The expected value of defecting is 4p/(p + 4(1-p), to within one part in the number of survey takers. Whether or not you defect makes no difference as to the proportion of people who defect.
The solution is to determine how likely it is that a random participant is going to defect, conditional on your choice of cooperate or defect. If you're playing with a total of N copies of yourself, you cooperate and get the maximal payout ($60/N). If you're playing against cooperate bots, you defect and get $60*4N/(N-1).
We can generalize this to partial levels. If you play with D defectors and C cooperators whose opinion you can't change, and X people who will cooperate when you cooperate (and defect when you defect), then the payouts are as thus:
C: (C + X)/(C + D + X) D: 4(C /(C + D + X)
You can solve for the break even point by setting C + X = 4 * C
So the answer is that you should defect, unless you think that for every person who is going to cooperate no matter what, there are at least three people who are thinking with similar enough reasoning to come up with the same answer you come up with (regardless of what answer that is).
I think you've got the denominators of your fractions wrong. There are 4 raffle tickets for everyone who defects. I get the values
C: (C + X)/(C + 4D + X) D: 4(C /(C + 4D + 4X)
which solves to a horrible quadratic surd.
If we wanted to we could combine your method with Zack's and assume that C people cooperate, D defect and X make the same choice I do, which is to cooperate with probability p. I think this gets kinda ugly though.
It's that time of year again.
If you are reading this post, and have not been sent here by some sort of conspiracy trying to throw off the survey results, then you are the target population for the Less Wrong Census/Survey. Please take it. Doesn't matter if you don't post much. Doesn't matter if you're a lurker. Take the survey.
This year's census contains a "main survey" that should take about ten or fifteen minutes, as well as a bunch of "extra credit questions". You may do the extra credit questions if you want. You may skip all the extra credit questions if you want. They're pretty long and not all of them are very interesting. But it is very important that you not put off doing the survey or not do the survey at all because you're intimidated by the extra credit questions.
It also contains a chance at winning a MONETARY REWARD at the bottom. You do not need to fill in all the extra credit questions to get the MONETARY REWARD, just make an honest stab at as much of the survey as you can.
Please make things easier for my computer and by extension me by reading all the instructions and by answering any text questions in the simplest and most obvious possible way. For example, if it asks you "What language do you speak?" please answer "English" instead of "I speak English" or "It's English" or "English since I live in Canada" or "English (US)" or anything else. This will help me sort responses quickly and easily. Likewise, if a question asks for a number, please answer with a number such as "4", rather than "four".
Last year there was some concern that the survey period was too short, or too uncertain. This year the survey will remain open until 23:59 PST December 31st 2013, so as long as you make time to take it sometime this year, you should be fine. Many people put it off last year and then forgot about it, so why not take it right now while you are reading this post?
Okay! Enough preliminaries! Time to take the...
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2013 Less Wrong Census/Survey
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Thanks to everyone who suggested questions and ideas for the 2013 Less Wrong Census/Survey. I regret I was unable to take all of your suggestions into account, because of some limitations in Google Docs, concern about survey length, and contradictions/duplications among suggestions. I think I got most of them in, and others can wait until next year.
By ancient tradition, if you take the survey you may comment saying you have done so here, and people will upvote you and you will get karma.