SavagePenguin
Put the win in penguin
Bronze Level
I've seen a few common mental pitfalls in poker and thought I'd comment a bit on them.
A Few Pitfalls in Thought
Being results oriented - Just because you lost doesn't mean it was a bad for that situation. Likewise, just because you won doesn't mean it was a good move for the situation. You cannot see the opposition's hole cards, so you can only base your moves on the known factors (reads, M, payout proximity, various aspects of position, your hand strength, etc.), so you can only go by what the likeliest outcome is based on known information.
For example, if everyone is deep stacked and a tight player raises UTG, and then a tight player in middle position re-raises, muck your J/J. Sometimes they'll both stack with their A/K's and so you would have been a whopping 67% favorite to triple up, but so what? You can't be expected to know what cards they have, so it was a good fold against what (at least) one of them was likely to have had.
So to sum it up, a good play is defined as one that will be most likely have the best outcome over the long run in similar situations, not one that just so happened to earn you chips in that hand.
Extrapolating short term results - Just because you are on a winning streak doesn't mean you are a winning player. Poker is swingy. A lot of players are quick to pat themselves on the back when they win (because the moves they thought were correct paid off) but then they curse their rotten cold spell when they start to lose.
The nature of the game is "limited knowledge of what's going on with bad plays showing a profit a decent percentage of the time." This means that everybody has winning and losing streaks regardless of their skill level Do not overestimate your poker prowess if you are on a winning streak, and do not beat yourself up if you're on a downswing, because you can make the right moves consistently and still have a long string of losses due to variance.
By no means am I advocating not examining your game if you are a consistent loser. By all means look at your game and try to find logical areas to improve. But be aware that if you’re mad because you never win a coin flip for stacks in the middle stages of a tournament your big problem is that you shouldn’t be taking such risks at that stage of the game, not bad luck.
Taking it personally - As I just stated, the nature of the game is swingy. Bad hands lose to good hands often. If you shove with A/K and get called by T/Qs, he's going to beat you nearly 40% of the time.
That means, for every ten times that happens you lose 4 of those times. That’s almost half. Throw in a little variance and you’re going to lose there a whole lot before you get a win.
I heard a story about a statistician at a college who likes to start the semester by having groups of students make up the results of 500 coin flips (I think I heard it on NPR). Once they do that, he has them flip a coin 500 times and record the results. They do not label the papers. Then he looks at the papers and is almost always able to differentiate the made-up results from the actual results. He is able to do this with such accuracy because random results are swingy. When people make up the results they tend even things out to stay truer to 50/50 results, but when you actually do it you'll flip tails seven times in a row, then heads, then tales another three times, etc. Random results are swingy. That's their nature. So statistically you are *supposed* to lose with A/A half a dozen times in a row on occasion. When I started poker, I had a had a 12 in a row suck streak where I lost 11/12 A/A hands, and the one I did win was when it was folded to me in the big blind. It was a blow to my spirit/ego/confidence.
When your hand loses to an inferior one try not fret it. You are supposed to lose a good percentage of the time. It is completely fair that you lose, and it's completely fair to lose with a superior hand many times in a row. That randomness is the nature of the game.
But it doesn't seem to ever go your way? That is because if you are a good player you will get sucked out on more than you suck out on people. That is the nature of the game as well. If you are a good player will get your money in as an underdog far a less often, so your opportunities to suck out on people are far less than their opportunities to suck out on you.
Something else to consider: Every time you win a hand you are essentially sucking out.
A lot of people do not consider that when you win a hand you are getting more chips than you deserve. Let’s say you are the short stack with 1,500 chips. The blinds are 100/200 and you shove with A/Ko on the button. The BB with 20,000 chips calls with 4/6s. (What an idiot, right? No, not really.) Neither of you hits and you take the 3,100 pot. Most people don’t think anything about their win, because this was what “should” have happened. They take their 3,100 chips and don’t think twice about the results.
You were only 58.84% to win that hand. So if the results were fair, you would have only received 1,830 chips (58.84% of the 3,100 pot, plus another 6 for the times you split). You taking the entire pot there deprived the villain of the 1,269 in equity that he had in that pot. So he is getting an “unfair” return for his investment.
(I’d like to note that you average a 330 chip profit in that situation when he calls, and the pot was 300 when you shoved.)
