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Help: Making a 2-Table Tourny Set work for 3 Tables (2 Viewers)

Taxi500

Taxi500

3 of a Kind
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My home game has expanded much more rapidly than expected. I bought a two table set back when that would be a stretch to achieve but a year ahead and I will likely have 3 tables full for our semi-annual tourny.

Edit: Assume 24 players - I consider "Full" 8 players because of my table's sizes.

PCF: How do I make this set work for 3 tables?

T25 x 160
T100 x 160
T500x 120
T1k x 160
T5k x 80
T25k x 20
 
What LeLe said, buy more chips.

Would help to know how you've been running your tournament. Starting stack size, how long are blinds? It's possible you might be able to fudge, but, likely you just need more chips.
 
Bare-bones T25-base set for 3 tables (30 players, 8/8/4/7 = 10k stacks):

240 x T25
240 x T100
120 x T500
210 x T1000
20 x T5000
--‐-----‐----
830 chips, using T5000 for all color-ups

Re-buys, add-ons, and/or stacks larger than 10k requires more T5000 chips.


You need 210 more chips, minimum:

80 x T25
80 x T100
50 x T1000
 
Modifying @BGinGA 's numbers above for 24 players at 8/8/4/7 you have (approx):

200x T25
200x T100
100x T500 (total T50K)
175x T1000 (total T175K)
25x T5000 (total 125K)

which should cover your 3 tables and have enough for colour ups and 11 rebuys
so you're about 1 rack away with 2 barrels T25, 2 barrels T100, 1 barrel T1000
you're golden with the T5K and you won't really need the T25K
 
If you can get 140x T25 and 140x T100, then you can have 12/12/5/6 stacks for less change-making in the early rounds, and not require any more higher denoms (if they would otherwise be very expensive or hard to get).
 
Oh God I forgot about rebuys too.. alright boys.. next SPW group buy and I'm smashing the buy button. Appreciate all your insights.
 
I have probably mentioned a dozen times, get more chips than you think you need.

More people need to mash the like button on those posts, because the point just isn't getting through.

@ekricket is right. You can stretch a set with uneven stacks, but more change making. It isn't ideal, and may make your game less fun, but it's better than turning players away.
 
Poker Zombie said:
I have probably mentioned a dozen times, get more chips than you think you need.

More people need to mash the like button on those posts, because the point just isn't getting through.

@ekricket is right. You can stretch a set with uneven stacks, but more change making. It isn't ideal, and may make your game less fun, but it's better than turning players away.
Click to expand...
With you here: In my defense when I bought these a year ago it was a stretch for me to get more than 12 guys and this set was liberal with chip stacks... now they're chomping at the bit and much more guys want to play.
 
Taxi500 said:
With you here: In my defense when I bought these a year ago it was a stretch for me to get more than 12 guys and this set was liberal with chip stacks... now they're chomping at the bit and much more guys want to play.
Click to expand...
I remember playing 5 handed, because that's all we could get.

We're consistently pushing 20 now.

#FirstWorldProblems
 
I think I can thread the needle here and squeeze out 24 stacks using the status quo.

You can do 20 stacks of 8/8/4/7 with what you have now and have 40*T500 and 20*T1000 left over.

With those leftovers you can build 4 stacks of 10*T500 and 5*T1000 to hit 24 stacks on the nose.

So I would give your first 20 players the standard 8/8/4/7 stacks and then the last 4 players get the 0/0/10/5 stacks.

This might still be tight for change making but may be just barely doable.

But personally, I think just adding 80 each of T25, T100, and T1000 will take your set to the perfect space for 30 buy ins along the lines of what @BGinGA suggested.

Even just doing 40 of each gets to you 25 buy ins as @allforcharity said.

Bottom line, you are short on chips, but not ridiculously so.
 
