Visible Aces vs. any two cards

[Nichlemn]

Inspired by Question 19 of the donkey test.

Two players are playing heads up Pot Limit Hold'em with effectively unlimited stacks. However, one of them is always dealt A[img]/images/graemlins/spade.gif[/img]A[img]/images/graemlins/club.gif[/img] face up. The other is dealt random cards face down as usual.

a) Who do you think has the advantage, all other things being equal? Is this a particularly big edge - or small enough so a particularly skilled player could win with either two over a weak player in the long run?
b) What is your basic strategy if you are playing the random cards?
c) What is your basic strategy if you are playing the Aces?

Obviously b) and c) depend on the opponent's tenancies, so it could be a matter of trying to find the Nash equilibrium if possible. Your strategy for playing the Aces could simply be what you think is the best counter strategy for the way you would suggest playing the random cards.

[AaronBrown]

It depends crucially on the blind/limit structure. In no-limit, AA always goes all-in preflop, and wins on average.

You never have much more than a 20% chance against AA. Say the other player plays 20% of his hands, with an average preflop winning chance of 20%. That means he wins 4% of the hands. With $1/$2 blinds, and $2/$4 limit bets; out of 25 hands, he'll pay $30 for the 20 hands he folds plus $20 to see 5 flops. Even if he knew for sure whether he would win or lose after the flop, it only costs A's $10 to call to the river. That means the random hand player makes $18 on a win, not enough to make up for the $50 he spent.

Those are just made-up numbers, and the random hand player can improve slightly with game theory, but it will still be a losing proposition. If he could bet $100 on the river, it's a different story.

[tarheeljks]

[ QUOTE ]
If he could bet $100 on the river, it's a different story.

[/ QUOTE ]

yeah, but couldn't the player w/AA negate this by always pushing pf? (in a nl game)

[Lyric]

The random cards would win.

After the flop, random cards can bluff, but AA can never bluff. Random cards win.

[AaronBrown]

[ QUOTE ]
yeah, but couldn't the player w/AA negate this by always pushing pf? (in a nl game)

[/ QUOTE ]
Yes, I said that at the start. In no-limit, AA wins. In standard limit, AA wins. Only in a structure that allowed large river bets, relative to the blind, pre-flop, flop and turn bets, do the random cards have chance.

[jogsxyz]

[ QUOTE ]
[ QUOTE ]
yeah, but couldn't the player w/AA negate this by always pushing pf? (in a nl game)

[/ QUOTE ]
Yes, I said that at the start. In no-limit, AA wins. In standard limit, AA wins. Only in a structure that allowed large river bets, relative to the blind, pre-flop, flop and turn bets, do the random cards have chance.

[/ QUOTE ]

That's not necessarily true. Look at the toy game in MOP 19.3. Only the clairvoyant is dealt a card. Standard deck. Ace or king, Clare(clairvoyant) wins, else the defender wins. Clare is dealt 2/13(or ~ 15.385%) winners. Both players ante 5 units. Three streets of pot size bets(my own restriction).

Since the defender cannot profit by betting, only Clare may bet each street. Defender may call or fold. Clare is the favorite in this toy game, even though she is dealt only 15.385% winners.

[Nichlemn]

The format is pot limit.

So, let's assume the second play always played any two cards.
They would always limp in the SB, Aces will always raise to 3x BB, the any two would always call.

If Aces call down pot sized bets to the river, the final pot is:

6 BBs after preflop betting
18 BBs after flop betting
54 BBs after turn betting
162 BBs after river betting

(If the river bet was removed (let's say because with no cards to come it's even more tilted in favour of the any two cards) does this make the implied odds poor enough to make the Aces the favourite?)

Now even though this makes it appear to have great implied odds, it's not that simple to win that pot with the random cards. If the random card player only ever bet with a hand that beat Aces, the Aces would always fold and lose the minimum only when outflopped. If the random card player always tried to bluff when the Aces were unimproved, the Aces could simply call to the river and win over 80% of the time. Therefore, some combination in between may work.

A starting point:

- Obviously, always bet with a bigger made hand and give up if they flop a set, maybe bluffing against a tiny percentage of the time with a scary board.

- Usually bluff/semi bluff on monotone red flops. You are just as likely to have flopped two pair or trips than on any other flop, but there's a reasonable probability you will already have a flush, and the Aces will be forced to fold on any fourth card of that suit (semibluffing potential, though with no implied odds).

- The same deal with any three connected cards. You could just as easily have two pair/trips/a straight made already, and if not, a fourth connected card later makes it near impossible for the Aces to call.

- With either of the above, you could alter the strategy if the Ace player will always or almost always bet a drawy board to protect itself, or never bet for fear of it. If they make a PSB on such a flop, you have the potential for a 486 BB pot if you were to raise them with the goods. If they never bet but would often call, it may be best to take a free card.

- In general, a lot of optimal bluffing % can be made up with semibluffs - any pair or any draw, giving you at least 4 outs, is much better than betting with nothing and having no outs if the opponent decides to call you down that time.

- Paired boards are roughly the same as nonpaired boards in aggregate. Pokerstoving Aces vs. random cards on paired boards vs. nonpaired boards give basically the same equity. While paired boards allow for a single hole card to make trips, they counterfeit any two pairs.

[apfel]

it does not make sense to raise with the aces preflop.the potsize preflop is not important for the aces.i dont know if the random cards should raise preflop.(pocket pairs,red suited connectors)

[onesandzeros]

Dependant on stack sizes.

[tarheeljks]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
yeah, but couldn't the player w/AA negate this by always pushing pf? (in a nl game)

[/ QUOTE ]
Yes, I said that at the start. In no-limit, AA wins. In standard limit, AA wins. Only in a structure that allowed large river bets, relative to the blind, pre-flop, flop and turn bets, do the random cards have chance.

[/ QUOTE ]

That's not necessarily true. Look at the toy game in MOP 19.3. Only the clairvoyant is dealt a card. Standard deck. Ace or king, Clare(clairvoyant) wins, else the defender wins. Clare is dealt 2/13(or ~ 15.385%) winners. Both players ante 5 units. Three streets of pot size bets(my own restriction).

Since the defender cannot profit by betting, only Clare may bet each street. Defender may call or fold. Clare is the favorite in this toy game, even though she is dealt only 15.385% winners.

[/ QUOTE ]

my bad aaron, missed that part

@jog: why are you assuming that AA will not call down?