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Subject: What's the best way to achieve this mechanism? rss

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Andreas T Chan
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I have a game idea where each period you earn a number of attempts (x) each of which has probability p of succeeding, where p = 1/y, and y is an integer. Y will normally be from 8 to 14, but it might vary slightly outside that. X will normally be 5 to 15, but it might vary slightly outside that.

So, you might have 10 attempts with a 1/11 chance each of succeeding. Or 12 attempts with a 1/8 chance, or 12 with a 1/14 chance and so on.

Edit: the number of successes matters.

The 'cleanest' way I can think of is to include lots and lots of custom dice in the game. Or you could include 1 of each type (8 sided, 9 sided etc.), and then roll them a lot, but that doesn't sound like a lot of fun.

A nice alternative I thought of is that each player gets their own set of cards with number 1, 2 ... (say) 20 on them. Once they find out their probability of success p, they select cards 1, 2 ... y. They shuffle them and deal them out x times. If the first card they deal is a 1, it counts as a 'success', if the second card they deal is a 2, it counts as a success and so on. The observations aren't i.i.d, so it's not ideal, but for y > 6 that doesn't bother me too much.

I think I'm overlooking something and that you guys have a much neater idea. Please prove me right!
 
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Seth Iniguez
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Not particularly elegant, but you could have a chart that shows the percentage chance of each possible event, and a single set of percentile dies to check the resolution.
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Jeremy Lennert
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Does the exact number of successes in a single period matter, or only whether or not you succeed at least once?

Do you have to use those exact probabilities? Seems like a strange and highly specific requirement; if I were you, I'd ponder whether that's truly important or just something that sounded cool. (One thing that leaps out at me is that those aren't equal steps; the difference between 1/8 and 1/9 is considerably bigger than the difference between 1/13 and 1/14.)

Would it be acceptable to have a system that produced the same average number of successes but with a different overall distribution?

Off hand, I'd say the odds of any game that requires a d13 being commercially viable are slim. The card idea is fairly clever, though.
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Maarten D. de Jong
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In general complex or non-standard PDFs cannot be approximated well by human players. They either involve a percentile chart as already suggested (which is what wargames usually do in one form or another: your alternative solution is rather like a one-sided Battle Deck), or lots of mindless and unexciting manipulation of ordinary materials.

My suggestions: make the probabilities far simpler to manage so the tediousness of doing a full simulation is removed; create an app, not a board game (computers can do the boring math in the blink of an eye); or bite the bullet and include percentile charts.
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Richard Irving
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Dice don't come in every number of sides. D8, D10, D12 are pretty common. D14 much less so.

D9, D11, D13 do not exist--but can possibly be simulated by rolling one die larger and rerolling any time the maximum shows up (but that could a bit tedious.).
 
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Andreas T Chan
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Antistone wrote:
Does the exact number of successes in a single period matter, or only whether or not you succeed at least once?


The exact number matters.

Antistone wrote:
(One thing that leaps out at me is that those aren't equal steps; the difference between 1/8 and 1/9 is considerably bigger than the difference between 1/13 and 1/14.)


The concavity is deliberate and adds to the game.

Antistone wrote:
Would it be acceptable to have a system that produced the same average number of successes but with a different overall distribution?

Maaaaybe.


Antistone wrote:
Off hand, I'd say the odds of any game that requires a d13 being commercially viable are slim.


Agreed. Especially if it required you to roll it multiple times. Or to roll multiple d13s!

Antistone wrote:

The card idea is fairly clever, though.


Thanks.

cymric wrote:
In general complex or non-standard PDFs cannot be approximated well by human players.


Isn't this part of the appeal? Would you want all poker players to know the exact probabilities?

cymric wrote:
My suggestions: make the probabilities far simpler to manage so the tediousness of doing a full simulation is removed; or bite the bullet and include percentile charts.


i) My op was terse, but the player isn't going to be faced with a simple 'Which is more likely for me to get two successes with, 8 chances with 1/12 each probability of success or 5 chances with 1/7 ...' ... there's going to be lots of other variables, and it's an interactive game ... so his true thinking will be something like

'Well, I could draw this card. If it's what I want, it will lower my probability of success from 1/7 to 1/8, but I'll get 2 more chances. But if it's not what I want then ... and of course this is all assuming my opponent's hand is what I think it is ... if it isn't then I want to draw three cards ... but if I draw three cards and his hand is what I think it is ...' etc.