Last edited:
allforcharity said:
Modifying @BGinGA 's numbers above for 24 players at 8/8/4/7 you have (approx):

200x T25
200x T100
100x T500 (total T50K)
175x T1000 (total T175K)
25x T5000 (total 125K)

which should cover your 3 tables and have enough for colour ups and 11 rebuys
so you're about 1 rack away with 2 barrels T25, 2 barrels T100, 1 barrel T1000
you're golden with the T5K and you won't really need the T25K
Click to expand...
Pm incoming about this
I got the 2 table set up for 12/12/5/6 and would like to make it 3
I play with alot of new guys and have tried the 8/8/4/7
early on it has never worked good unless its all the poker vets playing and its easy. I feel like with newer people making change really slows down the game and some people are like is this right everytime LOL
I was thinking
So were talking about
300 x T25
300 x T100
120 x T500
144 x T1k
25 x T5k
 
So, the post I made about chip breakdown is for 12/12/5/6 for two tables where the host wants to cover 3 full tables with the minimum extra expenditure for higher denoms.

Of course, it is best if you can keep 12/12/5/6 for all 3 tables, but it usually costs a fair bit more with all the extra T25/T100 required, but if you've got the money then go for it.

I have a set that covers 10 tables (100 players) with 12/12/5/6(/2) to cover T10-20k stacks, but I'm nuts!
 
allforcharity said:
So, the post I made about chip breakdown is for 12/12/5/6 for two tables where the host wants to cover 3 full tables with the minimum extra expenditure for higher denoms.

Of course, it is best if you can keep 12/12/5/6 for all 3 tables, but it usually costs a fair bit more with all the extra T25/T100 required, but if you've got the money then go for it.

I have a set that covers 10 tables (100 players) with 12/12/5/6(/2) to cover T10-20k stacks, but I'm nuts!
Click to expand...
OMG thats Amazing 100 player Tourney would be so cool
 
In the olden days, a tournament, even the first ones on TV like Late Night Poker, used to have smaller stacks. Often the same as the buy in, $1500. It used to be nice that the chip values weren't complete nonsense, even if they weren't literally true.

You can run a tournament starting with 4*25, 4*100 and 2*500 each, blinds start at 25/25. You can handle 40 runners, and colour up.

And it does actually work fine, a lot of the fashion for deep starting stacks and a fast shallow finish isn't necessarily a better game.
 
You can just about see those tiny stacks in this grainy photo.

1000025932.png


People joke about short stacks, but the reality is little happens for most players until a tournament has around 40bb, this starts at 60bb.

Late reg preference shows people have realized the early levels are low value, just get it going.
 
Dbxela said:
You can run a tournament starting with 4*25, 4*100 and 2*500 each, blinds start at 25/25. You can handle 40 runners, and colour up.

And it does actually work fine, a lot of the fashion for deep starting stacks and a fast shallow finish isn't necessarily a better game.
Click to expand...

BGinGA said:
No, it doesn't.
Click to expand...

I did actually try an come up with an "impossible" change situation with the 4/4/2 starting stack, and the only impossible situations I could find require moving a player off of a table. So if you never move players this is technically "doable." But nearly every pot will require change for multiple players. Most hosts here see benefit in trying to limit that, though we recognize there is no way to eliminate change making entirely without going to absurd counts.

Since we have established a lower boundary for required change, I got to thinking about what the highest boundary would be.

To 100% prevent any change-making, the TD could issue an entire starting of T1500 using T25 chips, 60 per player.

Or for a different example, if it's a T10K tournament for example, you COULD issue 4 racks of T25 per player, or you could calculate about how many hands you expect T25 chips to be in play, multiply that by 3 and that would be the maximum needed. So if you play 4 levels with T25 in play and 10 hands per level, that's 40 hands, or 120 T25 chips to guarantee (Though really, this would be assumption that every hand a player puts in a total that ends in 75 on every hand, it's possible to make multiple bets in a hand that end in 75, I suppose so maybe again just issue 4 racks of T25 after all to be 100% sure. )

So having considered a lower boundary and upper boundary for number of small chips in a starting stack, I starting thinking about how often we talk about more small chips reducing change making. I think everyone understand the trend by instinct, but we haven't made an effort to quantify this on PCF.

