Quote Originally Posted by elipsesjeff
1) Pocket pairs occur less frequency and are ahead of all non-paired unsuited hands preflop.
Hi Jeff,

some words to Pokerstove. Pokerstove is a hot/cold calculator, that means that all hands are simulated through showdown. No one is allowed to bet, raise or fold. Now some words to enumeration: any possible combination of both players predefined cards are outplayed versus any possible board combination. The possible combination of clashes of player 1 and 2 vs any possible board is more than 25 Million different combinations. But 25,585,246,368 is the maximum number of n! (n factorial) with regards to both players hands and all possible boards combinations that can occur as iterative combinations.

Now, do me one favor: check the hands with Pokerstove yourself and SEE that result for your self, the enumeration takes less than a second on my very fast computer.

Problem for 1) is that with Pokerstove calculations Pokerstove really STICKS to the hands that are to be enumerated as n! (n factorial) combinations versus all possible board combinations - see hot/cold explanation.

Here one example:

Player One will only push Pocket Pairs bigger than pocket Tens, Player Two lowers his range to all Pocket Pairs bigger than 77.

The result is once again enumerated, NOT simulated.
The result is that Player One has dominant equity:

2,208,872,160 games 0.050 secs 44,177,443,200 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 62.993% 61.68% 01.32% 1362371700 29056578.00 { TT+ }
Hand 1: 37.007% 35.69% 01.32% 788387304 29056578.00 { 77+ }

You see in the brackets that Player One only plays TT, JJ, QQ,KK,AA
and Player Two 77,88,99,TT,JJ, QQ,KK,AA

Player Two has a wider range here and therefore should both player have their range, Player One has dominant equity because he ALWAYS has at least TT for the calculations. Simple as that.

Now back to our ORIGINAL situation the problem is that HERE Player One has at least TT for the pocket pairs and Player Two goes as low as 77 as in the above example other cards exluded.

The nice thing about pokerstove is that ALL calculations can be recalculated by yourself with Pokerstove, enumeration will lead to the exact same result with the exact same amont of game down to the last game, THAT is why the result is definitive and recalculatable.

Now back to the second part of that sentence:
Quote Originally Posted by elipsesjeff
Thus, those that shove more are likely to be ahead, but they're never ahead by more than a few percentage points.
Pocket Pairs bigger than TT (TT, JJ, QQ,KK,AA) that are played out vs ANY suited connector BIGGER THAN 54s (AKs, KQs, QJs, JTs, T9s, 98s, 87s, 76s, 65s, 54s) are a MASSIVE favourite, 75% vs 25% FOR the Pocket Pairs and AGAINST the Suited Connectors
Again, Pokerstove ASSUMES that ONLY these hands are "clashed" against each other (once again, Pokerstove is a hot/cold calculator), it employs in its calculation the requencydistribution and mesures this n! (n facotiral) versus all possible board combinations, which are in this case more than ONE BILLION COMBINATIONS:

1,869,835,968 games 0.010 secs 186,983,596,800 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 75.275% 75.01% 00.26% 1402639668 4883634.00 { TT+ }
Hand 1: 24.725% 24.46% 00.26% 457429032 4883634.00 { AKs, KQs, QJs, JTs, T9s, 98s, 87s, 76s, 65s, 54s }

As you can see, what you wrote does not fit the result. The nice thing is, as I said that you can REPEAT and RECALCULATE that with Pokerstove yourself, your will come ENUMERATED to the EXACT SAME RESULT. ALWAYS!

Quote Originally Posted by elipsesjeff
2) The change in EV comes from the domination of kickers, like AJ to AT etc. You failed to include several hands that are obviously shoves in hand 2's range, including AQs, AJs as well as AJo+. You should see a slight increase in equity but whenever you're talking about All-ins preflop with two people with wide ranges you're never going to have a large EV difference.
Well Jeff, THAT is what I did NOT include in the calculations. As you can see, Pokerstove ALWAYS puts in the handrange that has been calculated.

Opern Pokerstove, Copy-Paste the line under here for Player one:
TT+,ATs+,KTs+,QTs+,JTs,ATo+,KTo

and als copy-paste the line under here for Player Two:
77+,AKs,KQs,QJs,JTs,T9s,98s,87s,76s,65s,54s

Then click on the button in Pokerstove of Player 1 and you will see that he plays exactely 25 hands and ONLY these 25 hands ALL the time, THESE are his range. Click OK, otherwise Pokerstove will loose the setting. The occurence of hands which are his "get hands dealt frequency" is here14.3%, which means he will be able to play 14.3% of his dealt hands always sticking to this range.

Then click on the button in Pokerstove of Player 2 and you will see that he plays exactely 18 hands and ONLY these 18 hands ALL the time, THESE are his range. Click OK, otherwise Pokerstove will loose the setting. he occurence of hands which are his "get hands dealt frequency" is here6.6%, which means he will be able to play 6.6% of his dealt hands always sticking to this range.

Now, make sure you have "Enumerate All" as calculation setting for Pokerstove. Click on "Evaluate" and you will get EXACTELY this result down to the last number down to the last game:

25,585,246,368 games 0.010 secs 2,558,524,636,800 games/sec

Board:
Dead:
{ TT+, ATs+, KTs+, QTs+, JTs, ATo+, KTo+, QTo+, JTo }
equity win tie pots won pots tied
49.124% 48.21% 00.92% 12334149208 234464490.00
{ 77+, AKs, KQs, QJs, JTs, T9s, 98s, 87s, 76s, 65s, 54s }
equity win tie pots won pots tied
50.876% 49.96% 00.92% 12782168180 234464490.00

Now, again: EVERYTHING you thought of I thought of before, too! But that is NOT the case and THAT is why it is worth to take a second look at this Paradoxon. Please anyone reading this, check it yourself with Pokerstove and be perplexed like I am.

One more info, should it make you take this paradoxon more serious:
I used to teach econometrics and statistics for Master students on a private university. No need to brag here about myself but the point is I crunch numbers for a living. If it perplexes me it SHOULD perplex YOU also, because in Poker we are crunching numbers to make some decent money out of it!

So someone, give this some thought again please, I am getting crazy about this, there IS NO sufficient explanation for me so far.