Simple Solver Quiz

Poker Discussion
1

This is not simple in terms of the answer being easy, and it won’t really teach you anything about poker, but is hopefully an interesting way to demonstrate what a solver is actually solving and how.

Here’s the setup:

It’s a two-player game with 3 cards in the deck - A, K, Q
Both players get dealt one hole card (the unused card is hidden)
Both players put $1 into the pot to start the hand

I can bet $1 or check
If I bet, you can call, raise to $2 or fold.
If I check, you can bet $1 or call.
I can only call or fold if action gets back to me.
High card wins if it gets to showdown

What percentage of the time do you take each action with each hole card on your turn?

(There’s 5 total lines for you to consider, Bet/Raise, Bet/Call, Bet/Fold, Check/Bet and Check/Call)

Note that this is not my idea, it’s a fairly well-known example, and you should be able find the answer with a bit of googling, but it’s going to be a better demonstration if you just take your best guess at first.

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@lihiue

This sounds like a similar topic and thread which was started by -Blackwidow- a couple of years ago. Maybe someone or even Blackwidow can find it.

3

1/3 of the time we’ll have Q, 1/3 K, and 1/3 A. Ditto for Villain.

Naively:
If we have A we’d like to raise every time
If we have Q we’d like to fold every time
If we have K we’d like to bet every time, and fold whenever we are raised.

–
It seems everything revolves around the presence of the K, which due to the nature of the setup is perfectly balanced between wanting to bet/raise and wanting to call/fold.

We can sometimes win by bluffing with a Q as opponent will have the K instead of the A 50% of the time.

This is as far as I’ve gotten lol.

4

Sorry, I might have made the first post confusing by using “you” and “I” instead of hero and villain. I’ve edited it to hopefully make it clearer.

You’re supposed to be playing as the hero, so your options are raise/call/fold if the villain bet initially, or bet/check otherwise.

It’s not actually important (ie a solver won’t reason like this and still gets the right answer), but:

As the villain, there is no reason to ever not bet/raise an ace, as checking back or calling ends the hand.

Even more obviously, you’d never call with a Q

A bit less obvious, but it also makes no sense to bet/raise with a K. You will always get called by an A, and a Q will never call

So really a lot of the options have a clear solution up front.

1 Like
5

ah ok thanks yes I was confused, get it now lol

6

Is there a free poker solver app out there that does this?

7

wouldn’t there be an option to bet the Q % of the time and fold if raised

I am not big on solvers but 66% of the time the villain would not have an A
or is that not in the calculation ?
So wouldn’t you just bet every time its your option to bet first

A- Bet/Raise100% of the time.
K- Bet/Call 100% of the time
Q- Bet 100% of the time, If you get raised, fold.

2 Likes
8

You’re on the right rack @_Rain, we can’t actually come up with a strategy for the hero without considering the villains strategy first.

So, let’s say that villain (first to act), does as you say - bet/calls 100% with the A or K, and bet/folds pure with the Q.

Consider the options for hero if they hold a Q:

  • Hero can raise, but villain must be holding an A or K if hero has the Q, so hero always gets called and loses - EV = -$2
  • Hero can call. Again, they will always lose, EV = -$1
  • Hero can fold. EV = 0

If Hero holds the A:

  • Hero can raise. They will always win the $3 already in the pot. Half the time the villain will call with a K, half the time they will fold with a Q, so EV = 50% * $1 + $3 = $3.50
  • Hero can call. They always win, so the EV is just $3.
  • Hero could fold, 0EV

Heroes response with when the have an A (100% raise) or Q (100% fold) is fairly obvious.
What about when Hero has a K, and is it possible for Hero to have a winning strategy against this villain?

Villain Card Hero Card Hero Profit
A K ??
A Q -$1
K A $3
K Q -$1
Q A $2
Q K ??
Average Profit: ??
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image
image255×181 1.92 KB

yes/no?

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10

Yes. If Hero always raises with a K, which is the most profitable line (or really the least losing line, because it does lose money, but less than calling or folding)
It looks like the average profit got rounded down to zero though, it’s actually $0.33 per hand.

11

Nice work @_Rain and I think I am starting to see the point of the exercise: I’m reading it as a nice illustration of how the most profitable overall strategy may involve subsets of actions that “feel bad” or seem rash. Most people are naturally somewhat risk averse, so the naive inclination is “I’ll play my A aggressively and my K cautiously,” but this is less profitable than “I’ll play both my A and my K aggressively” as the latter captures more value vs. Villain’s Q, and if villain has the A you’re screwed anyway.

