Avg Finish question for "Higher" STEPs

[Apathy]

Hi I have a question about how much it would cost you to work you way up the STEP ladder on the party higher steps starting at step 1.

Assumptions:

-Everyone plays exactly equal and everyone has an equal finsh distrubution for every level.

-All tiny cash prizes will be ignored to make it easier.

- You have an infinite bankroll and play an infinite # of tourneys, starting at step 1 and then playing until you either bust OUT, or make it to step 5.

The question is how much on average will it cost to get there. Step 5 is 15,000+500 to enter so obviously the answer will be much higher then 15,500, but how much higher?


Here is a discrption of the structure for the STEPs:


STEP 1: 30+3
1st - FR (Freeroll) STEP 2
2nd-6th - FR STEP 1
7-10 - OUT

STEP 2: 100+10
1st - FR STEP 3
2nd-4th - FR STEP 2
5-7th - FR STEP 1
8-10 OUT

STEP 3 500+50
1st- FR STEP 4
2nd-3rd FR STEP 3
4-8 - FR STEP 2
9 - FR STEP 1

STEP 4 - 3000+300

1st- FR STEP 5
2-5 FR STEP 4
6-7 FR STEP 3
8- FR STEP 2

STEP 5 - $15000+500


Thank you for your time and if I am missing any details or anything let me know.

I think this can be solved with a simple program or excel sheet but I'm not great with that stuff so hopefully someone can help me out!

Thanks again.

[mannika]

On average, it will take you 723 shots at buying in on Step 1 in order to make it to Step 5. This equals 723*33 = $23,859.


I solved this just by using some algebra:

First, I found equivalent terms for the value of each step in terms of the other steps.

S4 = (1/4)S5+(1/2)S3+(1/4)S2
S3 = (1/7)S4+(5/7)S2+(1/7)S1
S2 = (1/7)S3+(3/7)S1
S1 = (1/5)S2


Then I substituted those equations into one another to get:
S2 = (5)S1
S3 = (32)S1
S4 = (198)S1
S5 = (723)S1

Giving the Step 5 tournament the equivalent value of 723 Step 1 tournaments, and therefore $23,859.

[Apathy]

Thanks a lot for your help here. You can see discussion about this topic in this:
HERE (http://forumserver.twoplustwo.com/showflat.php?Cat=&Board=singletable&Number=2224976 &page=0&view=collapsed&sb=5&o=14&fpart=1)

Please join in the thread.

[Nottom]

[ QUOTE ]
First, I found equivalent terms for the value of each step in terms of the other steps.

S4 = (1/4)S5+(1/2)S3+(1/4)S2
S3 = (1/7)S4+(5/7)S2+(1/7)S1
S2 = (1/7)S3+(3/7)S1
S1 = (1/5)S2

[/ QUOTE ]

How did you arrive at these terms?

[mannika]

[ QUOTE ]
[ QUOTE ]
First, I found equivalent terms for the value of each step in terms of the other steps.

S4 = (1/4)S5+(1/2)S3+(1/4)S2
S3 = (1/7)S4+(5/7)S2+(1/7)S1
S2 = (1/7)S3+(3/7)S1
S1 = (1/5)S2

[/ QUOTE ]

How did you arrive at these terms?

[/ QUOTE ]

I'll do the S4 as an example. There are 8 places you can finish in S4, of which we will assume all are equally likely. Thereofre, there is a 1/8 chance that you make it to S5, a 4/8 chance you get another S4, a 2/8 chance that you drop down to S3, and a 1/8 chance that you drop down to S2.

Therefore:
S4 = (1/8)S5 + (4/8)S4 + (2/8)S3 + (1/8)S2
(4/8)S4 = (1/8)S5 + (2/8)S3 + (1/8)S2
S4 = (1/4)S5 + (1/2)S3 + (1/4)S2

[Dudd]

Maybe I'm missing something basic here, but wouldn't you have a 1/10 chance of getting a step five out of a step 4? I know that one of the assumptions is that the small cash prizes will be neglected, but wouldn't that assumption just lead to a 2/10 chance of getting nothing? Hell, it isn't that hard to throw in a bunch of constants for the cash prizes, so I'll do that as well:

S1= (1/10)S2+(5/10)S1+(1/10)25
S2= (1/10)S3+(3/10)S2+(3/10)S1+(1/10)21
S3=(1/10)S4+(2/10)S3+(5/10)S2+(1/10)S1
S4=(1/10)S5+(4/10)S4+(2/10)S3+(1/10)S2+(1/10)90

Therefore,
S2=10((.5)S1-2.5)=5S1-25
S3=7S2-3S1-21=(35)S1-175-3S1-21=(32)S1-196
S4=8S3-5S2-S1=(256)S1-1568-25S1+125-S1=(230)S1-1443
S5=6S4-2S3-S2-90=1380(S1)-8658-(64)S1+392-5S1+25-90=(1311)S1-9075

Therefore, One Step Higher 5 entry costs 1380(33)-9075=$36,465. That seems to be more like what I would have expected. On an entirely unrelated note, does anyone know when Party's going public on the London stock exchange?

[Che]

In an attempt to follow the logic of your calculations, I have calculated the "value" of steps 2 through 4 as follows:

S2/S5=(5)S1/(723)S1=5/723
723(S2)=5(S5)
S5=S2(723/5)=144.6(S2)

S2 buyin is $110 so an S5 is worth $15,906 if you buyin at S2.

Similarly for Steps 3 and 4 I find:

S5 if buyin at S3=$12,426
S5 if buyin at S4=$12,050

This implies that buying in at either step 3 or step 4 is superior to buying in at step 5 (ignoring opportunity cost) with step 4 as the best option.

When we include the possibility of receiving cash (as pointed out in the other thread) even buying in at step 2 becomes less expensive than buying in directly at step 5 (again, ignoring opportunity cost).

That doesn't make sense. What am I missing here?

Thanks,
Che

[mannika]

Here are the UPDATED and CORRECT calculations for the cost of the Step Higher SNGs at Party. These take into account the correct assumptions and all small cash giveaways.

First, using the payout structure, I have found values for S2-S5 in terms of S1.

S2 = (5)S1 - 25
S3 = (32)S1 - 196
S4 = (230)S1 - 1443
S5 = (1311)S1 - 8331

From this we can conclude that the average cost of an entry into the Step 5 tournament by starting at Step 1 each time is (1311)*(33) - 8331 = $34,932

To find the cost of entry into the Step 5 at different levels, we use the equations above and substitute them into each other.

Using the S1-S2 and the S1-S5 relationship,
(262.2)S2 - 1776 = S5
Therefore, average cost of entry into Step 5 from Step 2 = $27,066

Using the S1-S3 and the S1-S5 relationship,
(40.96875)S3 - 301.125 = S5
Therefore, average cost of entry into Step 5 from Step 3 = $22,232

Using the S1-S4 and the S1-S5 relationship,
(5.7)S4 - 333.9 = S5
Therefore, average cost of entry into Step 5 from Step 2 = $18,476

And of course, the direct buyin to Step 5 is $15,500.


So here are the final results for the cost of buying into Step 5 from all the different steps, assuming everyone plays equally.

Step 1: $34,932
Step 2: $27,066
Step 3: $22,232
Step 4: $18,476
Step 5: $15,500


Thanks to Dudd and Che for pointing out the inherant errors in the previous assumptions.