In the same way that you *could* do complicated maths when deciding whether to draw in TTR, but in practice you'll use heuristics.

cymric wrote:
create an app, not a board game (computers can do the boring math in the blink of an eye)


My (nascent) plan is to release it as a board game and then as an app. But I definitely appreciate the suggestion and it's something I'm willing to consider.

cymric wrote:
or bite the bullet and include percentile charts.


How do these work? You randomly choose a number from 1 to 100 (via some mechanism) and then look in a chart to see how many successes it corresponds to?
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Trevyn Hey
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AndreasTChan wrote:
cymric wrote:
or bite the bullet and include percentile charts.


How do these work? You randomly choose a number from 1 to 100 (via some mechanism) and then look in a chart to see how many successes it corresponds to?


Edit: Maybe I misunderstood, and the number of successes matters? Below is only for a single success.

I think what was intended was this:
Along one side you have the number of chances.
Along adjacent side you have the probability.

So a chart would be:
_ 1/2 1/3 1/4 1/5 1/6 ...
1 50% 33% 25% 20% 17%
2 75%
3
4
5
6
7
...

And in the center is the probability of success for each combination.

If your d100 roll is at or below the listed value, you succeed (or you could flip the chart and try to roll above)
Edit: Added some example percentages.


If you want multiple successes, you might have to break up the charts. I'm not great with statistics, so there is probably an easier way.

1/2 Probablilty
# of Successes across the top
# of Dice along the side


_ 1 2 3 4 5
1 50 0 0 0 0 0
2 50 75
3 17
4
5

So if you rolled between 51 and 75 with "2 chances" at "1/2 probability" you would get 2 successes.
 
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Jeremy Lennert
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AndreasTChan wrote:
How do these work? You randomly choose a number from 1 to 100 (via some mechanism) and then look in a chart to see how many successes it corresponds to?

That's the general idea for a table, yes: settle on some standard dice you're willing to roll, then create a table that maps the output of those dice onto the actual distribution you want (in this case, the number of successes) as closely as possible.

D100 is a popular choice for the distribution because most people are more comfortable with rolling 2d10 and multiplying one of them by 10 than they are with, say, rolling 2d20 and mutliplying one of them by 20.

For example, if you use d100 and you're creating a table column for 2 chances with 1-in-7 odds, you'd probably have 2 table entries saying "2" (because 1-in-49 is pretty close to 2%), 24 or 25 entries saying "1", and the rest (73 or 74) saying "0".

Unfortunately, in your case you'd need a 3-dimensional table. (Or just the "one chance" slice of it, and roll on it multiple times.) Realistically, you would present that 3D table as a series of 2D tables, but still, that's a lot of tables. I don't think a lot of players would be happy with that.


If you're dead set on these mechanics, then so far your card idea is my favorite option.
 
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rri1 wrote:
Dice don't come in every number of sides. D8, D10, D12 are pretty common. D14 much less so.

D9, D11, D13 do not exist--but can possibly be simulated by rolling one die larger and rerolling any time the maximum shows up (but that could a bit tedious.).


Any number of sides, 3+, can be on barrel log dice.
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James Hutchings
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You could have custom dice which give the same spread of results as rolling several dice.

For example, an eight-sided dice can simulate tossing three coins, and counting the heads, if the faces are marked as follows:

1 face showing 3.

3 faces showing 2.

3 faces showing 1.

1 face showing 0.
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Kristian Järventaus
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Have a bunch of d16 (don't exist or very uncommon, but your numbers are pretty awkward for normal dice. ).

If Y is 9 and number of dice is 7, roll seven of them and reroll ones that come up 10+. If you roll a 1, you succeed.
 
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Maarten D. de Jong
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AndreasTChan wrote:
cymric wrote:
In general complex or non-standard PDFs cannot be approximated well by human players.

Isn't this part of the appeal? Would you want all poker players to know the exact probabilities?

You misunderstand me. The PDF of Poker is not the result of someone sitting down with a list of probabilities and then approximating it with a deck of cards. No, it was quite the reverse: the deck of cards was a given and then a list of combinations was created with intuitively rarer and rarer combinations. You face the challenge of doing it the other way around, and then my statement applies.

Quote:
cymric wrote:
My suggestions: make the probabilities far simpler to manage so the tediousness of doing a full simulation is removed; or bite the bullet and include percentile charts.

i) My op was terse, but the player isn't going to be faced with a simple [...] In the same way that you *could* do complicated maths when deciding whether to draw in TTR, but in practice you'll use heuristics.

