A tournament may have what is known as an add-on option. Under certain conditions, a player may elect to add on to his stack without additional fee. The added money then increments the prize pool and the player presumably has a better chance for cashing because of the increased stack. We assume here that the amount of added chips is the same as the initial buy-in at the same cost less fee. Of course, the player’s investment is higher so the benefit of the increased stack may not overcome his higher investment. In fact, if one subscribes to the ICM model, additions to the stack do not provide a 1 to 1 return on the investment for added chips lose value as the stack size increases.

Consider a 25 player MTT with a 9 +1 entry fee. Prizes are given to the top 5 players with the prize structure being 100, 60, 30, 20, 15. Suppose at the beginning of the tourney there is an option to add-on the same number of chips as the buy-in at the same cost with no additional fee. Assume 10 of the 25 players choose that option. Using ICM theory, I’ll explore the ramifications of this particular scenario, which readers should recognize as being just one sample case.

Before any add-ons are considered, each player’s $EV is the rake equal to -1.0. This is based on the ICM assumption that players are of equal capability. If 10 players take the add-on, they double their stack and the prize pool is increased accordingly. But, because ICM theory says you don’t get a 1 to 1 return for additional stack, the $EV of the add-on players reduces to -1.27 a loss of 27%; not good. Now it is generally thought that the add-on is generally more favorable to better players. We explored this by assigning a capability factor to a player by multiplying his stack size by this factor. If all 10 of the add-on players were equally better than the rest of the field, this factor is about 1.04, meaning the add-on players must have an edge of at least 4.0% to avoid a lower $EV with add-on than if no add-on was taken. If only one player was better, his edge has to be only about 1.75% for break-even.

What about add-on later in the tournament, probably a more realistic situation? Well, here you would think that the better players have larger stacks as the tournament progresses and ICM tells us that the larger stacks gain less in EV with added chips than smaller stacks. So, unless the better player has a relatively small stack due to bad luck or bad cards, the add-on might not be very beneficial to him.

In reviewing several internet discussions on this topic it seems most posters favor add-ons because larger stacks increase the probability of cashing. Rarely is the increased cost considered. What does the above analysis tell us? Well, it seems add-on, when looked at strictly from an ICM-math POV, is not always a particularly good option especially for large stacks. You don’t get a $1 worth of $EV value for a $1 add-on. Sure, the math I applied doesn’t account for the good aspects of increasing your stack – not having to play scared for fear of busting out, being able to bully short stack players, being more flexible in your decision-making, etc. The analysis supports the notion that add-on is more advantageous (or less deleterious) for the better player if stacks are equal. On the other hand, knowing that an add-on or rebuy is available, may result in a player playing more loosely or aggressively than he might otherwise, and that may be either good or bad.

Granted that only a very specific case was considered and a 25 player tourney is relatively small, so ICM at the relatively small tournament beginning is a more important factor than it would be for a much larger tournament. Nevertheless, if ICM is applicable, there may be a negative $EV impact of taking the add-on. Hopefully this posting will encourage a more quantitative look at the issue, where more realistic scenarios are considered in a more comprehensive way.

Well I think you are right in that if the add-on is the same cost it is likely not worth it in $ev unless some other factor is present like knowledge of the players, seat position/good table or stack playability improvement.

In the past I think most add-ons gave you the same chips but without paying any rake so this also increased it's value. Now there are a a few different pay regimes and often you get more chips for the same cost.

These add-on re-buys don't seem as common to me, I think they have been replaced by so many more re-entry/late reg ones and these can also hold strange $ev both positive and negative.

The only add-ons I can see in the Pokerstars lobby atm seem to be $11+r, and a $1.10+r.
The $11 gives 1.5 the starting chips as add-on and the $1.10 give a huge 10x starting (presumably for this the $ev way is to simply try to get to the add-on period cheaply and definitely take the addon )

Just to be clear, my example case did not have any rake for the add-on.

Thanks for the comment. I was surprised in not having any responses for several days.

So was I.

Anyway, I thought I would try a simulation to see the effect of just icm has on the add-on for a moderate sized Mtt.

I now realize I should have had more starting players to simulate a typical re-buy/add-on tourny. I only set it to 180 starters and the add-on period having 120 left (no re-buys), I set the payouts to the Pstars 180 MttSng, 27 paid.

I haven't got a fast way to calculate icm for large fields, I am using a python port of sicm from a previous 2+2 thread. This is a monte-carlo approximation and so does vary a bit even using quite a large number of trials (for these sized fields and payouts - as you can see in the variance of the plot).

I set the add-on to 1.5 x the starting stack for 1 full buy-in (including the rake). I'll try another sometime for 1x and 1 buy-in no rake.

I set the stacks to approximately a Gaussian dist shifted and normalised to give a starting stack of 1000 chips.

I calculated the icm value of all these added-on stacks and then went through this list taking off the add-on one by one and adjusting the prize-pool by 1 buy-in. So for each possible stack I calculated the icm with no add-on while every single other stack does have the add-on and compared the value to the original calc.

I set the buy-in to $0.91 and the rake to $0.09.
Here is a plot of the results:

The stacks shown in the plot are the pre add-on ones, the y-value is the added value you gain from taking the add-on.

As you can see it is easy to see the trend. If I have this correct it is at least break even up to the point where a player has about 5x starting stack. I suspect quite a bit of the value of the add-on is from paying for a 1.5x stack when already 1/3 the players have already dropped out of the tournament without getting any payoff.

Probably a more realistic simulation would be to set the starting players to perhaps 250 and still have the add-on players at 120, and same payout structure, this would be equivalent to having 70 re-buys before the add-on period starts. I suspect this will make the add-on more valuable.

