Interesting Hazard Variant

Not sure if this is the correct forum for this, but I have an interest in the history of games, and have been reading up on hazard and its variants throughout time (culminating in craps, of course). I came up with this hypothetical proto-hazard variant:

Wagers are even money between the 'caster' and any takers.
First roll the dice until a 4,5,6,7,8,9,10 is rolled which establishes the 'main' point. A roll of 2,3,11,or 12 is rolled again.
After a main is established, roll the dice again. A roll of 2,3,11 or 12 loses for the caster EXCEPT that if the main is 7 then 11 wins for the caster, or if the main is 6 or 8 then 12 wins for the caster. Repeating the main also wins for the caster. (e.g. if the main is a 7, then 7 or 11 wins for the caster, 2,3,12 looses). Any other roll establishes the casters 'chance'. The caster must roll again until either he rolls the 'chance' or the 'main'. He wins if rolls the 'chance' before the 'main'.

This is similar to hazard except that mains of 4 and 10 are allowed (they are not in hazard). What is the probability of the caster winning this game? The answer is surprising (to me): Exactly 1/2!

Quote: DrEntropy

This is similar to hazard except that mains of 4 and 10 are allowed (they are not in hazard). What is the probability of the caster winning this game? The answer is surprising (to me): Exactly 1/2!

I guess I need to say more about why I think this is interesting, since no one responded! Hazard itself, played "French style" which means you choose the main randomly by rolling until you get a 5,6,7,8, or 9 has a disadvantage for the 'caster' of -1.8%. (See wikipedia article if you are not familiar with this ancestor of craps). De Moivre in his "Doctrine of Chances" suggests making it much more closer to an even proposition by agreeing that the caster lose only a 1/3 of his stake when he rolled a 2 (crabs) on the second roll (the one that determines his "chance"). This makes it pretty even, about .02% off. However, I think it is interesting that you can make it a exactly fair game by just allowing 4 and 10 to be 'main' points as well, as described above. However I can find no evidence that the variant I describe was ever played.

Perhaps the reason is that many players involved in side betting prefered to wager after the main was chosen. In such a case, a main of 4 or 10 has too much of a player advantage. The other mains have only a minor player disadvantage.

DrEntropy,
There is a "Game Inventors' Corner" forum area at this site, for new game or variation ideas.....

Quote: Paigowdan

DrEntropy,
There is a "Game Inventors' Corner" forum area at this site, for new game or variation ideas.....

If the admin wants to move it, I won't object. Of course a perfectly fair game wont be very interesting to casinos!

Quote: DrEntropy

Of course a perfectly fair game wont be very interesting to casinos!

DrEntropy, you have - as the game's designer - every right and option to apply a house edge mechanism to provide for that casino edge and interest, to produce a "casino-worthy" game.

I have sent you an email PM of several ways to implement a house edge mechanism for this game that would provide a 2.78 to 3.5% house edge, appropriate for an even money to 2:1 payout bet. You can discuss this with Wizard Mike of MathExtremist, and either one can generate a math report for you.

I cannot describe them here, as that would be public disclosure, and would nullify their use in a patent, should you wish to later try to sell the game as a casino product. I wish no claim to it if you wish to seriously pursue.) Idea #3 I feel is best.

Again, new Dice games are hard to sell, but if you believe in it, and if it's fun to play, you may of course pursue it. You never know. At the very least, you would see how a game designer adds an edge to a game, while making it an unobtrusive as possible.

Quote: Paigowdan

DrEntropy, you have - as the game's designer - every right and option to apply a house edge mechanism to provide for that casino edge, to produce a "casino-worthy" game.

I appreciate the information Dan! However, my purpose in posting this variant of hazard was not to produce a new casino game. I have been playing with different variants of Hazard to explore the history of the game that led to craps. In this process I stumbled upon the variant I posted and was surprised to find it was a perfectly fair game. There is no obvious symmetry. I thought it was weird :). It is NOT a historical variant as far as I can discover.

