The Akq Game

[Dictius 0]

There are three cards in the deck. A, K and Q. There is a certain amount of money in the pot already from blinds/antes, let's call it P.Your are heads up, in position in the hand, and your opponent always checks first.You have the option of checking behind and going to showdown or betting $1. Your opponent can only call or fold, he can not re raise.The next hand you switch positions, and the game continues for a long time.What is your strategy if you are dealt A?What is your strategy if you are dealt K?What is your strategy if you are dealt Q?

[NonZeroPossibility 0]

You're making the blinds/ante "P" but the bet is $1... I think we need an exact dollar amount on the blinds to answer it correctly... but maybe not. I'll try:Dealt A=raise every time obvDealt K=check every time. Chances are 50/50 you are beat, but he's never calling with a Q and you never win if he has ADealt Q=This one is confusing me. I think we need the specifics on $ amount for this one. At first I thought check every time was the obv decision, but if villain will lay down a K to a bet, then I think we need to bet every time.1 question tho: When we are OOP does villain check every time we check?

[potatoman 0]

Edited:Okay, assuming we bet first:I bet the A. I check the king. Then it's show down right?I bet the queen half the time . Using this strategy means my opponent can do the following:He calls with the A, K and folds the queen.When I hold Q's 10 times, I lose 5 extra bets.When I hold A's ten I win 5 betsWhen I hold K's I break even.If he calls with the A, and folds the K and Q. That means, I can bet everything win 2/3 of the pots.

[simo_8ball 0]

Firstly, we never bet a King because either he folds the Queen or he calls with the Ace. We always check it down.Secondly, checking the ace is stupid because we have the nuts. So, we always bet the Ace. He will fold the Queen and have to make a decision with the King.For a pot size from 0 to 1, it is never profitable to bluff a Q. Half our opponents hands are aces, so half the time we win less than 1 and half the time we lose 1. For pot sizes of <1, we just bet the ace and check everything else.We need to look at the game for pot sizes 1 or over.If we never bet a Q, he can always fold the K. If we always bet a Q, he can always call with the K (half the time he loses 1 to an ace, half the time he wins P+1 against a Q).I'll have a go at the payoff matrix later.

[Dictius 0]

Maybe there's some confusion in the question here.There is only one street of betting and there are only three cards in the deck so if you have the Q, the other guy has the A or K.The size of the money in the pot I have called a general P instead of one specific value, because the problem is interesting if there can be different amounts in the pot. Try considering what you would do if the money in the pot already was $1, $2 or $100 if you like.If you are OOP, obviously you would never call with the nut low (Q).When we are OOP the villain does not always check behind.

[mtdesmoines 3]

This is gonna draw math and stat freaks from all the other forums you know ....

[Temporary Nuts 1]

bet every street until you bust and be happy you can stop playing this stupid game.

[Sheiky 0]

Are you David Sklansky in disguise?

[Frez 0]

This theoretical game is so old I'm surprised it hasn't been discussed here before.If you want some discussion, post it in the general strat forum, not here since it's clearly not a NLHE q.This is all about bluffing frequency, and the only bluffing opportunity is betting the Q. If all you do for hours is bet your Aces, he soon learns he can fold all his Kings safely. Throw in one bet with a Q, and you start getting paid off a little.Of course as soon as you make the pot any reasonable size then calling is almost always right.

[NonZeroPossibility 0]