I am a database developer by trade, so I decided to use this skill with some basic statistics knowledge to completely over-engineer this question. Get comfortable...

Assume a 8-handed table using 4 green (T25) chips per player, so there are 32 on the table in all. On a given round, what percentage of the time would expect a at least one player to be short on the number of green chips required to construct all bets (meaning have fewer than 3 in their stack).

Using database code, I was able to generate a list of the possible ways assuming each player has a random number of green chips between 0 and 32 and the sum of all players chips equals 32. There are 3319 such ways without consideration of the order of players.

SQL: 
    SELECT DISTINCT
        d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
        , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortCount
        , d1, d2, d3, d4, d5, d6, d7, d8
        --, *
    INTO #SumOf32
    from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d8 (d8)
    WHERE
        d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 32
        AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8

Using this we can estimate how often a player would need change by counting how often players are short of the minimum number of chips requisite to construct all bets. In the case of T25, a player with fewer than 3 chips cannot construct all possible bets without change. I calculated that there are only 22 combinations out of the 3319 in which all 8 players would have at least 3 green chips. Or 0.67% of the time. That means for every 150 hands in which there is a bet ending in 75, in 149 instances, at least one player will possibly require change. (Say this player takes the action roughly 1 in 3 times, this means about 49 out of 50 hands in practice require change.)

1717601148582.png


(On a side note, since there are no results where all 8 players are short, I have very technically proven 4/4/x is possible provided 32 chips are on the table, no players moved. I have just quantified how much change-making is required and it's staggering.)

SQL: 
declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf32) -- 3319

SELECT ShortCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf32
GROUP BY ShortCount
ORDER BY ShortCount

So maybe we should throw out some of the extremely lopsided combinations. Let's say no player can acquire more than 16 greens without immediately making change with another player. So if a player instead has a random number between 0 and 16 chips, This reduces the possible combinations to 2725. However, that only moves the needle to how many hands there will be with no short players players up to 0.80% from 0.67%. We are now talking 124 hands out of 125 in which at least 1 player will possibly require change. (And again if we assume a given player will take the action required to put chips in 1 out of 3 times, this is still 39 times in 40 where change making will be required.)

1717601986210.png


SQL: 
DROP TABLE IF EXISTS #SumOf32_16Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8
    --, *
INTO #SumOf32_16Cap
from
    (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d1 (d1)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d2 (d2)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d3 (d3)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d4 (d4)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d5 (d5)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d6 (d6)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d7 (d7)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d8 (d8)
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 32
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf32_16Cap) -- 2755

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf32_16Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount

Now let's run this experiment for 64 greens on the table. (8 per stack,) For consistency, let's scale the max up to 32 so no player has more than half the greens in play to keep the high end outliers out.

We are now up to 102206 ways to distribute 0 to 32 chips to 8 players with a total of 64 on the table.

SQL: 
DROP TABLE IF EXISTS #SumOf64_32Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8
    --, *
INTO #SumOf64_32Cap
from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d8 (d8)
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 64
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf64_32Cap) -- 102206

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf64_16Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount

1717604615607.png


So we are up to about 9.4% of instances where no player possibly needs change when a bet ends in 75. Up considerably from 0.8% before. Over 11x as often no change required.

But what this illustrates is doubling chips yields about an 11x improvment in this situation. Suffice it to say at this point, it's pretty obvious this isn't the pointless extravagance that @DBexla is portraying.

But still we are looking at a little over 90% of instances requiring at least one player possibly needing change. (And if you want to assume that a player short of the required chips is also taking the given action 1/3 of the time, we are at about 70% of hands in practice require a change transaction of some sort.)

So let's take this one step further and put 96 greens on the table (12 per player. I am going to keep the max per player at 32 greens, since I found that increasing this to 48 will add about 26x the complexity to the query and force to run several hours.)

We are now up to 587819 ways to distribute 0 to 32 chips to 8 players with a total of 96 on the table.

SQL: 
-- Seriously how many of you opened all four of these?


