House edge and play time (for a same buy-ins)?

What exactly is the relationship between house edge and play time? Specifically, if game A has a HE of 2% and game B 1%, and a player buys in $100 each to play. Will he be able to play longer with game B? If two games have the same HE, does this mean $100 buy-in would last about the same play time for both games? We assume same speed of dealing and resolving hands for the games, for now, also leaving drinking and chatting out of the picture.

This example is somewhat misleading, because at a buy-in of $100, there's a good chance its going to be gone very quickly if the HE is 1% or 2%. Depends on minimum bet size. But the 1% and 2% HE figures are the expected loss for funds wagered, so if you think you will be able to wager $1,000 with your $100 buy in (say 100 $10 bets), then your expected loss will be .01 x $1000 which comes out to $10, or .02 x $1000 which comes out to $20. So if you are able to get 100 $10 bets in with your initial $100 buy in, the HE says that on average, you will be able to play one additional hand at the 1% game versus the 2% game. Say 100 bets vs. 99.

Variance comes into play in such a short timeframe, and so the actual results will likely vary significantly from expectations over such a short sample size.

Thanks for the comments. Let's assume buy-in $1,000 and play a minimum bet of $10 each hand.

Some people consider the House Edge to be the "hidden PRICE" for engaging in the game. I quoted hidden because to many people it is. They don't know it and they don't want to know it. What the casino looks at is your Expected Value to them which is how much you will lose and that is based on amounts bet, time and the House Edge of a particular game.

Casinos want consistent players but don't always get them. So time at table times amount bet is a good indication of how much they want you but its based on estimates and chance observations of your betting pattern.

The House Edge is the built-in price that differentiates a casino from some foolish charity. In a casino they take you for 5.26 percent in some foolish charity they just spin the wheel for entertainment and give you a fair deal.

If really depends on the game.

A game like Pai-Gow, you could play all day.

A game like craps (where you can get a bet out on every roll), it could go very quickly.

Keep in mind, unless you're goal is to be at the casino all day, you shouldn't try to "last a while"(exception to AP blackjack players)....

When I go to the casino, I try to wager about 1/10th my bankroll. If i lose it all in 10 rolls, then it's bad luck. I would say I am almost just as likely to double my money in 10 rolls (assuming I'm making good bets)...

TL;DR: Play Pai Gow Poker if you want to last. It's slow and you tie a lot (low game speed and variance).

The house edge is nowhere near as important a factor in your play time as money bet per hour variance are.

Money Bet Per Hour
For the sake of clarity (and easier typing) let's say "Money Bet Per Hour" is equal to: [Bets Per Hour] * [Average Bet]. This is called the "Handle" which is the total amount bet over a given session. For a $10 bettor on a 50 hands-per-hour BJ game playing for 2 hours, the handle would be $1000 (10 * 50 * 2).

Since we'll assume that your average bet will be the same on any game, such as you'll play $10/hand on Pai Gow Poker and $10/hand on Casino War and that ties just extend your play time, we can say that decisions per hour is the important metric here.

Assume Pai Gow Poker and Casino War have the same variance and you're betting the same flat bet on both. A full Pai Gow Poker table might deal 40 hands per hour while a solo game of Casino War might deal about 120 hands per hour. The Pai Gow table only has a decision 58% of the time while the War table has a decision 92% of the time. This means the decisions per hour on that Pai Gow table are roughly 23 while the decisions per hour on that War table are roughly 110.

In the above example, even though Pai Gow Poker has a worse house edge than War (8 Deck w/Bonus), 2.73 vs 2.34 respectively, you're going to tend to lose much faster on the War game due to decisions per hour.

Variance
The variance is roughly how far from the house edge you'll tend to swing. Imagine an incredibly boring slot machine that pays you back 95 cents every time you spin for $1. Since we know the payback will always be $0.95, never more or less, we can say the variance is 0. Take the following examples:

The variance for an even money payout on a $10 coin flip bet would be 100.
The variance for paying $350 (true odds) on a $10 bet to roll a 12 on two dice would be 3500.