Yes, but again that was not the point I was making. You want to create a complex PDF, but with the simple randomising materials you have available—dice, cards—that PDF is cumbersome to generate. So while the game may mathematically be just as you plan, it is simply a chore to play because players will spend a lot of time throwing dice and drawing cards for no apparent purpose.

Hence the above suggestion: simplify the PDFs until you are left with something that can be approximated with just a few dice rolls or card draws so that players are free to ponder the interesting parts of your design instead of being robots.

Quote:
cymric wrote:
create an app, not a board game (computers can do the boring math in the blink of an eye)

My (nascent) plan is to release it as a board game and then as an app. But I definitely appreciate the suggestion and it's something I'm willing to consider.

Well, I only suggested it for the reasons of getting the PDF as you intended. Computers don't mind doing this sort of stuff, and the users won't notice the drudgery the machine has done for them.

Quote:
cymric wrote:
or bite the bullet and include percentile charts.

How do these work? You randomly choose a number from 1 to 100 (via some mechanism) and then look in a chart to see how many successes it corresponds to?
[/q]
Precisely. Whether you have an ordinary PDF or its cumulative variant is up to you. The advantage is that the chart is simply doing all the hard work, and that changing the numbers there is easy as pie.
 
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Oliver Edleston
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Use the card based solution. It's perfect.

Assume p is 1/11 (so 11 cards).
Take cards numbered 1 to 11 and shuffle.
Chance that the first card drawn is 1 is clearly 1/11.
Chance that the second card drawn is 2 is:
1/11 * 0/10 (first card is 2) +
10/11 * 1/10 (first card is not 2 and the second card is 2) = 0 + 1/11 = 1/11
Chance that the third card drawn is 3 is:
1/11 * 0/9 + 1/11 * 0/9 (chance that first or second card is 3, as shown by the above results) +
9/11 * 1/9 (first 2 cards are not 3 and the third card is 3) = 0 + 0 + 1/11 = 1/11
etc.

The only issue is when you have more chances than the probability denominator, meaning you have to lay out all the "chances" then collect them up, shuffle and repeat for the last few chances. So 15 chances with p = 1/11 means 11 cards get laid out, the successes counted, then the 11 cards are collected re-shuffled and the first 4 are laid out for the remaining chances.
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Kevashim wrote:
Use the card based solution. It's perfect.

Assume p is 1/11 (so 11 cards).
Take cards numbered 1 to 11 and shuffle.
Chance that the first card drawn is 1 is clearly 1/11.
Chance that the second card drawn is 2 is:
1/11 * 0/10 (first card is 2) +
10/11 * 1/10 (first card is not 2 and the second card is 2) = 0 + 1/11 = 1/11
Chance that the third card drawn is 3 is:
1/11 * 0/9 + 1/11 * 0/9 (chance that first or second card is 3, as shown by the above results) +
9/11 * 1/9 (first 2 cards are not 3 and the third card is 3) = 0 + 0 + 1/11 = 1/11
etc.

The only issue is when you have more chances than the probability denominator, meaning you have to lay out all the "chances" then collect them up, shuffle and repeat for the last few chances. So 15 chances with p = 1/11 means 11 cards get laid out, the successes counted, then the 11 cards are collected re-shuffled and the first 4 are laid out for the remaining chances.

Or a d12 with 12 being a reroll. (cheaper than making d11 barrel dice)
 
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Mike L.
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Have you tried looking at the probabilities of success for each of your cases and using statistics and math (I know they are scary) to determine an equivalent set of dice rolls (it will probably require multiple different kinds of dice for your distribution) to get the probabilities you are looking for?

You can probably cut things down from 14 dice down to 2 or 3. If you are bad at math, ask around here, there are a lot smart people.
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Lucas Smith
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you will be able to modify pretty much every random experiment: rolling dice, drwing cards, taking things out of an urn (bag), flipping coins (not so good), you could even build you own "machine" like a roullett,...

maybe the card method you thought of, is already the best! you could modify this to the urn model: chips with the numbers 1-Y; you take X chips out, every chips matching to the order (1=>1st, 2=2nd) is a success.

I think -for me- it would depend on the game: Should it be a childrens game (then I'd maybe take the urn model) or a dice game (obvious one), a strategy game, where this is only a little detail (take the quickest one then!),a game including a funny "machine",....


