On Societal Contribution

A week ago, I attended my 2^nd cousin’s wedding. It was a great event, and I’m really happy for him and his wife. I only really see the family on my father’s side once every couple years or so, so I don’t have a close relationship with any of them. Anyway, the groom’s father is a pastor. At the end of dinner I went to congratulate him and say goodbye, and the first words to come out of his mouth when I shook his hand were, “Cut out the gambling. I don’t care how much money you make from it, you’re smart enough to do something better than that.” How amiable. I congratulated him and left after saying a couple more goodbyes.

It made me pretty angry that that was all he had to say to me, and after leaving I realized that he hadn’t talked to me all day even though he had checked up on my brother and sister. After I got over that, I got to thinking about whether I contribute less to society than I would with a more conventional job. I can think of two realistic ways one’s work can benefit society:

1.      Goods or services produced/provided while working

2.      Unselfish use of income (or knowledge, or whatever other gains)

Occupations like farming, construction, or teaching (I feel like the Christian community views pastors much the same as teachers) are really obvious examples of occupations that provide a beneficial good or service. On the opposite end of the spectrum, poker pros don’t really provide a good or service. You could argue that a poker pro provides entertainment in the same way an artist or a performer does, but whatever.

Some will argue that poker pros are, in part, responsible for the gambling addictions of others and actively make the world a worse place, like illegal drug dealers. I disagree. How about video game developers? Should we outlaw video games because they’re responsible for burn out gamers who didn’t live up to their potential? Movie producers? Plenty of people forego school and work so they can chase the dream of being a star actor.

The more money one makes at their job, the more they have to spare on others. Even if you don’t donate to charities or support small businesses that you couldn’t support if you were poorer, you are still contributing through something that’s mandatory: income tax. So now the question arises: how much more income tax does one have to pay to overcome the effective societal burden of not producing or providing anything?

If I were working in finance or data analysis or something, I wouldn’t be paying much in taxes compared to what I’m paying now. I’d be making a contribution to society with my work, but it’s really hard to calculate how much. Imagine I decide to only play poker for the next 5 years, and end up giving the government $100k more in income tax than I would doing something else.

In 2012, the U.S. is spending the most on Social Security ($610B), Homeland Security ($499B), and Health and Human Services ($315B) (usaspending.gov). In total, the U.S. will be spending $3.7T in 2012. So that’s 16% on SS, 13.5% on security, and 8.5% on medical. Over those 5 years, I’d be contributing an additional $16,000 to social security, $13,500 to homeland security, and $8,500 on medical spending, along with whatever else the government decides to use the other $62,000 on. Is that more of a contribution to society than I would’ve made with the services I’d have provided had I done something else? I don’t know.

I’m not saying everyone should aim to make as much money as possible, or proclaiming that poker is more altruistic than teaching (lol?). I’m only trying to debunk the notion that gambling is an evil occupation that hurts our society. Even poker pros who aren’t as capable as others, and definitively contribute less than they would at a more conventional job, shouldn’t be made villains. What happened to doing something you enjoy for a living? Is that only okay if it’s something less taboo than gambling? What about other entertainment providers like fiction authors or actors, are they selfish? Where do you draw the line? Is it not completely arbitrary?

Comments welcome. Would love to discuss this more.

John

GTO River Bet Sizing

Game theory. People don’t get it. I don’t really get it, but I probably get it more than 90%+ of poker pros. Part of becoming good at something is being humble enough to realize you suck and don’t know anything. The inspiration for this post can be found here:

http://forumserver.twoplustwo.com/56/medium-stakes-pl-nl/size-matters-1219472/,

where almost nobody gives a reason for their preferred bet size beyond “oh just bet what he calls duehh”. Let’s try to figure out the most profitable optimal betting strategy is for a similar situation, that is, a betting strategy that doesn’t let our opponent improve by deviating from his optimal strategy and that makes us the most money. If the players are bluffing and calling with optimal frequencies, the 1-period (as opposed to multiple periods) “game” of this river situation is in equilibrium and neither player can improve by changing their play.

There’s not really a need for a hand history, so I’ll just describe the hypothetical situation:

1.       There is $100 in the pot on the river.

2.       The players both have $200 remaining in their stacks.

3.       Villain is first to act and checks. Villain’s hand range is entirely made up of “bluff catchers”, or hands that beat all of hero’s bluffs and lose to all of hero’s value bets. Villain only ever calls or folds on the river.

4.       Hero gets to the river with 40 made hands that beat all of villain’s hands, and 60 missed draws that lose to all of villain’s hands. He needs to choose a bluff % and a bet amount.

Any equilibrium strategy for hero will result in villain being indifferent between calling and folding, meaning villain will call 50% of the time and fold 50% of the time. Let’s look at what that means from villain’s perspective.

The most villain ever has to win to call a bet is 50% of the time. If someone bets $100 into a $0 pot, you must have the best hand 50% of the time to call and break even. If someone bets pot, you must have the best hand 33% of the time to call and break even (.33*200 + .66*-100 = 0). If someone bets half pot, you need the best hand a quarter of the time, etc.