I'll try doing this soon and also the same for a no rake add-on but only starting stack sized add-on.

I think I now realize why the 1.5x starting stack with rake is more common as the site manages to grab more money even though the tourny may last a little longer

Here is a closer scenario to a real rebuy + add-on tournament. In this case the payout structure is still the same as a Pokerstars 180 but there are now 250 initial stacks, this would be like having the original 180 players starting and getting 70 re-buys before the add-on period.

This gives a bit of extra gain compared to only 180 starters.

Here is the case for 250 initial stacks as above but the add-on is only 1.0 x the starting stack but no rake charged, ie, in this case the add-on costs $0.91.

At this cost with a decent stack this doesn't seem a good deal as far as the basic icm result goes, each add-on chip costs more than the 1.5x for $1.00 case so drops the $value.

So atm for me I would tend to go with the add-on but I would take care to check the 'cost'.

I hardly ever play add-ons as I don't like very long lasting tournaments but I sometimes do play 180 MttSng's on PStars and these charge a full buy-in but you get 2000 chips when the normal starting stack is 1500, only 1.33x, this is approx in the middle per chip cost of the previous 2 cases. The add-on decision would also depend on how many re-buys had occurred as the more that have the more value due to the increased prize-pool per player left.

I think BaseMetal's last chart is equivalent to what I looked at. A one-time add-on of one buy-in, no rake. While I only looked at much smaller size tournaments, I came to the following conclusions.

In general,
. a player well-ahead in the tournament should not do the add-on without special justification
. a player way down in stack size will usually benefit in taking the add-on

Keep in mind, these results used the ICM model, which assume equal capability. If a player is way down in stack size because he is at a skill disadvantage, the add-on is probably a bad idea. If a player is well ahead because he has a skill advantage, he may not need the add-on.

Other cases I looked at showed varying patterns. In most cases though, out of 6 tiers I used for stack sizes, the player in the second tier benefited from having other players add to the prize pool by taking the add-on. He did not benefit when all the lower tier players took it.
So, add-on is something that at first seems like a good idea and may be good if you get a discount, but if not or if you have to pay rake or a premium in some way, it may not be beneficial.

Apologies if you guys have already discussed this ...

How critical to your own add-on decision is the decisions made by your opponents? You know, a real game theory situation.

How does your decision vary over the following scenarios:
- Assume 100% of your opponents take add-on
- Assume 75% of your opponents take add-on
- Assume 50% of your opponents take add-on
- Assume 25% of your opponents take add-on
- Assume 0% of your opponents take add-on.

Here are the results for a very special case. 25 players, all with same stack. Prize buy-in = 9, Fee = 1. The prize structure was
100
60
30
20
15
Total = 225, before any add-ons

I used the ICM simulation model first presented in 2P2 so results can be a bit off. For each villain add-on %, I considered two cases: Hero doesn’t add-on and Hero does add-on.

Hero EV

% V AddOn	No Hero AO	Yes Hero AO
0%	-1.00	-1.75
25%	-0.90	-1.40
50%	-0.81	-1.18
75%	-0.79	-1.07
100%	-0.79	-1.00

We can see the following holds for this example:
. With nobody taking the add-on with equal stacks and a fee of 1, hero (and everyone else) has EV = -1.0. The same is true if everybody adds-on. This should be fairly obvious.

. There is no case where someone has positive EV. The equal capability and starting stack equality are the main reasons.

. For a given villain add-on %, Hero does better in not taking the add-on. His EV actually increases with more villain add-ons but is always negative.

.The higher the villain add-on%, the better Hero does, but still always has negative EV.


As I indicated earlier, things won't be so "neat" if the stacks at add-on time are not equal and/or the ICM assumption of equal capability is questionable. The above does confirm, IMO, that add-on may not be such a great idea.

Here is a replacement for the case where there are 250 starters, 180 payout structure (equiv to 80 re-buys) and 120 add-ons of a starting stack for $0.91, ie no rake. I made a small mistake earlier.

There are a few extra players above the $0.91 break even mark because in the previous I accidentally took off a whole buy-in from the prize pool rather than the $0.91 when calculating the differences earlier.

Now with this particular stack distribution it seems to profitable to take the add-on while having a stack up to 1.3 x the starting stack.

If the top stack (~7x starting here) takes the add-on from a solely icm perspective this players loses about ~ 11 cents, ~15% of this extra cost. It is interesting to note that if the player didn't take it but all others did a few cents of value is added to his $ev without doing anything. As any player adds on a chunk of the $value goes to all the other players.

I did think that for this no rake add-on there would be as much money above the break even line as below but now I think due to the complication of how the money is spread around this is to be expected when comparing all add-on to the case when only one player doesn't.

While I was looking for errors in the script I did at least manage to speed up my python implementation of sicm about 20x mostly by using numpy . I spent a lot more time messing with this than debugging it's much more interesting.

Here is a replacement for the other 1.5x stack case with a better number of sims.

If you get a 1.5x stack it seems like it's basically always worth taking the add-on (well unless you have built up a 9x stack already).

I used to play in a very wild daily $5 tourney with unlimited rebuys. At the end of the 1 hour rebuy period, a rebuy would only get you 4 big blinds to play with, but there was a 20x starting stack add on. I thought it was a big mistake to decline the add on unless I already had a ton of chips.

Compare that extreme to a very small add on? Only the smallest of stacks should take a small add on.

So yeah I agree with the above conclusions.

didn't sklansky solve this by saying that you should rebuy/addon any time because chip EV is bigger than $ EV?

OP what's your opinion about how late can you rebuy? are 10BB worth it assuming no other tournaments exist for hero to jump to?