I think we're on the right track. Playing OOP needs a completely different strat because villain doesn't check when we check. Playing OOP, you just need to figure out what villain is betting with and adjust accordingly. The answer on OOP depends on your opponent, but here is a thought on playing in position:The A plays itself and the K plays itself. No consensus on the Q, but how about this:The Q is the hand in question. We're always losing if we check so I'm going to take a LHE type approach to this and test some numbers. OP says this game goes on for a long time... To make it easy, let's just say 1,000 hands with blinds at $1 and $1.How about for the first 100 hands, check every Q.Theoretically, we should be about even on chipstacks with villain after 100 hands playing this way as long as villain is folding every K when we bet the A.Opponent will realize you are playing straight up, not screwing around.Then start raising every Q for the next 100 hands. Change gears if you will.If we check a Q we lose $1 every time, but villain should be folding the K when we bet now so when we bet we're risking $1 to win $2. Example:Take 100 hands of being dealt Q, check every time and we lose $100 no matter what. Now take 100 hands of being dealt a Q and bet every time. Villain has an A in 50 of them. So we lose $100 on those 50 hands. Villain has a K in the other 50. If he folds every time, we win $50 so for those 100 hands we are only down $50. We lost $50 but that's better than losing $100. This would be ideal, but probably not realistic.....Say Villain calls half the time he has a K (25 hands) and folds half the time (25 hands). For 25 he folds, we win $25. For the $25 he calls, we lose $50, so...... -$100 (50 hands of villain calling with A) -$50 (25 hands of villain calling with K)+$25 (25 hands of villain folding with K)__________-$125 We are losing more money by betting than we would by checking if villain calls half the time or more when he has a K. Villain will catch on when you start bluffing Q's and start calling more often when he has a K and we have the A though.OK, so we break even when we have a K and check every time.So now let’s take 100 hands where we have the A. We bet every time and villain folds every time. We win $100. Obv we want to get paid off on the A, so we’re gonna have to bluff the Q at some point.Using the example from above, villain has a Q 50 times and folds all 50 times. We win $50. Now let’s say villain starts to call exactly half the time he has a K no matter what after he sees us bluffing the Q. We know we need to make at least $125 from the 100 hands when we have the A to offset the $125 we lost in betting every time with a Q. In the 25 hands that villain has a K and calls, we win $50. In the 25 hands where he has a K and folds, we win $25…. Back to even.It’s a 2-way street. It actually doesn’t matter what we do if villain stays consistent. We can check every Q or bet every Q, but the outcome will be the same if villain is consistent in his frequency of calling with the K. The only way to make a profit is by getting villain to fold the best hand and call with the worst, so my suggestion is to bet/check the Q until you see villain start calling more often with K’s. Then once he starts calling, stop bluffing for a bit until he stops calling. So how about first 100 hands, check ¾ of your Q and bet ¼. Then next 100 hands, bet ¾ of the Q and check ¼. Something like that should be optimal. I think….

[simo_8ball 0]

This is a game theory question. You can't try to outsmart your opponent. You're trying to come up with an exploitive strategy and then you're claiming it's an optimal strategy. Optimal and exploitive are opposite strategies.If your opponent figures out your pattern he can adjust and then play perfectly against you, losing less than he would if you were playing optimally.

[NonZeroPossibility 0]

Hi.Please re-read my post. I worked out the math and showed that it doesn't matter if we check every Q or bet every Q. If villain is consistent, the result is the same.And I said "Something like that should be optimal. I think…."I'm not at all trying to say I am right. I'm just saying it really doesn't matter what if we check every Q or bet every Q. Please prove me wrong, I'd really like to know the answer. But just making a post on here saying I am wrong with ABSOLUTELY NO explanation.... I mean come on, Simo, I know you're a smart guy. Show me something.

[NonZeroPossibility 0]

Oh yeah, how is the opponent going to figure out my pattern if I am still betting every single A? My post was only referring to play when we are in position and outsmarting villain is the only way to win at this game because we need him to fold the best hand and call with the worst. How the F else are you going to win?EDIT: Hopefully everyone realizes this math only works for when blinds are $1 and $1EDIT 2: If blinds are less than $1 and $1 you need to check every Q. If blinds are more than $1 and $1 you need to bet every Q.

[simo_8ball 0]

Oh yeah, how is the opponent going to figure out my pattern if I am still betting every single A? My post was only referring to play when we are in position and outsmarting villain is the only way to win at this game because we need him to fold the best hand and call with the worst. How the F else are you going to win?EDIT: Hopefully everyone realizes this math only works for when blinds are $1 and $1EDIT 2: If blinds are less than $1 and $1 you need to check every Q. If blinds are more than $1 and $1 you need to bet every Q.

It's not about your opponent figuring out your play.It's the fact that he might stumble across the perfect strategy to play against you.If for those 100 hands where you're betting 3/4 of your Qs he happens to be calling with all his kings, he beats you.With this problem you have to come up with one strategy for all hands. There is a precise % of Qs you should be betting so that it doesn't matter whether he calls or folds with kings. That is your optimal strategy.I'm just working on the matrices at the moment. I'll see if I can get an equation soon.

[NonZeroPossibility 0]

It's not about your opponent figuring out your play.It's the fact that he might stumble across the perfect strategy to play against you.If for those 100 hands where you're betting 3/4 of your Qs he happens to be calling with all his kings, he beats you.With this problem you have to come up with one strategy for all hands. There is a precise % of Qs you should be betting so that it doesn't matter whether he calls or folds with kings. That is your optimal strategy.I'm just working on the matrices at the moment. I'll see if I can get an equation soon.

Right, but if he is calling with all his K's, he's calling with every K when we have an A so we win an extra bet every time we have an A.It evens out.

[simo_8ball 0]

Right, but if he is calling with all his K's, he's calling with every K when we have an A so we win an extra bet every time we have an A.It evens out.

In that case you're saying it makes no difference how often we bet the queen.

[NonZeroPossibility 0]

In that case you're saying it makes no difference how often we bet the queen.

