Profitability of re-buys/add-ons when allowed

Question

If you are not familiar with the format: such tournaments are (at least) available online, mostly in form of satellites to other tournaments etc. Basically the standard satellite structure with 1 ticket per X dollars in the prize pool, sometimes with a guarantee of N tickets (i.e. if the prize pool is lower than the cost of such tickets, they are awarded anyway)

Pretty much for any such tournament I do not re-buy except in the earliest stages, or if I estimate to have an advantage over the opponents.

When is such logic incorrect and a re-buy/add-on would be profitable?

Side-notes/scenario analysis:

Busting out of such qualifier just now, I confidently clicked "leave tourney" when prompted - blinds were up to 2K/4K and, well, why would I pay $25 for 3K in chips???

Apparently, math disagrees (or at least my math) - at the time of my busting out in 11th, there was still only one $5K package guaranteed and I did not expect for the prize pool to grow (I was correct); there was 309K chips in play total. These were my options:

Leave tourney:  $0 for $0 expected profit, which is what I gracefully did
Single re-buy: $25 for 3K chips; (3K/312K)*$5K = $48.08; expected net profit $23.08 *lolwut*
Double re-buy: $50 for 6K chips; (6K/315K)*$5K = $95.24; expected net profit $45.24

Does this mean I should have re-bought and, in fact, continuing to do so would have remained profitable until (on the example of a single re-buy) the total number of chips in play exceeded 600K??

(3K/XK)*$5K > $25
3/X>25/5K
25X>15K
X>600

I am disappointed with the numbers above.

Answer 1

Math on standard re-buys is that they are basically always profitable if you were profitable to enter the tournament in the first place.

For flat payout structures rebuys are profitable because the $/chip value is higher for small stacks compared to big stacks. (Read up on ICM if you are not familiar with the concept.)

This does not apply to your tournament tough, since it is basically winner-takes all. But there appears to be a huge overlay in your tournament, so rebuys are definitely very profitable there as well.

Unsurprisingly, your math tells you that rebuying becomes break-even in winner-takes-all tournaments as soon as there is no more overlay.

Answer 2

A vast majority of the time it's going to be profitable to rebuy in any tournament. There is a stack below which rebuying is not going to be profitable, but it's not really possible to calculate this point exactly. The basic reasoning is that chip equity and money equity are not related linearly like you have calculated. For example, in a 1,000 player tournament with 100 BB stacks paying 10% of the field, someone with 50 BB doesn't have half the money expectation everyone else does (skill aside) - in fact their chances are close. ICM is a way to model this chip-money relationship near or in the money but is just a best guess to the actual money equity. In large fields it will mostly be subjectively obvious when your chances of making money start to diminish from the average. A rigorous answer would require a large sample of relative rebuy stacks and corresponding ROIs.

If you only wish to prove that rebuying always offers you a greater return than your initial entry, above some indeterminate stack level, contrive a tournament offering 10,000 BB per entry. Once the prize pool doubles and you haven't rebought, you have a 10,000 BB stack to battle an average stack of roughly 20,000 BB. Would you say there is enough room for play that your share of the prize pool is roughly the average share? If the answer is yes, you have your evidence for stacks at least 10,000 BB. Decrease that stack until the answer is no and you have your exception.

If this is hard to see, imagine a cash game with similar stacks - chips become more loosely related to relative return the fewer of them you are forced to commit. You can also think about this concept in terms of the number of all-ins various stacks can survive, where extrapolating tournament lifespans yields the same loose relationship to chips as does money equity.

Notice this logic shows that add-ons are worse the deeper you are but also have a profitable stackpoint (if you have 1 BB an add-on is roughly like a rebuy).