DROP TABLE IF EXISTS #SumOf96_32Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8
    --, *
INTO #SumOf96_32Cap
from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d8 (d8)
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 96
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf96_32Cap) -- 587819

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf96_32Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount
I will also note to those of you that are reading the code, I understand this approach is quick and crude, I am certainly aware there are ways to parametrize what I am doing for reusability. This was just a quick experiment.

1717607289139.png


So we rocket up to 27% of instances where no player possibly needs change when a bet ends in 75. Making the 12/x/x method nearly 3x better than 8/x/x and a staggering 34x better than 4/4/x.

So in 73% of instances there will still be one player that would need change if taking the action to put the chips in. If a player does this only 1/3 of the time as I speculated in the other scenarios, then in practice, fewer than 25% of instances require change making. So the difference between going 8/8/x to 12/12/x is that change making now takes place in a minority of rounds. I think this reveals why some hosts prefer the jump to 12/12/x (I would be one) over tolerating the extra required by 8/8/x. But both are clearly, massively better than 4/4/x.

To recap the trail, we have gone from change making being a near 100% certainly in a round with a bet ending in 7x in the 4/4/x scenario, to being only about a 25% occurrence in the 12/12/x scenario.
If we ramped this up to 128 chips on the table, the 16/16/x scenario, I would guess we would get another increase, but maybe not as significant as the increases in the two prior scenarios, and some point we will reach diminishing returns if we repeat this scenario enough. But I think this has gone way past keeping a long story short. Conclusion coming.

1717606282156.png

To close this, I admit my premises may not be perfect here. There are probably tweaks to the realistic max per player that could be more real-world, which would show a reduction in change making overall. I did deliberately use the word possible, we can quibble over what percentage of the time a player needing change for an action will actually take such an action, but that would affect all scenarios fairly equally. I figure the premises are close enough to at least fairly indicate the trend about how much change-making you can reduce by adding small chips to the starting stack, and that any tweaks probably affect all scenarios evenly, I think I am at least close on the factor of improvement.

Bottom line, short of the point of absurdity, more small chips always means less change making. This is my attempt to quantify how much less.
 
Last edited:
JustinInMN said:
I did actually try an come up with an "impossible" change situation with the 8/8/4 starting stack, and the only impossible situations I could find require moving a player off of a table. So if you never move players this is technically "doable." But nearly every pot will require change for multiple players. Most hosts here see benefit in trying to limit that, though we recognize there is no way to eliminate change making entirely without going to absurd counts.

Since we have established a lower boundary for required change, I got to thinking about what the highest boundary would be.

To 100% prevent any change-making, the TD could issue an entire starting of T1500 using T25 chips, 60 per player.

Or for a different example, if it's a T10K tournament for example, you COULD issue 4 racks of T25 per player, or you could calculate about how many hands you expect T25 chips to be in play, multiply that by 3 and that would be the maximum needed. So if you play 4 levels with T25 in play and 10 hands per level, that's 40 hands, or 120 T25 chips to guarantee (Though really, this would be assumption that every hand a player puts in a total that ends in 75 on every hand, it's possible to make multiple bets in a hand that end in 75, I suppose so maybe again just issue 4 racks of T25 after all to be 100% sure. )

So having considered a lower boundary and upper boundary for number of small chips in a starting stack, I starting thinking about how often we talk about more small chips reducing change making. I think everyone understand the trend by instinct, but we haven't made an effort to quantify this on PCF.

I am a database developer by trade, so I decided to use this skill with some basic statistics knowledge to completely over-engineer this question. Get comfortable...

Assume a 8-handed table using 4 green (T25) chips per player, so there are 32 on the table in all. On a given round, what percentage of the time would expect a at least one player to be short on the number of green chips required to construct all bets (meaning have fewer than 3 in their stack).

Using database code, I was able to generate a list of the possible ways assuming each player has a random number of green chips between 0 and 32 and the sum of all players chips equals 32. There are 3319 such ways without consideration of the order of players.