I ran a million trials on each assuming you'd quit when you were down $100 or had lasted 100 flips/rolls.
In the coin flip game with a variance of 100 the player lasted 100 decisions 72% of the time.
In the dice game with a variance of 3500 the player lasted 100 decisions only 15% of the time.

Summary
House edge isn't really a big indicator of how long you'll last. The decisions per hour of a game and the variance are the real factors in how long a bankroll will last assuming you have time goal to meet. This should be somewhat obvious, but if you're playing on a slow game your money will last longer than on a fast game of the same type. If you're playing a game that tends to payout a lot or nothing, like a single number on roulette, you'll tend to have smaller average play times for a given bankroll versus an even money game. If you want to maximize your play time look for a slow game that pays near even money, like Pai Gow Poker. A Craps line bet is a decent alternative.

Side Note
In your case you might want to focus more on expected loss than house edge. The expected loss for a given game is:

[Hands per Hour] * [Hours Played] * [Average Bet] * [House Edge] = [Expected Loss]

A change in any of those is directly proportionate to your expected loss. For example, playing an hour at $10/hand on a 1% game that's dealt twice as fast as normal will double your expected loss. Playing that same game at normal speed with 2% edge will have the same effect on your loss.

**EDIT**
I re-read your question and realized I didn't even answer it properly.

I ran a million decision trial with a 1% even money game and a 2% even money game. The goal was to hit 100 decisions or to be down $100. Here's the results:

$10 Bet, 1% Edge:
Lasted 100 decisions: 70%
Lasted 500 decisions: 31%

$10 Bet, 2% Edge:
Lasted 100 decisions: 67%
Lasted 500 decisions: 25%

$5 Bet, 1% Edge:
Lasted 100 decisions: 95%
Lasted 500 decisions: 57%

$5 Bet, 2% Edge:
Lasted 100 decisions: 95% (close to 1% edge, rounded up)
Lasted 500 decisions: 49%

Quote: PopCan

Assume Pai Gow Poker and Casino War have the same variance and you're betting the same flat bet on both. A full Pai Gow Poker table might deal 40 hands per hour while a solo game of Casino War might deal about 120 hands per hour. The Pai Gow table only has a decision 58% of the time while the War table has a decision 92% of the time. This means the decisions per hour on that Pai Gow table are roughly 23 while the decisions per hour on that War table are roughly 110.


**EDIT**
I re-read your question and realized I didn't even answer it properly.

I ran a million decision trial with a 1% even money game and a 2% even money game. The goal was to hit 100 decisions or to be down $100. Here's the results:

$10 Bet, 1% Edge:
Lasted 100 decisions: 70%
Lasted 500 decisions: 31%

$10 Bet, 2% Edge:
Lasted 100 decisions: 67%
Lasted 500 decisions: 25%

(1) Where did you get the % of time a decision is made for Pai Gow or War? Did you simulated? How about for blackjack?

(2) Which games did you use to "run a million trial"? Pai Gow, War or blackjack variants? I was thinking of the play time (money lasts) between say, regular blackjack (HE <=1%) and blackjack switch (HW =>2%), for example.

Quote: UCivan

(2) Which games did you use to "run a million trial"? Pai Gow, War or blackjack variants?

I think for that simulation, he was using a coin flip - but simulating coins with a 1% and 2% bias.

Regardless of how he did it, nobody is going to sit there for a million hands, so the simulation is kinda irrelvant.

To your original question, play a 1% or 2% game with your $100 or $1000 bankroll is also irrelevant because you're not playing a million hands. Variance has much more to do with how long you play. Therefore, a game with a very high push rate, and relatively low hands per hour is the key. That's why he suggested Pai Gow Poker.

