Mathematical rebuy analysis in the new Party and Empire rebuy tourneys

[allenciox]

Since I had a number of questions about proper rebuy in these tournaments, I decided to analyze it mathematically. Below is more detailed analysis, but basically:

If you are a decently good player in these (i.e about a 25% ROI): then

o you should always rebuy when you get to zero.
o you should always take the addon
o you should immediately rebuy only if the buyin amounts for the tournament are insignifant to you. You get slightly higher expectation for the immediate rebuy but at greater volatility for your money.
o If your starting stack decreases significantly, the earlier it is in the rebuy period, the more it benefits you to rebuy. If it is near the end of the rebuy period, do not rebuy.

Here is the detailed analysis:

Assumptions:

1) initial buyins, rebuys, and addons are all $10+1. For your buyin, just scale the results given below appropriately
2) initial buyins and rebuys net 1000 chips, addons net 1500 chips.
3) Rebuys may only be done within the first 60 minutes and only when stack <=1000, only one addon can be done, and that is 60 minutes into tournament.
4) You are a better than average player --- at the end of the rebuy perid, each chip you own is equivalent to 1.2 chips as a percentage of the prize pool. During the rebuy period, each chip up to 1000 in your stack is equivalent to 1.003 chips for each minute remaining in the rebuy period. Each chip between 1000 and 2000 in your stack is equivalent to 1.002 chips for each minute remaining in the rebuy period.
5) Your stack stays small enough that chip equity is proportional to tournament $ equity.
6) 100 people enter, 75 rebuys are done by other entrants, 25 people are eliminated and choose not to rebuy during the rebuy period, and 50 people addon. This rate of buyins and addons/retention of entrants seems pretty consistent with low buyin rebuy tournaments on Party and Empire.
7) You always take the addon at the end of the rebuy period.

Under these assumptions, at the end of the rebuy period, there are 250,000 chips in play, and $2250 in prize money not including any rebuys and addons you make.

Questions:

1) What is your expectation on your initial entry assuming that you do not rebuy, but you always take the addon?

E(initial buyin) = (1000*1.003^60 + 1500)*1.2/251500*2260 - 22 = $7.08

This is pretty good, if you are a decent player, you are essentially getting a 32% ROI.

2) what is your expectation if you immediately rebuy at the beginning?

E(w/initial rebuy) = (1000*1.003^60 + 1000*1.002^60 + 1500)*1.2/252500*2270 - 33 = $8.25

so your ROI goes down to about 25% with an initial rebuy, but your expected return goes up. So your expected gain comes with significantly greater risk.

3) Suppose you do not immediately rebuy, but you are down to 500 chips 30 minutes into the rebuy period. What should you do now (assuming no rebuy is necessary for rest of period)?

(in the following estimates, your original $11 entry is NOT included in the calculations, since it is a sunken cost)

E(no rebuy now) = (500*1.003^30 + 1500)*1.2/251500*2260 - 11 = $11.07
E(rebuy now) = (1000*1.003^30 + 500*1.002^30 + 1500)*1.2/252500*2270 - 22 = $11.71

You are getting even less value for your rebuy, than by rebuying at the beginning.

4) If you go to 0 with 30 minutes left,should you rebuy once or twice?

E(no rebuys) = 0

E(1 rebuy) = (1000*1.003^30 + 1500)*1.2/252500*2270 - 22 = $5.98

E(2 rebuys) = (1000*1.003^30 + 1000*1.002^30 + 1500)*1.2/253500*2280 - 33 = $6.45

The two rebuys are just marginally better expectation-wise, and have worse ROI. But it is clear you should never quit the tourney if you go to 0.

5) What about if you go to 0 right at end of rebuy period?

E(no rebuys) = 0
E(1 rebuy) = (2500)*1.2/252500*2270 - 22 = $4.97

E(2 rebuys) = (3500)*1.2/253500*2280 - 33 = $4.77

Here, it is better just to do the single rebuy.