Gedankenexperiment (Thought Experiment) #1

[PDosterM]

Dr. Savage,

Very good answer. Not only are the original hands listed in absolute value order, but in probability of winning order as well. I did not have in mind worrying about the probabilities of winning, but only their absolute poker ranks.

With that in mind, can you come up with a five-player solution? If the probabilities happen to come out in the correct order -- so much the better.

[DrSavage]

[ QUOTE ]
Dr. Savage,

Very good answer. Not only are the original hands listed in absolute value order, but in probability of winning order as well. I did not have in mind worrying about the probabilities of winning, but only their absolute poker ranks.

With that in mind, can you come up with a five-player solution? If the probabilities happen to come out in the correct order -- so much the better.

[/ QUOTE ]

I'd like some clarification on this.
You are saying that the rank of hands preflop should be made based on probability of them winning against a random field and not the other hands being in the same hand ?
How big is the field? Should we rank the hands on their probability of beating random hand headsup, or beating a field of 9 random hands or what? How exactly do you rank hands without the competition?
Example : AA, KK and JTs. AA destroys both , KK destroys JTs, but JTs has higher equity than KK if AA is also in the pot.
If it's done this way I don't really see any particular value in these examples. I find it more interesting to analyze each hand against the other hands in competition and then have the order reversed on the flop.
Also I still don't see what good does it do to be second on the flop, since B in your example and C in my example despit having made a second best hand have zero equity since they cannot improve while the other hands are drawing live.
It could be theoretically possible to come up with some sort of 6 hands example , I'd expect it to be smth. like str flush over flush over straight over set over pair over high card.

[Schneids]

Yeah, I'd like more clarification too about PF.

Are we supposed to pretend we are showing down the 2 hole cards, thus meaning that 22 beats AKs PF?

Or, the way how I've been determing PF rankings by going to twodimes and entering in all the hands that I'm going to use, then using their rankings to find the values of 'best' to 'worst' hand, in each set.

[PDosterM]

Schneids,

Not what I had in mind (because of the probabilities), but an inspired answer nonetheless. Very nice.

I doubt if we use probability of win as the sort criterion that we could prove what the theoretical maximum numbers of players is, other than by exhaustive enumeration of the (unreasonably large number of) combinations.

Can you come up with a five-player solution that is based on absolute poker ranks (i.e. bigger pairs beat lower pairs, pairs beat non-pairs, higher card beats lower card)?

But now you have me curious. Using your probability ranking instead of strict poker hand ranking, do you find a six-player solution?

[Festus22]

5 was all I could come up with:

Preflop (in ascending order in terms of rank)
6 [img]/images/graemlins/club.gif[/img] 2 [img]/images/graemlins/club.gif[/img]
7 [img]/images/graemlins/diamond.gif[/img] 6 [img]/images/graemlins/heart.gif[/img]
A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/heart.gif[/img]
3 [img]/images/graemlins/diamond.gif[/img] 3 [img]/images/graemlins/heart.gif[/img]
A [img]/images/graemlins/diamond.gif[/img] A [img]/images/graemlins/heart.gif[/img]

Flop comes 3 [img]/images/graemlins/club.gif[/img] 4 [img]/images/graemlins/club.gif[/img] 5 [img]/images/graemlins/club.gif[/img]

Now in decending order
6 [img]/images/graemlins/club.gif[/img] 2 [img]/images/graemlins/club.gif[/img] makes a straight flush
7 [img]/images/graemlins/diamond.gif[/img] 6 [img]/images/graemlins/heart.gif[/img] makes a straight
A [img]/images/graemlins/spade.gif[/img] 2 [img]/images/graemlins/heart.gif[/img] makes a lower straight
3 [img]/images/graemlins/diamond.gif[/img] 3 [img]/images/graemlins/heart.gif[/img] makes a set
A [img]/images/graemlins/diamond.gif[/img] A [img]/images/graemlins/heart.gif[/img] makes a pair

[pudley4]

4 [img]/images/graemlins/club.gif[/img]4 [img]/images/graemlins/heart.gif[/img]
A [img]/images/graemlins/club.gif[/img]7 [img]/images/graemlins/heart.gif[/img]
K [img]/images/graemlins/club.gif[/img]8 [img]/images/graemlins/heart.gif[/img]
Q [img]/images/graemlins/club.gif[/img]T [img]/images/graemlins/heart.gif[/img]
J [img]/images/graemlins/club.gif[/img]9 [img]/images/graemlins/heart.gif[/img]
J [img]/images/graemlins/diamond.gif[/img]4 [img]/images/graemlins/diamond.gif[/img]
9 [img]/images/graemlins/diamond.gif[/img]6 [img]/images/graemlins/diamond.gif[/img]

flop: T [img]/images/graemlins/diamond.gif[/img]8 [img]/images/graemlins/diamond.gif[/img]7 [img]/images/graemlins/diamond.gif[/img]

44 becomes bottom pair
A7 becomes 3rd pair
K8 becomes 2nd pair
QT becomes top pair
J9 becomes straight
J4 becomes flush
96 becomes straight flush

[PDosterM]

In the spirit of a gedankenexperiment (which does not necessarily have to lead to anything), I was asking the question with the intent that strict poker rankings (pair beats non-pair) would be used preflop. You have, however, convinced me that your interpretation (using probabilities of winning) is certainly valid – and perhaps even more interesting.

So I remain intrigued by answers to either interpretation.

[Schneids]

[ QUOTE ]

But now you have me curious. Using your probability ranking instead of strict poker hand ranking, do you find a six-player solution?


[/ QUOTE ]
Qh Jh 25.61%
Ac Kc 22.96
6s 6c 16.52
8s 4s 14.43
Kd Td 12.95
4d 3d 5.86

Flop: 7d 6d 5d
straight flush, flush, straight, set, ace high, queen high.

Note that as soon as you introduce a straight flush into the picture, nearly all other hands are drawing 100% dead, so, we have some ties in the likelihood of them winning at the river (ie 0%). That might be violating one of the principles you defined beforehand, regarding ties.

**Also, keeping this poker related, interesting to see that 84s has more value in this 6-handed pot than KTd. Kind of counter intuitive till you notice it's pretty much drawing for tens and hoping they'll hold up, since so many diamonds are already in play.**

Second-edit: Esssh, doing it 7-handed this way is gonna be a beast.

[Schneids]

Basically, made a few changes to my 6-hand one and added in another pair. Took a few minutes to find a way to get a second pair in here, keep QJh a top the rankings, and then keep the second flush maker lower than a straight maker, PF. Good times [img]/images/graemlins/grin.gif[/img]

Qh Jh 22.76%
Ac Tc 21.37
9c 9h 18.60
6s 6c 13.07
8s 4s 7.71
Jd 8d 7.55
4d 3d 5.86

Flop: 7d 6d 5d
straight flush, flush, straight, set, pair, ace high, queen high.

Note that as soon as you introduce a straight flush into the picture, nearly all other hands are drawing 100% dead, so, we have some ties in the likelihood of them winning at the river (ie 0%). That might be violating one of the principles you defined beforehand, regarding ties.

[gunboat]

[ QUOTE ]
correct. but he is in last place on that flop. that is the key to the question i believe

[/ QUOTE ]

Hand one has two outs while hand 2 is drawing dead. Since there is no prize for second best, hand one is clearly better than hand 2 after the flop.