Now we’ll examine the EV of different bet sizes. Let’s start with a half pot bet. If we decide that half pot ($50) will be our bet size, we need to have a betting range containing 75% value hands and 25% bluffs to make villain indifferent between calling and folding. Since only 40% of our getting-to-the-river range is made up of value hands, now we figure out what 40 is 75% of and fill the rest in with bluffs. 40/.75 = 53.333. Round that to 53…53-40 = 13 bluffs. If we bet half pot with 40 value hands and 13 bluffs, player B makes the same amount of money calling as he does folding: $0. With the other 47 bluffs, we have to just check back and lose, or else villain can call every time.

Half pot bet

Villain EV for calling = .25*150 + .75*-50 = $0 = villain EV for folding

.75*-50 is the probability that we lose the pot times the amount we lose. .25*150 is the probability that we win the pot times the amount we win.

It’s slightly more complicated to calculate hero’s EV for playing this strategy. Four things can happen:

1.       Bet, get called, win

2.       Bet, get called, lose

3.       Bet, villain folds

4.       Check and lose

We’re checking and losing with 47 hands (also 47% of our range) so that part is easy: .47*0 = 0. When we bet, which we do 53% of the time, we know that villain folds half of the time: .5*.53*100 = +26.5. The other half of the time we bet, he calls. 75% of the time he calls, we win, and 25% of the time he calls, we lose. .5*.53*(.75*150 + .25*-50) = +26.5.

Hero EV for this strategy = .47*0 (check back) + .5*.53*100 (bet, fold) + .5*.53*(.75*150 + .25*-50) (bet, call) = +$53

Just as a demonstration, let’s see what happens if we decide to bluff more and villain changes his strategy to calling every time. Let’s say we decide to value bet 40 hands, bluff 40 hands, and check back 20 hands.

Villain EV for calling every time vs. new strategy = .5*150 + .5*-50 = +$50.

Hero EV for new half pot strategy that lets villain call every time = .2*0 (check back) + .8*(.5*150 + .5*-50) (bet, call) = +$40.

So when we deviate from optimal bluffing frequency and start bluffing too much against a perfect villain, villain’s EV goes from $0 to +$50, and ours goes from +$53 to +$40. Our reaction to him calling every time would be to bluff less and less until we end up back in the first situation.

Let’s move on to different bet sizes. For a full pot bet, villain is indifferent between calling and folding when he has the best hand 33% of the time, so player A will be bluffing 33% of the time. 40/.66 = 60, so our betting range will be made up of all 40 value hands as well as 20 bluffs. We check back 40 missed draws.

Full pot bet

Villain EV = .33*200 + .66*-100 = $0

Hero EV = .4*0 (check back) + .5*.6*100 (bet, fold) + .5*.6*(.66*200 + .33*-100) (bet, call) = +$60

Looks like against a perfect player, in this specific bluff catching scenario, a full pot betting strategy wins more than a half pot strategy. Let’s look at all in for 2x pot now. Villain needs to put $200 in to win the $300 sitting in the middle, so he needs to win 40% of the time to break even on a call. 40/.6 = 67, so hero will be value betting 40 hands, bluffing 27 hands, and checking back 33 hands.

2x pot bet

Villain EV = .4*300 + .6*-200 = $0

Hero EV = .33*0 (check back) + .5*.67*100 (bet, fold) + .5*.67*(.6*300 + .4*-200) (bet, call) = +$67

Betting as large as possible yields the biggest profit against a perfect opponent in this situation. When I figure out GTO (game theory optimal) solutions like this, I like to use them as starting points in my thought process in real situations, and adjust according to mistakes I believe my opponent will make. If I’m playing against someone who will call a large bet too often in this spot because lots of draws missed, I will still bet as big as possible but will check back all of my bluffs. If I’m playing against a very good player who may call and may fold, I simply revert back to GTO strategy and become unexploitable. If I’m bluffing an optimal frequency for my chosen bet size, the very good player can’t do anything to win money against me. I don’t care how much he calls or folds, he’s still not beating me for more money by adjusting his play.

Sometimes in a real game, you can’t make such rock solid assumptions. If I’m betting $1,000 into a $100 pot it could go from optimal to a massive losing play if there’s any chance my opponent slowplayed a monster hand (one that beats some or all of my value betting hands). Most situations aren't as clear cut as in this example. Sometimes the villain check/raises, sometimes villain has monsters, air, and mediocre hands all in his range, etc. 

That’s it for now. Any feedback is appreciated.

 John




Edit: I've been reminded that, while it is correct in this case, it isn't always most profitable to make the opponent indifferent between calling and folding. I might not have gotten that across. The key is figuring out what maximizes value against a perfect opponent. In this case they are the same thing.


Edit2: First comment corrected me. Villain calls 33% for 2x pot, 50% for 1x pot, etc. If you plug in the numbers the end results are the same but that is an important point.