the Ellsberg paradox (gambling related)

[EXTRAPOLATER]

This was described very poorly on a documentary that I saw yesterday. Thusly:

In fist A there are 50% red marbles and 50% blue marbles.
In fist B there are an unknown number of red or blue marbles.
Which fist are you like to place a bet on?

For myself, I would rather gamble more on available information than unavailable information. No "odds" were offered in the documentary but I would be more tempted to play the 50% odds--maybe at +120, for example--than I would be to play a total unknown. Obviously the documentary was limited in its description, as the following will suggest.

The wikipedia site has the following entry:
http://en.wikipedia.org/wiki/Ellsberg_paradox

I think that I'd prefer gamble A over gamble B.

Gamble D looks better than gamble C.

I be (still) drinkin' so maybe I can't assess things properly. The main thrust of their argument is something called "ambiguity aversion." The write-up is short so maybe you can get it.

I need to re-examine when more coherent, but the original example presented (from the documentary) seems obvious to me. They claimed that fist B had the same odds (hypotheically 50-50) as fist A but with A you have a concrete probability while with fist B you are betting on a total lack of information; lack of this information does NOT imply a 50-50 probability, to my understanding. My analysis of the other example, from wikipedia, might be out to lunch, but with A you have a 33% chance of winning while with B you have a 0-66% chance of winning. With gamble C you have a 33-100% chance of winning while with gamble D you have a true 66% chance of winning. Just my thunks. Interesting problem for any concerned with probabilities.

I gotta kill another 4 or 5 hours so thought I'd spread some insanity.

Maybe I'll hold off betting under the influence in the future. Maybe the best advice possible regarding cognitative ecomoronics.

:mj06:

 

[Kanuck]

I WANT WHAT YOU'RE DRINKING

 

[EXTRAPOLATER]

You doing the march May 5th?
We could party.

 

[Kanuck]

Sorry I swing the other way.

I now get that avatar you have picked :mj07:

 

[EXTRAPOLATER]

You "get it," aye?

Some things are easier to get than others.

I get your meaning.

 

[Kanuck]

unscramble this handle.

PSYKPI

 

[krc]

Thanks.

just spent 10 min making my head hurt



:SIB :SIB


krc

 

[EXTRAPOLATER]

any senses of humour that emerge make everything worthwhile.

thanks

:toast:

 

[EXTRAPOLATER]

"lack of this information does NOT imply a 50-50 probability"


maybe I'm wrong about this.
If a fist holds an unknown number of either red or blue marbles is it a 50-50 proposition?
I guess it must be

some kinda error-potential possibility needs to be considered, here, methinks, to compare it to a 5-red marble, 5-blue marble scenario, which provides a "true" 50%

for most of our purposes "true" remains ()(open to questions)()

I guess that's what we all combat, one way or another (meant for gambling but ()(open to questions)())

want that on my tombstone

I think we all should

 

[EXTRAPOLATER]

Not that anyone gives a shit

(I sure freakin' don't at this stage)

but this is the same Daniel Ellsberg who leaked the Pentagon Papers.

?is it still March?

let's play ball

 

[JOSHNAUDI]

"lack of this information does NOT imply a 50-50 probability"


maybe I'm wrong about this.
If a fist holds an unknown number of either red or blue marbles is it a 50-50 proposition?
I guess it must be

some kinda error-potential possibility needs to be considered, here, methinks, to compare it to a 5-red marble, 5-blue marble scenario, which provides a "true" 50%

for most of our purposes "true" remains ()(open to questions)()

I guess that's what we all combat, one way or another (meant for gambling but ()(open to questions)())

want that on my tombstone

I think we all should

I think the reason for the paradox can be stated as the chooser will make the selection based on the best known odds. With the 2 situations we know that there is a 33% chance that you will pick a red ball (choice A) and we also know that there is a 66% chance that we will pick a yellow or black ball with (choice D).

Choice B offers an unknown % (0-66%)
as well as
Choice C (33%-99%)

With your other example of red and blue balls - There is a percentage - we just don't know what it is and there for cannot assume that it is 50% we have to keep it labeled as unknown. Further - there is no leading eveidence to sway our opinion.