SQL: 
    SELECT DISTINCT
        d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
        , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortCount
        , d1, d2, d3, d4, d5, d6, d7, d8
        --, *
    INTO #SumOf32
    from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d8 (d8)
    WHERE
        d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 32
        AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8

Using this we can estimate how often a player would need change by counting how often players are short of the minimum number of chips requisite to construct all bets. In the case of T25, a player with fewer than 3 chips cannot construct all possible bets without change. I calculated that there are only 22 combinations out of the 3319 in which all 8 players would have at least 3 green chips. Or 0.67% of the time. That means for every 150 hands in which there is a bet ending in 75, in 149 instances, at least one player will possibly require change. (Say this player takes the action roughly 1 in 3 times, this means about 49 out of 50 hands in practice require change.)

View attachment 1339761

(On a side note, since there are no results where all 8 players are short, I have very technically proven 4/4/x is possible provided 32 chips are on the table, no players moved. I have just quantified how much change-making is required and it's staggering.)

SQL: 
declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf32) -- 3319

SELECT ShortCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf32
GROUP BY ShortCount
ORDER BY ShortCount

So maybe we should throw out some of the extremely lopsided combinations. Let's say no player can acquire more than 16 greens without immediately making change with another player. So if a player instead has a random number between 0 and 16 chips, This reduces the possible combinations to 2725. However, that only moves the needle to how many hands there will be with no short players players up to 0.80% from 0.67%. We are now talking 124 hands out of 125 in which at least 1 player will possibly require change. (And again if we assume a given player will take the action required to put chips in 1 out of 3 times, this is still 39 times in 40 where change making will be required.)

View attachment 1339767

SQL: 
DROP TABLE IF EXISTS #SumOf32_16Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8
    --, *
INTO #SumOf32_16Cap
from
    (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d1 (d1)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d2 (d2)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d3 (d3)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d4 (d4)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d5 (d5)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d6 (d6)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d7 (d7)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d8 (d8)
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 32
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf32_16Cap) -- 2755

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf32_16Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount

Now let's run this experiment for 64 greens on the table. (8 per stack,) For consistency, let's scale the max up to 32 so no player has more than half the greens in play to keep the high end outliers out.

We are now up to 102206 ways to distribute 0 to 32 chips to 8 players with a total of 64 on the table.

SQL: 
DROP TABLE IF EXISTS #SumOf64_32Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8
    --, *
INTO #SumOf64_32Cap
from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d8 (d8)
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 64
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf64_32Cap) -- 102206

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf64_16Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount

View attachment 1339786

So we are up to about 9.4% of instances where no player possibly needs change when a bet ends in 75. Up considerably from 0.8% before. Over 11x as often no change required.

But what this illustrates is doubling chips yields about an 11x improvment in this situation. Suffice it to say at this point, it's pretty obvious this isn't the pointless extravagance that @DBexla is portraying.

But still we are looking at a little over 90% of instances requiring at least one player possibly needing change. (And if you want to assume that a player short of the required chips is also taking the given action 1/3 of the time, we are at about 70% of hands in practice require a change transaction of some sort.)

So let's take this one step further and put 96 greens on the table (12 per player. I am going to keep the max per player at 32 greens, since I found that increasing this to 48 will add about 26x the complexity to the query and force to run several hours.)

We are now up to 587819 ways to distribute 0 to 32 chips to 8 players with a total of 96 on the table.

SQL: 
-- Seriously how many of you opened all four of these?


DROP TABLE IF EXISTS #SumOf96_32Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8
    --, *
INTO #SumOf96_32Cap
from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d8 (d8)
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 96
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf96_32Cap) -- 587819

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf96_32Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount
I will also note to those of you that are reading the code, I understand this approach is quick and crude, I am certainly aware there are ways to parametrize what I am doing for reusability. This was just a quick experiment.

View attachment 1339799

So we rocket up to 27% of instances where no player possibly needs change when a bet ends in 75. Making the 12/x/x method nearly 3x better than 8/x/x and a staggering 34x better than 4/4/x.

