7

Suppose that in a Texas hold'em heads up both player 1 and 2 know from the beginning that player 1 has an Ace. (This cannot happen in a normal game, of course.) Who has an advantage, player 1, since she has an Ace, or player 2, since he knows a card?

I am interested in all variants of the general problem as well, i.e., when having a better hand is less advantageous than knowing that hand. In the above problem, take for example NLHE with even stack sizes, player 1 also knowing that player 2 knows and player 2 also knowing that player 1 knows that he knows and so on, i.e., they have full information regarding this. Assume anything non-specified as you like, I really don't think that the blinds and whoever starts matter too much - or do they? Of course the blinds need to be small compared to the stack sizes, otherwise player 2 does not have much freedom and the Ace has a higher change to win.

  • 1
    Does she know that she will always have an ace and does she know that you know that she will always have an ace? Player 2 seems to have a huge advantage here, mostly in terms of being able to lose less by easily laying down hands when the board contains an ace or is favorable to an ace. If the player knows you know this, the advantage will be minimal, but if the player doesn't know that you know this, then I think you would be able to clean up against her. – Chris Farmer Dec 29 '14 at 18:06
  • We need more info! Does the player numbering also imply position? Can we assume even stack sizes? Is this LHE or NLHE? Also, is there anything preventing the "non-Ace" player from picking a tight, tight range? If this player knows an Ace is always there, then they can adjust their own range. If so, then it comes down to bet-sizing and pot control - "playing the player". Please flesh out the question a little more so we can help you better – Unknown Coder Dec 29 '14 at 18:31
  • I haven't done the math, but I suspect the advantage would be to the player on the button, regardless of which is which. – Lee Daniel Crocker Dec 29 '14 at 23:15
  • 2
    @LeeDanielCrocker thats as a possibility but it all depends on more information. But I think the advantage comes from knowing the one card more than position. If you know one card then it drastically reduces your own, range, right? – Unknown Coder Dec 30 '14 at 1:37
  • @Jim: I tried to give more info. – domotorp Dec 30 '14 at 6:57
4

For the purposes of this answer, Player 1 is the one with the Ax hand. Player 2 is the player with the non-Ace hand. I'm also assuming that Player 2 has a random hand - in other words, sometimes their starting hand will also include an Ace.

This is an interesting question. My conclusion is that Player 2, knowing that Player 1 has an Ace, has the advantage, but it won't be easy to win!

Furthermore, the advantage comes from the one card knowledge, as opposed to something else like position.

If Player 2 knows the opponent has an Ace, then it would be imperative to narrow down their range drastically and only play things like Broadway cards with an Ace in the hand. Anything else and they are taking too many risks.

However, Player 2 narrowing down the range in this manner gives Player 1 a lot of information too! Player 2 needs to avoid a scenario where Player 1 always knows that they are on an Ace. So Player 2 would have to do some range balancing in order to throw Player 1 off their cards.

And that's really where the advantage comes in - Player 2 can balance their range. But Player 1, with the Ace, is basically "locked-in" to their range! This is the angle that needs to be exploited in order for Player 2 to win.

So, it's not easy, but in order to win, Player 2 would need to be willing to showdown relatively frequently with trashy hands and be prepared to lose with some of those hands. They would need to keep a tight range that keeps all the Broadways with an Ace and occasionally also show down with J2o and T4s

Player 2 would also consider playing other things that flop well and completely miss the range for Player 1. For example something like 8♥ 9♥ wouldn't be a terrible hand if the flop gives Player 2 multiple draws and/or a pair. If you can get to a board like that, then the hands are worthwhile to consider.

Depending on skill, Player 2 can also bluff at a lot of boards. The board texture is going to be either very good or very scary for Player 1. So, as stated above, a board with good draws, that miss the Ax range, are good to make a move at and represent those hands. All your low straights and majority of flushes have the potential to be hands where Player 2 can push Player 1 off their Ace. So Player 2 could consider playing all monochrome boards and connected boards that provide multiple straight possibilities. Again, Player 1 is somewhat locked-in to the range (albeit a high one!) but that means that the low stuff is going to miss them too! That's the part that Player 2 can exploit and potentially win over the long run.

Having said all that, in the process of writing all this, I've also come up with countermoves for Player 1. Basically, they would need to play very aggressive pre-flop. You would need to make it very, very expensive for Player 2 to enter the pot. You need to avoid those board textures that give a lot of draws; best way to do that is raise, raise, raise pre-flop! Player 1 would NEVER just check or call pre-flop. In this process, they would win by blinding-off their opponent and giving Player 2 no cheap flops. The Ace is most valuable pre-flop. After that, the equity is more likely to drop than anything else.