Okay, then...I will note here the mechanisms. I had to ask you first, in case there was any remote commercial pursuit on your part, I would have to give the consideration.

You can:
1. Charge a commission like standard pai gow or Baccarat; less desirable, as it would really slow the game down, and annoy the players, as less than full even money makes the game feel like "less of a value."
2. Insert a "Bar the 12" type of clause to the game, where you'd push on a number rolled that the casino would otherwise lose. Craps uses the "Bar 12" to provide an edge on the don't side. On a 2 or 12, there's one way to make with 2 dice = 2.78%, but you'd need a mechanism that'll push when the player would otherwise win. You can, - out of ALL the combinations that a player can win on a point that is made, push on a player's point of a 4 or 10 that was made hardways, as they are harder to make numers anyway. It would push instead of lose for the house in ONLY that case as a rare exception, providing the house edge, and one which is in line with standards, and very fair to the player, too.
3. You can require that 2x odds be played, but be paid out at "place bet" payouts insted of true odds, which would supply a house edge to the main bet with "house edge odds" attached on a true odds pass line bet - the "reversed positions" but equivelant result of the current crap game's mechanism, with a slightly lower HE.

Also interesting to note is how to turn the current version of craps into a true odds game on every bet - a reversal of this process (mentioned earlier in an old post of mine):
1. Place bets pay at true odds, like line odds.
2. Buys/Lays = no commission (same thing);
3. Hardways are 10 TO 1/8 TO 1, not FOR.
4. Prop bets: 2/12 at 35:1, 3/11 at 17:1, sitto for hop bets
5. Don't pass/DC = 1/2 pay on 12 or 2, AND
6. Pass line/come bets = 1/2 lose on 12 or 2.

Quote: Paigowdan

5. Don't pass/DC = 1/2 pay on 12 or 2, AND
6. Pass line/come bets = 1/2 lose on 12 or 2.

These two modifications almost make it even (-0.025% if I remember correctly. Yes that percent! So very close).
It might interest you to know that this modification is mentioned in deMoivre's "Doctrine of Chances". Well, it is mentioned for hazard with the main always 7, but that is the same as craps. It is corollary three of problem 56.

delete

Quote: DrEntropy

These two modifications almost make it even (-0.025% if I remember correctly. Yes that percent! So very close).
It might interest you to know that this modification is mentioned in deMoivre's "Doctrine of Chances". Well, it is mentioned for hazard with the main always 7, but that is the same as craps. It is corollary three of problem 56.

Interesting to note that in Scarne's Complete Guide to Gambling he mentioned - for ALL the incredible amount of math he did - that he was unaware of how to bring the PL/DP to no effective house edge. He didn't read/consider "Doctorine of Chances" for Hazard in that area, a game which he also covered briefly in his book.

I hadn't read it yet, but what I simply did was this:

Pass Line = 1.41% HE.
Don't Pass = 1.36% House edge.

Now, with knowing that the bar-12 on the Don't Pass swings the HE by 2.78% from its natural player's advantage of 1.41% without it, (reverse of the PL), then half of 2.78% is 1.39%; so, a half win on the DP crap-12 has half the effect of the bar-12's 2.78% swing, or bring the DP line neutral, essentially. Figure 1.39%. 1.36%-1.39% = -0.03% with a trivial rounding error, but effectively neutralizing the HE there.

Same for making the Pass Line lose half on the bar 12 (or 2). 1.41%-1.39% of +0.02% or 0.0002, effectively neutral.

If you had a nickel DP bet and threw a comeout crap-12, and got paid two white ($2), that'll do.

The River Palms in Laughlin used to have pink $2.50 checks in their racks when I dealt there in '06, perfect for this as it is for paying a $5 BJ or $5 odds on a 5 or 9 the $7.50. The $2.50 chips would be great for Three card poker or Decues wild Hold 'em Fold 'em, turning a $15 or $20 game into a more doable $7.50 or $10 game at nickel houses.