When to Rebuy in Multi-table Poker Tournaments
by Pikachu
(From Blackjack Forum , Fall 2006)
© Blackjack Forum 2006
In a recent post on rebuy analysis on the Poker Forum of this Web site, I ended with the conclusion that a player without skill should not rebuy in a percentage payback tournament, and a player with skill should rebuy if he’s got a certain (undefined) edge. While this served to illustrate the point that a winner-take-all example is not an appropriate model for multi-table tournaments, it wasn’t a very useful post. Now I’ll provide some examples from a multi-table tournament format that take a player’s skill level into consideration.
For these examples I make the following assumptions:
- 100 player field
- $100 buyin gets 100 in tournament chips
- A $100 rebuy gets an additional 100 tournament chips
- All 99 opponents have the same number of chips (makes the calculations easy)
- The “edge” translates into the increased probability of taking 1st place, and a lesser increased probability of lower finishes. So an unskilled player will have a 1% chance of finishing in 1st, while a player with a 10% edge will have a 1.1% chance of finishing in 1st. Note that this players ROI will be less than 10%. And yes, I might be defining it strangely.
- The following payout structure:
| Place | % Pool |
|---|---|
| 1 | 29.0% |
| 2 | 18.5% |
| 3 | 12.0% |
| 4 | 10.0% |
| 5 | 8.0% |
| 6 | 6.5% |
| 7 | 5.5% |
| 8 | 4.5% |
| 9 | 3.5% |
| 10 | 2.5% |
For the base case, an unskilled player who hasn’t rebought:
| Rebuys 0 | ||
|---|---|---|
| Edge 0.00% | ||
| Place | Prob | Return |
| 1 | 1.00% | $ 29.00 |
| 2 | 1.00% | $ 18.50 |
| 3 | 1.00% | $ 12.00 |
| 4 | 1.00% | $ 10.00 |
| 5 | 1.00% | $ 8.00 |
| 6 | 1.00% | $ 6.50 |
| 7 | 1.00% | $ 5.50 |
| 8 | 1.00% | $ 4.50 |
| 9 | 1.00% | $ 3.50 |
| 10 | 1.00% | $ 2.50 |
| EV $0.00 |
If he rebuys we get:
| Rebuys 1 | ||
|---|---|---|
| Edge 0.00% | ||
| Place | Prob | Return |
| 1 | 1.98% | $ 58.00 |
| 2 | 1.96% | $ 36.63 |
| 3 | 1.94% | $ 23.52 |
| 4 | 1.92% | $ 19.40 |
| 5 | 1.90% | $ 15.36 |
| 6 | 1.88% | $ 12.35 |
| 7 | 1.86% | $ 10.34 |
| 8 | 1.84% | $ 8.37 |
| 9 | 1.82% | $ 6.44 |
| 10 | 1.80% | $ 4.55 |
| EV ($5.04) |
As expected, an unskilled player should not rebuy. He doubles his buy-in but does not double his chances of finishing in the money.
Now, consider a player with an 10% edge:
| Rebuys 0 | Rebuys 1 | ||||
|---|---|---|---|---|---|
| Edge 10.00% | Edge 10.00% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 1.10% | $ 31.90 | 1 | 2.18% | $ 63.80 |
| 2 | 1.10% | $ 20.33 | 2 | 2.15% | $ 40.21 |
| 3 | 1.10% | $ 13.17 | 3 | 2.13% | $ 25.77 |
| 4 | 1.10% | $ 10.97 | 4 | 2.10% | $ 21.21 |
| 5 | 1.10% | $ 8.76 | 5 | 2.07% | $ 16.76 |
| 6 | 1.09% | $ 7.11 | 6 | 2.05% | $ 13.45 |
| 7 | 1.09% | $ 6.01 | 7 | 2.02% | $ 11.23 |
| 8 | 1.09% | $ 4.91 | 8 | 2.00% | $ 9.07 |
| 9 | 1.09% | $ 3.82 | 9 | 1.97% | $ 6.97 |
| 10 | 1.09% | $ 2.72 | 10 | 1.94% | $ 4.91 |
| EV $9.71 | EV $13.38 | ||||
| ROI 9.71% | ROI 6.69% |
Without rebuying that 10% edge translates into a 9.71% ROI, or $9.71 in EV. Rebuying increased EV (though it lowers ROI). That 10% edge should not be at all difficult to attain (as anyone who has played a low-limit fast tournament can attest to). In all cases where the player has an edge, his ROI when rebuying will be lower than when not rebuying, though the EV will be higher. This assumes no juice on the tournament.