Now, I put “unfair” in quotes because over time it all balances out. The more this situation happens the closer he comes to winning 40.72% of the time. So in the long run, he gets his 40.72% of all those pots he played. (But the times he wins he gets cussed at.)
Big fish in a small pond - Let’s say you are a successful $.10 & $.25 tournament player. You are at a level where a large percentage of the players have a poor understanding of the game. They don't understand position let alone the gap concept, they make stupid bluffs, they take unnecessary risks with their big stacks, they stack light, etc. It's not very difficult to be a winning player at these levels, and because people play so poorly you can get away with a lot more, stacking off lighter, stacking with one pair, etc. and still make a profit.
If you try to move up to $2 games the pool of fish that will stack off with their wild draws and bluffs will decrease and the people who don't respect an early position raise or a re-raise will decrease. Basically, play becomes more sophisticated. If you move up to $5 games there are even fewer fish. I'm not saying they're geniuses by any means, but they are far from the blind monkeys throwing chips around that you see in the lowest of the micro limit games and freerolls.
This is why you post the buy-in when you post a hand history in the analysis section. Plays that work in the lowest levels are suicide when you try them against players who have a clue about what's going on.
Because of this, getting advice on hands for play money and freerolls (especially early hands in freerolls) is tricky. When you play against people who think about hands you have to play different. On Facebook poker I wait until I have A/J+ and then shove because people call with any Ace or connectors or whatever.
So what does this mean? This means that as you move up you meet tougher and tougher players and you have to modify your play accordingly. Things that once worked will no longer work. +EV plays at one level may end up being bad habits at a higher level, so keep learning. Keep adapting your game as you face new environments.
Hands are about situations, not cards and moves – Let’s say I talk about min-raising from UTG at an aggressive table with A/A. That is a far cry from getting A/A in middle position, and min-raising after a couple people have already entered the pot ahead of you. In my case, I’m looking to get 3-bet. In the latter case, you’re making a move that’ll make a small pot with a lot of callers which is pretty much the worst situation you can be in with A/A.
That is, min-raising with A/A is *not* the important part. The important part is that I had a powerful hand and was in a situation where I was likely to get 3-bet, which gives me the power/choice to call or re-raise a likely aggressor, so I’m trying to get a big pot against only one or two players. In the second example, it’s just setting up a call-fest with all sorts of drawing hands.
Similar moves (like min-raising with A/A) mean different things in different situations. It’s all about your goal and the expected results for your move given all the information available (including your hand, all aspects of positions, player images, etc.). It is not just your hand in relationship to just your move.
I guess another way to word it is that a move has value according to the big picture, not according to just a few factors (like how much you bet with a big hand).
The value of hidden monsters – I have a friend who told me that he likes to play 7/2 on occasion because if it hits nobody can put him on it. I could never convince him that it was a bad play. He eventually quit playing poker online because he lost too much money.
When you play junk like that you run into a variety of obstacles. First and foremost, you rarely hit a big hand. Top pair has a lousy kicker, and you still have to worry about over-pairs, and over-cards on later streets and stuff. It’s just a dangerous situation to be in.
Secondly, if you hit a big hand you will rarely get paid off. Any hand that gives you a monster will mostly likely totally miss everybody else. So the times you do hit you don’t get paid off.
Some people can play a lot of junk and make it work. You have to be very very good to do it successfully, *and* you have to play against people who are smart enough to understand what’s going on because you can’t accurately predict the actions of the ignorant. You can’t read or bluff a player who makes whimsical plays because they do not have consistent (or logical) reactions to the situations presented to them.
How do I play J/J?
The old joke is, "There are three ways to play J/J... and all of them are wrong."
Play it just about the same way you play T/T. Nobody complains about T/T. T/T doesn't have the pretty picture so it's easier to let go of.
Poker is rigged!
Nobody can prove otherwise if you want to believe that.
The fact is, tens of thousands of players use tracking software, recording billions of hands, and cards show up as often as they should. Humans are designed to see patterns and anomies, so when randomness generates the illusion of a pattern we pick it up, and when something weird happens we make note of it. It is common to see these weird things and assume that they are proof of foul play.
The thing is there are many thousands of ways for weird things to happen in each hand of poker. When one of those thousands of “10,000-to-1-against” things happens we pick up on it because your brain was designed to pick up on oddities. These are to be expected, and not necessarily indicative of foul play.
