Slashdot's poll of the moment is just fantastic, combining psychology and reasoning in a very cool way. Here's the question:

# How Many People Will Select The Same Option As You?

- 0%
- 1-25%
- 26-50%
- 51-75%
- 76-99%
- 100%
- Just CowboyNeal [
this is the traditional joke Slashdot poll answer]

At first pass, you might figure that there are 6 possible responses, so if people respond randomly then about 16 percent will choose each answer, so the correct response would be 1-25 percent.

But of course, if everyone used that same logic, then many more than 25 percent of respondents would choose that answer. The next logical response would be to move on to 26-50 percent.

But once again, if everyone used that logic, the correct response would be higher still. What's the logical result of this progression? Let's see if we can capture it in our own version of the poll. I've been having trouble with online polls lately so I'm going to place this one below the fold.

So, did we do better than Slashdot readers? Discuss.

*Update:* See the analysis of the results here.

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Another good place to find thought-provoking online polls is at Irregular Webcomic, run by an extraordinarily geeky Australian named David Morgan-Mar. Sample question: "Will this be the most lopsided poll victory ever?"

Seems like it's bad we can check the results before we submit

:0)

Looks like the same as slashdot: both 1-25 and 26-50 were correct answers when I took the poll.

I guess this could be considered a variant of what Herbert Clark called a Schelling game; a coordination problem where people have to solve a problem without communicating.

People's answers should converge on a Schelling point, what Schelling himself describes as "focal point[s] for each person's expectation of what the other expects him to expect to be expected to do."

The obvious Schelling point here is the 100% option, because if everyone selects 100% everyone will be right. Now, this is the Internet so naturally, all bets are off, but in a more controlled setting I think virtually everybody would pick that option.

Karl: Wouldn't that be the case only if we could change our vote any number of times?

Hmm. 0% is wrong the moment anyone clicks on it...

100% would be right if everybody did.

Now assume for arguments sake that the bins were infinitesimally small.

Then even if everyone decided it was going to be very very close to 100%, the odds on any too people choosing the same bin would be small since the bins are so small. ie. Some may choose 99.0-99.1% others 99.1-99.2%.

Then clearly the odds are way less than 99% that anyone will choose exactly the same as me.

ie. The more bins there are, the more logical it is to choose lower probability bins.

So both 2 and 3 are correct, but 1 is actually the more accurate since there is only a 1% window.

Guess most of us think we're just a little unique and not quite normal.

Guess most of us think we're just a little unique and not quite normal.

I think SlashDot got "Cogitive Dailyed", it took forever to load...

haha, this reminds of the overwrought logic that leads vizzini to his death in the Princess Bride. too funny. =D

Is this a converging series or not?

Joel H: Not really.

Consider a situation where a limited number of people take this poll knowing that they will win a prize if and only if they all answer correctly. Again, 100% is the obvious Schelling point. Everybody knows that if everyone else selects 100% they will win the prize by choosing 100% as well, and so everyone should end up picking that option.

A slashdot reader mentioned that this is an example of a Keynesian beauty contest.

What I think would be more interesting, however, is to run the poll once, and allow everyone to see the results, and then run the poll again. Since everyone knows what the optimal strategy is, they will be tempted to all select the 26-50% option. But since everyone knows that everyone else will be tempted to select this option, some of them will realize that this is no longer the best strategy. So what is? Is there any chance we can have another copy of the poll placed below the previous one to find out? :)

(I meant to say, they will all be tempted to select either the 1-25% or 26-50% options, since both turned out to be correct.)

Obviously, the answer is 0%.

The mere act of selecting an option does not constitute anyone's entire answer to the question. It seems unlikely that any of us would be trying to accomplish or communicate the same exact thing with our selections.

But even if you choose to interpret "answer" in an absurdly narrow way, notice that exactly one person can choose 0% and be correct. I bet the survey tabulator doesn't subtract one response from the output, though. Ignoring that inaccuracy, notice also that 0% looks like the most accurate response of all of them. I chose it, and at the moment I'm only 3% off.

I guess my reasoning was faulty. I figured that everyone

oughtto pick 100%, but since this is the Internet and there are always weirdos, 75-99% was the safer option. Instead, only 1-25% and 26-50% are currently winners.Fascinating poll, though.

The idea that we

oughtto pick 100% reminds me of some of those experiments where people were paired up and they had a choice to forfeit their money or keep it, and if they both forfeited it they'd get a little more money back, if only one chose to keep it he'd get the other's too, and if they both chose to keep it neither of them would get to... the obvious choice is to forfeit every time and everyone wins, but that's not what actually happens.I participated in the slashdot poll earlier today and enjoyed reading the discussion of the poll there. I'm still puzzled by the people there (and some people here) who insist that 100% or 75-99% are good options. That simply makes no sense to me.

What sane person would ever think that out of hundreds or thousands of votes that every single one would choose 100%?