While edges far in excess of 10% are easily had, it may be interesting to see what edge is required to overcome the intrinsic disadvantage of rebuying in a percentage payback tournament. Brute force says:
| Rebuys 0 | Rebuys 1 | ||||
|---|---|---|---|---|---|
| Edge 5.77% | Edge 5.77% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 1.06% | $ 30.67 | 1 | 2.09% | $ 61.35 |
| 2 | 1.06% | $ 19.56 | 2 | 2.07% | $ 38.70 |
| 3 | 1.06% | $ 12.68 | 3 | 2.05% | $ 24.82 |
| 4 | 1.06% | $ 10.56 | 4 | 2.02% | $ 20.45 |
| 5 | 1.06% | $ 8.44 | 5 | 2.00% | $ 16.17 |
| 6 | 1.05% | $ 6.85 | 6 | 1.98% | $ 12.99 |
| 7 | 1.05% | $ 5.80 | 7 | 1.95% | $ 10.86 |
| 8 | 1.05% | $ 4.74 | 8 | 1.93% | $ 8.78 |
| 9 | 1.05% | $ 3.68 | 9 | 1.91% | $ 6.75 |
| 10 | 1.05% | $ 2.63 | 10 | 1.89% | $ 4.76 |
| EV $5.61 | EV $5.61 | ||||
| ROI 5.61% | ROI 2.80% |
In the case of a player with a 5.77% edge, he hasn’t gained or lost EV by rebuying, but merely increased his variance. It is worth noting that a player wanting to maximize his EV/VAR should not rebuy unless his edge is greater than 103%. This suggests that a skilled player with a sufficiently small bankroll would not want to rebuy in many cases.
For a decently bankrolled player with a 30% edge our payback table looks like this:
| Rebuys 0 | Rebuys 1 | ||||
|---|---|---|---|---|---|
| Edge 30.00% | Edge 30.00% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 1.30% | $ 37.70 | 1 | 2.57% | $ 75.40 |
| 2 | 1.30% | $ 23.98 | 2 | 2.53% | $ 47.33 |
| 3 | 1.29% | $ 15.51 | 3 | 2.49% | $ 30.20 |
| 4 | 1.29% | $ 12.88 | 4 | 2.45% | $ 24.76 |
| 5 | 1.28% | $ 10.27 | 5 | 2.41% | $ 19.48 |
| 6 | 1.28% | $ 8.32 | 6 | 2.37% | $ 15.56 |
| 7 | 1.28% | $ 7.02 | 7 | 2.33% | $ 12.95 |
| 8 | 1.27% | $ 5.72 | 8 | 2.29% | $ 10.41 |
| 9 | 1.27% | $ 4.44 | 9 | 2.25% | $ 7.96 |
| 10 | 1.26% | $ 3.16 | 10 | 2.21% | $ 5.59 |
| EV $28.99 | EV $49.65 | ||||
| ROI 28.99% | ROI 24.82% |
Rebuy!
Consider what happens when we add a $9 entry fee. Assume rebuys are juice free. Playing with a 10% edge (pre-juice) is not so great anymore:
| Juice 9 | Juice 9 | ||||
|---|---|---|---|---|---|
| Rebuys 0 | Rebuys 1 | ||||
| Edge 10.00% | Edge 10.00% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 1.10% | $ 31.90 | 1 | 2.18% | $ 63.80 |
| 2 | 1.10% | $ 20.33 | 2 | 2.15% | $ 40.21 |
| 3 | 1.10% | $ 13.17 | 3 | 2.13% | $ 25.77 |
| 4 | 1.10% | $ 10.97 | 4 | 2.10% | $ 21.21 |
| 5 | 1.10% | $ 8.76 | 5 | 2.07% | $ 16.76 |
| 6 | 1.09% | $ 7.11 | 6 | 2.05% | $ 13.45 |
| 7 | 1.09% | $ 6.01 | 7 | 2.02% | $ 11.23 |
| 8 | 1.09% | $ 4.91 | 8 | 2.00% | $ 9.07 |
| 9 | 1.09% | $ 3.82 | 9 | 1.97% | $ 6.97 |
| 10 | 1.09% | $ 2.72 | 10 | 1.94% | $ 4.91 |
| EV $0.71 | EV $4.38 | ||||
| ROI 0.66% | ROI 2.09% |
The rebuy is a must in this situation. Because there is no juice on the rebuy, the rebuy chips are essentially sold at a discount. Even the small discount makes a large difference in EV and ROI.
Take a look at the 30% edge player when juice is added:
| Juice 9 | Juice 9 | ||||
|---|---|---|---|---|---|
| Rebuys 0 | Rebuys 1 | ||||
| Edge 30.00% | Edge 30.00% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 1.30% | $ 37.70 | 1 | 2.57% | $ 75.40 |
| 2 | 1.30% | $ 23.98 | 2 | 2.53% | $ 47.33 |
| 3 | 1.29% | $ 15.51 | 3 | 2.49% | $ 30.20 |
| 4 | 1.29% | $ 12.88 | 4 | 2.45% | $ 24.76 |
| 5 | 1.28% | $ 10.27 | 5 | 2.41% | $ 19.48 |
| 6 | 1.28% | $ 8.32 | 6 | 2.37% | $ 15.56 |
| 7 | 1.28% | $ 7.02 | 7 | 2.33% | $ 12.95 |
| 8 | 1.27% | $ 5.72 | 8 | 2.29% | $ 10.41 |
| 9 | 1.27% | $ 4.44 | 9 | 2.25% | $ 7.96 |
| 10 | 1.26% | $ 3.16 | 10 | 2.21% | $ 5.59 |
| EV $19.99 | EV $40.65 | ||||
| ROI 18.34% | ROI 19.45% |
The $9 entry fee comes directly out of our EV. Notice the ROI when rebuying is higher because of the discount.
