The Half-Street Fixed-Limit AKQ Game

I'm typing all of this out here so I can link to it later when I need to show this to someone. It's a toy poker game that can be used to illustrate a large number of important ideas and concepts. I'm also going to change things up a little from how this is normally presented for the sake of clarity.

The Rules

We're heads-up with two players that we will call Hero and Villain, with Hero always in position. We use a deck that contains one each of three cards (an Ace, a King, and a Queen). Each player posts a $0.50 ante and is then dealt a card out of the 3-card deck.

After the cards are dealt, Villain checks in the dark. Hero then has to the choice to make a bet of size B or check to see a showdown. If Hero bets, then Villain has the options to either fold or call to see a showdown. At showdown the high card wins.

Hero's Strategy

If Hero is dealt an Ace, he always bets because there is a non-zero chance that Villain will call, making betting better than calling. (Similarly, you wouldn't check behind the nuts in poker on the river). If Hero is dealt a King, he always checks because he can't get his opponent to fold a better hand (Villain never folds an Ace for obvious reasons) or call with a worse hand (Villain never calls with a Queen for obvious reasons). If Hero is dealt a Queen, he will have the option to bet with it some percentage of the time since he can possibly get his opponent to fold a King.

Villain's Strategy

Villain only gets to make a decision if he's facing a bet. With an Ace he always calls. With a Queen he always folds. With a King, he has the option to call some percentage of the time since he can possibly pick off his opponent's bluff.

Looking at these two strategies show an example of looking at a player's entire range and deciding how to play the entire range, an important skill.

Essence of the Game

The real point of this game is that Hero and Villain only make 1 decision each. Hero decides how often to bet a Queen, and Villain decides how often to call with a King. Everything else is forced.


So talk about this a bit and see what you can figure out and I'll post more on this later.

I find that villain needs to fold a king more than 2b/(1+b) of the time to be + or neutral ev if we bet b when we held a queen.

It would appear that villain needs to fold a king a fairly large amount of the time for a bet of any size to be +ev though and this seems pretty hard to accomplish in such a game if we always bet every queen and ace.

Now if we only bet a queen x% of the time...hrm...

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Originally Posted by JKDS

It would appear that villain needs to fold a king a fairly large amount of the time for a bet of any size to be +ev though and this seems pretty hard to accomplish in such a game if we always bet every queen and ace.

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if B=$0.01, he doesn't need to fold very much at all.

In addition to this being a good exercise, I'm going to introduce this game at the next party I go to. I figure by playing proper strategy I can make a fair amount of money at this. :)

I'm pretty sure to win you need to randomise.

I think for any given percentage of time we chose to bet the q, there is an optimal level for villain to call that makes it a draw.

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Originally Posted by DanAronG

I'm pretty sure to win you need to randomise.

I think for any given percentage of time we chose to bet the q, there is an optimal level for villain to call that makes it a draw.

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This could be said about any variation of poker.

Example of Exploitive Bluffing

Suppose the bet size is set to $0.40 and Villain calls with a King 25% of the time he holds one. This means that Villain is folding 37.5% of the time. The pot is $1, and if Hero bets a Queen he needs Villain to fold 0.40/(1+0.40) = 0.4/1.4 = 29% of the time to be profitable. In a vacuum, Hero can bet a Queen profitably.

Example of Exploitive Calling

Suppose the bet size is set to $0.75 and Hero bets with a Queen 50% of the time he holds one. This means that when Villain holds a King he will have 33% equity against Hero's range. After Hero's bet, the pot will be $1.75 and Villain will be calling $0.75. For the call to be profitable, Villain needs to have more than 0.75/(1.75+0.75) = 0.75/2.50 = 30% equity. Since he has more than 30% equity against Hero's range, a call is profitable in a vacuum.


So now figure out unexploitable strategies for Hero and Villain.

this is an awsome exercise. i'm still trying to figure it out which is tuff here at work, but I can't wait to see some answers.

not sure if this changes much, but what about the possibility of a tie? how often will both hero and villain hold a Q and make it a 0 ev? or does it not even matter?

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Originally Posted by Santo2True

not sure if this changes much, but what about the possibility of a tie? how often will both hero and villain hold a Q and make it a 0 ev? or does it not even matter?

