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.

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

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)