Yes there are cheaters online. People multi-account, collude, use bots, and use hand history software to gain unfair insight into their opponents play. But this doesn’t make the game unwinnable. A villain pretty much has to be a smart/winning player to make proper use of the illegally obtained hand histories. Collusion is fairly obvious to spot and several CC’ers have been contacted by the poker sites and had money returned because the site’s automated software caught the colluders and returned money to the effected players. And as far as multi-accounters go, if the accounts aren’t playing together their advantage is that they might have a history on you while you won’t have their hand history matched to their alternate account. And if they are playing together the advantage would be that he could collude with himself and he’d know an extra 3.84% of the deck that was out-of-play because the other account had it (which will only benefit him in rare situations).
To put it bluntly, if you put some serious time into reading poker books and improving your game, and following proper bankroll management without tilting you are going to be a winning micro-stakes poker player. If you can’t do win at the micro stakes it is not because the games are rigged; it is because you are doing something wrong.
Having the best hand is over rated.
You buy into a single table S&G for $10+$1.
1st pays $45
2nd pays $27
3rd pays $18
Total prize pool = $90
4 players are left (you are on the bubble)
Chip stacks are about even (average 3,375), and the blinds aren’t huge.
Player 1 folds.
Player 2 folds.
Player 3 accidently exposes 7h 8h and shoves
You look down and see that you have Ac Ks.
What do you do? You insta-shove, right?
Here’s where ICM (Independent Chip Model) comes into play.
Basically, each chip you have is worth something in relationship to the prize pool.
With 4 of you left and similar stacks, you are all playing with about 1/4th of the $90 prize pool, or about $22.50 in expected payout. (Note that this is not profit. Winnings and profit are different, as you paid $11 to play.)
When the SB shoves he’s putting you in a decision for your tournament life.
If you call and win you are guaranteed a $7 profit ($18 - $11), and have about 6,750 chips (half the chips in play!). Winning = $7 + 6,750 chips.
If you call and lose you lose your $11 buy-in for an $11 loss.
To fully understand the value of winning, we need to know what the value of having half the chips in play is,
If we run this situation through an ICM calculator, 6,750 chips are worth about 38.3% of the prize pool, or a payout of $34.50. Thanks to your win (or loss) the two villains at the table with half your stack, are now playing with about 30.8% of the prize pool, or and expected payout of $27.75 each. (Does that seem weird to you? If you think the chip leader has a bigger expected return than 38.3% to 30.8%, remember that if you double up either of the other two, you are basically switching places with them, and if the other two go all in against each other you’re at an even chip stack with the survivor. So if half the time you coin flip with someone for stacks, and half the time they coin flip against each other for stacks, you’ll end up with ¾ of the chips 25% of the time, ¼ of the chips 25% of the time, and ½ the chips 50% the time, so ¾ of the time you’ll eventually have an equal stack or worse. Having half the chips in play isn’t a god-like advantage my any means.)
A $34.50 average payout for winning the hand sounds nice huh?
Of course, Ac Ks only beats 7h 8h 58% of the time. 42% of the time you lose $11.
58% of the time a call has an expected tournament payout of $34.50.
100% of the time folding here keeps you at an expected tournament payout of $22.50.
58% of $34.50 for the call is an average payout of $20.00.
100% of $22.50 for the times you fold is an average payout $22.50.
So the most profitable move? Fold your two over cards and make an extra $2.50 on average.
(I also recommend grumbling to yourself, but not telling anybody what you folded because a lot of other players are going to call you a fool.)
$2.50 is 22.7% of your buy-in, so plays like this can make quite a difference in your long-term ROI (Return On Investment).
But in the above example, how can I know if he has 7h 8h if he doesn’t show?
That hand was an example to make things simpler. Poker is about putting your opponents on ranges of hands, not actual hands.
If you have Ac Ks and you know that the villain will shove with any pocket pair, and any two Broadway cards, then you will win 58% of the time as well.
If your opponent is only going to shove with A/Q+ and J/J+, then you know you are about 50% to win so the fold is easier.
If he shoves any Ace, any Broadway, and any pair you’re 63% to win ($21.83 average return, which is still less than $22.50)
(I'd like to note that ICM does *not* take into account a players skill or the bullying advantage that having a big stack may have. It basically assumes everyone plays the same, and calculates their chances of winning accordingly.)