Also think about the components: Are there any other cards? if yes the cards variant might be cheaper. Are there any other die? if yes the cards variant might be cheaper.

I think for the maths it doesnt matter, you can modify every "experiment", so if you can't decide, take the cheapest one.-

Good luck!
 
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John Breckenridge
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I picture a spinner on a field of concentric rings, with each ring sliced into more sections than the one inside it, one section in each ring colored and marked with its number, so your player would get X spins to hit the target in ring Y.
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Mark J
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Doing it with n-sided dice would, I think, make the game prohibitively expensive. At the very least, you would need one die of each possible size, from D8 to D14 or whatever the minimum and maximum are. As others have noted, finding a D9 or a D11 die might be difficult. And that assumes that the player will roll the same die repeatedly. If you want the player to just roll once, you'd need 15 dice of EACH size -- assuming 15 is the maximum number of rolls -- or 15 * 7 = 105 dice!

A logically simple solution would be to have a deck of 1 success card and 13 fail cards. Then take the success card, and a number of fail cards equal to 1 less than Y. Shuffle and flip one card. Note whether it's a success or a failure. Then shuffle it back into the deck and flip another. Etc, for X shuffles and picks. (You have to shuffle the picked card back into the deck or the probabilities for the second draw won't be the same as the probabilities for the first draw.) I think actually doing this would be a pain, though.

Your idea of numbered cards, etc: Works if x<y. If x>y you have to deal out the entire deck, count the successes, then reshuffle and do another deal of x-y cards. Maybe multiple cycles if x>2*y, etc.

Mechanically, you could count the cards as you play them. That is, say "1" and turn over a card, if it's a 1 that's a success. Then say "2" and turn over the next card. Etc. My gut feel is that that would be awkward, that players would get confused and, for example, say "4", turn over a 5, and then have 5 in their head and say "6" as they turned the next card. Trying to deal with two sets of numbers at the same time tends to be confusing to human beings.

Probably better would be to lay them out in a row, better still if you had a matt with numbered spaces. Then you could look for matches between the number on the matt space and the number on the card, and count the number of successes.

But as others have said, this all sounds like too much trouble to me. I'd be much inclined to figure out what probability you want, and then have a chart where you roll a regular 2D6 or some such and read off the success or failure. Or change it from X rolls with probability 1/Y to X-prime rolls all with probability 1/6 and roll X-prime D6's.
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Mark J
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AndreasTChan wrote:

cymric wrote:
In general complex or non-standard PDFs cannot be approximated well by human players.


Isn't this part of the appeal? Would you want all poker players to know the exact probabilities?
[/q]

Umm, yes, I would.

If you make the probabilities difficult to calculate, that doesn't mean that no one will know the probabilities. It just means that you reward the person who does a bunch of tedious calculations, because now he has an advantage.

Personally, I dislike games that reward players for doing tedious calculations. I want to do my best to win a game, but at the same time I want it to be fun. So if the way to win is to do something really really boring, I am stuck between trying to win and trying to have fun. I'd rather not play that game. If I wanted to do tedious calculations, I'd get a job as an accountant or tax preparer.

If I was playing your game, I'm sure that if I thought it enough fun to play repeatedly, I would quickly do the calculations and make myself a chart with a row for each X and a column for each Y and the probability of success and/or the expected number of successes in each cell. And unless I was playing for money or something, I'd consider it elementary good sportsmanship to share that chart with other players.
 
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Mark J
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jbrecken wrote:
I picture a spinner on a field of concentric rings, with each ring sliced into more sections than the one inside it, one section in each ring colored and marked with its number, so your player would get X spins to hit the target in ring Y.


Excellent idea! Unlike having to assemble a deck of cards with the right numbers, the spinner is just ... there. You just have to look at the right ring for each spin.
 
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Mike L.
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saneperson wrote:
jbrecken wrote:
I picture a spinner on a field of concentric rings, with each ring sliced into more sections than the one inside it, one section in each ring colored and marked with its number, so your player would get X spins to hit the target in ring Y.


Excellent idea! Unlike having to assemble a deck of cards with the right numbers, the spinner is just ... there. You just have to look at the right ring for each spin.


Unfortunately, spinning something a dozen times is just as obnoxious as rolling a dice a dozen times. It also introduces bias, as a player can become skilled at spinning. Especially if they have to do it hundreds of times to play a game.
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