So, this may be a long-winded post, but bear with me. My brother and I recently had a discussion about whether Add-ons were worth it - the one's where you can add-on after late registration is over - and we came up with some interesting findings.

We based our findings in math, but I'm definitely not sure we got the math right. I used my understanding of ICM, which I'm new to and very may well have gotten wrong. In fact, what I'm calculating might not even be related to ICM. It's very simplistic, but I'll explain more.

So, to start, let's assume a few things. Tournament buy-in is $5.50 (so $5 goes to the prizepool). Add-on is also $5.50, for 1.5x the original buy-in chips (5000 chips for buy-in, 7500 for add-on). You can also rebuy for $5.50 unlimited (5000 chips) until late reg ends.

For the purposes of a test case, we assumed 200 people registered, there were 200 rebuys, and we're at the end of late registration, so they'll be no more. That means there's 2,000,000 chips in play (200 x 5000) + (200 x 5000) and a total prize pool of $2000.

So, now we have to decide - are we going to add-on? Well, this is where things get interesting... I'm going to assume 4 different test cases, with all of the possible extremes:

• Nobody adds-on, and we have a short stack,
• Everybody adds-on (150 players including us), and we have a short stack,
• Nobody adds-on, and we have a huge stack,
• Everybody adds-on (150 players including us) when we have a huge stack.

(NOTE: This is where I'm really unsure of my logic/math - so if this is wrong, don't bother reading the rest, because I'm making the same calculations going forward. Feel free to correct!)

First test case:

We have 10000 chips, nobody adds-on, except potentially us. Add-on is 7500 chips. We go from 10000 / 2,000,000, or 0.005 of the total chip count, to 17500 / 2,007,500, or 0.00871 of the total chip count. Multiply by the prizepool, or 2000 before and 2005 after, and we get a "value" of our chips of $10 before, and $17.46 after. Given a $5.50 add-on, I'm viewing this as "profitable".

Second:

10000 chips, everybody adds on.
10000/(2,000,000 + (149 * 7500)) = 10000/3,117,500 = 0.0032
17500/(2,000,000 + (150 * 7500)) = 17500/3,125,000 = 0.0056
Value of chips (w/o addon) = 0.0032 * ($2000 + (149 * 5)) = 0.0032 * 2745 = $8.78 vs.
Value of chips (w/ addon) = 0.0056 * ($2000 + (150 * 5)) = 0.0056 * 2750 = $15.4

Again, "profitable" by this formula.

These make intuitive sense to me. We have a small stack, so whether a lot of people add-on or none do, we gain a significant portion of our stack and thus increase our value by enough to make it profitable.

What is strange to me, is how the numbers work out for a huge (even comically huge, which I'll show below) stack. I would assume that because the add-on is such a small percentage of our stack, it would no longer be profitable to add-on. But that doesn't seem to be the case.

Third:

Assuming a comically huge stack of 1,000,000 chips at the end of late reg.

1,000,000 chips, nobody adds on.

1,000,000/2,000,000 = 0.5
1,007,500/2,007,500 = 0.5019
Value of chips (w/o addon) = 0.5 * ($2000) = $1000 vs.
Value of chips (w/ addon) = 0.5019 * $2005 = $1006.3

Still profitable...

Fourth and final

1,000,000 chips, everybody adds-on

1,000,000/(2,000,000 + (149 * 7500)) = 1,000,000/3,117,500 = 0.3208
1,007,500/(2,000,000 + (150 * 7500)) = 1,007,500/3,125,000 = 0.3224
Value of chips (w/o addon) = 0.3208 * ($2000 + (149 * 5)) = 0.3208 * 2745 = $880.60 vs.
Value of chips (w/ addon) = 0.3224 * ($2000 + (150 * 5)) = 0.3224 * 2750 = $886.60

Still profitable?!

THE END. Somebody tell me why I'm stupid I know ICM is much more complicated than the above, but as far as I'm aware that's due to the calculations required when you've already made the money, and 1 chip has a significant value. If you haven't made the money, and you assume equal skill levels across the field, isn't what I've done above accurate for each scenario?

First, I didn't read your post.

So... If all players make the addon you will be at bb disadvantage
You become a short stack on a table full of deeper stacks

There is no math that matters then

Except thats not always the case?

If you have 3x the chips of everyone at the table, and everyone addons, you don't automatically become a short stack.

I'm looking to understand the math behind add-ons and when it's worth it and when it isn't. That's the basic point of this post. If you don't want to read it, then don't reply /shrug

If you are chip leader with 3x is pretty obvious you don't news the addon to put pressure on others
But most of the time as chip leader you will have 1.5-2 X the average stack
You have to make the addon too, if most of the table make it
Because you don't want to others compete for the chip lead.

Use common sense, you don't need math here

Glad to have your 2cents. Would still like to know if there's a way of concretely figuring it out. Thanks though!

Firstly, props for putting in the effort in running through your thoughts in so much detail.

Secondly, which would you rather have... 5k chips for $5.5 or 7.5k chips for $5.5?

Thirdly, ICM here is minimal if you are assuming your equity based on the different probabilities of you finishing in each possible position from your stack (in the vast majority of payout structures).

Yah, I guess that makes sense. And I'll take the 7.5k for $5.50, thanks :P

I guess I'm just confused on two things: One, how does 7500 chips increase my equity if I have 1,000,000 chips? And two, is the math I'm doing logical, or am I off-base?

I appeciate the effort, provable answers on this subject would be valuable rather than the historical answer of "moar chips is bettah".

Check out this current thread in theory https://forumserver.twoplustwo.com/1...y-not-1738589/