The 75-99% is silly both because it assumes a high level of similar minded people (and practically ignores random votes) and because you can get the exact same guess by choosing 1-25%. If 75-99% is true then the 1-25% option will also be true in all but the most unlikely circumstances. But, the 1-25% option can also be true while 75-99% is false. So 75-99% is NEVER a good option to choose no matter what you think others will vote for.

In the end, the only real options to choose from are 1-25, 26-50, and 51-75. I personally felt that 51-75 still assumed much too high a level of similarity between voters, so I chose 26-50. On slashdot 1-25 ended up a few percent below 25. 26-50 ended up a few percent above 26. Both were just barely true.

Yay. I picked 26-50 and was 'right'. I think. What is 'right'?

If everyone were ruled completely by logic, the right answer would be 100%. Here's the proof: If any of the available answers can be deduced by logic as the correct one to choose, then a logical person should choose it. Hence, if everyone were logical (my premise above), then everyone would choose that answer. If everyone chose that answer, 100% would have chosen it, hence that answer must be the 100% one.

The only escapes from this conclusion are (a) that there is more than one logical answer, or (b) not everyone will answer logically. Taking (a) a step further, if there are two logical answers, then the "right" answer is 50% (26-50 category), but this is self-contradictory since you end up with only one right answer by assuming there are two. The same if there are three logical answers (33%, so chose 26-50 again), 4 right answers (1-25), 5 (1-25), or 6 (1-25). Therefore, (a) is invalid. It can't be used to escape the fact that 100% must be the only logical answer.

This leaves the only logical escape from 100% being the correct answer as (b) not everybody will answer logically. Once that happens, you can't predict where it will go, so, the best guess is random (1-25).

The funny thing is, despite 100% being the only answer that can be logically derived, some comments suggest this is an insane choice. It seems it isn't a good choice, but only if you accept that not everybody is logical. Ironically, if everybody were logical, the insane choice would be the right one.

I'm answered a different value here than I did for the Slashdot poll. My reasoning is that there is a significant difference in behavior between the two audiences. The Slashdot audience is more likely to not care or do a random answer; this audience is more likely to give the matter careful thought. Thus, I gave a higher category here (which will be correct if others think it through the same way I do). In a smaller community on a math-oriented site I'd guess even a HIGHER category.

@Chad: How can you rule out that there is more than 1 logical answer? Suppose that 70% choose the category '51-75' and 30% choose the category '26-50'. Then EVERYONE in the poll has given the correct answer, just as in the 100% case. In addition this solution is much more stable, since a single person choosing '0%' will not alter the correctness of any answer but his own.

26-50 seems a way better choice as most Keynesian Beauty contest type polls predict people predict other people's behaviours (expectations of, if you want to split straws, ad infinitum) up to three degrees. Which made me choose this one. Hey?

There is the psychological factor too, because in the question it's implied how we perceive ourselfs in relation to others, it's not a matter of what answer is right, but a matter of how we see ourselfs in comparisson with others, no the other we see and live with, but the others that we donÂ´t see but refer to as "the people".

Of the people who select the same answer as you .... That would be 100% ... Doesn't matter which answer as long as it's yours.

[Comment removed. I'm happy to hear criticism, but if you don't explain why you disagree with something -- and use offensive language to boot -- you'll be deleted. Our readers aren't interested. -- DM]if everybody would just pick 100% we could all win, whats wrong with you people?

Isn't this poll just a subtler way of asking: "What percentage of the people taking this poll do you think will act completely rationally?"

This explains the follow-on poll results as well as the differences between this poll and the one on Slashdot, as the author of this poll clearly primed the takers of the poll towards rational action, and much more strongly so in the second poll than the first.

Now that this link is on reddit, I expect that additional polls might converge more towards the Slashdot results... of course that's just my way of biasing any such poll :-).

This question cannot be deduced logically to 100%. If I assume everyone will go to 100%, I can choose 0%, making me right and everyone who chose 100% wrong. Since I will not be the only one to figure this out, more than just me will choose 0%, making it wrong. This will go right up through the percentages back to 100% and start over again. This brings the question to a best guess. eliminate 100% and 0% since a single random vote would kill it and you are left with four options. I chose 1-25, which seems to be a correct choise currently. I guessed a large portion would follow the logic up to 100%, and some would go that high and guess lower for random votes, but not 75%+. Since that would take a decent percentage, I eliminated 50%+ since the majority of the remainder would have to choose it, leaving 1-25 and 25-50. Both seem logical, and it is possible for both to end up 'correct', I went low, but at that point it felt like a 50-50 change. While this looks like a logic question, it is a combination of logic and human nature to not be logical.

I wish your link to the "poll of the moment" had been to the voting booth page, not the results page. Now I can't honestly try voting in the poll, since I've been influenced by knowing the results. :(