If I was to tell you I have 9 balls in my hand that are either red or blue - then you cannot make the assumption that you have a 50% chance of selecting a red ball, just based on the fact that it is impossible to have a 50/50 chance with 9 balls. You either have, 11%, 22%, 33%, 44%, 56%, 67%, 78% or 89% chance and all are just as likely to be the real odds.
There is no leading evidence to help you make your decision. But if I gave you the same scenerio and told you that at least 4 balls were red. What would your decision be? Mine would be red because I know I can eliminate the 11%, 22%, and 33%. Which is exactly the same kind of leading evidence given in situation A and B of the original question.

I think we are saying the same thing - that when you have a choice between known odds and unknown odds we tend to pick the known.

What I find interesting is this known floor vesrus unknown ceiling and I believe that if tweaked we could get to a range that is very unpredictable.

Back to the original A versus B selection - Originally you had an urn containing 30 red balls and 60 balls that were either black or yellow. your choice was you would be given $100 if you picked A - red ball or B - black ball.

What if the urn contained (1,2,3...29) red balls and 60 balls that were either black or yellow.
At 1 red ball it's pretty easy to assume that everyone would pick black.
At 29 red balls I would still pick red knowing that I still had a 32.5% (29/89)chance of picking a red ball versus an unknown number of black balls.

What about 20 red balls and 60 yellow or black balls, what would be your choice? Red or black?

 

[EXTRAPOLATER]

I'm not sure what the proper answer is for your last scenario. I think I'd be tempted to try for a black ball but I don't know if that's the best answer. Obviously if there was a line, say +100, then I wouldn't try either scenario (25% red balls so need +300 to break even there). If a twist was thrown in such as to save your life then you must choose the correct ball colour then we'd have another matter, and I'd be tempted to go black but, again, not sure if that's right.

Black balls would be at a 0-75% range.
Would that create an average of 37.5% (or is it 37% due to the zero?)--thereby making it a better option than red?
One way to assess it.
Not sure if it's right.

Anybody remember the sequences and series formula for that black ball problem. Add 0,1,2,...up to 75 and then divide by the 76 possibilities should provide the right answer. I certainly don't want to add them all up. Would be 37.5% in this case, wouldn't it? 0+75 / 2.

 

[JOSHNAUDI]

No right or wrong answer - just interesting when it comes down to choice

For your answer - assume that you have no risk, you are on a game show and will receive lovely parting gifts whether you choose correctly or not. You only get one choice (one instance) and must declare
A. If I draw a red ball I receive $100
B. If I draw a black ball I receive $100

The information you are given is
There are 20 red balls, and 60 balls that are either yellow or black. The balls are well mixed and their is an equal chance that any one of the 80 balls can be drawn.

You must now make your choice and explain why you made your choice.

------

You can deduce.

If you choose red ball - you have a 25% (20/80)chance of winning $100

If you choose black ball you have an unknown percentage between - 1.25% (1/80) and 73.75% (59/80) of winning $100.

You also know that you have an unknown percentage between - 1.25% (1/80) and 73.75% (59/80) of not being able to win no matter what your choice with a yellow ball emerging.

Are you willing to trade in a known 25% for an unknown 1-73.75%

--------
I would choose the black balls - thinking that there is more to gain than lose in this situation - risk reward possbile chance to gain 48% versus losing 24%

but

I believe some people will still choose red balls because they believe a known 25% chance is better than risking getting a 1.25% chance.

-------

 

[EXTRAPOLATER]

Took me a puff before I could think about this clearly. I would say that we are right to go for the black balls but that the justification is that the probability is higher. Like I finally got around to saying--though you properly refined is somewhat by assuming that there will be at least one ball either yellow or black--is that it would be the mean probability amongst all possible situations. Refined to 1/80 to 59/80 that gives me a 37.18% chance of pulling a black ball (just under the 37.5 posted earlier).

