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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%