  • Yes, I had similar thoughts. I just don't think it's worth for Player 1 to raise pre-flop with any hand - that would just be equivalent for her paying a high blind in each game. So probably Player 1 also has to bid according to her cards, what makes the game quite interesting. I guess one should try it in real life and see the outcome... – domotorp Dec 30 '14 at 11:01
  • @domotorp Did you get the players backwards? Player 1 (with the Ace) should ALWAYS raise with ANY hand pre-flop. They DO NOT want to give a cheap flop. Player 1 has the highest equity at that point, they are never checking or calling pre-flop. Always raising and being aggressive about it. – Unknown Coder Dec 30 '14 at 11:40
  • I did not make a mistake - in fact in any game for any player always raising is equivalent to playing higher blinds all the time. Imagine that the blinds are around 1 chip, the stack sizes are 999999 and Player 1 always raises to 1000. Then Player 2 can keep on folding until she gets an Ace and another good card himself, so he will have the advantage after the flop. – domotorp Dec 30 '14 at 12:34
  • @domotorp I'm sure you understand that even though there are only 2 players there are still 2 blinds! (the small blind is on the button). So, no, Player 2 cannot just sit back and wait for an Ace. Besides, as I wrote above, that can be deadly because then Player 2 is also giving information about their hand away! Player 2 would still need to take an active role in many hands to (1) stay ahead of the blinds and (2) balance their range. Therefore, it's advantageous for Player 1 to raise, raise, raise and never give away a cheap flop. – Unknown Coder Dec 30 '14 at 16:54
  • I think he can wait as the blind is too small compared to his stack size. But of course, to avoid giving information, he can sometimes bluff or play with other good cards. – domotorp Dec 30 '14 at 20:43
1

Call player2 (not ace) Hero

The problem for Hero is a random hand is a dog to even the worse possible hand Villain has - A2o.

The only hands that are ahead of Ax are AA-22 and AK-AT. That is only 9.3% - call it 10%.

Say Villain raises to 3 BB every time pre-flop
Hero has a hand to defend one in every 5 orbits
Hero goes all in and Villain folds every time
Each orbit cost 1.5BB

EV = -5*1.5BB + 3BB = -4.5BB

Hero has to invest 7.5BB in blinds plus to pick up the Villain's pre-flop raise to 3BB.

Villain will call with some premium hands like AKs where the are ahead of Hero's range so Hero EV is even less.

So the Hero bluff raises pre-flop 1 in 5 orbits
Hero bluffing 1/2 the time they go all in pre-flop
Villain still folds to every time

EV = -2.5*1.5 BB + 3BB = -.75 BB  

Even with excessive bluffs and Villain folding every time a negative EV.

Hero would have to bluff so much that Villain can call and be ahead.

With only 10% of the hands ahead of a lone ace there is no strategy to beats a known lone ace.

Hero would need more like 25-30% of the hands to be ahead of a lone ace to have a GTO edge.

The ace (Villain) should actually charge more like 6BB to see a flop. Ace should push pre flop with AKs. If the Hero is playing more than 1/10 hands then start pushing with AQs and AKo pre flop. If the Hero is playing more than 1/5 of the hands then push with AT+. Villain does NOT want see a flop as if an ace hit Hero knows to get away.

  • I like your arguments and I believe that it has many interesting observations, but I don't think this is a complete analysis. I mainly doubt that sentence "Hero would have to bluff so much that Villain can call and be ahead." – domotorp Mar 12 '17 at 20:31
  • Sometimes I can bluff with raising just a little, sometimes I can raise much more. I've posted a related question, which should make the issue clear: poker.stackexchange.com/questions/8794/… – domotorp Mar 12 '17 at 20:52
  • It is a complete analysis you just do not understand it. You cannot bluff more than 50% of the time expect to them to fold. You bluff a different amount does fix that. – paparazzo May 5 '17 at 19:03
  • I'm sorry, it might indeed be my lack of poker jargon to understand your answer. If you can expose the gist of your argument by leaving an answer on poker.stackexchange.com/questions/8794/…, I would appreciate. – domotorp May 5 '17 at 19:21
  • Jargon? I have no desire to help you. You just dismissed what you did not understand. – paparazzo May 5 '17 at 19:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.