Expectations Galore: Final Musings

I want to conclude this series on expectation with a discussion about the implications for the serious player. Most people playing the various games offered by casinos are not thinking about expectation. They are in the casino for a wide spectrum of reasons and their decisions on which games to play are ruled by considerations far removed from expectation. On the other hand, there are a handful of people who actually think about the house edge and decide which games to play based on the information they can glean about expectation.

In a previous article for Poker Digest, I informally proved there is no deterministic betting strategy which can beat a game with negative expectation at each round of the game. Where does that leave us? The conclusion is obvious. First, if you want to win in the long-run, do not play games for which the expectation is negative at each round of the game.

Unfortunately, the preceding sentence covers almost all games.

``Wait a minute'' you say, ``you only wrote `almost all games'. Does that mean there are games for which the expectation is positive for some rounds?''

That is the entire basis for counting strategies in blackjack. Bet only a small amount when the expectation for a given round is negative, but bet large amounts when the expectation is positive for a given round. The number of rounds in blackjack for which the expectation is positive is large enough that a player can gain an overall positive expectation in blackjack. There are a variety of counting strategies which give a blackjack player an overall positive expectation.

Of course, once the casinos became aware of the advantage a counter has, they took countermeasures. Some of the actions they took included using more decks (this in itself does not negate the advantage a counter has), not dealing a large proportion of the cards (this is a serious blow to counting), restricting how much a bettor can increase her bet (this also damages a counter's advantage), and outright banning of a counter from playing (a serious blow). I believe there are players currently beating blackjack using counting strategies, but they are wisely lying low and not drawing attention to themselves.

As a base point, there is a so-called basic strategy for players who do not wish to delve into any form of counting. Determining the precise house edge is an intractable computational problem so that people resort to simulation methods. A good simulation is very accurate. Steve Jacobs has published (available via the Internet, for example, at ConJelCo's web site) definitive information on the house edge for basic strategy in a variety of situations.

Over the years there has been a lot written on how to win at roulette. As we saw in the preceding article, deterministic betting schemes are not worth the paper on which they are written. Nevertheless, the game is beatable. After all, the game is simply a matter of physics, so given the rotational speeds of the ball and wheel, one can predict with high probability a small area of the wheel where the ball will come to rest. The catch, of course, is that no casino is going to allow a player to set up the equipment necessary to make the appropriate measurements. Apparently there have been a few people who have been successful at doing this simply working with their eyes and minds. A casino can prevent this technique from working by not allowing any bets to be placed after the ball has been spun.

There have been people who have beaten roulette by detecting mechanical flaws in the wheel--some of which have been introduced by confederates which is, of course, blatantly illegal--after recording a very large sample of the numbers which have come up in the game. However, given modern roulette wheels, I don't think it is worth the effort.

This brings us back to poker from which we shall not stray in the immediate future. Can we say anything meaningful about a poker player's expectation? This will be the subject of a future column or two. For now let me simply say it is a very complicated issue. As humans there is a tendency to want to attach a single number to each player and want that single number be meaningful. But each time we play, the players are likely not the same people; even with the same people, the seating arrangements will be different; and people are in different emotional states in different sessions. There is a problem with the accuracy of players' records. In spite of this, there are players who appear to be winners and players who appear to be losers. So it would be interesting to attempt to quantify this.