A Few Pitfalls in Thought
Being results oriented - Just because you lost doesn't mean it was a bad for that situation. Likewise, just because you won doesn't mean it was a good move for the situation. You cannot see the opposition's hole cards, so you can only base your moves on the known factors (reads, M, payout proximity, various aspects of position, your hand strength, etc.), so you can only go by what the likeliest outcome is based on known information.
For example, if everyone is deep stacked and a tight player raises UTG, and then a tight player in middle position re-raises, muck your J/J. Sometimes they'll both stack with their A/K's and so you would have been a whopping 67% favorite to triple up, but so what? You can't be expected to know what cards they have, so it was a good fold against what (at least) one of them was likely to have had.
So to sum it up, a good play is defined as one that will be most likely have the best outcome over the long run in similar situations, not one that just so happened to earn you chips in that hand.
Extrapolating short term results - Just because you are on a winning streak doesn't mean you are a winning player. Poker is swingy. A lot of players are quick to pat themselves on the back when they win (because the moves they thought were correct paid off) but then they curse their rotten cold spell when they start to lose.
The nature of the game is "limited knowledge of what's going on with bad plays showing a profit a decent percentage of the time." This means that everybody has winning and losing streaks regardless of their skill level Do not overestimate your poker prowess if you are on a winning streak, and do not beat yourself up if you're on a downswing, because you can make the right moves consistently and still have a long string of losses due to variance.
By no means am I advocating not examining your game if you are a consistent loser. By all means look at your game and try to find logical areas to improve. But be aware that if you’re mad because you never win a coin flip for stacks in the middle stages of a tournament your big problem is that you shouldn’t be taking such risks at that stage of the game, not bad luck.
Taking it personally - As I just stated, the nature of the game is swingy. Bad hands lose to good hands often. If you shove with A/K and get called by T/Qs, he's going to beat you nearly 40% of the time.
That means, for every ten times that happens you lose 4 of those times. That’s almost half. Throw in a little variance and you’re going to lose there a whole lot before you get a win.
I heard a story about a statistician at a college who likes to start the semester by having groups of students make up the results of 500 coin flips (I think I heard it on NPR). Once they do that, he has them flip a coin 500 times and record the results. They do not label the papers. Then he looks at the papers and is almost always able to differentiate the made-up results from the actual results. He is able to do this with such accuracy because random results are swingy. When people make up the results they tend even things out to stay truer to 50/50 results, but when you actually do it you'll flip tails seven times in a row, then heads, then tales another three times, etc. Random results are swingy. That's their nature. So statistically you are *supposed* to lose with A/A half a dozen times in a row on occasion. When I started poker, I had a had a 12 in a row suck streak where I lost 11/12 A/A hands, and the one I did win was when it was folded to me in the big blind. It was a blow to my spirit/ego/confidence.
When your hand loses to an inferior one try not fret it. You are supposed to lose a good percentage of the time. It is completely fair that you lose, and it's completely fair to lose with a superior hand many times in a row. That randomness is the nature of the game.
But it doesn't seem to ever go your way? That is because if you are a good player you will get sucked out on more than you suck out on people. That is the nature of the game as well. If you are a good player will get your money in as an underdog far a less often, so your opportunities to suck out on people are far less than their opportunities to suck out on you.
Something else to consider: Every time you win a hand you are essentially sucking out.
A lot of people do not consider that when you win a hand you are getting more chips than you deserve. Let’s say you are the short stack with 1,500 chips. The blinds are 100/200 and you shove with A/Ko on the button. The BB with 20,000 chips calls with 4/6s. (What an idiot, right? No, not really.) Neither of you hits and you take the 3,100 pot. Most people don’t think anything about their win, because this was what “should” have happened. They take their 3,100 chips and don’t think twice about the results.
You were only 58.84% to win that hand. So if the results were fair, you would have only received 1,830 chips (58.84% of the 3,100 pot, plus another 6 for the times you split). You taking the entire pot there deprived the villain of the 1,269 in equity that he had in that pot. So he is getting an “unfair” return for his investment.
(I’d like to note that you average a 330 chip profit in that situation when he calls, and the pot was 300 when you shoved.)