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there is only one card of each type

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Originally Posted by Ope

there is only one card of each type

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Yeah there are only 3 cards in the deck, so there can't be any ties. If you change it to say a 6 card deck with two of each rank or something like that, then some of the dominated strategies break down (ie: betting with a K sometimes can become correct depending on the bet size and opponent tendencies).

Optimal solution is when our opponent always has 0EV so can never gain an edge over us.

Lets say we bet a Q x% of the time. Half the time we have an A, other half a Q.

EV of calling with a K for a our opponent.

0.5*(-B) + 0.5*x%*(B+P) = 0
-B + x%*(B+P) = 0
x% = B/(B+P)

Optimal calling y% means it doesn't matter if he checks or bets a Q.

Check = 0EV
Bluff = 0.5*-B + 0.5*y%*-B + 0.5*(1-y%)*P

Basically first term is when you have an A, second a K which you call with and the third term is when he makes you fold a K.

Bluff = -0.5*B - 0.5*y%*B + 0.5*P - 0.5*y%*P

Set bluff = check

0 = -B - y*%*B + P - y%*P

Solve for y%

y%(B+P) = P - B

y% = P - B/P+B

Assuming B is a fixed constant as the pot tends to infinity the King should always call and the Queen should never bluff.

If you want a graphical representation google "function grapher" and plot 1/x+1 and x-1/x+1 on the same graph - the point where they intersect is the optimal strategy.

oh wtf... i read that wrong sorry, now it makes a lot more sense to me, i thought it was a deck but with only those three face cards.

now i actually understand, ha

Very interesting spoon - you ever take any upper-level game theory courses? This sounds very much like an imperfect information bayesian game, wherein each player (1 and 2) receive a "signal" as to whether they are weak, normal, or strong (Q, K or A) before any actions take place.

Given this, there's probably a mathematically rigorous mixed-strategy solution to be had. That said, I think a few of the posts above are bang-on. I might be able take some time in the (hopefully near) future to see if I can express this a bit better. Good stuff though.

It's also interesting to analyse a similar game with a bigger deck. Say each player has a number between 0 and 100 instead of an A, K or Q. For a given bet size there's a Nash equilibrium (i.e. neither player can change their strategy to improve their EV) here. For example for a pot sized bet Hero bets the bottom ninth and top two ninths of their hands, and Villain calls with the top four ninths. The more mathematically inclined among you can probably solve this game for a full street of action (i.e. Hero is oop, Villian is in position and can choose to check behind or bet when Hero checks), although it will take an hour or two unless you're cleverer than me.

The game theory discussion in a simple toy game like this is cool but what I'm really posting this for is for people to understand some important principles that have practical use at the table.

This reminds rather heavily of a section in Theory of Poker where Sklansky explains that we should randomise our Hero Q betting and Villain K calling with the exact percentage one that is based on the size of the bet.

Specifically you should make the odds against you bluffing the same as your pot odds. So if the pot is $200 and the bet size is $100 your pot odds are 3-to-1 and the odds against you bluffing should be 3-to-1 against. So you should be bluffing 25% of the time.

In the AKQ example when you are Hero with Q you should then bluff 33% of the time. That's because, when you bet, 25% of your bets should be bluffs - and the 75% of them should not be bluffs. They should be A betting for value. So 75% of all bets are 100% of all A - and the corresponding percentage of Q is 25/75=33.33%.

I've done up a little spreadsheet with all this preprogrammed in and I notice a few oddities (after weeding out too many small errors).

When the Q Bet % is the ideal one defined by game theory the K Call % can be anything and the EV for both sides remains the same. Exact same.
In this game, apparently, if Hero is playing correctly Villain can make no difference to the outcome by picking a calling frequency - and if Hero is not playing correctly Villain's best strategy is exploitative (either 100% call or 0% call).
It's only when he thinks hero might be close to right that he can minimise his losses by picking the right calling frequency.

In ToP Sklansky seems to have done lots of EV calculations for the bluffing bit, but not so many for calling the bluffing. He says that when bluffing you should match pot odds with odds against bluffing and when calling bluff he says you should do the same - but he does it differently. For bluffing he considers the pot including the call. For calling the bluff he considers the pot without a call. His experimental EV calculations leads him aright with the bluffing part.

