The parable of thirty dollars

“Why do Vegas resorts always smell like a pool inside,” Mark asked as his group of friends entered the Venetian.  Kevin was going to explain how the ventilation systems add aromatic oils to the circulation.  Humans are wired for smell and the resort likes relaxed guests.  Kevin’s attention, however, was drawn to the grand hallway with its perfectly polished marble floors from which fifty foot vertical roman columns rose to the grand arched ceiling covered in Italian renaissance murals.  They had all walked the Venetian’s grand entry a couple of dozen times in the past but it never gets old.

Mark wasn’t here for the scenery though.  His room cost a fraction of the retail price so long as he played four hours of Texas Hold’em a day.  The funny thing about that is that the hotel wont make enough money off of Mark’s play to cover his discount.  Unlike every other casino game, poker is a competition between the players, not the house.  The house only gets paid a fee for dealing cards.  Because of this, most casinos have done away with poker altogether.  It’s just not that profitable.  At least the Venetian still provided it.  Maybe the resort figured that guys like Mark bring their spouses or friends who help make up the loss on the room.  Regardless, Mark’s plan was to grind at the poker table until he won enough to cover the entire cost of his trip.  The mental focus and the competition does more for his mood than the aromatic oils.

Mark sat at a table with nine other players.  The dealer pulled a deck from the shuffler and sent two cards face down to each player.  The players carefully lifted one corner from the cards a half inch off the table.  This would let them see what they were dealt without their opponents seeing.

Mark was “under the gun”.  This means that during this hand, he would have to act first in each of the four rounds.  He’d be the first to check, bet, or fold.  Being under the gun would put him at a disadvantage.  The other players will get to see what he does before they make their moves.  They’ll get clues about what his hand might be before they play.  They’ll adjust their play to what Mark does.

Normally, Mark folds whenever he’s under the gun.  It’s not worth playing with such a positional disadvantage.  He’ll just wait a few hands until it’s his turn to be “on the button”.  At that point, his position will give him the advantage.  Besides, Mark believes its good policy to always fold the first half hour of hands.  He wants to watch the other players to profile their playing style.  The more information he gathers, the stronger he competes.

This time, however, Mark was dealt two queens.  He was dealt one of the strongest hands in Texas Hold’em.  The strength of his cards could compensate for his positional weakness.

He decides to play this hand after all.  Mark considers the scenarios that would put his queens at risk.  His first concern is the number of players at the table.  Big pairs like queens aren’t great against multiple players.  The more players, the higher the likelihood that someone will get lucky and beat his otherwise dominating hand.  He wants to be heads up against one other player, two at the most.  That maximizes his chances of winning.

At this table, $10 is considered a good sized bet.  Mark wants to scare off as many other players as possible so he bets $25.  Since the other players haven’t seen Mark play yet, they don’t know if his big bet means that he has a strong hand.  He might be one of those guys who always bets too much.  Still, at this point there is more value to watching him to find out as opposed to paying him to find out.  The next five players fold.  The sixth player, however, calls his bet.  He’s on the button and must be holding something he thinks has potential to beat a monster hand.  Maybe he thinks he’ll get a chance to bluff as a contingency.  Mark’s best guess in that his opponent is holding an ace with some other high card.  The remaining two players fold.  Mark’s big bet worked.  He’s playing heads up just like he wanted.

It’s time now for “the flop” where the dealer turns over the first three community cards.  Each player’s hand is made up of his two cards plus the three community cards on the table.  The community cards this time are the ace of clubs, ten of diamonds, and two of clubs.

Ouch, the one card Mark definitely didn’t want to see on the flop was an ace.  Remember, that’s the card he thought his opponent most likely held.  A pair of aces beats a pair of queens.  At this point, Mark thinks he likely has the losing hand but there are two more cards to come.  If one of them is a queen, he’d be in a great position to elicit a big bet from his opponent and win the hand.  However, there’s only a 8% chance that one of the next two cards is a queen.

Mark doesn’t want to bet but he worries that he has to anyway.  Not betting now after his big pre-flop bet would send a strong signal that Mark doesn’t have an ace.  If his opponent figured that out, he could use it against him.  Mark decides to semi-bluff here and bet another $50.  Maybe his opponent has a pair of Kings and this will prompt him to fold.  Or maybe this will make his opponent think Mark is holding a pair of aces which will could come in handy as the hand progresses.  No such luck.  His opponent called his bet.  If Mark thought this guy had an ace before, he really believes it now.  The pot has grown to $153.

Now comes “the turn” where the dealer turns up the fourth community card.  It’s the jack of clubs.  This changes things.  With that jack, it’s now possible for a player to have a straight or a flush.  If player had a king and a queen, then their hand would be ten, jack, queen, king, ace (a straight).  If a player had two club cards then their hand would be 5 club cards (a flush).  A straight or a flush would beat a pair of aces or just about anything else his opponent might have.  The turn just gave both Mark and his opponent reason to proceed with caution.

Mark double-checks his hand.  One of his cards is the queen of clubs.  He thinks that to have the winning hand, he needs the fifth and final card to be either a queen (making 3 of a kind for him), a king (making a straight for him) or any club card (making him a flush).  Mark knows there is a 28% chance that the last card will be one of those.  Based on this and the possibility that his opponent might also be able to make a straight or a flush, Mark decides he has around a 20% (1 in 5) chance of ending up with the winning hand.

