Useless Statistical Almanach n°4

February 13, 2005

Is there such thing as useless or worthless information?

I’d like to bring your attention to 2 examples from the games world again:

Game 1 (a classic, and an easy one)

You are given the choice between 3 cards, which are face down, one of which is red and the other two black. You are asked to find the red one.

After you have made your choice (but not looked at it yet), the dealer reminds you that out the two remaining cards, at least one is black, and he shows one of the two cards, which is indeed black.

Should you change your choice of card if given the opportunity? How much more would you be willing to bet if you had that possibility?

Game 1 “The Grass is Greener”, from Brad De Long (but don’t go peek there yet!)

You are given the choice between two envelopes and told that there is money in both envelopes, with one amount being double the other amount.

If you choose one envelope, you can always think the following: the other envelope has a 50% probability of having double the money, and a 50% of having half, so it’s “expected value” is 0.5*2 + 0.5*0.5, i.e. 1.25 the expected value of this envelope. Trouble is, the same reasoning applies with the other envelope?
Do you have too much information or too little? What would you think if the person givng you the envelopes had pointed out that above reasoning AFTER you had chosen one envelope?

Posted by on February 13, 2005 at 09:55 UTC | Permalink

Comments

As for game 1 of the two (both of them are labelled “Game 1”):

You should change to the other card. Since the dealer knows where the red card is located, he deliberately always picks a black card, which means that the chances have still not changed. Your chance of being right has not changed from one third to one half, it is still one third. The other card, the one which the dealer did not turn over, now represents all the rest of the chances, so the chance that it is the red card is two thirds, or double your current chance. (If you don’t believe it, try playing this game with a friend a few dozen times. You will win more often if you change cards.)

Personally, I would not bet any more on new card than I had previously bet. And if this is really a bet, my bet would have been nothing. In the real world, since there is an obvious solution to the problem, it means the person running the game must be cheating or else they would go out of business.

As for the second game: I suspect this is an example of applying the wrong analysis to the problem. There are situations in geometry where if you attempt the wrong method of solving a problem, you can get pulled into an infinite construction which, while being completely correct, don’t help you find an answer without reference to analytical mathematics which are not allowed under the “rules” of geometry; I suspect this is the probabilistic equivalent. Given only the information as stated, there is no advantage in either envelope. And, in general, if any expected payoff analysis leads you back to the initial state, it is a sign that there is no advantage in picking any of the steps in the analysis.

In the real world, the envelopes probably have the same amount in them, and the person running the game will claim that you picked the smaller one. The setups in logic puzzles just don’t happen in the real world.

Posted by: Blind Misery | Feb 13 2005 18:13 utc | 1

game 1
——
this is a nice one. i am given information which decreases the uncertainty of the lottery increases my mathematical expectation of winning from 1/3 to 1/2 w/o penalty. it is not necessary to bet anew.
game 2
——
> 0.5*2 + 0.5*0.5, i.e. 1.25
first of all, that should be 1/2*2 + 1/2*1, i.e. 1.5
(if one pays attention to the description of the problem).
this game – or lottery *) – is equivalent to the second lottery of the first example but with the difference that here a utility function is introduced. while the probabilities and the ME remain the same, the issue at hand is that the player does not have enough information to build an objective utility function, he is thus trying to use a subjective utility function (i.e. greed got hold of him).
in both examples the player is give the chance to change his bet. this is the usual question presented to participants of tv shows. this is just a useless run-in with the second theorem of gambling theory which states that no advantage is to be had from betting on a sub-sequence of trials in a lottery.
that was probably sloppy, and i havent yet looked at the page after the link.
FYI, there is a small jewel of a book which is a must read for anyone into gambling or stock trade, it is called ‘the theory of gambling and statistical logic’ (2nd edition) written by richard epstein. the book is easy to understand with school math and it makes for a good bed read.
*) ‘lottery’ is what games of chance – those which are based on uncertainty – are called in gambling theory.

