After my 1000th, 2000th and 3000th number crunching posts, I skipped 4000 but here's my 5000th post about ICM benefits of entering (or re-entering) late to a tournament with a shorter stack than the average.

Ok, let's start with a thought experiment. Imagine a $10+$1 9-man SNG, paying prizes of $45, $27, $18. What if a crazy site allowed us to enter late after some the other players have knocked each other out, would we want to do it?

To take the most extreme case, if we were allowed to wait until the other players had played down to a winner, and then play him heads-up, even with the consequent 8-1 chip deficit, it would obviously be in our favour, as we would win either $45 or (much more often) $27 dollars for our $10 entry fee, easily defeating the $1 rake (in fact ICM says our stack is worth $29).
What about 3-handed - well assuming even stacks, we would be the short stack in a 4-4-1 battle, something which is worth $22.60. All the following calculations assume even stacks for the other players, which will understate the effect of ICM somewhat.

Our equity when entering X-handed is as follows
2 29.00
3 22.60
4 12.46
5 11.17
6 10.63
7 10.33
8 10.13
9 10.00

The fifth number is probably the most important result, as it shows that if we take the final spot in this tournament when 4 of the original 8 have been eliminated, then ICM pays our $1 rake for us. We're going to call it the ICM rake-paid point or ICM RPP. This tournament has an ICM RPP of 5. We can also look at the other numbers to see if we'd rather take the ICM advantage or just play. So by waiting till 2 people are knocked out, ICM increases the value of our stack by 3.3%. If we are good at the early stages of the tournament with 75BB or 50BB stacks we might prefer to just play. If not then it might be best to enter late.

Now for something closer to reality. A $3.50 rebuy with 200 re-buys fired. This is the same as the MTTSNGs which run but with no account taken for add-ons.

28 4.88
29 4.50
30 4.31
31 4.18
32 4.09
33 4.03
34 3.97
35 3.93
36 3.89
37 3.86
38 3.82
39 3.79
40 3.77
41 3.74
42 3.73
43 3.71
44 3.69
45 3.68
46 3.66
47 3.66
48 3.62
49 3.61
50 3.61
54 3.57
63 3.52
64 3.51
65 3.51
66 3.51
67 3.49
68 3.49
69 3.49
70 3.49
71 3.47
72 3.47
99 3.41
117 3.39
126 3.38
135 3.37
144 3.37
153 3.34
162 3.33
171 3.33
180 3.33

ICM RPP of 66. Conclusion. The ICM advantages of rebuying don't offset the rake expended; at a typical 150 players remaining ICM pays for about a quarter of the rake - it may therefore help to offset the lost chance for the player (or rather this bullet) to exercise his edge in the early part of the tournament. Though its outside the scope of this post, it's also worth pointing out that the later bullets fired are more likely to result in a chance to get cheap chips in the add-on.

As the final example I'm going to take a satellite. Should we late enter (or rebuy) in a satellite? These figures are based on the bounty-builder satellite from 23 January, paying twelve $44 seats and consolation prizes of $7.50 and $4.50. The buy in was $4.40 and there 135 entries.

15 8.00
16 7.14
17 6.63
18 6.27
19 5.99
20 5.78
21 5.60
22 5.45
23 5.34
24 5.23
25 5.14
26 5.06
27 5.00
28 4.93
29 4.88
30 4.83
31 4.78
32 4.74
33 4.71
34 4.67
35 4.64
36 4.60
37 4.58
38 4.55
39 4.53
40 4.50
41 4.49
42 4.47
43 4.45
44 4.43
45 4.41
46 4.39
47 4.38
48 4.37
49 4.36
50 4.34
54 4.29
72 4.17
99 4.07

So the ICM RPP for this satellite tells us we are rake-free if we enter when the player pool is about three and a half times the number of places available to win. This relationship is a rough guide for different shapes of satellite (except those where only a small number qualify or where the rake is not 10%). Use about 2.5x for a satellite where a fifth of entries qualify, 3.5x for satellites where a tenth get through and still only 4.5x where one entry in a hundred qualifies.

First...Thank you so much for taking the time to write this keep those coming.

I was recently asked how the ICM RPP number is derived.

Basically, it's the number of people including ourselves there would have to be left for our own stack to be worth more than we would pay for it if we late entered or re-entered at that point (assuming even stacks) so it depends on the tournament structure and rake.

So in the example of a $10+1 tournament paying out in a 5-3-2 ratio that started with 8 other players - if we enter with 4 remaining (with us as the fifth player), and the stacks are 2000-2000-2000-2000-1000, then our own stack would be worth $11.16 dollars - i.e. more than we paid for it. This doesn't apply if we entered as the sixth player (joining 5 other players with 1600 each), so the ICM RPP is 5.

There isn't a formula to calculate ICM RPP directly, you just evaluate your stack using ICM for the various hypothetical situations, as in the tables above.

cool post, i missed this one when you wrote it. I actually just dug up your post on bubble factors from 2015. nice work