Quote: UCivan

(1) Where did you get the % of time a decision is made for Pai Gow or War? Did you simulated? How about for blackjack?

I got the decision rates from the Wizard of Odds pages on Pai Gow and BJ.

Quote: UCivan

(2) Which games did you use to "run a million trial"? Pai Gow, War or blackjack variants? I was thinking of the play time (money lasts) between say, regular blackjack (HE <=1%) and blackjack switch (HW =>2%), for example.

Quote: DJTeddyBear

Regardless of how he did it, nobody is going to sit there for a million hands, so the simulation is kinda irrelvant.

As DJTeddyBear said I was using a coin flip with a 1% and 2% bias. I also stated it incorrectly. I ran one million SESSIONS where the goal was to play 100 hands before being down $100. The "Lasted x decisions" percentages were the percentage of sessions where the goal of playing 100 hands was reached, regardless of win/loss. I think it's therefore a somewhat relevant indicator of how house edge affects your play time.

Quote: DJTeddyBear

Variance has much more to do with how long you play. Therefore, a game with a very high push rate, and relatively low hands per hour is the key. That's why he suggested Pai Gow Poker.

Yup, I'm a bit long-winded but that was my point. Variance and decisions per hour are more important than house edge to judge your play time.

Quote: PopCan

A full Pai Gow Poker table might deal 40 hands per hour while a solo game of Casino War might deal about 120 hands per hour. The Pai Gow table only has a decision 58% of the time while the War table has a decision 92% of the time. This means the decisions per hour on that Pai Gow table are roughly 23 while the decisions per hour on that War table are roughly 110.

Thanks PopCan for that detailed response, I learned several things as a result of your thorough explanation.

Question for you:

With regard to the techical term "Decisions Per Hour" of a game, does a "decision" only occur when a player wins or loses their entire main bet?

I ask because you indicate that in Casino War, there is only a decision 92% of the time (I am assuming that the remaining 8% refers to ties between the dealer and player's hand). But when a player ties the dealer, they have the option in Casino War to surrender 1/2 their wager or put up an additional bet and "Go to War". Eventually the hand is settled either as a win (of only the additional war wager), a loss of both wagers or parital loss (if they surrender and decide not to go to War).

Is this 8% original tie hand not a "decision" hand because the original wager may end up pushing (even if the War additional wager is won/paid). I guess I had thought that unless the ultimate outcome of a hand was a push (i.e. a split in PGP or a tie in BJ), every hand would be considered a "decision" and be factored into the "decision per hour" calculation.

Thanks for sharing your knowledge!

Ultimately the goal is to estimate win per hour, aka theoretical win (or just "theo"). There are two ways of going about doing that:
hands per hour * bet per hand * EV per hand
decisions per hour * bet per decision * EV per decision

Unless there is some reason to consider certain hand outcomes "not decisions", then the two formulae are equivalent. It's when you start mucking around with the distinction between pushes vs. ties vs. no-action that things get complicated. For example, a push in blackjack is not normally considered "no-decision" -- the EV on blackjack takes pushes into account (and they occur slightly more than every 12 hands). At the same time, the number of "decisions per hour" in blackjack is every hand dealt, pushes or not. On the other hand, the EV on the don't pass craps bet usually *doesn't* take pushes into account, and that's why you tend to see two numbers for don't pass: 1.4% and 1.36%. The Wizard calls these "Per bet resolved (not counting pushes)" and "Per bet made (counting pushes)", respectively.

What's important is not mixing things up when you do the computations. If you're using EV per bet resolved, then you also need to use decisions per hour. If you're using EV per bet made, you need to use hands per hour. As long as you're consistent, either approach leads to the same answer.

I'm not sure of your stopping conditions. A common stopping condition is to double your bankroll or to go bankrupt (whichever comes first). The parameter of interest to most people is the probability of doubling before you go bankrupt.

The highest value possible is to bet all your money on a single bet. Then the probability is the same as the house edge. Of course the variance is also a maximum, while the play time is as short as possible.