So in 73% of instances there will still be one player that would need change if taking the action to put the chips in. If a player does this only 1/3 of the time as I speculated in the other scenarios, then in practice, fewer than 25% of instances require change making. So the difference between going 8/8/x to 12/12/x is that change making now takes place in a minority of rounds. I think this reveals why some hosts prefer the jump to 12/12/x (I would be one) over tolerating the extra required by 8/8/x. But both are clearly, massively better than 4/4/x.

To recap the trail, we have gone from change making being a near 100% certainly in a round with a bet ending in 7x in the 4/4/x scenario, to being only about a 25% occurrence in the 12/12/x scenario.
If we ramped this up to 128 chips on the table, the 16/16/x scenario, I would guess we would get another increase, but maybe not as significant as the increases in the two prior scenarios, and some point we will reach diminishing returns if we repeat this scenario enough. But I think this has gone way past keeping a long story short. Conclusion coming.

View attachment 1339795

To close this, I admit my premises may not be perfect here. There are probably tweaks to the realistic max per player that could be more real-world, which would show a reduction in change making overall. I did deliberately use the word possible, we can quibble over what percentage of the time a player needing change for an action will actually take such an action, but that would affect all scenarios fairly equally. I figure the premises are close enough to at least fairly indicate the trend about how much change-making you can reduce by adding small chips to the starting stack, and that any tweaks probably affect all scenarios evenly, I think I am at least close on the factor of improvement.

Bottom line, short of the point of absurdity, more small chips always means less change making. This is my attempt to quantify how much less.
Click to expand...
Soooo, that made my head hurt, Justin. But what I’m taking away is…Always MOAR Chips.
 
JustinInMN said:
I did actually try an come up with an "impossible" change situation with the 8/8/4 starting stack, and the only impossible situations I could find require moving a player off of a table. So if you never move players this is technically "doable." But nearly every pot will require change for multiple players. Most hosts here see benefit in trying to limit that, though we recognize there is no way to eliminate change making entirely without going to absurd counts.

Since we have established a lower boundary for required change, I got to thinking about what the highest boundary would be.

To 100% prevent any change-making, the TD could issue an entire starting of T1500 using T25 chips, 60 per player.

Or for a different example, if it's a T10K tournament for example, you COULD issue 4 racks of T25 per player, or you could calculate about how many hands you expect T25 chips to be in play, multiply that by 3 and that would be the maximum needed. So if you play 4 levels with T25 in play and 10 hands per level, that's 40 hands, or 120 T25 chips to guarantee (Though really, this would be assumption that every hand a player puts in a total that ends in 75 on every hand, it's possible to make multiple bets in a hand that end in 75, I suppose so maybe again just issue 4 racks of T25 after all to be 100% sure. )

So having considered a lower boundary and upper boundary for number of small chips in a starting stack, I starting thinking about how often we talk about more small chips reducing change making. I think everyone understand the trend by instinct, but we haven't made an effort to quantify this on PCF.

I am a database developer by trade, so I decided to use this skill with some basic statistics knowledge to completely over-engineer this question. Get comfortable...

Assume a 8-handed table using 4 green (T25) chips per player, so there are 32 on the table in all. On a given round, what percentage of the time would expect a at least one player to be short on the number of green chips required to construct all bets (meaning have fewer than 3 in their stack).

Using database code, I was able to generate a list of the possible ways assuming each player has a random number of green chips between 0 and 32 and the sum of all players chips equals 32. There are 3319 such ways without consideration of the order of players.

SQL: 
    SELECT DISTINCT
        d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
        , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortCount
        , d1, d2, d3, d4, d5, d6, d7, d8
        --, *
    INTO #SumOf32
    from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d2 (d2)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d3 (d3)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d4 (d4)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d5 (d5)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d6 (d6)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d7 (d7)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d8 (d8)
    WHERE
        d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 32
        AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8

Using this we can estimate how often a player would need change by counting how often players are short of the minimum number of chips requisite to construct all bets. In the case of T25, a player with fewer than 3 chips cannot construct all possible bets without change. I calculated that there are only 22 combinations out of the 3319 in which all 8 players would have at least 3 green chips. Or 0.67% of the time. That means for every 150 hands in which there is a bet ending in 75, in 149 instances, at least one player will possibly require change. (Say this player takes the action roughly 1 in 3 times, this means about 49 out of 50 hands in practice require change.)