Next, consider the situation where our player has managed to double his stack through play, and is considering an add-on. His 100 chip increase has left each other player 1.01 chips shorter.
| Increased Stack 100 | Increased Stack 100 | ||||
|---|---|---|---|---|---|
| Juice 9 | Juice 9 | ||||
| Rebuys 0 | Rebuys 1 | ||||
| Edge 10.00% | Edge 10.00% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 2.20% | $ 63.80 | 1 | 3.27% | $ 95.70 |
| 2 | 2.17% | $ 40.20 | 2 | 3.19% | $ 59.64 |
| 3 | 2.15% | $ 25.76 | 3 | 3.12% | $ 37.78 |
| 4 | 2.12% | $ 21.20 | 4 | 3.04% | $ 30.74 |
| 5 | 2.09% | $ 16.74 | 5 | 2.97% | $ 24.01 |
| 6 | 2.07% | $ 13.43 | 6 | 2.90% | $ 19.04 |
| 7 | 2.04% | $ 11.22 | 7 | 2.83% | $ 15.72 |
| 8 | 2.01% | $ 9.06 | 8 | 2.76% | $ 12.54 |
| 9 | 1.99% | $ 6.95 | 9 | 2.69% | $ 9.51 |
| 10 | 1.96% | $ 4.90 | 10 | 2.62% | $ 6.63 |
| EV $104.26 | EV $102.32 | ||||
| ROI 95.65% | ROI 48.96% |
With a 10% edge, having doubled his stack, the player should now decline the add-on.
If the player manages to triple up before the add-on, he needs to play with an edge over 20% to make the rebuy a +EV play:
| Increased Stack 200 | Increased Stack 200 | ||||
|---|---|---|---|---|---|
| Juice 9 | Juice 9 | ||||
| Rebuys 0 | Rebuys 1 | ||||
| Edge 20.54% | Edge 20.54% | ||||
| Place | Prob | Return | Place | Prob | Return |
| 1 | 3.62% | $ 104.87 | 1 | 4.77% | $ 139.83 |
| 2 | 3.52% | $ 65.12 | 2 | 4.59% | $ 85.77 |
| 3 | 3.43% | $ 41.10 | 3 | 4.41% | $ 53.48 |
| 4 | 3.33% | $ 33.32 | 4 | 4.24% | $ 42.82 |
| 5 | 3.24% | $ 25.93 | 5 | 4.07% | $ 32.90 |
| 6 | 3.15% | $ 20.48 | 6 | 3.91% | $ 25.66 |
| 7 | 3.06% | $ 16.84 | 7 | 3.75% | $ 20.84 |
| 8 | 2.98% | $ 13.39 | 8 | 3.60% | $ 16.35 |
| 9 | 2.89% | $ 10.12 | 9 | 3.45% | $ 12.19 |
| 10 | 2.81% | $ 7.02 | 10 | 3.31% | $ 8.35 |
| EV $229.19 | EV $229.19 | ||||
| ROI 210.27% | ROI 109.66% |
From this, we can conclude that even a player with a moderate edge who is getting a small discount on his chips should not rebuy if he has managed to sufficiently increase his stack. A player an edge of 50% should still take the rebuy if his stack is less than 4.5 times greater than his initial stack. A 100% edge player should rebuy with a stack less than 7.5 times his initial stack. For very skilled players rebuys should still be made except in extreme cases.
These examples demonstrate that rebuys should not always be taken. The more skill a player has, the more often he should rebuy. The more chips the player has accumulated the less often he should add-on.
Always rebuying and adding on is a mistake. Most of the time, rebuying and adding on is a good move for the skilled, well bankrolled player. ♠
Note from Arnold Snyder on Rebuys and Add-ons
In The Poker Tournament Formula, I advise that unskilled players should not make rebuys. I advise that skilled players should always add-on or rebuy as soon as possible, in order to always have as many chips in front of you as possible.
In Appendix A of The Poker Tournament Formula, I advise players that it is futile to try to profit from poker tournaments with a small edge, and I give an example of 10% as too small an edge to play with. I also advise players that my win rate, using the exact basic strategy provided in The Poker Tournament Formula in Skill Level 2-4 tournaments, was consistently over 200%.
Higher edges will be available with fast play properly adjusted for tournament structure in slower tournaments with higher patience factors and skill levels, as defined in The Poker Tournament Formula.
And one other point. I consider Pikachu’s analysis to be of value to players in exceptional circumstances, which is why I’ve published it here. But it does not take into account the concept of an implied discount on chips due to the increased utility of chips in a big stack. Due to the implied discount, I would rebuy and add-on much more frequently than Pikachu suggests. In fact, I would almost always rebuy and add-on. ♠