As long as villain is consistent in his frequency of calling with his K's we lose a bet when we have the Q but we gain a bet when we have the A.Of course, this only works for blinds of $1 and $1

[simo_8ball 0]

As long as villain is consistent in his frequency of calling with his K's we lose a bet when we have the Q but we gain a bet when we have the A.Of course, this only works for blinds of $1 and $1

So you're saying, at a pot size of $2, it makes no difference how often we bet a Queen?

[simo_8ball 0]

Villain			Ace			 King			Queen				   Call	Fold	Call	Fold	Call	FoldHeroAce	   Bet	  XXX	 XXX	 +1	  0	   +1	  0		  Check	XXX	 XXX	 0	   0	   0	   0							King	  Bet	  -1	  +P	  XXX	 XXX	 +1	  0									  Check	0	   0	   XXX	 XXX	 0	   0							Queen	 Bet	  -1	  +P	  -1	  +P	  XXX	 XXX									  Check	0	   0	   0	   0	   XXX	 XXX

We can see that betting the Ace is always better than checking, we can get rid of the check line.Because villain will never fold an Ace or call with a Queen (looking down the columns we can see the merits of the two lines) we can remove those.We can also see from there that betting a king is worse than checking, so we can remove that as well.

Villain			Ace			 King			Queen				   Call	Fold	Call	Fold	Call	FoldHeroAce	   Bet	  XXX	 XXX	 +1	  0	   +1	  0							King	  Check	0	   0	   XXX	 XXX	 0	   0							Queen	 Bet	  -1	  +P	  -1	  +P	  XXX	 XXX									  Check	0	   0	   0	   0	   XXX	 XXX

This is the final payoff matrix.Damn it takes forever to get that aligned properly.I'll look for the equation later.

[BaseJester 1]

The key to the AKQ is tells. All the game theory goes out the window when you can't control your heart-rate.

[LJB723 0]

The key to the AKQ is tells. All the game theory goes out the window when you can't control your heart-rate.Bitch.

FYP

[Dictius 0]

Nice work simo, I'm pretty sure that's correct.I think the optimal solution is Always bet with an A, always check with the K as you are only going to get called by by the A and the Q will fold and bet some % of the time with a Q.In game theory, the optimal solution is when the villain does not care if he calls or folds, he will always have 0EV, so he can never gain an edge over us.What percentage of the time do we bet so the villain has zero ev when he calls with the king?Lets say we bet the Q x% of the time.EV of villain calling with the king is0.5*(-1) + 0.5*x*(1+P) = 0Half the time we have the Ace and villain loses $1, half the time we have the Q and we decided to bet it x% of the time and the villain gains 1+P if he calls and for the optimal solution, villain has 0evso solving this givesx = 1/(1+P)so the % of the time we should bet the Q optimally is dependent on the size of the pot.Does this look correct?

[simo_8ball 0]

Nice work simo, I'm pretty sure that's correct.I think the optimal solution is Always bet with an A, always check with the K as you are only going to get called by by the A and the Q will fold and bet some % of the time with a Q.In game theory, the optimal solution is when the villain does not care if he calls or folds, he will always have 0EV, so he can never gain an edge over us.What percentage of the time do we bet so the villain has zero ev when he calls with the king?Lets say we bet the Q x% of the time.EV of villain calling with the king is0.5*(-1) + 0.5*x*(1+P) = 0Half the time we have the Ace and villain loses $1, half the time we have the Q and we decided to bet it x% of the time and the villain gains 1+P if he calls and for the optimal solution, villain has 0evso solving this givesx = 1/(1+P)so the % of the time we should bet the Q optimally is dependent on the size of the pot.Does this look correct?

D'oh.Yeah, that's right. I was staring blankly at the matrix, not seeing where to get the equation. I am so bad at maths now it's scary.That is our optimal bluffing % with queens.Optimal calling % with Kings. We need him to be indifferent between checking and betting a Q.Checking = 0Bluffing = [ 0.5 x lose to aces ] + [ 0.5 x Call% x Lose to kings ] + [ 0.5 x (1-Call%) x win pot when K folds ]Bluffing = [ 0.5 x -1 ] + [ 0.5C x -1 ] + [ (0.5 - 0.5C) x P ]Bluffing = -0.5 - 0.5C + 0.5P - 0.5CPSetting bluffing equal to checking:0 = -0.5 - 0.5C + 0.5P - 0.5CP0 = -1 - C + P - CPC + CP = P - 1C ( 1 + P ) = P - 1C = (P - 1) / (P + 1)So, the optimal calling frequency with a king is (P-1)/(P+1), which is (P-1) multiplied by the bluffing frequency.What it says overall is that as the pot increases to infinity, kings should always call, and queens should never bluff.