I'm bored: so poker riddle

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RCC: Act II
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I'm bored: so poker riddle

Post by RCC: Act II » Fri Jan 05, 2018 6:09 pm

You are trying to get a seat to the $10,000 main event at the world series of poker. To this end you have entered a tournament that awards a seat to the top two finishers.

The tournament is down to three players. You have 9,990 chips. A second player has 9,990 chips. The third has 20 chips.

The blinds are 50-100. Player 3 is on the button, and folds his hand. Player two is in the small blind (has 100 in the pot), laughs maniacally and goes all in, betting the 9,890 he has remaining. He turns over a seven of clubs and a deuce of hearts. He laughs again and tears up the seven of clubs. The floorman is an idiot and rules that the hand is live and he only can use the deuce, while you can use both cards.

You are next to act. You have 200 already in the pot and can call for your last 9790. Are there any hands you do not call with?

(the spoiler tag is a thing...)

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Re: I'm bored: so poker riddle

Post by ed » Fri Jan 05, 2018 7:07 pm

I have no idea what you are going on about. Really. i understand the words but not what they mean when you string them together.

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Re: I'm bored: so poker riddle

Post by Rob Lister » Fri Jan 05, 2018 7:31 pm

Fold. YOu're not trying to win, only to place. If you have a two, call. Anything else, he can win with the 2. The third guy doesn't have enough to cover the blinds on the next hand. Why did he fold?

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Re: I'm bored: so poker riddle

Post by RCC: Act II » Fri Jan 05, 2018 9:36 pm

Rob Lister wrote:Fold. YOu're not trying to win, only to place. If you have a two, call. Anything else, he can win with the 2. The third guy doesn't have enough to cover the blinds on the next hand. Why did he fold?
The third guy's fold is arguably optimal with all hands.

I'm an idiot and screwed up the numbers.

The sure solution is a deuce and any heart. That combo can't lose.

It turns out that your answer of any other deuce is arguably break even.

Your 9790 stack represents a 97.9% chance of winning, so it is worth about $9790 in real money assuming equal skill. If you call and win, you get $10,000. Call and lose, you get zero.

So winning nets you $210 in value, and losing costs you $9790 in value, meaning you need to win 98% of the time, which, me being a moron, is roughly the chances of the lone deuce making a flush.

(When it is that close it becomes a question of relative skill which makes it a likely fold because player 2 is a moron and you have a long term skill advantage over him so gambling is not the best move, but that is getting a bit too upstream.

If he were to play both the deuce and the seven, then you shouldn't even look at your hand. Which is the more real world application...

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Re: I'm bored: so poker riddle

Post by Rob Lister » Fri Jan 05, 2018 10:41 pm

RCC: Act II wrote:
Rob Lister wrote:Fold. YOu're not trying to win, only to place. If you have a two, call. Anything else, he can win with the 2. The third guy doesn't have enough to cover the blinds on the next hand. Why did he fold?
The third guy's fold is arguably optimal with all hands.
I get the rest. I didn't even consider suits. I shoulda. Why is the button's best bet to fold? Is it because one of the two of you might lose it all in that one hand?

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Re: I'm bored: so poker riddle

Post by RCC: Act II » Fri Jan 05, 2018 11:30 pm

Rob Lister wrote:
RCC: Act II wrote:
Rob Lister wrote:Fold. YOu're not trying to win, only to place. If you have a two, call. Anything else, he can win with the 2. The third guy doesn't have enough to cover the blinds on the next hand. Why did he fold?
The third guy's fold is arguably optimal with all hands.
I get the rest. I didn't even consider suits. I shoulda. Why is the button's best bet to fold? Is it because one of the two of you might lose it all in that one hand?
Pretty much, but that assumes one or both of you guys go nuts because neither of you should even look at your hands. If they aren't looking, he should play everything with at least 34% equity against one random hand because the small blind should always fold. That is everything but 32 and 42 offsuit.

The more chance the go nuts thing is likely, the narrower the guy should go. Against two guys that are capable of getting it in when both have at least kings, folding the borderline offsuit trash could be a good idea. If they are total muppets that can't fold a pair or AK, fold everything.

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Re: I'm bored: so poker riddle

Post by Beleth » Tue Jan 30, 2018 7:11 pm

I'm approaching this by going to the next hand(s) and working backwards.

