Thoughts on VP$IP as a function of M during a MTT

[trojanrabbit]

[ QUOTE ]
Wow. Did you really overcome the hand ranking and overcall limitations you mentioned a few months ago? If so, kudos, that's quite an accomplishment.

(still waiting on Amazon for my book)

[/ QUOTE ]
No problems with hand rankings, but overcalls are still ignored. I have studied the overcall problem and it only has a minor effect, especially for an M of 4+.

I can understand the frustration with Amazon. They've sold out 3 times. This last time after they announced immediate availability they sold out in 12 hours. I guess that's good and bad news.

As far as an example, here's the equilibrium solution for pushing with an M of 6, with no antes:

<font class="small">Code:</font><hr /><pre>
Position No Antes
Small Blind 22+,A2+,K2+,Q6o+,Q2s+,J8o+,J3s+,T7o+,T4s+,97o+,95s +,87o,85s+,76o,74s+,64s+,53s+,43s (59.9%)
Button 22+,A2+,KTo+,K5s+,QTo+,Q8s+,JTo,J8s+,T7s+,97s+,86s +,76s (32.7%)
Cut-Off 22+,A4o+,A2s+,KTo+,K9s+,QJo,Q9s+,JTo,J8s+,T8s+,98s (27.0%)
Hijack 22+,A9o+,A2s+,KJo+,K9s+,QJo,Q9s+,J8s+,T8s+,98s (20.7%)
3-off 22+,ATo+,A7s+,A5s,KJo+,K9s+,Q9s+,J9s+,T9s (16.7%)
4-off 33+,AJo+,A8s+,KQo,K9s+,Q9s+,J9s+,T9s (13.9%)
5-off 55+,AJo+,A9s+,KQo,K9s+,QTs+,JTs,T9s (12.1%)
6-off 66+,AJo+,A9s+,KTs+,QTs+,JTs (10.1%)
7-off 77+,AQo+,ATs+,KTs+,QTs+,JTs (8.4%)
</pre><hr />
For those nit-picky game theorists, this does consider mixed strategies and these are the hands that are pushed at least 50% of the time.

Kill Everyone lists out the solutions for the other stack sizes and also does it with and without antes for comparison. I also list what the equilibrium calling ranges are as well for each case. And... there is some nice discussion about it including some shortcuts for helping you memorize approximate solutions for use at the table.

Hope that's enough of a teaser. [img]/images/graemlins/wink.gif[/img]

Tysen

[Nougat]

I certainly don't want you to give away too much of the goods if you're trying to make money off book sales but I do have a question about the push equilibrium hand rankings you just listed in this thread. How stable is this Nash Equilibrium?

[trojanrabbit]

[ QUOTE ]
I certainly don't want you to give away too much of the goods if you're trying to make money off book sales but I do have a question about the push equilibrium hand rankings you just listed in this thread. How stable is this Nash Equilibrium?

[/ QUOTE ]
There is no dependence on arbitrary hand rankings - the strategy for each hand is considered individually. For the conditions specified (i.e. mixed solutions okay but no overcalls) the equilibrium is 100% stable.

Tysen

[Nougat]

Wow. That's quite an impressive achievement and certainly useful to a great many people including myself. Congrats.

[rama96ab]

This is very interessting for me - Excellent thanks!

I've ordered the book today to read more...

I discovered a few thing with the equilibriums with M=6 which you posted. Compare with the function 1/X, where X is active players left in the hand. First-in from BTN , where X=3 gives 33% pushing hands, CO X=4 gives 25% pushing range..etc.. this function is VERY equal to the Killer pushing ranges in all position with M=6!

With the M=6 this gives an 20% VP$IP (BB=9% range!) (If first-in in all positions). Compared to Harrington's pushing ranges Killer is much tighter! Going through lot's of his examples and "plotting" Harrington's hand ranges, his willing to puch with a lot more hands.. (Ex. any to cards first-in on BTN with M=5).

My origial ideas with M=6 was 10% (my share on a 10 person table) plus 1/M = 1/6 = 17%, which sums up to 27% VP$IP.

I've now tighted this in my Excel sheet to watch's Killers recommeded ranges - great! (Anyone interested to see let me know) - I got 2nd place trying in 1 SNG so this showed the correctness ;o)

My new challenge is plotting a few other point into the function F(M) = VP$IP to model the function when M differs from 6. Say M = 3 and M = 25.. or whatever - at least one point above and one point below with M = 6. You wrote Killer "only" put ranges for M&lt;=8, so I guess I need to make some assumption with higher M's!

Let's do a thought about the high M. Playing with an M = 33 you're pretty functional poker player (using Harrington's term) so let's assume this is the point where you want to hit VP$IP = 10% which should be your share in a 10 man table. What I do is to take each possitions pushing ranges and normalized by 10% / 20% - So I'd play half the range with M=33 - dropping below I'd have convergence to 20% as M -&gt; 6.

With a very high M (&gt;33%) i'd then play less my share of 10% etc. which seem pretty intuative to me
and in much other literature - play tight early in tournemants.

Any thought on a good "break-even" M where you hit 10% VP$IP - M= 33 good? M=20 or 50?

What does Killer say with M=3 and M=8 - just to get an idea of the function F(M) = VP$IP? I have an intuition that the function is something like a*Ln(M)+b = VP$IP, especially to get f(33)=10% and a appropriate convergence to f(6)=20% !

Thank for the respnses so far..

Best regards
Rama96ab