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Best of Donald Catlin

Gaming Guru

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Hold Percentage, Can You Affect It?

2 September 2000

If you are a regular reader of this column, then you are familiar with the notion of house edge. This idea was introduced a year ago in my July 1999 article, Right Question - Right Answer (Hopefully). As we saw there, and have seen many times in subsequent articles, the house edge is a viable statistic for ranking a casino game.

To quickly review, if one calculates the negative of the ratio of the player's expected return divided by the player's expected bet, then the house edge is just this fraction expressed as a percentage. This statistic represents the average loss per dollar wagered that the player should expect and certainly seems like a reasonable measure of a game. What is more, it has the attractive feature that, in a game with repeated independent trials, the figure remains constant no matter how the player bets or when the player stops playing. In other words, for a given game as just described, the house edge is a fixed number that measures just how good or bad such a game is.

I am sure you are all familiar with the oft-quoted figures of 1.414% for the Pass Line at craps, 5.26% for roulette, and 1.06% for the Banker in baccarat; these are all house edges. Slot machines and video poker are reported by giving expected paybacks, such as "98% return to the player." This is just another way of reporting the house edge. The payback is just 100% minus the house edge, so a 98% return would be the same as a 2% house edge.

Casinos regularly check their slot machines to ensure that the house edge is, in fact, holding. They do this by keeping track of the number of coins paid out and dividing that number by the number of coins played; this fraction as a percentage should be close to the payback percentage. The apparently surprising thing is that for table games a different statistic is used. We'll discuss why this is so later; but first let me explain this new statistic; it is called the hold percentage. The hold percentage is defined as the portion of the player's buy-in that is held by the casino. In other words, it is the amount that the player loses divided by the total amount of money brought to the table by said player. An example will help us to see how the casino estimates this statistic.

Suppose that the dealer's rack holds $15,000 in chips and the drop box is empty at the beginning of a shift. During the shift there are $5000 in fills, so that the casino's total chip contribution to the table is $20,000. You will note that as people arrive at the table to play with cash, their cash is traded for chips from the rack and the cash goes to the drop box. Although some casinos allow money to play, if you watch carefully that cash generally makes its way to the drop box. If people arrive with chips, these do not end up in the drop box, but such a buy-in is noted by the floor personnel and may be accounted for at the end of the shift.

Anyway, suppose that at the end of the shift there is $12,000 in the rack and $11,000 in cash in the drop box. Further suppose that the floor person notes that $3000 in chips were used for buy-ins. Altogether, the casino now possesses $23,000 in cash and chips. Since they started with $20,000, they have made a $3000 profit. The $11,000 in cash and the $3000 reported by the floor person represent $14,000 worth of buy-ins during the shift. The ratio 3,000/14,000 expressed as a percentage, 21.43%, is the hold percentage for that particular shift.

I am sometimes asked by gaming developers or casino managers if I can calculate a theoretical hold percentage for a particular game. I have to tell them that I cannot. There is no theoretical hold percentage. The hold percentage depends very much on the whims of the player (which answers the question posed in the title of this article).

To see this, let us look at a few extreme scenarios. In the first, there is only one player during the whole shift. This particular bloke buys in to the game for $1000, plays once for $5, loses and leaves. Well, the table has made a profit of $5 and has had a total buy-in of $1000. The table hold is 0.5%. Now consider the case where there is only one player during the whole shift, but this player buys in for $5 and loses it. The table hold is 100%. Finally, consider the case wherein a player buys in for $10 and gets up and leaves when he loses his first bet. If there are no other players, then the table hold this time is 50%. Notice that in all three scenarios the table has made only $5.

Now, no one would lend any credence to a statistic that was obtained using just one trial. The point of the preceding examples is simply to show how similar casino profits can lead to very different hold percentages. If one were to use the above three scenarios to estimate house edge, the calculation for all three would be 5/5 or 100%. This is a silly figure, of course, but the point is that it is the same in all three cases.

Does this mean that hold percentage is not a reliable statistic? Not at all. The hold percentage answers the casino's question "What fraction of the player's buy-in are we going to keep?" This is a perfectly reasonable question as long as we understand that the answer to it, the hold percentage, depends in part on how the player acts. Hence it is not a statistic that measures a game's profitability solely in terms of the game's rules the way the house edge does.

Why not just use the house edge like the slots do? After all, it seems like a more stable statistic. Think about it. A slot machine can electronically keep track of every coin played. How would you keep track of every bet by every player on a table game? This would drive a floor person crazy. Of course, you could design a table with electronic sensors and use special chips, but I suspect that table players would shun such a set up figuring that their play was being tracked by the casino for sinister motives (especially blackjack players). Thus the casino keeps track of its table games in a manner that it can realistically handle.

Now here is a surprise. Although I stated earlier that there is no theoretical hold percentage for a game, if one makes one additional assumption about how a rational player would act playing a particular game, then a theoretical hold percentage can be calculated. I'll explain how this is done next month. Get ready for some more mathematics. See you then.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers