Friday, February 20, 2015

On the Envelope Problem

A continuation of a discussion from Facebook, with hopefully enough context that others can catch up using this post.  (or pose questions in the comments that I can respond to).



The Envelope Problem:

Omega (who never lies) walks up to you and states "these envelopes contain the values X and 2X" He then SHUFFLES them up [this informs us our subsequent switching strategy should be irrelevant], slides one over to you. He states "I have given you an envelope containing either X or 2X. You can open it, and then decide whether you'd like to switch for the other envelope." You open it up, see a $100 bill. Do you switch? Some decision theory has to take you from probabilistic knowledge about the world to an action you should take, if you care about acting (VNM) rationally.

Premises and Conclusion of EV Calculus Using the Same EV Calculus Rules We Normally Use:

1) Your envelope contains $100.
2) the other envelope contains $50 with .5 probability, and $200 with .5 probability
3) Half the time, switching costs me $50.  Half the time, switching gains me $100
4) therefore, I stand to gain an average of $25 by switching.

Which numbered part do you think is false?

If you think it's 2) because the state of the world is already fixed by Omega, rather than variable, then I offer this: when something has a fixed state, but your knowledge of it is probabilistic, ev calculus treats it as if it is not determined but will be at the stated probabilities.  If you disagree, then your next task is to explain how we calculate the odds of being dealt an ace from a shuffled but now fixed-state deck of playing cards.

We can solve this using logic other than EV calculus, namely an observation that since the envelopes were shuffled and fixed, switching can't change our actual EV unless we know something new about the other envelope, and we do not.  We know as much as we did before our envelope was opened.  But why are steps 1-4 above so persuasive under the same logic we correctly apply to cards being dealt from a shuffled deck, despite leading to the wrong conclusion (that switching is +ev)?

"When we switch we stand to finish with 1.5x, if we don't switch, we stand to finish with 1.5x" is incomplete work.  You observe $100 in your envelope.  So at the very least you should be able to redo your math and get us an actual dollar amount EV.  It has to be $100 since your premise is switching doesn't matter.  That means X = $66.66 in the earlier formulation of "we are guaranteed 1.5x."  Well, if X = $66.66, your envelope must contain either $66.66 or $123.33 by reference to the rules of the game.  So your solution is not well formed, it is not coherent.

[EDIT 1 Below]

Stolzman is arguing that EV calculus doesn't apply here, but that's not an issue.  Well, it's a major issue if EV calculus gives you neither the correct answer nor an indication as you work through steps 1-4 that you shouldn't be using this method.  In other words, it doesn't just fail to return an answer, it returns the wrong answer!  Only because we have access to other logic do we even realize it has failed.  This is a big problem.

[EDIT 2 Below]

Well, I correctly predicted Step 2 is where the action is, but people don't seem to agree with my note on it above.  Yes, the state of the universe is fixed.  No, your knowledge about the state of the universe does not therefore use only 1 and 0 as probabilities.  If a coin is biased (unfair) but you do not know in which direction, your probability estimates of heads and tails are each .5.  Probability is in the mind.  http://lesswrong.com/lw/oj/probability_is_in_the_mind/

Omega may have filled the envelopes with $50 and $100 because his son was born in the galaxy his people call "50zba100".  Or he may have filled them with $100 and $200 for any other whismsical reason.  We do not have access to his algorithm.  But since he has not given any hints as to which he picked, why should you not switch?  Why should you not act under P(50) = .5 and P(100) = .5, which most closely approximates your state of knowledge?  Furthermore, NOT switching is implying that $50&$100 is LESS likely than $100&$200 (otherwise you're a fool).  That is your implied conclusion when you do not switch, having seen $100.  How did you arrive there with such confidence?

Thursday, February 5, 2015

Introducing: Team Work!

For Magic: The Gathering's Pro Tour Fate Reforged, a few players who don't have the time to show up early and test on-site decided to test online and communicate via Facebook.  The men and women who investigate these vicious felonies are members of an elite squad known as "Team Work."  This is their story.

Not all were able to attend the Pro Tour or were qualified, but they still helped prepare the team.

Team Work:

Bob Maher


Pro Tour Hall of Fame
Pro Tour Champion
Slightly outplayed Brian Davis once
Immortalized as a blob on Modern Masters Dark Confidant


Ben Stark


Pro Tour Hall of Fame
Pro Tour Champion
Fashion consultant for Men's Denim Cut-Offs magazine

Paul Rietzl


Pro Tour Hall of Fame
Pro Tour Champion
Dice Roll Angling National Champion 2006 & 2009
Portuguese Vendetta Holder, 2011 - present

Sam Black


Platinum Pro
Pro Tour Top 8 Competitor
Designed a deck good enough to win Tom Martell a Pro Tour
Tells interesting stories

Matt Sperling


Gold Pro
Pro Tour Top 8 Competitor
One SCG Open Top 8
Author of this blog post and most underrated player in the world

David Williams


Pro Tour Top 8 Competitor
World Series of Poker Bracelet Winner
Broke People Vendetta Holder, 2004 - present

Gary Wise


Pro Tour Hall of Fame
Pro Tour Champion
Almost All of You Had No Idea About the Above Two Facts
Always knows what to Shock

Lucas Siow


GP Top 8 Competitor
Canadian

Andrew Baeckstrom


U.S. National Team Member
GP Top 8 Competitor
Solid Internet Connection

Orrin Beasley


GP Top 8 Competitor
Team's Best Mustache and Google+ Profile Image
Working his way up to Team Manager

Justin Cohen


PTQ Champion
Here's a name you already recognize (assuming Justin himself is reading this)

Matt Severa


Grand Prix Top 8 Competitor
Does Jazz Hands above his cards each turn for good luck

David Heineman


PTQ Champion
Looks like one of the guys who invented fantasy baseball

Keep an eye on those up and comers from Madison (Justin and Severa and David)

Ben Rasmussen


GP Top 8 Competitor
Is that a leather polo??

Adrian Sullivan


Top 20 Deckbuilder All Time (Next Level Deckbuilding)
GP Top 8 Competitor
Plays his cards upside down
(I pulled the image from Tinder)

Brian Kowal


GP Top 8 Competitor
Underrated deck builder
Your grandpa might recognize the name (if he's from Wisconsin)


Thursday, January 15, 2015

2015 Magic: The Gathering Pro Tour Season Goals

I've never blogged about goals before, but I'm feeling optimistic about this season and I want to record my goals now while I have the wind at my back, so that when I hit the inevitable setback I can look back on something concrete and immutable and not let myself adjust my goals or commitment downward without being cognizant of the fact that I'm lowering the bar I set for myself.





Goal: Make Platinum.  

Top 8'ing another Pro Tour would be amazing.  Making the World Championships would be amazing.  I'm not saying Platinum is the only carrot out there, but it's the one I see right now on the end of my stick.  Plus if I make Platinum, making Worlds will be a likely consequence a good percentage of the time.

Platinum this year requires 48 pro points.

How Am I Doing So Far:

PRO TOUR KHANS OF TARKIR: (bolded and all-caps because the Pro Tours really matter a ton).  3 points.  Yikes.  Made day 2 but flamed out.

Grand Prix of Latter Day Saints: 1 point.

Grand Prix Roscoe's and Broken Dreams - 3 points.

Grand Prix Why the Fuck is this Location Chosen in January I - 6 points.

Grand Prix Why the Fuck is this Location Chosen in January II - 3 points.

Total so far: 16 points.

As good as I could hope for after 4 GPs and 1 PT given that I crapped out at said PT.  In each leg of the upcoming season (3 legs left, one for each PT), I'll need to do better at the PT because 48 is a shit ton of points needed and because only 6 GP finishes will count at the end of the year, meaning I can't get 16 points this way every leg.

I really will hate the World Championships and the Magic World Cup for as long as I don't get to participate.  I will switch to loving these events once I get to play and enjoy the head start my peers enjoy now.  If I get way too enthusiastic defending them once I've made it in to either (assuming I ever do), you can direct people here.  (Self-service aside I think these PT point bonus that go into "next year" are fine - the rewards were earned and it helps top pros stay at the top which helps the brand.  I get it, I'm just Sick of It TM).

----------------------------------------------------------------------

Projecting Specific Goals:

REMAINING 3 PRO TOURS: One 11-5 (10 points), two 10-6 (6 points each) for a total of 22 points.  Consistency will be rewarded, and of course if I spike something better I give myself additional wiggle room, but I want to set out the goals that I know won't require me to run like Shahar.

Remaining GPs:  Upgrade my 1 point finish to a 3 point finish (netting me 2 points).  Make 2 additional top 8s or secure 1 GP win (8+ points).   Total of 10 points.  This one seems pretty hard given the low volume of GPs I go to.  I will have to stay hot, and there's no reason to think I'll keep running well over a handful of tournaments just because I have been.  What I'm feeling is the benefit of setting this out in writing is that I probably have to book an extra trip or two, and I can't "blow one off" drinking on Friday night or not caring what deck I play.

Total projected: 32 points.  (making 48 points, holy shit it works!)  

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Minor Goal: Stop Bitching About Being the Guy With a Full Time Job and a Wedding Coming Up Etc. Which Means I Can't Play Magic 24/7

Many other pros have other commitments, and the full time players have their own crosses to bear (staying motivated to challenge themselves in practice, feeling the pressure of having to do well in the actual events, and others), and this narrative is just getting kind of tired.  I want to consider myself a top 25 player at year-end, and not because I decided to grade on a curve.  This whole thing originates as a kind of defense mechanism to "Sure, Owen Turtenwald is 2-4x better than me, but that's because of [excuses]."  I'm getting over that; Owen is better than me because he prepares better and plays better.  The details of why that is the case might suggest things I can do to improve, but I will no longer cite them to other people in order to make it seem like "well, we're all pros, and who knows what would happen if I played Magic full time" (all the evidence suggests Owen would remain the best in the world).  I'll keep putting "real life" well ahead of Pro Magic, but I've found ways to serve both (including a new testing team that will test online).

Nobody likes the guy at the YMCA who would have played shooting guard in the NBA if his high school coach didn't hate him or if his girlfriend hadn't gotten pregnant.  Even if it's true 0.1% of the time you hear it, everyone wisely rounds that 0.1% off and it just comes off sad.  My goal is to never become the Magic equivalent.

If anyone wants to post their own season goals and current progress in the comments, I'd be genuinely curious as to who is aiming for Gold, Platinum, to Qualify for 2 PTs, to cash 2 GPs, to learn competitive Hearthstone, whatever.  I'm not yanking your chain when I say my goal last year was to try and spike a good finish and if I couldn't, then quietly phase competitive Magic out of my life a bit.  I spiked that PT T8 out of a combination of good prep and good luck, and so I have a new attitude and a new set of goals.  I wish I had written more about where my head was at last year, so that's the motiviation for this post, and thanks for reading.