View attachment 1339761

(On a side note, since there are no results where all 8 players are short, I have very technically proven 4/4/x is possible provided 32 chips are on the table, no players moved. I have just quantified how much change-making is required and it's staggering.)

SQL: 
declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf32) -- 3319

SELECT ShortCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf32
GROUP BY ShortCount
ORDER BY ShortCount

So maybe we should throw out some of the extremely lopsided combinations. Let's say no player can acquire more than 16 greens without immediately making change with another player. So if a player instead has a random number between 0 and 16 chips, This reduces the possible combinations to 2725. However, that only moves the needle to how many hands there will be with no short players players up to 0.80% from 0.67%. We are now talking 124 hands out of 125 in which at least 1 player will possibly require change. (And again if we assume a given player will take the action required to put chips in 1 out of 3 times, this is still 39 times in 40 where change making will be required.)

View attachment 1339767

SQL: 
DROP TABLE IF EXISTS #SumOf32_16Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8 
    --, *
INTO #SumOf32_16Cap
from
    (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d1 (d1)
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d2 (d2) 
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d3 (d3) 
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d4 (d4) 
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d5 (d5) 
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d6 (d6) 
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d7 (d7) 
    CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16)) d8 (d8) 
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 32
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf32_16Cap) -- 2755

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf32_16Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount

Now let's run this experiment for 64 greens on the table. (8 per stack,) For consistency, let's scale the max up to 32 so no player has more than half the greens in play to keep the high end outliers out.

We are now up to 102206 ways to distribute 0 to 32 chips to 8 players with a total of 64 on the table.

SQL: 
DROP TABLE IF EXISTS #SumOf64_32Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8 
    --, *
INTO #SumOf64_32Cap
from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d2 (d2) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d3 (d3) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d4 (d4) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d5 (d5) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d6 (d6) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d7 (d7) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32)) d8 (d8) 
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 64
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf64_32Cap) -- 102206

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf64_16Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount

View attachment 1339786

So we are up to about 9.4% of instances where no player possibly needs change when a bet ends in 75. Up considerably from 0.8% before. Over 11x as often no change required.

But what this illustrates is doubling chips yields about an 11x improvment in this situation. Suffice it to say at this point, it's pretty obvious this isn't the pointless extravagance that @DBexla is portraying.

But still we are looking at a little over 90% of instances requiring at least one player possibly needing change. (And if you want to assume that a player short of the required chips is also taking the given action 1/3 of the time, we are at about 70% of hands in practice require a change transaction of some sort.)

So let's take this one step further and put 96 greens on the table (12 per player. I am going to keep the max per player at 32 greens, since I found that increasing this to 48 will add about 26x the complexity to the query and force to run several hours.)

We are now up to 587819 ways to distribute 0 to 32 chips to 8 players with a total of 96 on the table.

SQL: 
-- Seriously how many of you opened all four of these?


