three chips on a single number, a single chip on three numbers, a single chip on four numbers, and so on. It looks all higgledy-piggledy, but it turns out the expectation for his wager has a nice simple answer. (We are assuming the roulette wheel is an American wheel and the player does not place any chips on the single way of betting five numbers at a time.) Namely, the player's expectation is -n/19, where n is the total number of chips he has wagered. So if a player follows a deterministic betting strategy in roulette for m rounds, the total expectation must be negative because each individual round has negative expectation as we just observed. Therefore, some of the possible outcomes after m rounds must make a negative contribution to the expectation. This means there is some positive probability of being at one of these negative locations after m rounds. Since a player must terminate the game at some point, there is no guarantee he can quit at a positive location.
The computations in roulette are as simple as it gets so one can work out exact values. The computations for other games can get uglier, but all we really care about is the difference between positive and negative. Any game which has negative expectation at each round behaves the same with respect to an overall strategy. That is, after m rounds for any value of m, there is a positive probability the player will be behind. Therefore, there cannot be a betting strategy which guarantees the player wins. The best one can do is produce a strategy which has a high probability of showing a profit. However, there is a trade-off. If the strategy loads a positive finish with high probability, then there will be a small probability of a large loss because the overall expectation is negative and the negative contribution to the overall expectation must go somewhere.
So the next time you read about some hotshot's betting strategy which always wins for negative expectation games, realize you possibly are dealing with an evil person.