Now, I put “unfair” in quotes because over time it all balances out. The more this situation happens the closer he comes to winning 40.72% of the time. So in the long run, he gets his 40.72% of all those pots he played. (But the times he wins he gets cussed at.)
Big fish in a small pond - Let’s say you are a successful $.10 & $.25 tournament player. You are at a level where a large percentage of the players have a poor understanding of the game. They don't understand position let alone the gap concept, they make stupid bluffs, they take unnecessary risks with their big stacks, they stack light, etc. It's not very difficult to be a winning player at these levels, and because people play so poorly you can get away with a lot more, stacking off lighter, stacking with one pair, etc. and still make a profit.
If you try to move up to $2 games the pool of fish that will stack off with their wild draws and bluffs will decrease and the people who don't respect an early position raise or a re-raise will decrease. Basically, play becomes more sophisticated. If you move up to $5 games there are even fewer fish. I'm not saying they're geniuses by any means, but they are far from the blind monkeys throwing chips around that you see in the lowest of the micro limit games and freerolls.
This is why you post the buy-in when you post a hand history in the analysis section. Plays that work in the lowest levels are suicide when you try them against players who have a clue about what's going on.
Because of this, getting advice on hands for play money and freerolls (especially early hands in freerolls) is tricky. When you play against people who think about hands you have to play different. On Facebook poker I wait until I have A/J+ and then shove because people call with any Ace or connectors or whatever.
So what does this mean? This means that as you move up you meet tougher and tougher players and you have to modify your play accordingly. Things that once worked will no longer work. +EV plays at one level may end up being bad habits at a higher level, so keep learning. Keep adapting your game as you face new environments.
Hands are about situations, not cards and moves – Let’s say I talk about min-raising from UTG at an aggressive table with A/A. That is a far cry from getting A/A in middle position, and min-raising after a couple people have already entered the pot ahead of you. In my case, I’m looking to get 3-bet. In the latter case, you’re making a move that’ll make a small pot with a lot of callers which is pretty much the worst situation you can be in with A/A.
That is, min-raising with A/A is *not* the important part. The important part is that I had a powerful hand and was in a situation where I was likely to get 3-bet, which gives me the power/choice to call or re-raise a likely aggressor, so I’m trying to get a big pot against only one or two players. In the second example, it’s just setting up a call-fest with all sorts of drawing hands.
Similar moves (like min-raising with A/A) mean different things in different situations. It’s all about your goal and the expected results for your move given all the information available (including your hand, all aspects of positions, player images, etc.). It is not just your hand in relationship to just your move.
I guess another way to word it is that a move has value according to the big picture, not according to just a few factors (like how much you bet with a big hand).
The value of hidden monsters – I have a friend who told me that he likes to play 7/2 on occasion because if it hits nobody can put him on it. I could never convince him that it was a bad play. He eventually quit playing poker online because he lost too much money.
When you play junk like that you run into a variety of obstacles. First and foremost, you rarely hit a big hand. Top pair has a lousy kicker, and you still have to worry about over-pairs, and over-cards on later streets and stuff. It’s just a dangerous situation to be in.
Secondly, if you hit a big hand you will rarely get paid off. Any hand that gives you a monster will mostly likely totally miss everybody else. So the times you do hit you don’t get paid off.
Some people can play a lot of junk and make it work. You have to be very very good to do it successfully, *and* you have to play against people who are smart enough to understand what’s going on because you can’t accurately predict the actions of the ignorant. You can’t read or bluff a player who makes whimsical plays because they do not have consistent (or logical) reactions to the situations presented to them.
How do I play J/J?
The old joke is, "There are three ways to play J/J... and all of them are wrong."
Play it just about the same way you play T/T. Nobody complains about T/T. T/T doesn't have the pretty picture so it's easier to let go of.
Poker is rigged!
Nobody can prove otherwise if you want to believe that.
The fact is, tens of thousands of players use tracking software, recording billions of hands, and cards show up as often as they should. Humans are designed to see patterns and anomies, so when randomness generates the illusion of a pattern we pick it up, and when something weird happens we make note of it. It is common to see these weird things and assume that they are proof of foul play.
The thing is there are many thousands of ways for weird things to happen in each hand of poker. When one of those thousands of “10,000-to-1-against” things happens we pick up on it because your brain was designed to pick up on oddities. These are to be expected, and not necessarily indicative of foul play.
