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Subject : A closer look at the MTT format and playing with an edge posted by Pikachu on Aug-28-2006 08:58pm |
| My last post 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 MTTs, it wasn’t a very useful post. Now I’ll provide some examples from an MTT format that takes 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 buyin but does not double his chances of finishing ITM. 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 bank rolled 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. inc stack 100 . inc 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%, 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: inc stack 200 . inc 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 are 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. |
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