Posted by: name | Feb 13 2005 20:17 utc | 2

Name:
In the first game, the information given increases the chance of winning to 2/3, if you change the bet. It was simply a 2/3 chance that you didn´t pick the winning ticket in the first case.
As I am not as convinced as Blackie about the mathematical correctness of games in the real world (I have met some dealers who didn´t have a clue about statistics), I would bet about 5 euros. Not more then I can easily spare, should I be wrong about the skill of the lottery manager. Or just pick the losing side.

Posted by: A swedish kind of death | Feb 15 2005 12:42 utc | 3

Lots of interesting charts and statistics, but you have to scroll down the page of Feb. 17, to “Dirty Pictures v. Dirty Laundry” as I was not able to link to them directly – here

Posted by: Fran | Feb 18 2005 7:11 utc | 4

Mil $ stats problem I’m vigorously trying to resolve – Is it worth chasing break-even?? Any ideas/suggestions would be most appreciated.
Hello All,
I think I have an
interesting question to pose to you if you might
have a mere couple minutes to consider it.
In Las Vegas as you probably know wagering on
sports is legal. Basic wagering on baseball and
hockey is unlike football where there is a spread.
For these sports there is a “Money Line” and
for Favorites the convention is they are designated
by a minus (-) sign. -200 means you wager $20 to
profit $10. If you wager standard $20 wagers
you must win 66.67%(rounded) of the time to break
even over the long term.
What I do is parlay my wagers so if two wins
occur in a row I put $5 in a “Profit Pool” and I
have two more wagers of $20. Thus I have a tree
system – what I call Fav Tree.
IFF! (If and only if) my favorite percentage is a
mere couple hundreths over standard wagering
break-even my “yield” … “blows away” (in terms of
profit / money wagered) standard wagering profits
over the long term. IFF you can be a mere couple
hundreths over break-even I can readily demonstrate
by simulation your yield will be one beautiful thing.
The problem is picking consistently over the
long term even a smidgen over break-even for any
given money line -AND- Fav Tree won’t work below
break-even … because that is negative expectation
territory. No matter what gyrations I do in
“The Land Of Negative Expectation” my long term
expectation is a loss.
From my historical money line studies/observations
especially for baseball (MLB) it seems there’s always
one or two money lines over the long term that have
a “good population” that shows just a tad over
standard wagering Favorite Money line break-even
for the entire season … thus positive expectation.
From my (corrected) Fav Tree simulator I see that
I need a mere few hundreths over “standard break-even”
in the -110 to -200 Money Line range to realize a
MUCH! higher yield than standard wagering after many
iterations. Higher Money Lines don’t work because
there is either no or not enough money left over
for the “Profit Pool” on a split.
My idea is after the first week or so is to
“follow” which money lines show a tendency
for a winning percentage to remain the slightest
over break-even with a “good number” of
“occurrences” that make up the ratio.
I realize the danger that past performance is
no indication of future results and a given
money line may indeed start to “crap out” (no
pun intended) at any juncture. What was a
good Favorite winning percentage at a given
Money Line can “go south” at any time.
However that said, I also see that often the Money
Lines with the winning percentage over break even
with a decent number of “occurrences” can get to
points where one or two games don’t change their
status of being over break-even.
My idea is to “follow the flow” that overall I’ll
always be on the whole a mere few hundreths over
break-even … that’s all I need.
I was wondering how sound my logic is and is there
any mathematical formulae, principles and/or
criteria where it is valid/viable to do such
Favorite winning percentage “chasing”; perhaps
wagering a little at first and progressively/
proportionally (?) more as the winning percentages
for all the money lines becomes increasingly
established as the season wears on.
Thanks in advance for your consideration. I welcome
and look forward to your correspondence.
Regards,
Joel Shapiro
P.S. FWIW If you’re intersted in experiencing my
Money Line Fav Tree simulator and the phenomena I
hope I’ve sufficiently described in this document
firsthand you can download it from my web page at
http://home.rochester.rr.com/grassroots1
[simulator link]
self extracting .zip file.
[Ctrl][l] (Lower case L) hotkey combination to
initiate macro.
Bolded blue cells are user defined.
You should be able to figure out the sheet and
what’s going on from context.
JRS

Posted by: Joel Shapiro | Apr 5 2005 9:45 utc | 5

Back to Main