DROP TABLE IF EXISTS #SumOf96_32Cap

SELECT DISTINCT
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 AS Total
    , CASE WHEN d1 < 3 THEN 1 ELSE 0 END + CASE WHEN d2 < 3 THEN 1 ELSE 0 END + CASE WHEN d3 < 3 THEN 1 ELSE 0 END + CASE WHEN d4 < 3 THEN 1 ELSE 0 END + CASE WHEN d5 < 3 THEN 1 ELSE 0 END + CASE WHEN d6 < 3 THEN 1 ELSE 0 END + CASE WHEN d7 < 3 THEN 1 ELSE 0 END + CASE WHEN d8 < 3 THEN 1 ELSE 0 END AS ShortPlayerCount
    , d1, d2, d3, d4, d5, d6, d7, d8 
    --, *
INTO #SumOf96_32Cap
from
        (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d1 (d1)
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d2 (d2) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d3 (d3) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d4 (d4) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d5 (d5) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d6 (d6) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d7 (d7) 
        CROSS APPLY (VALUES (0),(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11),(12),(13),(14),(15),(16),(17),(18),(19),(20),(21),(22),(23),(24),(25),(26),(27),(28),(29),(30),(31),(32),(33),(34),(35),(36),(37),(38),(39),(40),(41),(42),(43),(44),(45),(46),(47),(48)) d8 (d8) 
WHERE
    d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 96
    AND d1 >= d2 AND d2 >= d3 AND d3 >= d4 AND d4 >= d5 AND d5 >= d6 AND d6 >= d7 AND d7 >= d8


declare @Total AS INT = (SELECT COUNT(*) FROM #SumOf96_32Cap) -- 587819

SELECT ShortPlayerCount,
    COUNT(*) AS ShortCombinations,
    (COUNT(*) * 1.0 / @Total) * 100 AS ShortPercentage
FROM #SumOf96_32Cap
GROUP BY ShortPlayerCount
ORDER BY ShortPlayerCount
I will also note to those of you that are reading the code, I understand this approach is quick and crude, I am certainly aware there are ways to parametrize what I am doing for reusability. This was just a quick experiment.

View attachment 1339799

So we rocket up to 27% of instances where no player possibly needs change when a bet ends in 75. Making the 12/x/x method nearly 3x better than 8/x/x and a staggering 34x better than 4/4/x.

So in 73% of instances there will still be one player that would need change if taking the action to put the chips in. If a player does this only 1/3 of the time as I speculated in the other scenarios, then in practice, fewer than 25% of instances require change making. So the difference between going 8/8/x to 12/12/x is that change making now takes place in a minority of rounds. I think this reveals why some hosts prefer the jump to 12/12/x (I would be one) over tolerating the extra required by 8/8/x. But both are clearly, massively better than 4/4/x.

To recap the trail, we have gone from change making being a near 100% certainly in a round with a bet ending in 7x in the 4/4/x scenario, to being only about a 25% occurrence in the 12/12/x scenario.
If we ramped this up to 128 chips on the table, the 16/16/x scenario, I would guess we would get another increase, but maybe not as significant as the increases in the two prior scenarios, and some point we will reach diminishing returns if we repeat this scenario enough. But I think this has gone way past keeping a long story short. Conclusion coming.

View attachment 1339795

To close this, I admit my premises may not be perfect here. There are probably tweaks to the realistic max per player that could be more real-world, which would show a reduction in change making overall. I did deliberately use the word possible, we can quibble over what percentage of the time a player needing change for an action will actually take such an action, but that would affect all scenarios fairly equally. I figure the premises are close enough to at least fairly indicate the trend about how much change-making you can reduce by adding small chips to the starting stack, and that any tweaks probably affect all scenarios evenly, I think I am at least close on the factor of improvement.

Bottom line, short of the point of absurdity, more small chips always means less change making. This is my attempt to quantify how much less.
Click to expand...
Excellent, simply excellent!

I would like to see the corresponding numbers for 16/16/x and 20/20/x (T25-base), along with a similar analysis of 10/10/x vs 15/9/x (or 15/13/x) stacks of T5-base and 10/6/x vs 15/5/x stacks for T100-base (and even 6/12/x vs 8/11/x vs 10/10/x for T500-base might be enlightening, too).

I suspect that 20/20/x stacks is where diminishing returns really kicks in. Will be interesting to see the differences in 10/x vs 15/x sizing compared to your 8/x vs 12/x numbers.

My starting stack composition recommendations have long been a "minimum 8/8/x to maximum 20/20/x, with optimum 12/12/x to 16/16/x per stack", with the reasoning that 8/8/x slows the game play (excessive change-making) as does 20/20/x (excessive counting/re-stacking).

Would love to see your analysis produce hard numbers that "prove" it all.
 

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