That was a pain in the ass--can't remember the sequences and series formula. Instead of adding up 1 through 59 and dividing by 59 different universes (one way to look at it), I added up 29 pairs of 60 (1740, as 1+59=60, 2+58=60, etc.) and added half of the central number (30/2=15), then divided this number (1755) by 59 possible situations (29.745...) and then divided by my 80 total number of balls, for 0.3718...or my 37%

I surely miss the KISS rule but I hope that makes sense.
Need another puff now.

 

[EXTRAPOLATER]

Don't ask me why I divided 30/2 (I know but nevermind).
Total should be 1770/59/80
=37.5%

I think this is right.

 

[JOSHNAUDI]

thanks for putting the math to it and proving the black ball is the best choice for 20 red balls and 60 black or yellow balls

i thought about the original 30 red balls and 60 black or yellow balls question today - and the thing that popped into my mind is that if you choose the black ball then you are playing a 2 game parlay / The first game is a coin flip on whether there will be more black balls than red balls (technically more red than yellow) , then the 2nd game is that a black ball will actually be picked. The 2 game parlay portion is just to define odds of winning with a black ball.

The other thing here is implied odds on unknown variables. After picking, the urn can be opened and the red, black and yellow balls can be counted to see what the real odds were.

good stuff - appreciate the post

 

[EXTRAPOLATER]

thanks Josh, I just got one or two more things to say on the matter.

I actually didn't read the entry (though short) fully before posting all this but just read the description of the mind-game. I got the meaning of the "ambiguity aversion" but didn't know if they spoke of it really. Clearly they don't. Anyway, my initial picks matched those that people normally give, supposedly due to this amb.aversion.

They, strangely, finish with this:

In light of the ambiguity in the probabilities of the outcomes, the agent is unable to evaluate a precise expected utility. Consequently, a choice based on maximizing the expected utility is also impossible. The info-gap approach supposes that the agent implicitly formulates info-gap models for the subjectively uncertain probabilities. The agent then tries to satisfice the expected utility and to maximize the robustness against uncertainty in the imprecise probabilities. This robust-satisficing approach can be developed explicitly to show that the choices of decision-makers should display precisely the preference reversal which Ellsberg observed (Ben-Haim, 2006, section 11.1).

===

The first bit looks like mumbo-jumbo, they way I see it. Are they suggesting that we cannot, in fact, compute the probabilities. I'd disagree.
Most importantly, I don't get the last sentence that some (unknown--not explained) argument can be made to demonstrate the choice opposite of those made. As in, one choice (AorB--CorD) is supposed to be better, or is the combo what doesn't make sense, or WTF?
I'm missing something, apparently, but is this the clear kind of language being used to "explain" things these days? A twisted branch of philosophy or psychology, unless I really missed something.

I think I now get that the probabilities, in their original examples, are the same. I would still choose, for example (in the original--5 blue 5 red or unknown (say 10 total) yellow and unknown black) the blue/red option of 50% as opposed to the yellow/black option of a multiverse 50% but I can only justify it emotionally, perhaps, rather than rationally:
(assuming 1-9 black marbles--min.1 each colour)
I get a true 50% in fist A (5-5)
In first B I could get as low as 11.1% odds or as high as 88.9% odds (you can see how that totals a cool 100 and divided by 2 gives me 50%).
I value "not getting ripped off" (not 11.1%) more than I value "getting excess value" (88.9%) so I would take the true odds to avoid getting so ripped off. I can't justify the choices, personally, any other way. Both options--each choice in the paradox--has the same actual probabilities. This is why I'm confused by the conclusion.

Probably total overkill here but I'm just feeling clued (out).
The wikipedia entry maybe isn't the best source. I should check elsewhere.

Better yet focus on baseball.

Part of my problem is that I spoke with my Pops about it today and, while I respect his opinion, he gave me an argument for yellow-(or)-black that just didn't make any sense to me. Bordered on either existentialism or phenomenology or perhaps he just wants to believe that (was beyond me so I'm not sure) There is no advantage. I think others mentioned that. I can see no other reason to choose save for a value judgement.

Hate to use the word "value" that way.
I use it way differently handicapping as to ethics but I'm using the latter here.

Is it still freaking March?
Let's play ball