Assume you fold this hand so it's 9790 (you) to 10,190 to 20. You are on the button. Player 3 (small blind) puts in his 20. Player 2 (big blind, and who has regained his sanity) puts in 200. You call the 200, and for the rest of the cards it's checks all around so it ends up being a 3-sided coin flip. If player 3 goes out, great, you are one of the top two left. But that only happens 2/3 of the time, and the next hand is more or less a repeat of this one (player 3 blinds his 60, you and player 2 call, then check check check) so player 3 is knocked out 2/3 of that time too.

So if you fold this hand (call this hand Hand 1), player 3 only has a 1/9 (11%) chance of still being in the game at the end of Hand 3. It's hard to predict the future any further out, because player 3 is on the button again in Hand 4 and is back in the game with 180 chips.

So by that logic, don't you increase your odds of coming first or second by playing any hand that has a higher than 1/9 chance of beating a lone 2H? Or am I looking at this too simplistically?
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Re: I'm bored: so poker riddle

Post by RCC: Act II » Wed Feb 07, 2018 3:40 pm

Beleth wrote:I'm approaching this by going to the next hand(s) and working backwards.

Assume you fold this hand so it's 9790 (you) to 10,190 to 20. You are on the button. Player 3 (small blind) puts in his 20. Player 2 (big blind, and who has regained his sanity) puts in 200. You call the 200, and for the rest of the cards it's checks all around so it ends up being a 3-sided coin flip. If player 3 goes out, great, you are one of the top two left. But that only happens 2/3 of the time, and the next hand is more or less a repeat of this one (player 3 blinds his 60, you and player 2 call, then check check check) so player 3 is knocked out 2/3 of that time too.

So if you fold this hand (call this hand Hand 1), player 3 only has a 1/9 (11%) chance of still being in the game at the end of Hand 3. It's hard to predict the future any further out, because player 3 is on the button again in Hand 4 and is back in the game with 180 chips.

So by that logic, don't you increase your odds of coming first or second by playing any hand that has a higher than 1/9 chance of beating a lone 2H? Or am I looking at this too simplistically?
Yeah, a bit,

He has an 1/9 chance of getting to 180 chips, and 180 chips is still at a dire disadvantage. He needs to get to 6666 before he likely reaches parity and a 2/3 chance of winning. Realistically, 180 chips gives about a 1/50 chance of winning, so really we are looking at needing to be a 450/1 favorite to make a call correct, which means only call with the lock. This more or less ends up in the same place as what I was saying.

(I missed before when I said that a deuce and a heart is a lock. The other heart has to be the three, four, or five of hearts to block the 2h from a straight flush. My guess is that a 2 and any other heart is still a better than a 450-1 favorite tho...)

To digress though:

As player 3 accumulates chips, it gets complicated fast. Every extra chip a player gets is worth less in real money terms than the ones he already has because of the payout structure. The value of a chip is not at all linear in that it only has value as far as it makes it more likely you won't come in last. In fact, the value of a stack varies greatly depending on the distribution of the other stacks:

If he has 6666 chips, and the others have 6667 and 6667, he has a 2/3 chance of winning 10K, meaning the chips are worth $6666.

Now, if he has 6666, and the other players have 13,332 and 2, his chances of winning are more like 99.9%, making that same 6666 stack worth about $9990 in real money terms.

This effect is generally referred to as the independent chip model, or ICM for short. It is absurdly hard to calculate assuming players of equal skill, and just horrifying when that assumption is removed. ICM effects are present in any non-winner take all tournament where first place is less than the total value of all the chips. In a satellite like this example where a number of identical prizes are given, ICM effects are huge.

So, if our guy triples up twice, he finds himself with 9 big blinds. At that point it isn't a lock that he gets a fair three way coin flip. Until he manages to reach some level of parity he is effectively playing against two people who are far more concerned with denying him chips than getting each other's chips. This isn't illegal collusion unless the players are explicitly agreeing to collude and sharing hand strength or signaling or something. It just so happens that their honest best interest is to play that way.

This in itself doesn't change anything if all the players adjust optimally, but usually there is someone that completely misses the ICM bus and exploiting that can change the chances quite a bit.