Right? Something like that?

12

That’s certainly a reasonable and correct take away - but not the point of this exercise - I’ll be getting to that soon…

The next question is, can villain adjust against our strategy so that they are now winning, or at least losing less?

We haven’t explored the part of the game tree where villain checks initially, but let’s ignore that and change the rules to say that villain always has to bet.

Villain now only has one lever to pull. They can’t start folding an A when we raise, or calling with a Q, both are clearly going to be losing more, so all they can do is call less when they have a K. Villain actually loses the least if they pure fold a K to a raise, losing $0.17 per hand.

If villains’ strategy against a raise is now:
A - 100% call
K - 100% fold
Q - 100% fold

How does hero adjust their strategy to take advantage, which is currently:
A - 100% raise
K - 100% raise
Q - 100% fold

14

Sorry, this is nonsense. Hero raising with the K is less bad than folding, but it’s obviously not as good as just calling, which is 0EV.

The number @_Rain gave for raising with a K are correct, but if you call, you get this:

Villain Card Hero Card Hero Profit
A K -$2
A Q -$1
K A $3
K Q -$1
Q A $2
Q K $2
Average Profit: $0.50

Luckily it doesn’t change anything for the villains’ response. Hero’s strategy is now:
A - 100% raise
K - 100% call
Q - 100% fold

Villain should still fold the K when facing a raise (because they’re always beaten by an A), so their strategy is still:
A - 100% call
K - 100% fold
Q - 100% fold

And the question remains, can the hero do anything now on their turn to exploit the villains adjustment?

For reference, the profit after villain starts folding a K to the raise is:

Villain Card Hero Card Hero Profit
A K -$2
A Q -$1
K A $2
K Q -$1
Q A $2
Q K $2
Average Profit: $0.33
15

I’m going to try resurrecting this thread with this video: Why Randomizing Kills Your Winrate

It might not seem directly related, but it’s great, and the toy game actually demonstrates the why for a lot of the points he raises.

We’re pretty close to getting to:

  • why the solver mixes
  • why if you get your frequencies wrong you become exploitable (even though both options are the same EV)
  • how the solver actually works
16

Just going to drop this here without comment for now:

The two highlighted box should be editable, I will explain what this means at a later date.

17

Sorry, I forgot to reply to this thread. Hopefully people are still interested, because we’re about to get into the good stuff (finally).

Quick recap, Hero has decided to always raise the A, always call the K and always fold the Q. In response, villain will always fold to raise.

Hero can now start bluffing raising their Q. If villain didn’t adjust, Hero could bluff raise their Q 100% of the time, but obviously villain with have the A sometimes and so isn’t going to fold to the raise in that case. Once villain knows that hero is bluff raising with the Q, they need to start calling sometimes with the K.

It turns out that if Hero bluffs with a Q 20% of the time, Hero always makes $0.37 per hand. That’s true if villain calls with a K 100% of the time, or folds 100% of the time, or anywhere in between.

It may seem that villain is now indifferent when facing a raise and holding a K, but they’re actually not. If they fold 100%, hero can start deviating again and start bluffing the Q 100%. So villain can’t do any better than losing $0.37 per hand, but they can do worse. It turns out villain must call with a K 20% of the time as well. Now, Hero wins 37c if they always fold the Q, 37c if the always raise the Q, and everything in between.

We now have an equilibrium. If hero deviates from raising the Q 20% of the time, villain can reduce their average loss. If villain deviates from calling with the K 20% of the time, Hero can increase their average profit.

If you’re wondering why 20%, it’s because of the pot odds we’ve set up in this toy game. The interesting things is that you can get to that value through trial and error, without ever having to calculate anything, and that is in fact what a solver does. (It is actually what I did too - I couldn’t be bothered working out what the answer should be and just plugged numbers into the google sheet I posted earlier)

Here’s the important thing to note: If Hero bluffs their Q at 19% instead of 20%, villain should always fold their K. If villain calls with a K only 19% of the time, Hero should always bluff raise the Q.

That is, an equilibrium is really a very finely balanced seesaw (teeter totter for 'merican’s), and if someone even just farts on the other end, the balance shifts completely.

A solver is always on the upside of that shift, but might only be 20% away from the middle, where we can be on the very far end.

I’ll leave it at that for now, but I have more points to make, some of them not even flatulence related.