The longest play would be to flat bet at minimum allowable rate. Of course the maximizes the chance of going bankrupt before doubling. But it would also maximize the play time for that bankroll, for that game. In choosing a game, then decisions per hour would be the operative parameter (as stated above). For practical purposes, you should take your bankroll and bet minimum bet on the pass line in craps and never take advantage of free odds. That way you will need over 3 rolls of the dice to resolve each bet. If you pick a table with novice dice throwers, you can drag it out all night.

Quote: UCivan

What exactly is the relationship between house edge and play time?

HE (house edge) is just one part of the Survival-based risk of ruin formula.
It is used to calculate the ev (expected value) so it can be then used with the standard deviation of the bet in a few terms of the formula.

"Survival-based risk of ruin is a function of bankroll, bet size, number of decisions, and the probabilities and payoffs associated with various possible results"
For the math, some excellent reading and a link to an Excel worksheet by Alan Krigman, see:
Edge, Volatility, and Risk of Ruin in Gambling
The free excel worksheet is worth the visit.

Enjoy!

added: formulas to calculate HE and standard deviation per unit bet (they are not in the worksheet)

For a wager that pays x:1 with a probability of winning p, the house edge and standard deviation (per unit bet and per 'root decision') are:
he = (x+1)*p - 1 
std = (x+1)*Sqrt[p*(1-p)]

Wow, that's a lot of long winded answers :-).
OP, the answer to your question is "yes".

Quote: MathExtremist

Ultimately the goal is to estimate win per hour, aka theoretical win (or just "theo"). There are two ways of going about doing that:
hands per hour * bet per hand * EV per hand
decisions per hour * bet per decision * EV per decision

Unless there is some reason to consider certain hand outcomes "not decisions", then the two formulae are equivalent. It's when you start mucking around with the distinction between pushes vs. ties vs. no-action that things get complicated. For example, a push in blackjack is not normally considered "no-decision" -- the EV on blackjack takes pushes into account (and they occur slightly more than every 12 hands). At the same time, the number of "decisions per hour" in blackjack is every hand dealt, pushes or not. On the other hand, the EV on the don't pass craps bet usually *doesn't* take pushes into account, and that's why you tend to see two numbers for don't pass: 1.4% and 1.36%. The Wizard calls these "Per bet resolved (not counting pushes)" and "Per bet made (counting pushes)", respectively.

Thanks ME, that clears it up for me! I have always thought about every hand being a decision and using the EV per hand to determine theo.

I agree that if you find a cheap enough Pai Gow table then that's the choice. However one interesting point about "decisions" is considering two bets at craps. The Pass bet has a lower house edge than a six (or eight) but because the place bet does not work on the come out, on average making place 6/8 bets has a lower cost per 100 rolls. Essentially the house edge doesn't operate on any come out roll. Thus you might be better with a £3 (paying £3.50) place bet at a full craps table than a £10 Pai Gow one. Also note that some craps bets, on average, take longer than one roll for resolution - so inherently last longer.

The other factor when comparing games and their "house edge" is whether additional bets are required to play the game. Personally I think the stated value should be the cost of sitting, at say a $1 table, and how much it costs you on average to play one hand. However others take into account how many additional bets are usually made, looking at total money wagered rather than cost per initial bet made.

I think at blackjack it's the expected profit/loss of making an initial $1 wager and playing one hand, accepting that sometimes you'll need to split or double, and the game will result in a variety of net results including zero. Some other games, rather than the p/l for making the initial $1 bet, look at how much, on average, you bet (e.g. in an Ante/Raise game) and then dividing the profit/loss by the average number of bets. This method is useful for determining whether, for instance, it's better to put $20 on Red/Black roulette or $10 on a Ante/2xRaise game where sometimes you fold [$10] and sometimes you raise [total $30], so the average wagered might be $20.

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