Poker Articles

A Bit of Maths: To add-on or not
By Paul Samuel
(The UK's answer to Mike Caro or Lassie)

1. Introduction

My normal game is Wednesday night at Luton. A £20 rebuy comp with add-ons followed by a cash game if things have not gone well or bed if they have.

I am reasonably successful, achieving about 60% of final tables with 40-50 runners as a rule but have a consistent problem presented to me

To add-on or not. That is the question.

You start with 1000TC and the blinds are 25,50 increasing on 45m, 45m, 20m, 20m...

Last week I finished the 90minute rebuy phase with about 10000TC (with rebuys) and asked my usual question.

Should I add-on ?

2. Analysis

Lets state the assumptions. Quickly in some cases in the hope you miss them.

Lets assume the tournament has 50 runners, pays 9 places and that I have a 60.00% chance of reaching the final.

Lets say that the prize pool after the add-ons (which occur after the rebuys are done) is £4,500 .

Here's a table:

Table 1 - Final table probability distribution

Pos	Prize Share	P-Pos
1	40.00%	5.00%
2	20.00%	6.00%
3	12.00%	10.00%
4	9.50%	15.00%
5	8.00%	17.50%
6	4.00%	20.00%
7	3.00%	17.50%
8	2.00%	6.00%
9	1.50%	3.00%

Just to explain, as an illustration what I am saying in the above is that the chance of myself winning the comp once I attain the final table is 5.00%.

This table gives an expected prize per tournament of £235 for yours truly from a standing start and an expectation of making £392 should I reach the final.

But hold on a moment !

This means that for an add-on to have a positive expectation we must increase our chance of attaining the final table by £20 / £392 or 5.10% .

Even if we allow that each additional 1000 of tournament chips increases our chance by this amount we see that very soon we run out of percentage !

Table 2 - Illustration of Probability Distribution

Chips	P-Win
0	0.00%
1000	9.00%
2000	17.19%
3000	24.64%
4000	31.43%
5000	37.60%
6000	43.21%
7000	48.32%
8000	52.97%
9000	57.21%
10000	61.06%
11000	64.56%
12000	67.75%
13000	70.65%
14000	73.30%
15000	75.70%
16000	77.89%
17000	79.88%
18000	81.69%
19000	83.34%

You see with the most evenly distributed number of increases imaginable, it is clear that 19000 tournament chips is the absolute maximum where an add-on MAY return a breakeven on my £20 investment !

However this is the most even distribution of probabilities; It is much more reasonable to assume that the shifts between each 1000 chip jump peak decline.

A curve like this might suffice:

You see how the probability of us reaching the final table flattens off as our chips increase.

See the table of final table probabilities with increases (delta's) to the right :-

Table 3 - Table of deltas

Chips	P-Win	Delta
0	0.00%
1000	9.00%	9.00%
2000	17.19%	8.19%
3000	24.64%	7.45%
4000	31.43%	6.78%
5000	37.60%	6.17%
6000	43.21%	5.62%
7000	48.32%	5.11%
8000	52.97%	4.65%
9000	57.21%	4.23%
10000	61.06%	3.85%
11000	64.56%	3.50%
12000	67.75%	3.19%
13000	70.65%	2.90%
14000	73.30%	2.64%
15000	75.70%	2.40%
16000	77.89%	2.19%
17000	79.88%	1.99%
18000	81.69%	1.81%
19000	83.34%	1.65%

Which concludes:-

3. Conclusion

In the case above the most chip we should add-on with is 7000.

Well I did'nt know that either !