With a potential straight or flush on the table, Mark decides it ok to not bet this time.  He signals that he “checks” (isn’t going to bet) and the play goes to his opponent.  Mark’s opponent isn’t sure whether mark checked because he’s nervous or because he’s laying some kind of trap.  He’s tempted to check (not bet) as well and see what the final card brings.  However, he decides that betting against Mark might help him find out where he stands.  It’s better to bet a little and lose now if he’s going to lose anyway.  So his opponent bets $30 back to Mark.

At this point, Mark has an important decision to make.  He could fold now and cut his losses.  After all, he thinks he’s 80% likely to end up with the losing hand.  He could call the $30 bet and hope for the best on the final card.  Or he could raise the bet and try to bluff his opponent off of his ace.

What should he do?

He’s most likely going to lose, right?  So take option A and fold?

It might seem counter intuitive but that would be a mistake.  It’s not because bluffing is better.  It’s not because Mark is worried that he hasn’t accounted for all of the possible outcomes.  This issue is that the pot is $183 but it only costs $30 to call the bet.

This means that by calling, Mark has a 20% chance of winning 6x his $30.  The 600% reward outweighs the 80% risk.  See why?  If he gets into this situation 100 times, then 80 times, he’ll lose $30 for a total loss of $2400.  However, 28 of the times, he’ll win $183 for a total gain of $3660.  His ends up $960 ahead after playing this scenario 100 times.  $3360 – $2400 = $960.

This is how Mark approaches poker.  Each move is an independent decision with its own cost-benefit-analysis.  It doesn’t matter what happened in the past.  He plays each bet as if he’ll run that scenario a million times keep the net.  That net gain or loss after playing the scenario a million times is called the expected value.  In any given hand, Mark’s actual gain or loss wont be the expected value.  However, over time, his aggregate gain or loss will become the expected value.  In this case, he’s most likely going to lose the hand but the expected value is positive.  In other words, over time the infrequent winners will pay for the frequent losers.  Thus, the critical question Mark asks himself is, “what move will maximize the expected value?”  Should he check, bet, fold, or raise?  It only matters which move maximizes the expected value.  It doesn’t matter how what his gut says.

Mark knew the $30 bet by his opponent was a miscalculation.  It gave Mark instant expected value.  He calls the $30 bet even though he’s most likely going to lose this hand.

The dealer turns over the final card, “the river”.  It was a seven of diamonds.  Mark didn’t get one of the cards he needed.  He knows that he has the losing hand now.  Of course, he’s not surprised.  He knew going in that this was a way more likely outcome.  After his opponent bets, Mark folds his hand.  His opponent wins.

Ok, this wasn’t the ending Mark hoped to see.  Still, he took comfort knowing that he played the hand correctly by calling that last bet.  He lost a battle but not the war.  The next time he’s in this situation where the reward outweighs the risk, he’ll do the very same thing — even if it means playing a hand most likely to lose.  Over time, as he does this, the relative few wins more than pay for the frequent losses.  Over time, he ends up ahead.  Predictably.

Isn’t that weird?

It’s hard for us to assess risk and to know how to trade reward for risk.  We’re so wired for self-preservation that fear tends to drive our behavior.  We tend to emotionally feel the same way about a 25% risk as we do about a 99% risk.  That’s why it feels wrong to play a hand that we know is most likely to lose.  We don’t emotionally connect with the fact that the greater reward makes playing that hand the most likely thing to bring a positive outcome over time.

Venture capitalists get this.  Their whole plan is to invest in lots of startups that are most likely going to fail.  Yet VC’s nonetheless increase their funds’ value over time.  How can this be?  Their few big winners cover their frequent losers.

Life is full of situations where short term risks, even ones unlikely to win, can pay off in the long term with a little persistence and a willingness to fail.  In 2016, the Oakland Raiders made 95% of their one point conversions but they made 71% of their two point conversions.  If those rates held, they could have scored around 8% more points in their season by just going for the two point conversion every time.  No team does that. Of course, there are other factors but maybe it’s worth reexamining the conventional wisdom for a team like that.

Maybe we’re missing opportunities because the risk motivates us more than the reward.  We don’t put ourselves out there.   We turn down a chance to grow because it’s uncomfortable.  We stay in our comfort zones.  We avoid change because we think of all the ways the change could be worse than the status quo.  Our emotions discount the ways it could be better.

Not only that, maybe we punish ourselves too much when we fail.  If you think about it, if we put ourselves in a position where failure was possible, it means we judged the reward to be worth the risk.  You can feel bad if you misjudged the risk or the reward.  You can feel bad about a mistake in your cost benefit analysis.  You can feel bad if you bet the farm on a losing hand instead of betting something you could afford.  You can feel back if you didn’t try to prevent the failure along the way.  But if you knew the risk, you knew the reward, and you did your best, then does it really make sense to feel bad when the failure occurs?  The failure was always on the table.  Why not feel proud for going for it?  If the reward really outweighs the risk and you consistently take the risk, you’ll eventually end up ahead.   This isn’t a self-help, positive thinking pep talk.  It’s cold, rational math.

Mark could have regretted calling that $30 bet but it was the right thing to do.  He took pride in making a good play without taking shame in the loss.  Sure, he lost that hand but when he eventually wins, it’s worth it.  Mark hasn’t paid for a Vegas vacation in the last decade.





2 Replies to “The parable of thirty dollars”

    1. Thanks, and that’s a good point. I needed Mark to lose in order to make the moral of the story. The alternate ending could be that Mark re-raises because he figures the fold equity maximizes EV. Or even better, he re-raises on the river, gets called, and loses anyway. Mark, btw, is a friend of mine and not an alias for me. He told me I should follow up this story with playing AQ under the gun to a later raise so that one small mistake compounds overtime.


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