Easy way to put that in a spreadsheet is: bet/(pot+bet)
The result is not how often your bet is a bluff, but how often you should bluff with your Q.

In the section on calling bluffs he's a bit confused and goes on about finding it in a similar manner and suggests that a $100 pot and $20 bet means you should call 5 times and fold once. It's not really that similar when the similar situation is one in which he would call 6 times and fold once. It's also incorrect as my spreadsheet EV has shown me. I kept wondering why by reducing the Hero Q bluffing % when my Villain K calling % should provoke an equilibrium was higher EV than the "optimal". I guessed that my formula was wrong and experimented until I found one that works.

I do not yet know why mathematically it is so, but I can note experimentally that the unexploitable strategy for K is to call (pot-bet)/(pot+bet) of the time when facing a bet.

Anyway, the real reason I was playing around with spreadsheets was because I wanted to find if there was a specific bet size that maximised EV for Hero. And there is. It's around 41.43% of the pot, which you then bluff with 29.29% of the time.

If you want to make it a party game, suggest fixing the bet size at 50% of the pot, because there both the unexploitable Hero Q bluff % and the unexploitable Villain K call % is 33.33% (yeah, exactly one third). Given a wrist watch (can be split into thirds as easily as halves or quarters) or some other object that gives true random thirds you would have a certain EV edge - as long as you make sure you play at least as many times as Hero as you play as Villain. Because villain really isn't in for much EV.

Another possible fixed bet size is 1/3 pot where Q bluff % is 25% and K call % is 50%.
1/4 pot: Q bluff % 20 - K call % 60
2/3 pot: Q bluff % 40 - K call % 20
full pot: Q bluff % 50 - K call % 0

With full pot it's 0 EV both ways. At 41.43% the EV is +2.86% of the pot for Hero per game. At 1/2 pot and 1/3 pot it is 2.78%, at 1/4 pot it is 2.50%, at 2/3 pot it is 2.22%.

Problem with playing full pot (and not caring which side you're on) is that optimal strategy is to always fold the K. That way it's very easy for someone to spot what you're doing and simply copy your actions. While you will still have an edge when you play Villain and someone tries to bluff (or not), if someone makes you play Hero you get the stress of doing the randomisation right while your opponent can easily play perfect against you.

@Erpel, 1/(sqr(2)+1) is an important number in the game theory surrounding poker. That's the % of the pot that maximizes EV in this game that you found.

Bump.

You are dealt A,K,Q. Your opponents bets B into a pot of $1. You call with an ace, fold with a queen. With a king when you call you must be correct B/B+1% of the time. We call if villain has a queen at least 1+B/B% of the time.



We want to bet a queen B/B+1% of the time.
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Is this equilibrium?

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Originally Posted by Imthenewfish

You are dealt A,K,Q. Your opponents bets B into a pot of $1. You call with an ace, fold with a queen. With a king when you call you must be correct B/B+1% of the time. We call if villain has a queen at least 1+B/B% of the time.



We want to bet a queen B/B+1% of the time.
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Is this equilibrium?

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<@spoonitnow> like ok the pot is 1 and the bet size is B
<@spoonitnow> assume we bet a queen x% of the time
<@spoonitnow> then for villain the EV of folding a king is 0
<@spoonitnow> if we bet a queen x% of the time
<@spoonitnow> then when we bet we have a ratio of ace:queen of 1:x
<@spoonitnow> so the % of time we have an ace is 1/(1+x) and the % of time we have a queen is x/(1+x)
<@spoonitnow> so the EV of villain calling a bet is (1/(1+x))(-B) + (x/(1+x))(1+B)
<@spoonitnow> set that equal to zero and solve for x
<@spoonitnow> note it's (1+B) b/c when we have a queen and villain calls he wins the pot of 1 plus our bet size of B
<@spoonitnow> so multiply both sides by (1+x)
<@spoonitnow> that gives
<@spoonitnow> 0 = -B + x(1+B)
<@spoonitnow> and
<@spoonitnow> x = B/(1+B)

And yes this is the kind of shit we sit around and talk about in IRC on a Sunday afternoon.