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Home » Forums » Questions and Answers » Math » Calculating Return for "Lock & Roll" 5X Pay Quarter Machine
Calculating Return for "Lock & Roll" 5X Pay Quarter Machine
| December 9th, 2009 at 4:34:04 PM permalink | |
| Wizard Administrator Member since: Oct 14, 2009 Threads: 313 Posts: 6795 | Thanks. I agree with miplet's figure now. Thanks to all who helped. Here is a preview of what I plan to write about the game. It's not whether you win or lose; it's whether or not you had a good bet. |
| December 20th, 2009 at 10:03:16 PM permalink | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 |
There is a line:
It should be 5000 * 5 coins or 5000 bets. Also it should be noted that there is optimal strategy with 100% return for jackpot size of 42,746. And here are the return tables with just slightly improved outlook for better readability and a bit extended: Return on the Deal
Return on the Draw
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| December 20th, 2009 at 10:05:07 PM permalink | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | Full strategy for jackpot size up to 9300 bets (aka "5k" strategy).
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| December 20th, 2009 at 10:06:58 PM permalink | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | JP size dependence (change in hand values):
Difference in strategy from the base level of JP=5k. Should lock only '5x' in case of:
{1} respin only for "5x * 5x" and "5x 5x *" | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| December 20th, 2009 at 10:08:52 PM permalink | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | There are total of 113 optimal strategies, each starts at a shown JP size. 'Small return' is for return of small prizes only, i.e. without a jackpot. '5k return' is return if "5k strategy" is used instead of optimal. (I printed the whole table because it'll be helpful to compute the return of a play, described in the next message)
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| December 20th, 2009 at 10:11:24 PM permalink | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | Is "optimal" strategy really optimal? At first, I have a question: does anybody know, what is the grown rate of a JackPot size? 1/25 of a bet? It's strange, but I haven't seen on these sites the solution to the next problem. It's, as I understand, must be the well-demanded question to any progressive skill-based game, notably Video Poker. So, assume there is only one player. And he plays continually until he wins a jackpot. So, the questions are: 1) at what jackpot size he must start to play to get zero profit on average; and 2) what strategy he should use in every single game? To simplify computations, we can divide each bet by income into 3 pockets: a) small return (SR), which player gets back immediately with the probability of 1; 2) increase in jackpot size (dJP), which player gets after he wins; and 3) house income (1-SR-dJP). So, to find out 0-profit starting point, we can calculate total house income over the play. Obviously, they are equal.
Luckily, for fixed strategies (which player does not change while JP grows) it's simple: player pays to the house (1-SR-dJP)/P on average whenever he starts. And here is the table for 'early' strategies:
But for "optimal" strategies the sum of payments depends on starting point. So we need a sort of recursion. Hopefully, it doesn't need too much time to compute, so:
Look at the third row. These values are from the previous table. Seems like fixed strategy is always better than "optimal"! In other words, for calculating optimal long-run strategy, we must somehow include in the paytable dJP! I don't know how... yet. :) To prove that fixed strategy can be better, assume dJP=.111306 and starting JP=0. Obviously, if we choose fixed "0"-strategy with SR=.888694, then at any JP we won't pay to a house even a penny on average. But at the "optimal" strategy our SR will be lower when we reach JP level of 179. The things are even worse. We can try "intermediate" strategy, where we still play optimal strategies, but change them more rarely. Starting with the same strategy, we contribute to "virtual JP" (which points to optimal strategy) only the part of dJP. To be precise, k*dJP: k=0 if for strictly fixed strategy, k=1 is for fully "optimal" in terms of standard return table. Thus, for dJP=.08 we get:
So, at any k > 0 we lose more, and there is no better strategy, than fixed. I can't believe it's true. It must be a mistake somewhere. But where? I'm sure, that SR may be viewed as a single payment, especially on a long-term basis. I'm sure, it changes only variance, but not the return we are expecting. PS. Maybe plain simulation without any assumptions could confirm or disprove such analysis, but right now I'm too sceptical doing so. The game's variance is huge, and each "play" consists of almost 400,000 single games on average. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| December 21st, 2009 at 10:35:42 PM permalink | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | Ok, I've done simulations. Not for "Lock & Roll" machine, but for the simplest one I could imagine. It has 2 identical reels with 3 stops on each: 1 for "Seven" and 2 for "Bar". It pays 1 for a combination of "Bar Bar" and a jackpot payed on "Seven Seven". With pencil and paper, I've found there are 4 strategies and corresponding numbers of combinations are:
Return equals to 1 if JP is (81-36*1)/21= 2.1429. Here is the-same-way-calculated 0-profit table for fixed and optimal strategies. Again, fixed strategy is better than "optimal".
To prove these calculations, here are simulation results of 1 mln plays each. Numbers are net profit of a player, including starting jackpot value (JP0). dJP is 0.10.
PS. In the most primitive form the game may be reduced to 1 reel with 2 stops (a sort of "heads or tails"). Strategies and combinations are obvious:
Zero-profit table:
Simulation of 1 mln plays agrees with that (dJP=.25):
Any suggestions? Any suppositions? Any thoughts? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| December 22nd, 2009 at 9:42:07 PM permalink | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | The house edge for 'Lock & Roll'. It's (1-SR-dJP) - JP0*P for fixed strategy:
And here is for optimal strategy:
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| December 23rd, 2009 at 9:55:33 PM permalink | |
| DesM Member since: Dec 17, 2009 Threads: 0 Posts: 8 | Well, the question is over, at last. The key word for a fixed strategy to be useful was 'a single player game'. It lets you play longer and thus you'll harvest more, even though it returns less per every single bet. So, if there are only a few players (or you are a member of a huge organized group), and your style of play can affect the longevity of it, then you should milk this cow gently and slowly using appropriate fixed strategy (or, since there are some other players, change it a bit over time). As the number of players increases, you should not waste your time. So seek for a bigger single return and get as much milk as your force lets you, till you don't harm the cow. PS. The lowest zero-profit point and thus suitable fixed strategy can be easily determined by the requirement: Return=1-dJP. |
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