Yes there are cheaters online. People multi-account, collude, use bots, and use hand history software to gain unfair insight into their opponents play. But this doesn’t make the game unwinnable. A villain pretty much has to be a smart/winning player to make proper use of the illegally obtained hand histories. Collusion is fairly obvious to spot and several CC’ers have been contacted by the poker sites and had money returned because the site’s automated software caught the colluders and returned money to the effected players. And as far as multi-accounters go, if the accounts aren’t playing together their advantage is that they might have a history on you while you won’t have their hand history matched to their alternate account. And if they are playing together the advantage would be that he could collude with himself and he’d know an extra 3.84% of the deck that was out-of-play because the other account had it (which will only benefit him in rare situations).
To put it bluntly, if you put some serious time into reading poker books and improving your game, and following proper bankroll management without tilting you are going to be a winning micro-stakes poker player. If you can’t do win at the micro stakes it is not because the games are rigged; it is because you are doing something wrong.
Having the best hand is over rated.
You buy into a single table S&G for $10+$1.
1st pays $45
2nd pays $27
3rd pays $18
Total prize pool = $90
4 players are left (you are on the bubble)
Chip stacks are about even (average 3,375), and the blinds aren’t huge.
Player 1 folds.
Player 2 folds.
Player 3 accidently exposes 7h 8h and shoves
You look down and see that you have Ac Ks.
What do you do? You insta-shove, right?
Here’s where ICM (Independent Chip Model) comes into play.
Basically, each chip you have is worth something in relationship to the prize pool.
With 4 of you left and similar stacks, you are all playing with about 1/4th of the $90 prize pool, or about $22.50 in expected payout. (Note that this is not profit. Winnings and profit are different, as you paid $11 to play.)
When the SB shoves he’s putting you in a decision for your tournament life.
If you call and win you are guaranteed a $7 profit ($18 - $11), and have about 6,750 chips (half the chips in play!). Winning = $7 + 6,750 chips.
If you call and lose you lose your $11 buy-in for an $11 loss.
To fully understand the value of winning, we need to know what the value of having half the chips in play is,
If we run this situation through an ICM calculator, 6,750 chips are worth about 38.3% of the prize pool, or a payout of $34.50. Thanks to your win (or loss) the two villains at the table with half your stack, are now playing with about 30.8% of the prize pool, or and expected payout of $27.75 each. (Does that seem weird to you? If you think the chip leader has a bigger expected return than 38.3% to 30.8%, remember that if you double up either of the other two, you are basically switching places with them, and if the other two go all in against each other you’re at an even chip stack with the survivor. So if half the time you coin flip with someone for stacks, and half the time they coin flip against each other for stacks, you’ll end up with ¾ of the chips 25% of the time, ¼ of the chips 25% of the time, and ½ the chips 50% the time, so ¾ of the time you’ll eventually have an equal stack or worse. Having half the chips in play isn’t a god-like advantage my any means.)
A $34.50 average payout for winning the hand sounds nice huh?
Of course, Ac Ks only beats 7h 8h 58% of the time. 42% of the time you lose $11.
58% of the time a call has an expected tournament payout of $34.50.
100% of the time folding here keeps you at an expected tournament payout of $22.50.
58% of $34.50 for the call is an average payout of $20.00.
100% of $22.50 for the times you fold is an average payout $22.50.
So the most profitable move? Fold your two over cards and make an extra $2.50 on average.
(I also recommend grumbling to yourself, but not telling anybody what you folded because a lot of other players are going to call you a fool.)
$2.50 is 22.7% of your buy-in, so plays like this can make quite a difference in your long-term ROI (Return On Investment).
But in the above example, how can I know if he has 7h 8h if he doesn’t show?
That hand was an example to make things simpler. Poker is about putting your opponents on ranges of hands, not actual hands.
If you have Ac Ks and you know that the villain will shove with any pocket pair, and any two Broadway cards, then you will win 58% of the time as well.
If your opponent is only going to shove with A/Q+ and J/J+, then you know you are about 50% to win so the fold is easier.
If he shoves any Ace, any Broadway, and any pair you’re 63% to win ($21.83 average return, which is still less than $22.50)
(I'd like to note that ICM does *not* take into account a players skill or the bullying advantage that having a big stack may have. It basically assumes everyone plays the same, and calculates their chances of winning accordingly.)
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